PyMeeus¶

Library of astronomical algorithms in Python.

PyMeeus is a Python implementation of the astronomical algorithms described in the classical book ‘Astronomical Algorithms, 2nd Edition, Willmann-Bell Inc. (1998)’ by Jean Meeus.

There are great astronomical libraries out there. For instance, if you’re looking for high precision and speed you should take a look at libnova. For a set of python modules aimed at professional astronomers, you should look at Astropy. On the other hand, the advantages of PyMeeus are its simplicity, ease of use, ease of reading, ease of installation (it has the minimum amount of dependencies) and abundant documentation.

Installation¶

The easiest way of installing PyMeeus is using pip:

pip install pymeeus

Or, for a per-user installation:

pip install --user pymeeus

If you prefer Python3, you can use:

pip3 install --user pymeeus

If you have PyMeeus already installed, but want to upgrade to the latest version:

pip3 install -U pymeeus

Properly Using PyMeeus¶

It is very common to try to run PyMeeus like this:

import pymeeus

mydate = pymeeus.Epoch(1992, 10, 13.0)

But if you do that, you’ll get an error like this:

Traceback (most recent call last):
  File "/home/user/test/test.py", line 3, in <module>
    epoch = pymeeus.Epoch(1992, 10, 13.0)
AttributeError: module 'pymeeus' has no attribute 'Epoch'

This issue points to a misunderstanding that is very common in the Python world. The keyword import is used to import MODULES… but PyMeeus is NOT a module: It is a LIBRARY composed of MULTIPLE modules (Angle, Epoch, Coordinates, etc). As of today, the library Pymeeus has 19 different modules (if you look into the directory where pip stores the library, you’ll find one “.py” file per module).

Therefore if you want to use, for example, the module Angle you should use:

import pymeeus.Angle

I.e., your module is pymeeus.Angle, and not just Angle.

But there is more! When you use import to fetch a module, you must then use the dot notation to access the components of the module (classes, functions, etc). For instance:

import pymeeus.Angle

i = pymeeus.Angle.Angle(11.94524)

In this case, you are telling the Python interpreter that you want to use the class Angle (with parameter ‘11.94524’) from the module Angle belonging to the library pymeeus.

There is, however, a more practical (and common) way to handle modules using the statement from <MODULE> import <COMPONENT>. For instance:

from pymeeus.Angle import Angle
from pymeeus.Epoch import Epoch, JDE2000
from math import sin, cos, tan, acos, atan2, sqrt, radians, log10

This way is preferred because, among other reasons, only the required components are loaded into memory instead of the whole module. Also, now the component is directly added to your execution environment, which means that you no longer need to use the dot notation.

Therefore, the script at the beginning would become:

from pymeeus.Epoch import Epoch

mydate = Epoch(1992, 10, 13.0)

Meta¶

Author: Dagoberto Salazar

Distributed under the GNU Lesser General Public License v3 (LGPLv3). See LICENSE.txt and COPYING.LESSER for more information.

Documentation: https://pymeeus.readthedocs.io/en/latest/

GitHub: https://github.com/architest/pymeeus

If you have Sphinx installed, you can generate your own, latest documentation going to directory ‘docs’ and issuing:

make html

Then the HTML documentation pages can be found in ‘build/html’.

Contributing¶

The preferred method to contribute is through forking and pull requests:

Fork it (https://github.com/architest/pymeeus/fork)
Create your feature branch (git checkout -b feature/fooBar)
Commit your changes (git commit -am 'Add some fooBar')
Push to the branch (git push origin feature/fooBar)
Create a new Pull Request

Please bear in mind that PyMeeus follows the PEP8 style guide for Python code (PEP8). We suggest you install and use a linter like Flake8 before contributing.

Additionally, PyMeeus makes heavy use of automatic tests. As a general rule, every function or method added must have a corresponding test in the proper place in tests directory.

Finally, documentation is also a big thing here. Add proper and abundant documentation to your new code. This also includes in-line comments!!!.

Contributors¶

Neil Freeman - Fixed undefined variable in Epoch.tt2ut
molsen234 - Fixed bug when using fractional seconds, minutes, hours or days
Sebastian Veigl - Added functionality for Jupiter’s moons
Sophie Scholz - Added functionality for Jupiter’s moons
Vittorio Serra - Added functionality for Jupiter’s moons
Michael Lutz - Added functionality for Jupiter’s moons
Ben Dilday - Added __hash__() method to class Epoch
Zivoslav - Bug report of winter solstice
Devid, Hugo van Kemenade - Test suggestions

What’s new¶

0.5.12
- Fixed a bug in the computation of the winter solstice. Added new tests and information about proper use of the library.
0.5.11
- Added parameter local to the Epoch class constructor and the methods get_date() and get_full_date().
0.5.10
- Added methods moon_librations() and moon_position_angle_axis().
0.5.9
- Added method moon_maximum_declination().
0.5.8
- Fixed several bugs in Epoch class, and added method doy().
0.5.7
- Added method moon_passage_nodes().
0.5.6
- Added method moon_perigee_apogee().
0.5.5
- Added method moon_phase().
0.5.4
- Added methods illuminated_fraction_disk() and position_bright_limb() to Moon class.
0.5.3
- Fixed error in the return type of method Sun.equation_of_time().
0.5.2
- Added methods to compute the Moon’s longitude of ascending node and perigee.
0.5.1
- Changes in the organization of the documentation.
0.5.0
- Added Moon class and position() methods.
0.4.3
- Added method ring_parameters() to Saturn class.
0.4.2
- Added method __hash__() to Epoch. Now Epoch objects can be used as keys in a dictionary.
0.4.1
- Added funtionality to compute the positions of Jupiter’s Galilean moons.
0.4.0
- Added methods to compute Saturn’s ring inclination and longitude of ascending node.
0.3.13
- Additional encoding changes.
0.3.12
- Deleted encoding keyword from setup.py, which was giving problems.
0.3.11
- Added encoding specification to setup.py.
0.3.10
- Fixed characters with the wrong encoding.
0.3.9
- Relaxed requirements, added contributor molsen234, and fixed format problems showed by flake8.
0.3.8
- Fixed undefined variable in Epoch.tt2ut.
0.3.7
- Fix bug when using fractional seconds, minutes, hours or days, plus documentation improvements.
0.3.6
- Add method to compute rising and setting times of the Sun.
0.3.5
- Add method magnitude() to planet classes.
0.3.4
- Add method to compute the parallax correction to Earth class.
0.3.3
- Add methods to compute the passage through the nodes.
0.3.2
- Add methods to compute the perihelion and aphelion of all planets.
0.3.1
- Fix errors in the elongation computation, add tests and examples of use of methods geocentric_position(), and tests and examples for Pluto class.
0.3.0
- Added Pluto class.

API¶

Core¶

Base¶

Basic, general functions and constants.

pymeeus.base.TOL = 1e-10¶: Internal tolerance being used by default

pymeeus.base.get_ordinal_suffix(ordinal)[source]¶

Method to get the suffix of a given ordinal number, like 1’st’, 2’nd’, 15’th’, etc.

Parameters:	ordinal (int) – Ordinal number
Returns:	Suffix corresponding to input ordinal number
Return type:	str
Raises:	TypeError if input type is invalid.

>>> get_ordinal_suffix(40)
'th'
>>> get_ordinal_suffix(101)
'st'
>>> get_ordinal_suffix(2)
'nd'
>>> get_ordinal_suffix(19)
'th'
>>> get_ordinal_suffix(23)
'rd'

pymeeus.base.iint(number)[source]¶

This method behaves in the same way as the INT() function described by Meeus in his book: Greatest integer which is not greater than number.

Parameters:	number (int, float) – Number or expresion
Returns:	Greatest integer which is not greater than number
Return type:	int
Raises:	TypeError if input type is invalid.

>>> iint(19)
19
>>> iint(19.95)
19
>>> iint(-2.4)
-3

pymeeus.base.machine_accuracy()[source]¶

This function computes the accuracy of the computer being used.

This function returns a tuple containing the number of significant bits in the mantissa of a floating number, and the number of significant digits in a decimal number.

Returns:	Number of significant bits, and of significant digits
Return type:	tuple

Angle¶

Class to handle angles.

class pymeeus.Angle.Angle(*args, **kwargs)[source]¶

Class Angle deals with angles in either decimal format (d.dd) or in sexagesimal format (d m’ s’’).

It provides methods to handle an Angle object like it were a simple float, but adding the functionality associated with an angle.

The constructor takes decimals and sexagesimal input. The sexagesimal angles can be given as separate degree, minutes, seconds values, or as tuples or lists. It is also possible to provide another Angle object as input.

Also, if radians=True is passed to the constructor, then the input value is considered as in radians, and converted to degrees.

__abs__()[source]¶

This method is used to obtain the absolute value of this Angle.

Returns:	A new Angle object.
Return type:	`Angle`

>>> a = Angle(-303.67)
>>> print(abs(a))
303.67

__add__(b)[source]¶

This method defines the addition between Angles.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(83.1)
>>> b = Angle(18.4)
>>> print(a + b)
101.5

__call__()[source]¶

Method used when object is called only with parenthesis.

Returns:	The internal value of the Angle object.
Return type:	int, float

>>> a = Angle(54.6)
>>> print(a())
54.6

__div__(b)[source]¶

This method defines the division between Angles.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	ZeroDivisionError if divisor is zero.
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(172.0)
>>> b = Angle(86.0)
>>> print(a/b)
2.0

__eq__(b)[source]¶

This method defines the ‘is equal’ operator between Angles.

Note

For the comparison, the internal tolerance value is used.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(172.01)
>>> b = Angle(172.009)
>>> a == b
False

__float__()[source]¶

This method returns Angle value as a float.

Returns:	Internal angle value as a float.
Return type:	float

>>> a = Angle(213.8)
>>> float(a)
213.8

__ge__(b)[source]¶

This method defines ‘is equal or greater’ operator between Angles.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(172.01)
>>> b = Angle(172.009)
>>> a >= b
True

__gt__(b)[source]¶

This method defines the ‘is greater than’ operator between Angles.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(172.01)
>>> b = Angle(172.009)
>>> a > b
True

__iadd__(b)[source]¶

This method defines the accumulative addition to this Angle.

Returns:	This Angle.
Return type:	`Angle`

>>> a = Angle(172.1)
>>> b = Angle(54.6)
>>> a += b
>>> print(a)
226.7

__idiv__(b)[source]¶

This method defines the accumulative division to this Angle.

Returns:	This Angle.
Return type:	`Angle`
Raises:	ZeroDivisionError if divisor is zero.
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(330.0)
>>> b = Angle(30.0)
>>> a /= b
>>> print(a)
11.0

__imod__(b)[source]¶

This method defines the accumulative module b of this Angle.

Returns:	This Angle.
Return type:	`Angle`

>>> a = Angle(330.0)
>>> b = Angle(45.0)
>>> a %= b
>>> print(a)
15.0

__imul__(b)[source]¶

This method defines the accumulative multiplication to this Angle.

Returns:	This Angle.
Return type:	`Angle`

>>> a = Angle(30.0)
>>> b = Angle(55.0)
>>> a *= b
>>> print(a)
210.0

__init__(*args, **kwargs)[source]¶

Angle constructor.

It takes decimals and sexagesimal input. The sexagesimal angles can be given as separate degree, minutes, seconds values, or as tuples or lists. It is also possible to provide another Angle object as input.

If radians=True is passed, then the input value is converted from radians to degrees.

If ra=True is passed, then the input value is converted from Right Ascension to degrees

Parameters:	args (int, float, list, tuple, `Angle`) – Input angle, in decimal or sexagesimal format, or Angle radians (bool) – If True, input angle is in radians. False by default. ra (bool) – If True, input angle is in Right Ascension. False by default
Returns:	Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(-13, 30, 0.0)
>>> print(a)
-13.5
>>> b = Angle(a)
>>> print(b)
-13.5

__int__()[source]¶

This method returns Angle value as an int.

Returns:	Internal angle value as an int.
Return type:	int

>>> a = Angle(213.8)
>>> int(a)
213

__ipow__(b)[source]¶

This method defines the accumulative power to this Angle.

Returns:	This Angle.
Return type:	`Angle`

>>> a = Angle(37.0)
>>> b = Angle(3.0)
>>> a **= b
>>> print(a)
253.0

__isub__(b)[source]¶

This method defines the accumulative subtraction to this Angle.

Returns:	This Angle.
Return type:	`Angle`

>>> a = Angle(97.0)
>>> b = Angle(39.0)
>>> a -= b
>>> print(a)
58.0

__itruediv__(b)[source]¶

This method defines accumulative division to this Angle (Python3).

Returns:	This Angle.
Return type:	`Angle`
Raises:	ZeroDivisionError if divisor is zero.
Raises:	TypeError if input values are of wrong type.
See:	__idiv__

__le__(b)[source]¶

This method defines ‘is equal or less’ operator between Angles.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(72.0)
>>> b = Angle(72.0)
>>> a <= b
True

__lt__(b)[source]¶

This method defines the ‘is less than’ operator between Angles.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(72.0)
>>> b = Angle(72.0)
>>> a < b
False

__mod__(b)[source]¶

This method is used to obtain the module b of this Angle.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(333.0)
>>> b = Angle(72.0)
>>> print(a % b)
45.0

__mul__(b)[source]¶

This method defines the multiplication between Angles.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(33.0)
>>> b = Angle(72.0)
>>> print(a * b)
216.0

__ne__(b)[source]¶

This method defines the ‘is not equal’ operator between Angles.

Note

For the comparison, the internal tolerance value is used.

Returns:	A boolean.
Return type:	bool

>>> a = Angle(11.200001)
>>> b = Angle(11.200000)
>>> a != b
True

__neg__()[source]¶

This method is used to obtain the negative version of this Angle.

Returns:	A new Angle object.
Return type:	`Angle`

>>> a = Angle(-11.2)
>>> print(-a)
11.2

__pow__(b)[source]¶

This method defines the power operation for Angles.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(12.5)
>>> b = Angle(4.0)
>>> print(a ** b)
294.0625

__radd__(b)[source]¶

This method defines the addition between Angles by the right

Returns:	A new Angle object.
Return type:	`Angle`

>>> a = Angle(83.1)
>>> print(8.5 + a)
91.6

__rdiv__(b)[source]¶

This method defines division between Angles by the right.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	ZeroDivisionError if divisor is zero.
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(80.0)
>>> print(350 / a)
4.375

__repr__()[source]¶

Method providing the ‘official’ string representation of the object. It provides a valid expression that could be used to recreate the object.

Returns:	As string with a valid expression to recreate the object
Return type:	string

>>> a = Angle(12.5)
>>> repr(a)
'Angle(12.5)'

__rmod__(b)[source]¶

This method defines module operation between Angles by the right.

Returns:	A new Angle object.
Return type:	`Angle`

>>> a = Angle(80.0)
>>> print(350 % a)
30.0

__rmul__(b)[source]¶

This method defines multiplication between Angles by the right.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(11.0)
>>> print(250.0 * a)
230.0

__round__(n=0)[source]¶

This method returns an Angle with content rounded to ‘n’ decimal.

Returns:	A new Angle object.
Return type:	`Angle`

>>> a = Angle(11.4361)
>>> print(round(a, 2))
11.44

__rpow__(b)[source]¶

This method defines the power operation for Angles by the right.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(5.0)
>>> print(24.0 ** a)
144.0

__rsub__(b)[source]¶

This method defines the subtraction between Angles by the right.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(25.0)
>>> print(24.0 - a)
-1.0

__rtruediv__(b)[source]¶

This method defines division between Angle by the right (Python3).

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	ZeroDivisionError if divisor is zero.
Raises:	TypeError if input values are of wrong type.
See:	__rdiv__

__str__()[source]¶

Method used when trying to print the object.

Returns:	Angle as string.
Return type:	string

>>> a = Angle(12.5)
>>> print(a)
12.5

__sub__(b)[source]¶

This method defines the subtraction between Angles.

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> a = Angle(25.4)
>>> b = Angle(10.2)
>>> print(a - b)
15.2

__truediv__(b)[source]¶

This method defines the division between Angles (Python 3).

Returns:	A new Angle object.
Return type:	`Angle`
Raises:	ZeroDivisionError if divisor is zero.
Raises:	TypeError if input values are of wrong type.
See:	__div__

__weakref__¶: list of weak references to the object (if defined)

static deg2dms(deg)[source]¶

Converts input from decimal to sexagesimal angle format.

Parameters:	deg (int, float) – Degrees decimal format.
Returns:	Angle in sexagesimal format, with ranges adjusted.
Return type:	tuple

Note

The output format is (Degrees, Minutes, Seconds, sign)

>>> print(Angle.deg2dms(23.44694444))
(23, 26, 48.999983999997596, 1.0)

static dms2deg(degrees, minutes, seconds=0.0)[source]¶

Converts an angle from sexagesimal to decimal format.

Parameters:	degrees (int, float) – Degrees. minutes (int, float) – Minutes. seconds (int, float) – Seconds. 0.0 by default.
Returns:	Angle in decimal format, within +/-[0:360) range.
Return type:	float

>>> print(Angle.dms2deg(-23, 26, 48.999983999997596))
-23.44694444

dms_str(fancy=True, n_dec=-1)[source]¶

Returns the Angle value as a sexagesimal string.

The parameter fancy allows to print in “Dd M’ S’’” format if True, and in “D:M:S” (easier to parse) if False. On the other hand, the n_dec parameter sets the number of decimals used to print the seconds. Set to a negative integer to disable (default).

Parameters:	fancy (int) – Format of output string. True by default. n_dec – Number of decimals used to print the seconds
Returns:	Angle value as string in sexagesimal format.
Return type:	string
Raises:	TypeError if input value is of wrong type.

>>> a = Angle(42.75)
>>> print(a.dms_str())
42d 45' 0.0''
>>> print(a.dms_str(fancy=False))
42:45:0.0
>>> a = Angle(49, 13, 42.4817)
>>> print(a.dms_str(n_dec=2))
49d 13' 42.48''

dms_tuple()[source]¶

Returns the Angle as a tuple containing (degrees, minutes, seconds, sign).

Returns:	Angle value as (degrees, minutes, seconds, sign).
Return type:	tuple

get_ra()[source]¶

Returns the Angle value as a Right Ascension in float format

Returns:	The internal value of the Angle object as Right Ascension.
Return type:	int, float

>>> a = Angle(138.75)
>>> print(a.get_ra())
9.25

get_tolerance()[source]¶

Gets the internal tolerance value used to compare Angles.

Note

The default tolerance value is base.TOL.

Returns:	Internal tolerance.
Return type:	float

ra_str(fancy=True, n_dec=-1)[source]¶

Returns the Angle value as a sexagesimal string in Right Ascension.

The parameter fancy allows to print in “Hh M’ S’’” format if True, and in “H:M:S” (easier to parse) if False. On the other hand, the n_dec parameter sets the number of decimals used to print the seconds. Set to a negative integer to disable (default).

Parameters:	fancy (int) – Format of output string. True by default. n_dec – Number of decimals used to print the seconds
Returns:	Angle value as Right Ascension in sexagesimal format.
Return type:	string
Raises:	TypeError if input value is of wrong type.

>>> a = Angle(138.75)
>>> print(a.ra_str())
9h 15' 0.0''
>>> print(a.ra_str(fancy=False))
9:15:0.0
>>> a = Angle(2, 44, 11.98581, ra=True)
>>> print(a.ra_str(n_dec=3))
2h 44' 11.986''

ra_tuple()[source]¶

Returns the Angle in Right Ascension format as a tuple containing (hours, minutes, seconds, sign).

Returns:	Angle value as RA in (hours, minutes, seconds, sign) format.
Return type:	tuple

rad()[source]¶

Returns the Angle value in radians.

Returns:	Angle value in radians.
Return type:	float

>>> a = Angle(47.762)
>>> print(round(a.rad(), 8))
0.83360416

static reduce_deg(deg)[source]¶

Takes a degree value in decimal format and converts it to a float value in the +/-[0:360) range.

Parameters:	deg (int, float, `Angle`) – Input degree angle in decimal format.
Returns:	Float value of the angle in the +/-[0:360) range.
Return type:	float

>>> a = 386.3
>>> b = Angle.reduce_deg(a)
>>> print(round(b, 1))
26.3

static reduce_dms(degrees, minutes, seconds=0.0)[source]¶

Takes a degree value in sexagesimal format and converts it to a value in the +/-[0:360) range (degrees) and [0:60) range (minutes and seconds). It also takes care of fractional degrees and minutes.

Parameters:	degrees (int, float) – Degrees. minutes (int, float) – Minutes. seconds (int, float) – Seconds. 0.0 by default.
Returns:	Angle in sexagesimal format, with ranges properly adjusted.
Return type:	tuple

>>> print(Angle.reduce_dms(-743.0, 26.0, 49.6))
(23, 26, 49.6, -1.0)

set(*args, **kwargs)[source]¶

Method used to define the value of the Angle object.

It takes decimals and sexagesimal input. The sexagesimal angles can be given as separate degree, minutes, seconds values, or as tuples or lists. It is also possible to provide another Angle object as input.

If radians=True is passed, then the input value is converted from radians to degrees

If ra=True is passed, then the input value is converted from Right Ascension to degrees

Parameters:	args (int, float, list, tuple, `Angle`) – Input angle, in decimal or sexagesimal format, or Angle radians (bool) – If True, input angle is in radians. False by default. ra (bool) – If True, input angle is in Right Ascension. False by default
Returns:	None.
Return type:	None
Raises:	TypeError if input values are of wrong type.

set_ra(*args)[source]¶

Define the value of the Angle object from a Right Ascension.

It takes decimals and sexagesimal input. The sexagesimal Right Ascensions can be given as separate hours, minutes, seconds values, or as tuples or lists.

Parameters:	args (int, float, list, tuple) – Input Right Ascension, in decimal or sexagesimal format.
Returns:	None.
Return type:	None
Raises:	TypeError if input values are of wrong type.

>>> a = Angle()
>>> a.set_ra(9, 14, 55.8)
>>> print(a)
138.7325

set_radians(rads)[source]¶

Method to define the value of the Angle object from radians.

Parameters:	rads (int, float) – Input angle, in radians.
Returns:	None.
Return type:	None
Raises:	TypeError if input value is of wrong type.

>>> a = Angle()
>>> a.set_radians(pi)
>>> print(a)
180.0

set_tolerance(tol)[source]¶

Changes the internal tolerance value used to compare Angles.

Parameters:	tol (int, float) – New tolerance value.
Returns:	None
Return type:	None

to_positive()[source]¶

Converts the internal angle value from negative to positive.

Returns:	This angle object.
Return type:	`Angle`

>>> a = Angle(-87.32)
>>> print(a.to_positive())
272.68

Coordinates¶

Module holding functions to handle coordinates.

pymeeus.Coordinates.NUTATION_ARG_TABLE = [[0, 0, 0, 0, 1], [-2, 0, 0, 2, 2], [0, 0, 0, 2, 2], [0, 0, 0, 0, 2], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [-2, 1, 0, 2, 2], [0, 0, 0, 2, 1], [0, 0, 1, 2, 2], [-2, -1, 0, 2, 2], [-2, 0, 1, 0, 0], [-2, 0, 0, 2, 1], [0, 0, -1, 2, 2], [2, 0, 0, 0, 0], [0, 0, 1, 0, 1], [2, 0, -1, 2, 2], [0, 0, -1, 0, 1], [0, 0, 1, 2, 1], [-2, 0, 2, 0, 0], [0, 0, -2, 2, 1], [2, 0, 0, 2, 2], [0, 0, 2, 2, 2], [0, 0, 2, 0, 0], [-2, 0, 1, 2, 2], [0, 0, 0, 2, 0], [-2, 0, 0, 2, 0], [0, 0, -1, 2, 1], [0, 2, 0, 0, 0], [2, 0, -1, 0, 1], [-2, 2, 0, 2, 2], [0, 1, 0, 0, 1], [-2, 0, 1, 0, 1], [0, -1, 0, 0, 1], [0, 0, 2, -2, 0], [2, 0, -1, 2, 1], [2, 0, 1, 2, 2], [0, 1, 0, 2, 2], [-2, 1, 1, 0, 0], [0, -1, 0, 2, 2], [2, 0, 0, 2, 1], [2, 0, 1, 0, 0], [-2, 0, 2, 2, 2], [-2, 0, 1, 2, 1], [2, 0, -2, 0, 1], [2, 0, 0, 0, 1], [0, -1, 1, 0, 0], [-2, -1, 0, 2, 1], [-2, 0, 0, 0, 1], [0, 0, 2, 2, 1], [-2, 0, 2, 0, 1], [-2, 1, 0, 2, 1], [0, 0, 1, -2, 0], [-1, 0, 1, 0, 0], [-2, 1, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 1, 2, 0], [0, 0, -2, 2, 2], [-1, -1, 1, 0, 0], [0, 1, 1, 0, 0], [0, -1, 1, 2, 2], [2, -1, -1, 2, 2], [0, 0, 3, 2, 2], [2, -1, 0, 2, 2]]¶: This table contains the periodic terms for the argument of the nutation. In Meeus’ book this is Table 22.A and can be found in pages 145-146.

pymeeus.Coordinates.NUTATION_COSINE_COEF_TABLE = [[92025.0, 8.9], [5736.0, -3.1], [977.0, -0.5], [-895.0, 0.5], [54.0, -0.1], [-7.0, 0.0], [224.0, -0.6], [200.0, 0.0], [129.0, -0.1], [-95.0, 0.3], [0.0, 0.0], [-70.0, 0.0], [-53.0, 0.0], [0.0, 0.0], [-33.0, 0.0], [26.0, 0.0], [32.0, 0.0], [27.0, 0.0], [0.0, 0.0], [-24.0, 0.0], [16.0, 0.0], [13.0, 0.0], [0.0, 0.0], [-12.0, 0.0], [0.0, 0.0], [0.0, 0.0], [-10.0, 0.0], [0.0, 0.0], [-8.0, 0.0], [7.0, 0.0], [9.0, 0.0], [7.0, 0.0], [6.0, 0.0], [0.0, 0.0], [5.0, 0.0], [3.0, 0.0], [-3.0, 0.0], [0.0, 0.0], [3.0, 0.0], [3.0, 0.0], [0.0, 0.0], [-3.0, 0.0], [-3.0, 0.0], [3.0, 0.0], [3.0, 0.0], [0.0, 0.0], [3.0, 0.0], [3.0, 0.0], [3.0, 0.0]]¶: This table contains the periodic terms for the coefficients of the cosine of the argument of the nutation, and they are used to compute Delta epsilon. Units are in 0.0001’’. In Meeus’ book this is Table 22.A and can be found in pages 145-146.

pymeeus.Coordinates.NUTATION_SINE_COEF_TABLE = [[-171996.0, -174.2], [-13187.0, -1.6], [-2274.0, -0.2], [2062.0, 0.2], [1426.0, -3.4], [712.0, 0.1], [-517.0, 1.2], [-386.0, -0.4], [-301.0, 0.0], [217.0, -0.5], [-158.0, 0.0], [129.0, 0.1], [123.0, 0.0], [63.0, 0.0], [63.0, 0.1], [-59.0, 0.0], [-58.0, -0.1], [-51.0, 0.0], [48.0, 0.0], [46.0, 0.0], [-38.0, 0.0], [-31.0, 0.0], [29.0, 0.0], [29.0, 0.0], [26.0, 0.0], [-22.0, 0.0], [21.0, 0.0], [17.0, -0.1], [16.0, 0.0], [-16.0, 0.1], [-15.0, 0.0], [-13.0, 0.0], [-12.0, 0.0], [11.0, 0.0], [-10.0, 0.0], [-8.0, 0.0], [7.0, 0.0], [-7.0, 0.0], [-7.0, 0.0], [-7.0, 0.0], [6.0, 0.0], [6.0, 0.0], [6.0, 0.0], [-6.0, 0.0], [-6.0, 0.0], [5.0, 0.0], [-5.0, 0.0], [-5.0, 0.0], [-5.0, 0.0], [4.0, 0.0], [4.0, 0.0], [4.0, 0.0], [-4.0, 0.0], [-4.0, 0.0], [-4.0, 0.0], [3.0, 0.0], [-3.0, 0.0], [-3.0, 0.0], [-3.0, 0.0], [-3.0, 0.0], [-3.0, 0.0], [-3.0, 0.0], [-3.0, 0.0]]¶: This table contains the periodic terms for the coefficients of the sine of the argument of the nutation, and they are used to compute Delta psi. Units are in 0.0001’’. In Meeus’ book this is Table 22.A and can be found in pages 145-146.

pymeeus.Coordinates.angular_separation(alpha1, delta1, alpha2, delta2)[source]¶

This function computes the angular distance between two celestial bodies whose right ascensions and declinations are given.

Note

It is possible to use this formula with ecliptial (celestial) longitudes and latitudes instead of right ascensions and declinations, respectively.

Parameters:	alpha1 (`Angle`) – Right ascension of celestial body #1, as an Angle object delta1 (`Angle`) – Declination of celestial body #1, as an Angle object alpha2 (`Angle`) – Right ascension of celestial body #2, as an Angle object delta2 (`Angle`) – Declination of celestial body #2, as an Angle object
Returns:	An Angle object with the angular separation between the given celestial objects
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> alpha1 = Angle(14, 15, 39.7, ra=True)
>>> delta1 = Angle(19, 10, 57.0)
>>> alpha2 = Angle(13, 25, 11.6, ra=True)
>>> delta2 = Angle(-11, 9, 41.0)
>>> sep_ang = angular_separation(alpha1, delta1, alpha2, delta2)
>>> print(round(sep_ang, 3))
32.793

pymeeus.Coordinates.apparent_position(epoch, alpha, delta, sun_lon)[source]¶

This function computes the apparent position of a star, correcting by nutation and aberration effects.

Parameters:	epoch (`Epoch`) – Epoch to compute the apparent position for alpha (`Angle`) – Right ascension of the star, as an Angle object delta (`Angle`) – Declination of the star, as an Angle object sun_lon (`Angle`) – True (geometric) longitude of the Sun
Returns:	A tuple with two Angle objects: Apparent right ascension, and aparent declination
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2028, 11, 13.19)
>>> alpha = Angle(2, 46, 11.331, ra=True)
>>> delta = Angle(49, 20, 54.54)
>>> sun_lon = Angle(231.328)
>>> app_alpha, app_delta = apparent_position(epoch, alpha, delta, sun_lon)
>>> print(app_alpha.ra_str(n_dec=2))
2h 46' 14.39''
>>> print(app_delta.dms_str(n_dec=2))
49d 21' 7.45''

pymeeus.Coordinates.apparent_vsop_pos(epoch, vsop_l, vsop_b, vsop_r, nutation=True)[source]¶

This function computes the apparent position of a celestial body at a given epoch when its VSOP87 periodic term tables are provided. The small correction to convert to the FK5 system is always included.

Parameters:	epoch (`Epoch`) – Epoch to compute the position, given as an `Epoch` object vsop_l (list) – Table of VSOP87 terms for the heliocentric longitude vsop_b (list) – Table of VSOP87 terms for the heliocentric latitude vsop_r (list) – Table of VSOP87 terms for the radius vector nutation – Whether the nutation correction will be applied
Returns:	A tuple with the geometric heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

pymeeus.Coordinates.circle_diameter(alpha1, delta1, alpha2, delta2, alpha3, delta3)[source]¶

This function computes the diameter of the smallest circle that contains three celestial bodies.

Parameters:	alpha1 (`Angle`) – Right ascension, as an Angle object, of celestial body #1 delta1 (`Angle`) – Declination, as an Angle object, of celestial body #1 alpha2 (`Angle`) – Right ascension, as an Angle object, of celestial body #2 delta2 (`Angle`) – Declination, as an Angle object, of celestial body #2 alpha3 (`Angle`) – Right ascension, as an Angle object, of celestial body #3 delta3 (`Angle`) – Declination, as an Angle object, of celestial body #3
Returns:	The diameter (as an Angle object) of the smallest circle containing the three bodies.
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> alpha1 = Angle(12, 41,  8.63, ra=True)
>>> delta1 = Angle(-5, 37, 54.2)
>>> alpha2 = Angle(12, 52,  5.21, ra=True)
>>> delta2 = Angle(-4, 22, 26.2)
>>> alpha3 = Angle(12, 39, 28.11, ra=True)
>>> delta3 = Angle(-1, 50,  3.7)
>>> d = circle_diameter(alpha1, delta1, alpha2, delta2, alpha3, delta3)
>>> print(d.dms_str(n_dec=0))
4d 15' 49.0''
>>> alpha1 = Angle(9,  5, 41.44, ra=True)
>>> delta1 = Angle(18, 30, 30.0)
>>> alpha2 = Angle(9,  9, 29.0, ra=True)
>>> delta2 = Angle(17, 43, 56.7)
>>> alpha3 = Angle(8, 59, 47.14, ra=True)
>>> delta3 = Angle(17, 49, 36.8)
>>> d = circle_diameter(alpha1, delta1, alpha2, delta2, alpha3, delta3)
>>> print(d.dms_str(n_dec=0))
2d 18' 38.0''

pymeeus.Coordinates.diurnal_path_horizon(declination, geo_latitude)[source]¶

This function returns the angle of the diurnal path of a celestial body relative to the horizon at the time of its rising or setting.

Parameters:	declination (`Angle`) – Declination, as an Angle object geo_latitude (`Angle`) – Geodetic latitude, as an Angle object
Returns:	Angle of the diurnal path of the celestial body relative to the horizon at the time of rising or setting, given as one `Angle` object
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> declination = Angle(23.44)
>>> latitude = Angle(40.0)
>>> path_angle = diurnal_path_horizon(declination, latitude)
>>> print(path_angle.dms_str(n_dec=1))
45d 31' 28.4''

pymeeus.Coordinates.ecliptic_equator(longitude, latitude, obliquity)[source]¶

This function returns the angle between the direction of the northern celestial pole and the direction of the north pole of the ecliptic, taking as reference the point whose ecliptic longitude and latitude are given.

Please note that if we make latitude=0, the result is the angle between the ecliptic (at the given ecliptical longitude) and the east-west direction on the celestial sphere.

Parameters:	longitude (`Angle`) – Ecliptical longitude, as an Angle object latitude (`Angle`) – Ecliptical latitude, as an Angle object obliquity (`Angle`) – Obliquity of the ecliptic, as an Angle object
Returns:	Angle between the direction of the northern celestial pole and the direction of the north pole of the ecliptic, given as one `Angle` object
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> lon = Angle(0.0)
>>> lat = Angle(0.0)
>>> eps = Angle(23.5)
>>> ang_ecl_equ = ecliptic_equator(lon, lat, eps)
>>> print(ang_ecl_equ.dms_str(n_dec=1))
156d 30' 0.0''

pymeeus.Coordinates.ecliptic_horizon(local_sidereal_time, geo_latitude, obliquity)[source]¶

This function returns the longitudes of the two points of the ecliptic which are on the horizon, as well as the angle between the ecliptic and the horizon.

Parameters:	local_sidereal_time (`Angle`) – Local sidereal time, as an Angle object geo_latitude (`Angle`) – Geodetic latitude, as an Angle object obliquity (`Angle`) – Obliquity of the ecliptic, as an Angle object
Returns:	Longitudes of the two points of the ecliptic which are on the horizon, and the angle between the ecliptic and the horizon (in that order), given as three `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> sidereal_time = Angle(5.0, ra=True)
>>> lat = Angle(51.0)
>>> epsilon = Angle(23.44)
>>> lon1, lon2, i = ecliptic_horizon(sidereal_time, lat, epsilon)
>>> print(lon1.dms_str(n_dec=1))
169d 21' 29.9''
>>> print(lon2.dms_str(n_dec=1))
349d 21' 29.9''
>>> print(round(i, 0))
62.0

pymeeus.Coordinates.ecliptical2equatorial(longitude, latitude, obliquity)[source]¶

This function converts from ecliptical coordinates (longitude and latitude) to equatorial coordinated (right ascension and declination).

Parameters:	longitude (`Angle`) – Ecliptical longitude, as an Angle object latitude (`Angle`) – Ecliptical latitude, as an Angle object obliquity (`Angle`) – Obliquity of the ecliptic, as an Angle object
Returns:	Equatorial coordinates (right ascension, declination, in that order), given as two `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> lon = Angle(113.21563)
>>> lat = Angle(6.68417)
>>> epsilon = Angle(23.4392911)
>>> ra, dec = ecliptical2equatorial(lon, lat, epsilon)
>>> print(ra.ra_str(n_dec=3))
7h 45' 18.946''
>>> print(dec.dms_str(n_dec=2))
28d 1' 34.26''

pymeeus.Coordinates.equatorial2ecliptical(right_ascension, declination, obliquity)[source]¶

This function converts from equatorial coordinated (right ascension and declination) to ecliptical coordinates (longitude and latitude).

Parameters:	right_ascension – Right ascension, as an Angle object declination – Declination, as an Angle object obliquity (`Angle`) – Obliquity of the ecliptic, as an Angle object
Returns:	Ecliptical coordinates (longitude, latitude, in that order), given as two `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> ra = Angle(7, 45, 18.946, ra=True)
>>> dec = Angle(28, 1, 34.26)
>>> epsilon = Angle(23.4392911)
>>> lon, lat = equatorial2ecliptical(ra, dec, epsilon)
>>> print(round(lon(), 5))
113.21563
>>> print(round(lat(), 5))
6.68417

pymeeus.Coordinates.equatorial2galactic(right_ascension, declination)[source]¶

This function converts from equatorial coordinates (right ascension and declination) to galactic coordinates (longitude and latitude).

The current galactic system of coordinates was defined by the International Astronomical Union in 1959, using the standard equatorial system of epoch B1950.0.

Parameters:	right_ascension (`Angle`) – Right ascension, as an Angle object declination (`Angle`) – Declination, as an Angle object
Returns:	Galactic coordinates (longitude and latitude, in that order), given as two `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> ra = Angle(17, 48, 59.74, ra=True)
>>> dec = Angle(-14, 43, 8.2)
>>> lon, lat = equatorial2galactic(ra, dec)
>>> print(round(lon, 4))
12.9593
>>> print(round(lat, 4))
6.0463

pymeeus.Coordinates.equatorial2horizontal(hour_angle, declination, geo_latitude)[source]¶

This function converts from equatorial coordinates (right ascension and declination) to local horizontal coordinates (azimuth and elevation).

Following Meeus’ convention, the azimuth is measured westward from the SOUTH. If you want the azimuth to be measured from the north (common custom between navigators and meteorologits), you should add 180 degrees.

The hour angle (H) comprises information about the sidereal time, the observer’s geodetic longitude (positive west from Greenwich) and the right ascension. If theta is the local sidereal time, theta0 the sidereal time at Greenwich, lon the observer’s longitude and ra the right ascension, the following expressions hold:

H = theta - ra H = theta0 - lon - ra

Parameters:	hour_angle (`Angle`) – Hour angle, as an Angle object declination (`Angle`) – Declination, as an Angle object geo_latitude (`Angle`) – Geodetic latitude of the observer, as an Angle object
Returns:	Local horizontal coordinates (azimuth, elevation, in that order), given as two `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> lon = Angle(77, 3, 56)
>>> lat = Angle(38, 55, 17)
>>> ra = Angle(23, 9, 16.641, ra=True)
>>> dec = Angle(-6, 43, 11.61)
>>> theta0 = Angle(8, 34, 57.0896, ra=True)
>>> eps = Angle(23, 26, 36.87)
>>> delta = Angle(0, 0, ((-3.868*cos(eps.rad()))/15.0), ra=True)
>>> theta0 += delta
>>> h = theta0 - lon - ra
>>> azi, ele = equatorial2horizontal(h, dec, lat)
>>> print(round(azi, 3))
68.034
>>> print(round(ele, 3))
15.125

pymeeus.Coordinates.galactic2equatorial(longitude, latitude)[source]¶

This function converts from galactic coordinates (longitude and latitude) to equatorial coordinates (right ascension and declination).

The current galactic system of coordinates was defined by the International Astronomical Union in 1959, using the standard equatorial system of epoch B1950.0.

Parameters:	longitude (`Angle`) – Longitude, as an Angle object latitude (`Angle`) – Latitude, as an Angle object
Returns:	Equatorial coordinates (right ascension and declination, in that order), given as two `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> lon = Angle(12.9593)
>>> lat = Angle(6.0463)
>>> ra, dec = galactic2equatorial(lon, lat)
>>> print(ra.ra_str(n_dec=1))
17h 48' 59.7''
>>> print(dec.dms_str(n_dec=0))
-14d 43' 8.0''

pymeeus.Coordinates.geometric_vsop_pos(epoch, vsop_l, vsop_b, vsop_r, tofk5=True)[source]¶

This function computes the geometric position of a celestial body at a given epoch when its VSOP87 periodic term tables are provided. The small correction to convert to the FK5 system may or not be included.

Parameters:	epoch (`Epoch`) – Epoch to compute the position, given as an `Epoch` object vsop_l (list) – Table of VSOP87 terms for the heliocentric longitude vsop_b (list) – Table of VSOP87 terms for the heliocentric latitude vsop_r (list) – Table of VSOP87 terms for the radius vector tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied
Returns:	A tuple with the geometric heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

pymeeus.Coordinates.horizontal2equatorial(azimuth, elevation, geo_latitude)[source]¶

This function converts from local horizontal coordinates (azimuth and elevation) to equatorial coordinates (right ascension and declination).

Following Meeus’ convention, the azimuth is measured westward from the SOUTH.

This function returns the hour angle and the declination. The hour angle (H) comprises information about the sidereal time, the observer’s geodetic longitude (positive west from Greenwich) and the right ascension. If theta is the local sidereal time, theta0 the sidereal time at Greenwich, lon the observer’s longitude and ra the right ascension, the following expressions hold:

H = theta - ra H = theta0 - lon - ra

Parameters:	azimuth (`Angle`) – Azimuth, measured westward from south, as an Angle object elevation (`Angle`) – Elevation from the horizon, as an Angle object geo_latitude (`Angle`) – Geodetic latitude of the observer, as an Angle object
Returns:	Equatorial coordinates (as hour angle and declination, in that order), given as two `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> azi = Angle(68.0337)
>>> ele = Angle(15.1249)
>>> lat = Angle(38, 55, 17)
>>> h, dec = horizontal2equatorial(azi, ele, lat)
>>> print(round(h, 4))
64.3521
>>> print(dec.dms_str(n_dec=0))
-6d 43' 12.0''

pymeeus.Coordinates.illuminated_fraction(sun_dist, earth_dist, sun_earth_dist)[source]¶

This function computes the illuminated fraction of the disk of a planet, as seen from the Earth.

Parameters:	sun_dist (float) – Planet’s distance to the Sun, in Astronomical Units earth_dist (float) – Distance from planet to Earth, in Astronomical Units sun_earth_dist (float) – Distance Sun-Earth, in Astronomical Units
Returns:	The illuminated fraction of the disc of a planet
Return type:	float
Raises:	TypeError if input values are of wrong type.

>>> sun_dist = 0.724604
>>> earth_dist = 0.910947
>>> sun_earth_dist = 0.983824
>>> k = illuminated_fraction(sun_dist, earth_dist, sun_earth_dist)
>>> print(round(k, 3))
0.647

pymeeus.Coordinates.kepler_equation(eccentricity, mean_anomaly)[source]¶

This function computes the eccentric and true anomalies taking as input the mean anomaly and the eccentricity.

Parameters:	eccentricity (int, float) – Orbit’s eccentricity mean_anomaly (`Angle`) – Mean anomaly, as an Angle object
Returns:	A tuple with two Angle objects: Eccentric and true anomalies
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> eccentricity = 0.1
>>> mean_anomaly = Angle(5.0)
>>> e, v = kepler_equation(eccentricity, mean_anomaly)
>>> print(round(e(), 6))
5.554589
>>> print(round(v(), 6))
6.139762
>>> eccentricity = 0.99
>>> mean_anomaly = Angle(2.0)
>>> e, v = kepler_equation(eccentricity, mean_anomaly)
>>> print(round(e(), 6))
32.361007
>>> print(round(v(), 6))
152.542134
>>> eccentricity = 0.99
>>> mean_anomaly = Angle(5.0)
>>> e, v = kepler_equation(eccentricity, mean_anomaly)
>>> print(round(e(), 6))
45.361023
>>> print(round(v(), 6))
160.745616
>>> eccentricity = 0.99
>>> mean_anomaly = Angle(1.0)
>>> e, v = kepler_equation(eccentricity, mean_anomaly)
>>> print(round(e(), 6))
24.725822
>>> print(round(v(), 6))
144.155952
>>> e, v = kepler_equation(0.999, Angle(7.0))
>>> print(round(e(), 7))
52.2702615
>>> print(round(v(), 6))
174.780018
>>> e, v = kepler_equation(0.99, Angle(0.2, radians=True))
>>> print(round(e(), 8))
61.13444578
>>> print(round(v(), 6))
166.311977

pymeeus.Coordinates.length_orbit(e, a)[source]¶

This function computes the length of an elliptic orbit given its eccentricity and semimajor axis.

Parameters:	e (float) – Orbital eccentricity a (float) – Semimajor axis of the orbit, in Astronomical Units
Returns:	Length of the orbit in Astronomical Units
Return type:	float
Raises:	TypeError if input values are of wrong type.

>>> a = 17.9400782
>>> e = 0.96727426
>>> length = length_orbit(e, a)
>>> print(round(length, 2))
77.06

pymeeus.Coordinates.mean_obliquity(*args, **kwargs)[source]¶

This function computes the mean obliquity (epsilon0) at the provided date.

This function internally uses an Epoch object, and the utc argument then controls the way the UTC->TT conversion is handled for that object. If leap_seconds argument is set to a value different than zero, then that value will be used for the UTC->TAI conversion, and the internal leap seconds table will be bypassed.

Parameters:	args (int, float, `Epoch`, datetime, date, tuple, list) – Either `Epoch`, date, datetime or year, month, day values, by themselves or inside a tuple or list utc (bool) – Whether the provided epoch is a civil time (UTC) or TT leap_seconds (int, float) – This is the value to be used in the UTC->TAI conversion, instead of taking it from internal leap seconds table.
Returns:	The mean obliquity of the ecliptic, as an `Angle`
Return type:	`Angle`
Raises:	ValueError if input values are in the wrong range.
Raises:	TypeError if input values are of wrong type.

>>> e0 = mean_obliquity(1987, 4, 10)
>>> a = e0.dms_tuple()
>>> a[0]
23
>>> a[1]
26
>>> round(a[2], 3)
27.407

pymeeus.Coordinates.minimum_angular_separation(alpha1_1, delta1_1, alpha1_2, delta1_2, alpha1_3, delta1_3, alpha2_1, delta2_1, alpha2_2, delta2_2, alpha2_3, delta2_3)[source]¶

Given the positions at three different instants of times (equidistant) of two celestial objects, this function computes the minimum angular distance that will be achieved within that interval of time.

Note

Suffix ‘1 _’ is for the first celestial object, and ‘2 _’ is for the second one.

Note

This function provides as output the ‘n’ fraction of time when the minimum angular separation is achieved. For that, the epoch in the middle is assigned the value “n = 0”. Therefore, n < 0 is for times before the middle epoch, and n > 0 is for times after the middle epoch.

Parameters:	alpha1_1 (`Angle`) – First right ascension of celestial body #1, as an Angle object delta1_1 (`Angle`) – First declination of celestial body #1, as an Angle object alpha1_2 (`Angle`) – Second right ascension of celestial body #1, as an Angle object delta1_2 (`Angle`) – Second declination of celestial body #1, as Angle object alpha1_3 (`Angle`) – Third right ascension of celestial body #1, as an Angle object delta1_3 (`Angle`) – Third declination of celestial body #1, as an Angle object alpha2_1 (`Angle`) – First right ascension of celestial body #2, as an Angle object delta2_1 (`Angle`) – First declination of celestial body #2, as an Angle object alpha2_2 (`Angle`) – Second right ascension of celestial body #2, as an Angle object delta2_2 (`Angle`) – Second declination of celestial body #2, as Angle object alpha2_3 (`Angle`) – Third right ascension of celestial body #2, as an Angle object delta2_3 (`Angle`) – Third declination of celestial body #2, as an Angle object
Returns:	A tuple with two components: The first component is a float containing the ‘n’ fraction of time when the minimum angular separation is achieved. The second component is an Angle object containing the minimum angular separation between the given celestial objects
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> alpha1_1 = Angle(10, 29, 44.27, ra=True)
>>> delta1_1 = Angle(11, 2, 5.9)
>>> alpha2_1 = Angle(10, 33, 29.64, ra=True)
>>> delta2_1 = Angle(10, 40, 13.2)
>>> alpha1_2 = Angle(10, 36, 19.63, ra=True)
>>> delta1_2 = Angle(10, 29, 51.7)
>>> alpha2_2 = Angle(10, 33, 57.97, ra=True)
>>> delta2_2 = Angle(10, 37, 33.4)
>>> alpha1_3 = Angle(10, 43, 1.75, ra=True)
>>> delta1_3 = Angle(9, 55, 16.7)
>>> alpha2_3 = Angle(10, 34, 26.22, ra=True)
>>> delta2_3 = Angle(10, 34, 53.9)
>>> a = minimum_angular_separation(alpha1_1, delta1_1, alpha1_2, delta1_2,                                       alpha1_3, delta1_3, alpha2_1, delta2_1,                                       alpha2_2, delta2_2, alpha2_3, delta2_3)
>>> print(round(a[0], 6))
-0.370726
>>> print(a[1].dms_str(n_dec=0))
3' 44.0''

pymeeus.Coordinates.motion_in_space(start_ra, start_dec, distance, velocity, p_motion_ra, p_motion_dec, time)[source]¶

This function computes the star’s true motion through space relative to the Sun, allowing to compute the start proper motion at a given time.

Parameters:	start_ra (`Angle`) – Initial right ascension start_dec (`Angle`) – Initial declination distance (float) – Star’s distance to the Sun, in parsecs. If distance is given in light-years, multipy it by 0.3066. If the star’s parallax pie (in arcseconds) is given, use (1.0/pie). velocity (float) – Radial velocity in km/s p_motion_ra (`Angle`) – Proper motion in right ascension, in degrees per year. p_motion_dec (`Angle`) – Proper motion in declination, in degrees per year. time (float) – Number of years since starting epoch, positive in the future, negative in the past
Returns:	Equatorial coordinates (right ascension, declination, in that order) corresponding to the final epoch, given as two objects `Angle` inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> ra = Angle(6, 45, 8.871, ra=True)
>>> dec = Angle(-16.716108)
>>> pm_ra = Angle(0, 0, -0.03847, ra=True)
>>> pm_dec = Angle(0, 0, -1.2053)
>>> dist = 2.64
>>> vel = -7.6
>>> alpha, delta = motion_in_space(ra, dec, dist, vel, pm_ra, pm_dec,
...                                -1000.0)
>>> print(alpha.ra_str(False, 2))
6:45:47.16
>>> print(delta.dms_str(False, 1))
-16:22:56.0
>>> alpha, delta = motion_in_space(ra, dec, dist, vel, pm_ra, pm_dec,
...                                -4000.0)
>>> print(alpha.ra_str(False, 2))
6:47:39.91
>>> print(delta.dms_str(False, 1))
-15:23:30.6

pymeeus.Coordinates.nutation_longitude(*args, **kwargs)[source]¶

This function computes the nutation in longitude (Delta psi) at the provided date.

This function internally uses an Epoch object, and the utc argument then controls the way the UTC->TT conversion is handled for that object. If leap_seconds argument is set to a value different than zero, then that value will be used for the UTC->TAI conversion, and the internal leap seconds table will be bypassed.

Parameters:	args (int, float, `Epoch`, datetime, date, tuple, list) – Either `Epoch`, date, datetime or year, month, day values, by themselves or inside a tuple or list utc (bool) – Whether the provided epoch is a civil time (UTC) or TT leap_seconds (int, float) – This is the value to be used in the UTC->TAI conversion, instead of taking it from internal leap seconds table.
Returns:	The nutation in longitude (Delta psi), as an Angle
Return type:	`Angle`
Raises:	ValueError if input values are in the wrong range.
Raises:	TypeError if input values are of wrong type.

>>> dpsi = nutation_longitude(1987, 4, 10)
>>> a = dpsi.dms_tuple()
>>> a[0]
0
>>> a[1]
0
>>> round(a[2], 3)
3.788
>>> a[3]
-1.0

pymeeus.Coordinates.nutation_obliquity(*args, **kwargs)[source]¶

This function computes the nutation in obliquity (Delta epsilon) at the provided date.

This function internally uses an Epoch object, and the utc argument then controls the way the UTC->TT conversion is handled for that object. If leap_seconds argument is set to a value different than zero, then that value will be used for the UTC->TAI conversion, and the internal leap seconds table will be bypassed.

Parameters:	args (int, float, `Epoch`, datetime, date, tuple, list) – Either `Epoch`, date, datetime or year, month, day values, by themselves or inside a tuple or list utc (bool) – Whether the provided epoch is a civil time (UTC) or TT leap_seconds (int, float) – This is the value to be used in the UTC->TAI conversion, instead of taking it from internal leap seconds table.
Returns:	The nutation in obliquity (Delta epsilon), as an `Angle`
Return type:	`Angle`
Raises:	ValueError if input values are in the wrong range.
Raises:	TypeError if input values are of wrong type.

>>> depsilon = nutation_obliquity(1987, 4, 10)
>>> a = depsilon.dms_tuple()
>>> a[0]
0
>>> a[1]
0
>>> round(a[2], 3)
9.443
>>> a[3]
1.0

pymeeus.Coordinates.orbital_elements(epoch, parameters1, parameters2)[source]¶

This function computes the orbital elements for a given epoch, according to the parameters beeing passed as arguments.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object parameters1 (list) – First set of parameters parameters2 (list) – Second set of parameters
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

pymeeus.Coordinates.orbital_equinox2equinox(epoch0, epoch, i0, arg0, lon0)[source]¶

This function reduces the orbital elements of a celestial object from one equinox to another.

Parameters:	epoch0 (`Epoch`) – Initial epoch epoch (`Epoch`) – Final epoch i0 (`Angle`) – Initial inclination, as an Angle object arg0 (`Angle`) – Initial argument of perihelion, as an Angle object lon0 (`Angle`) – Initial longitude of ascending node, as an Angle object
Returns:	A tuple with three Angle objects: Final inclination, argument of perihelion and longitude of ascending node, in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch0 = Epoch(2358042.5305)
>>> epoch = Epoch(2433282.4235)
>>> i0 = Angle(47.122)
>>> arg0 = Angle(151.4486)
>>> lon0 = Angle(45.7481)
>>> i1, arg1, lon1 = orbital_equinox2equinox(epoch0, epoch, i0, arg0, lon0)
>>> print(round(i1(), 3))
47.138
>>> print(round(arg1(), 4))
151.4782
>>> print(round(lon1(), 4))
48.6037

pymeeus.Coordinates.p_motion_equa2eclip(p_motion_ra, p_motion_dec, ra, dec, lat, epsilon)[source]¶

It is usual that proper motions are given in equatorial coordinates, not in ecliptical ones. Therefore, this function converts the provided proper motions in equatorial coordinates to the corresponding ones in ecliptical coordinates.

Parameters:	p_motion_ra (`Angle`) – Proper motion in right ascension, in degrees per year, as an `Angle` object p_motion_dec (`Angle`) – Proper motion in declination, in degrees per year, as an `Angle` object ra (`Angle`) – Right ascension of the astronomical object, as degrees in an `Angle` object dec (`Angle`) – Declination of the astronomical object, as degrees in an `Angle` object lat (`Angle`) – Ecliptical latitude of the astronomical object, as degrees in an `Angle` object epsilon (`Angle`) – Obliquity of the ecliptic
Returns:	Proper motions in ecliptical longitude and latitude (in that order), given as two `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

pymeeus.Coordinates.parallactic_angle(hour_angle, declination, geo_latitude)[source]¶

This function computes the parallactic angle, an apparent rotation that appears because celestial bodies move along parallel circles. By convention, the parallactic angle is negative before the passage through the southern meridian (in the north hemisphere), and positive afterwards. Exactly on the meridian, its value is zero.

Please note that when the celestial body is exactly at the zenith, the parallactic angle is not defined, and this function will return ‘None’.

The hour angle (H) comprises information about the sidereal time, the observer’s geodetic longitude (positive west from Greenwich) and the right ascension. If theta is the local sidereal time, theta0 the sidereal time at Greenwich, lon the observer’s longitude and ra the right ascension, the following expressions hold:

H = theta - ra H = theta0 - lon - ra

Parameters:	hour_angle (`Angle`) – Hour angle, as an Angle object declination (`Angle`) – Declination, as an Angle object geo_latitude (`Angle`) – Geodetic latitude of the observer, as an Angle object
Returns:	Parallactic angle as an py:class:Angle object
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> hour_angle = Angle(0.0)
>>> declination = Angle(45.0)
>>> latitude = Angle(50.0)
>>> q = parallactic_angle(hour_angle, declination, latitude)
>>> print(q.dms_str(n_dec=1))
0d 0' 0.0''

pymeeus.Coordinates.passage_nodes_elliptic(omega, e, a, t, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of a given celestial object with an elliptic orbit.

Parameters:	omega (`Angle`) – Argument of the perihelion e (float) – Orbital eccentricity a (float) – Semimajor axis of the orbit, in Astronomical Units t (`Epoch`) – Time of perihelion passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> omega = Angle(111.84644)
>>> e = 0.96727426
>>> a = 17.9400782
>>> t = Epoch(1986, 2, 9.45891)
>>> time, r = passage_nodes_elliptic(omega, e, a, t)
>>> year, month, day = time.get_date()
>>> print(year)
1985
>>> print(month)
11
>>> print(round(day, 2))
9.16
>>> print(round(r, 4))
1.8045
>>> time, r = passage_nodes_elliptic(omega, e, a, t, ascending=False)
>>> year, month, day = time.get_date()
>>> print(year)
1986
>>> print(month)
3
>>> print(round(day, 2))
10.37
>>> print(round(r, 4))
0.8493

pymeeus.Coordinates.passage_nodes_parabolic(omega, q, t, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of a given celestial object with a parabolic orbit.

Parameters:	omega (`Angle`) – Argument of the perihelion q (float) – Perihelion distance, in Astronomical Units t (`Epoch`) – Time of perihelion passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> omega = Angle(154.9103)
>>> q = 1.324502
>>> t = Epoch(1989, 8, 20.291)
>>> time, r = passage_nodes_parabolic(omega, q, t)
>>> year, month, day = time.get_date()
>>> print(year)
1977
>>> print(month)
9
>>> print(round(day, 2))
17.64
>>> print(round(r, 4))
28.0749
>>> time, r = passage_nodes_parabolic(omega, q, t, ascending=False)
>>> year, month, day = time.get_date()
>>> print(year)
1989
>>> print(month)
9
>>> print(round(day, 3))
17.636
>>> print(round(r, 4))
1.3901

pymeeus.Coordinates.phase_angle(sun_dist, earth_dist, sun_earth_dist)[source]¶

This function computes the phase angle, i.e., the angle Sun-planet-Earth from the corresponding distances.

Parameters:	sun_dist (float) – Planet’s distance to the Sun, in Astronomical Units earth_dist (float) – Distance from planet to Earth, in Astronomical Units sun_earth_dist (float) – Distance Sun-Earth, in Astronomical Units
Returns:	The phase angle, as an Angle object
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> sun_dist = 0.724604
>>> earth_dist = 0.910947
>>> sun_earth_dist = 0.983824
>>> angle = phase_angle(sun_dist, earth_dist, sun_earth_dist)
>>> print(round(angle, 2))
72.96

pymeeus.Coordinates.planet_star_conjunction(alpha_list, delta_list, alpha_star, delta_star)[source]¶

Given the positions of one planet passing near a star, this function computes the time of conjunction in right ascension, and the difference in declination of the two bodies at that time.

Note

This function provides as output the ‘n’ fraction of time when the minimum angular separation is achieved. For that, the epoch in the middle is assigned the value “n = 0”. Therefore, n < 0 is for times before the middle epoch, and n > 0 is for times after the middle epoch.

Note

When the entries in the input values for the planet are more than three and pair, the last entry is discarted and an odd number of entries will be used.

Parameters:	alpha_list (list, tuple of `Angle`) – List (or tuple) containing the right ascensions (as Angle objects) for the planet (minimum 3 entries) delta_list (list, tuple of `Angle`) – List (or tuple) containing the declinations (as Angle objects) for the planet (minimum 3 entries) alpha_star (`Angle`) – Right ascension, as an Angle object, of the star delta_star (`Angle`) – Declination, as an Angle object, of the star
Returns:	A tuple with two components: The first component is a float containing the ‘n’ fraction of time when the conjunction occurs. The second component is an Angle object containing the declination separation between the given objects at conjunction epoch
Return type:	tuple
Raises:	ValueError if input values for planet have less than three entries or they don’t have the same number of entries.
Raises:	TypeError if input values are of wrong type.

>>> alpha_1 = Angle(15,  3, 51.937, ra=True)
>>> delta_1 = Angle(-8, 57, 34.51)
>>> alpha_2 = Angle(15,  9, 57.327, ra=True)
>>> delta_2 = Angle(-9,  9,  3.88)
>>> alpha_3 = Angle(15, 15, 37.898, ra=True)
>>> delta_3 = Angle(-9, 17, 37.94)
>>> alpha_4 = Angle(15, 20, 50.632, ra=True)
>>> delta_4 = Angle(-9, 23, 16.25)
>>> alpha_5 = Angle(15, 25, 32.695, ra=True)
>>> delta_5 = Angle(-9, 26,  1.01)
>>> alpha_star = Angle(15, 17, 0.446, ra=True)
>>> delta_star = Angle(-9, 22, 58.47)
>>> alpha_list = [alpha_1, alpha_2, alpha_3, alpha_4, alpha_5]
>>> delta_list = [delta_1, delta_2, delta_3, delta_4, delta_5]
>>> pc = planet_star_conjunction(alpha_list, delta_list,                                      alpha_star, delta_star)
>>> print(round(pc[0], 4))
0.2551
>>> print(pc[1].dms_str(n_dec=0))
3' 38.0''

pymeeus.Coordinates.planet_stars_in_line(alpha_list, delta_list, alpha_star1, delta_star1, alpha_star2, delta_star2)[source]¶

Given the positions of one planet, this function computes the time when it is in a straight line with two other stars.

Note

This function provides as output the ‘n’ fraction of time when the minimum angular separation is achieved. For that, the epoch in the middle is assigned the value “n = 0”. Therefore, n < 0 is for times before the middle epoch, and n > 0 is for times after the middle epoch.

Note

When the entries in the input values for the planet are more than three and pair, the last entry is discarted and an odd number of entries will be used.

Parameters:	alpha_list (list, tuple of `Angle`) – List (or tuple) containing the right ascensions (as Angle objects) for the planet (minimum 3 entries) delta_list (list, tuple of `Angle`) – List (or tuple) containing the declinations (as Angle objects) for the planet (minimum 3 entries) alpha_star1 (`Angle`) – Right ascension, as an Angle object, of star #1 delta_star1 (`Angle`) – Declination, as an Angle object, of star #1 alpha_star2 (`Angle`) – Right ascension, as an Angle object, of star #2 delta_star2 (`Angle`) – Declination, as an Angle object, of star #2
Returns:	A float containing the ‘n’ fraction of time when the alignment occurs.
Return type:	float
Raises:	ValueError if input values for planet have less than three entries or they don’t have the same number of entries.
Raises:	TypeError if input values are of wrong type.

>>> alpha_1 = Angle( 7, 55, 55.36, ra=True)
>>> delta_1 = Angle(21, 41,  3.0)
>>> alpha_2 = Angle( 7, 58, 22.55, ra=True)
>>> delta_2 = Angle(21, 35, 23.4)
>>> alpha_3 = Angle( 8,  0, 48.99, ra=True)
>>> delta_3 = Angle(21, 29, 38.2)
>>> alpha_4 = Angle( 8,  3, 14.66, ra=True)
>>> delta_4 = Angle(21, 23, 47.5)
>>> alpha_5 = Angle( 8,  5, 39.54, ra=True)
>>> delta_5 = Angle(21, 17, 51.4)
>>> alpha_star1 = Angle( 7, 34, 16.40, ra=True)
>>> delta_star1 = Angle(31, 53, 51.2)
>>> alpha_star2 = Angle( 7, 45,  0.10, ra=True)
>>> delta_star2 = Angle(28,  2, 12.5)
>>> alpha_list = [alpha_1, alpha_2, alpha_3, alpha_4, alpha_5]
>>> delta_list = [delta_1, delta_2, delta_3, delta_4, delta_5]
>>> n = planet_stars_in_line(alpha_list, delta_list, alpha_star1,                                  delta_star1, alpha_star2, delta_star2)
>>> print(round(n, 4))
0.2233

pymeeus.Coordinates.planetary_conjunction(alpha1_list, delta1_list, alpha2_list, delta2_list)[source]¶

Given the positions of two planets passing near each other, this function computes the time of conjunction in right ascension, and the difference in declination of the two bodies at that time.

Note

This function provides as output the ‘n’ fraction of time when the minimum angular separation is achieved. For that, the epoch in the middle is assigned the value “n = 0”. Therefore, n < 0 is for times before the middle epoch, and n > 0 is for times after the middle epoch.

Note

When the entries in the input values are more than three and even, the last entry is discarted and an odd number of entries will be used.

Parameters:	alpha1_list (list, tuple of `Angle`) – List (or tuple) containing the right ascensions (as Angle objects) for object #1 (minimum 3 entries) delta1_list (list, tuple of `Angle`) – List (or tuple) containing the declinations (as Angle objects) for object #1 (minimum 3 entries) alpha2_list (list, tuple of `Angle`) – List (or tuple) containing the right ascensions (as Angle objects) for object #2 (minimum 3 entries) delta2_list (list, tuple of `Angle`) – List (or tuple) containing the declinations (as Angle objects) for object #2 (minimum 3 entries)
Returns:	A tuple with two components: The first component is a float containing the ‘n’ fraction of time when the conjunction occurs. The second component is an Angle object containing the declination separation between the given objects at conjunction epoch
Return type:	tuple
Raises:	ValueError if input values have less than three entries or they don’t have the same number of entries.
Raises:	TypeError if input values are of wrong type.

>>> alpha1_1 = Angle(10, 24, 30.125, ra=True)
>>> delta1_1 = Angle( 6, 26, 32.05)
>>> alpha1_2 = Angle(10, 25,  0.342, ra=True)
>>> delta1_2 = Angle( 6, 10, 57.72)
>>> alpha1_3 = Angle(10, 25, 12.515, ra=True)
>>> delta1_3 = Angle( 5, 57, 33.08)
>>> alpha1_4 = Angle(10, 25,  6.235, ra=True)
>>> delta1_4 = Angle( 5, 46, 27.07)
>>> alpha1_5 = Angle(10, 24, 41.185, ra=True)
>>> delta1_5 = Angle( 5, 37, 48.45)
>>> alpha2_1 = Angle(10, 27, 27.175, ra=True)
>>> delta2_1 = Angle( 4,  4, 41.83)
>>> alpha2_2 = Angle(10, 26, 32.410, ra=True)
>>> delta2_2 = Angle( 3, 55, 54.66)
>>> alpha2_3 = Angle(10, 25, 29.042, ra=True)
>>> delta2_3 = Angle( 3, 48,  3.51)
>>> alpha2_4 = Angle(10, 24, 17.191, ra=True)
>>> delta2_4 = Angle( 3, 41, 10.25)
>>> alpha2_5 = Angle(10, 22, 57.024, ra=True)
>>> delta2_5 = Angle( 3, 35, 16.61)
>>> alpha1_list = [alpha1_1, alpha1_2, alpha1_3, alpha1_4, alpha1_5]
>>> delta1_list = [delta1_1, delta1_2, delta1_3, delta1_4, delta1_5]
>>> alpha2_list = [alpha2_1, alpha2_2, alpha2_3, alpha2_4, alpha2_5]
>>> delta2_list = [delta2_1, delta2_2, delta2_3, delta2_4, delta2_5]
>>> pc = planetary_conjunction(alpha1_list, delta1_list,                                    alpha2_list, delta2_list)
>>> print(round(pc[0], 5))
0.23797
>>> print(pc[1].dms_str(n_dec=1))
2d 8' 21.8''

pymeeus.Coordinates.precession_ecliptical(start_epoch, final_epoch, start_lon, start_lat, p_motion_lon=0.0, p_motion_lat=0.0)[source]¶

This function converts the ecliptical coordinates (longitude and latitude) given for an epoch and a equinox, to the corresponding values for another epoch and equinox. Only the mean positions, i.e. the effects of precession and proper motion, are considered here.

Parameters:	start_epoch (`Epoch`) – Initial epoch when initial coordinates are given final_epoch (`Epoch`) – Final epoch for when coordinates are going to be computed start_lon (`Angle`) – Initial longitude start_lat (`Angle`) – Initial latitude p_motion_lon (`Angle`) – Proper motion in longitude, in degrees per year. Zero by default. p_motion_lat (`Angle`) – Proper motion in latitude, in degrees per year. Zero by default.
Returns:	Ecliptical coordinates (longitude, latitude, in that order) corresponding to the final epoch, given as two `Angle` objects inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> start_epoch = JDE2000
>>> final_epoch = Epoch(-214, 6, 30.0)
>>> lon0 = Angle(149.48194)
>>> lat0 = Angle(1.76549)
>>> lon, lat = precession_ecliptical(start_epoch, final_epoch, lon0, lat0)
>>> print(round(lon(), 3))
118.704
>>> print(round(lat(), 3))
1.615

pymeeus.Coordinates.precession_equatorial(start_epoch, final_epoch, start_ra, start_dec, p_motion_ra=0.0, p_motion_dec=0.0)[source]¶

This function converts the equatorial coordinates (right ascension and declination) given for an epoch and a equinox, to the corresponding values for another epoch and equinox. Only the mean positions, i.e. the effects of precession and proper motion, are considered here.

Parameters:	start_epoch (`Epoch`) – Initial epoch when initial coordinates are given final_epoch (`Epoch`) – Final epoch for when coordinates are going to be computed start_ra (`Angle`) – Initial right ascension start_dec (`Angle`) – Initial declination p_motion_ra (`Angle`) – Proper motion in right ascension, in degrees per year. Zero by default. p_motion_dec (`Angle`) – Proper motion in declination, in degrees per year. Zero by default.
Returns:	Equatorial coordinates (right ascension, declination, in that order) corresponding to the final epoch, given as two objects `Angle` inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> start_epoch = JDE2000
>>> final_epoch = Epoch(2028, 11, 13.19)
>>> alpha0 = Angle(2, 44, 11.986, ra=True)
>>> delta0 = Angle(49, 13, 42.48)
>>> pm_ra = Angle(0, 0, 0.03425, ra=True)
>>> pm_dec = Angle(0, 0, -0.0895)
>>> alpha, delta = precession_equatorial(start_epoch, final_epoch, alpha0,
...                                      delta0, pm_ra, pm_dec)
>>> print(alpha.ra_str(False, 3))
2:46:11.331
>>> print(delta.dms_str(False, 2))
49:20:54.54

pymeeus.Coordinates.precession_newcomb(start_epoch, final_epoch, start_ra, start_dec, p_motion_ra=0.0, p_motion_dec=0.0)[source]¶

This function implements the Newcomb precessional equations used in the old FK4 system. It takes equatorial coordinates (right ascension and declination) given for an epoch and a equinox, and converts them to the corresponding values for another epoch and equinox. Only the mean positions, i.e. the effects of precession and proper motion, are considered here.

Parameters:	start_epoch (`Epoch`) – Initial epoch when initial coordinates are given final_epoch (`Epoch`) – Final epoch for when coordinates are going to be computed start_ra (`Angle`) – Initial right ascension start_dec (`Angle`) – Initial declination p_motion_ra (`Angle`) – Proper motion in right ascension, in degrees per year. Zero by default. p_motion_dec (`Angle`) – Proper motion in declination, in degrees per year. Zero by default.
Returns:	Equatorial coordinates (right ascension, declination, in that order) corresponding to the final epoch, given as two objects `Angle` inside a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

pymeeus.Coordinates.refraction_apparent2true(apparent_elevation, pressure=1010.0, temperature=10.0)[source]¶

This function computes the atmospheric refraction converting from the apparent elevation (i.e., the observed elevation through the air) to the true, ‘airless’, elevation.

Note

This function, by default, assumes that the atmospheric pressure is 1010 milibars, and the air temperature is 10 Celsius.

Note

Due to the numerous factors that may affect the atmospheric refraction, especially near the horizon, the values given by this function are approximate values.

Parameters:	apparent_elevation (`Angle`) – The elevation, in degrees and as an Angle object, of a given celestial object when observed through the normal atmosphere pressure (float) – Atmospheric pressure at the observation point, in milibars temperature (:float) – Atmospheric temperature at the observation point, in degrees Celsius
Returns:	An Angle object with the true, ‘airless’ elevation of the celestial object
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> apparent_elevation = Angle(0, 30, 0.0)
>>> true_elevation = refraction_apparent2true(apparent_elevation)
>>> print(true_elevation.dms_str(n_dec=1))
1' 14.7''

pymeeus.Coordinates.refraction_true2apparent(true_elevation, pressure=1010.0, temperature=10.0)[source]¶

This function computes the atmospheric refraction converting from the true, ‘airless’, elevation (i.e., the one computed from celestial coordinates) to the apparent elevation (the observed elevation through the air)

Note

This function, by default, assumes that the atmospheric pressure is 1010 milibars, and the air temperature is 10 Celsius.

Note

Due to the numerous factors that may affect the atmospheric refraction, especially near the horizon, the values given by this function are approximate values.

Parameters:	true_elevation (`Angle`) – The elevation, in degrees and as an Angle object, of a given celestial object when computed from celestial coordinates, and assuming there is no atmospheric refraction due to the air pressure (float) – Atmospheric pressure at the observation point, in milibars temperature (:float) – Atmospheric temperature at the observation point, in degrees Celsius
Returns:	An Angle object with the aparent, ‘with air’ elevation of the celestial object
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> true_elevation = Angle(0, 33, 14.76)
>>> apparent_elevation = refraction_true2apparent(true_elevation)
>>> print(apparent_elevation.dms_str(n_dec=2))
57' 51.96''

pymeeus.Coordinates.relative_position_angle(alpha1, delta1, alpha2, delta2)[source]¶

This function computes the position angle P of a body with respect to another body.

Parameters:	alpha1 (`Angle`) – Right ascension of celestial body #1, as an Angle object delta1 (`Angle`) – Declination of celestial body #1, as an Angle object alpha2 (`Angle`) – Right ascension of celestial body #2, as an Angle object delta2 (`Angle`) – Declination of celestial body #2, as an Angle object
Returns:	An Angle object with the relative position angle between the given celestial objects
Return type:	`Angle`
Raises:	TypeError if input values are of wrong type.

>>> alpha1 = Angle(14, 15, 39.7, ra=True)
>>> delta1 = Angle(19, 10, 57.0)
>>> alpha2 = Angle(14, 15, 39.7, ra=True)
>>> delta2 = Angle(-11, 9, 41.0)
>>> pos_ang = relative_position_angle(alpha1, delta1, alpha2, delta2)
>>> print(round(pos_ang, 1))
0.0

pymeeus.Coordinates.straight_line(alpha1, delta1, alpha2, delta2, alpha3, delta3)[source]¶

This function computes if three celestial bodies are in a straight line, providing the angle with which the bodies differ from a great circle.

Parameters:	alpha1 (`Angle`) – Right ascension, as an Angle object, of celestial body #1 delta1 (`Angle`) – Declination, as an Angle object, of celestial body #1 alpha2 (`Angle`) – Right ascension, as an Angle object, of celestial body #2 delta2 (`Angle`) – Declination, as an Angle object, of celestial body #2 alpha3 (`Angle`) – Right ascension, as an Angle object, of celestial body #3 delta3 (`Angle`) – Declination, as an Angle object, of celestial body #3
Returns:	A tuple with two components. The first element is an angle (as Angle object) with which the bodies differ from a great circle. The second element is the Angular distance of central point to the straight line (also as Angle object).
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> alpha1 = Angle( 5, 32,  0.40, ra=True)
>>> delta1 = Angle(0, -17, 56.9)
>>> alpha2 = Angle( 5, 36, 12.81, ra=True)
>>> delta2 = Angle(-1, 12,  7.0)
>>> alpha3 = Angle( 5, 40, 45.52, ra=True)
>>> delta3 = Angle(-1, 56, 33.3)
>>> psi, om = straight_line(alpha1, delta1, alpha2, delta2, alpha3, delta3)
>>> print(psi.dms_str(n_dec=0))
7d 31' 1.0''
>>> print(om.dms_str(n_dec=0))
-5' 24.0''

pymeeus.Coordinates.times_rise_transit_set(longitude, latitude, alpha1, delta1, alpha2, delta2, alpha3, delta3, h0, delta_t, theta0)[source]¶

This function computes the times (in Universal Time UT) of rising, transit and setting of a given celestial body.

Note

If the body is circumpolar there are no rising, transit nor setting times. In such a case a tuple with None’s is returned

Note

Care must be taken when interpreting the results. For instance, if the setting time is smaller than the rising time, it means that it belongs to the following day. Also, if the rising time is bigger than the setting time, it belong to the previous day. The same applies to the transit time.

Parameters:	longitude (`Angle`) – Geodetic longitude, as an Angle object. It is measured positively west from Greenwich, and negatively to the east. latitude (`Angle`) – Geodetic latitude, as an Angle object alpha1 (`Angle`) – Apparent right ascension the previous day at 0h TT, as an Angle object delta1 (`Angle`) – Apparent declination the previous day at 0h TT, as an Angle object alpha2 (`Angle`) – Apparent right ascension the current day at 0h TT, as an Angle object delta2 (`Angle`) – Apparent declination the current day at 0h TT, as an Angle object alpha3 (`Angle`) – Apparent right ascension the following day at 0h TT, as an Angle object delta3 (`Angle`) – Apparent declination the following day at 0h TT, as an Angle object h0 (`Angle`) – ‘Standard’ altitude: the geometric altitude of the center of the body at the time of apparent rising or setting, as degrees in an Angle object. It should be -0.5667 deg for stars and planets, -0.8333 deg for the Sun, and 0.125 deg for the Moon. delta_t (float) – The difference between Terrestrial Time and Universal Time (TT - UT) in seconds of time theta0 (`Angle`) – Apparent sidereal time at 0h TT on the current day for the meridian of Greenwich, as degrees in an Angle object
Returns:	A tuple with the times of rising, transit and setting, in that order, as hours in UT.
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> longitude = Angle(71, 5, 0.0)
>>> latitude = Angle(42, 20, 0.0)
>>> alpha1 = Angle(2, 42, 43.25, ra=True)
>>> delta1 = Angle(18, 2, 51.4)
>>> alpha2 = Angle(2, 46, 55.51, ra=True)
>>> delta2 = Angle(18, 26, 27.3)
>>> alpha3 = Angle(2, 51, 7.69, ra=True)
>>> delta3 = Angle(18, 49, 38.7)
>>> h0 = Angle(-0.5667)
>>> delta_t = 56.0
>>> theta0 = Angle(11, 50, 58.1, ra=True)
>>> rising, transit, setting = times_rise_transit_set(longitude, latitude,                                                          alpha1, delta1,                                                           alpha2, delta2,                                                           alpha3, delta3, h0,                                                           delta_t, theta0)
>>> print(round(rising, 4))
12.4238
>>> print(round(transit, 3))
19.675
>>> print(round(setting, 3))
2.911

pymeeus.Coordinates.true_obliquity(*args, **kwargs)[source]¶

This function computes the true obliquity (epsilon) at the provided date. The true obliquity is the mean obliquity (epsilon0) plus the correction provided by the nutation in obliquity (Delta epsilon).

This function internally uses an Epoch object, and the utc argument then controls the way the UTC->TT conversion is handled for that object. If leap_seconds argument is set to a value different than zero, then that value will be used for the UTC->TAI conversion, and the internal leap seconds table will be bypassed.

Parameters:	args (int, float, `Epoch`, datetime, date, tuple, list) – Either `Epoch`, date, datetime or year, month, day values, by themselves or inside a tuple or list utc (bool) – Whether the provided epoch is a civil time (UTC) or TT leap_seconds (int, float) – This is the value to be used in the UTC->TAI conversion, instead of taking it from internal leap seconds table.
Returns:	The true obliquity of the ecliptic, as an Angle
Return type:	`Angle`
Raises:	ValueError if input values are in the wrong range.
Raises:	TypeError if input values are of wrong type.

>>> epsilon = true_obliquity(1987, 4, 10)
>>> a = epsilon.dms_tuple()
>>> a[0]
23
>>> a[1]
26
>>> round(a[2], 3)
36.849

pymeeus.Coordinates.velocity(r, a)[source]¶

This function computes the instantaneous velocity of the moving body, in kilometers per second, for an unperturbed elliptic orbit.

Parameters:	r (float) – Distance of the body to the Sun, in Astronomical Units a (float) – Semimajor axis of the orbit, in Astronomical Units
Returns:	Velocity of the body, in kilometers per second
Return type:	float
Raises:	TypeError if input values are of wrong type.

>>> r = 1.0
>>> a = 17.9400782
>>> v = velocity(r, a)
>>> print(round(v, 2))
41.53

pymeeus.Coordinates.velocity_aphelion(e, a)[source]¶

This function computes the velocity of the moving body at aphelion, in kilometers per second, for an unperturbed elliptic orbit.

Parameters:	e (float) – Orbital eccentricity a (float) – Semimajor axis of the orbit, in Astronomical Units
Returns:	Velocity of the body at aphelion, in kilometers per second
Return type:	float
Raises:	TypeError if input values are of wrong type.

>>> a = 17.9400782
>>> e = 0.96727426
>>> va = velocity_aphelion(e, a)
>>> print(round(va, 2))
0.91

pymeeus.Coordinates.velocity_perihelion(e, a)[source]¶

This function computes the velocity of the moving body at perihelion, in kilometers per second, for an unperturbed elliptic orbit.

Parameters:	e (float) – Orbital eccentricity a (float) – Semimajor axis of the orbit, in Astronomical Units
Returns:	Velocity of the body at perihelion, in kilometers per second
Return type:	float
Raises:	TypeError if input values are of wrong type.

>>> a = 17.9400782
>>> e = 0.96727426
>>> vp = velocity_perihelion(e, a)
>>> print(round(vp, 2))
54.52

pymeeus.Coordinates.vsop_pos(epoch, vsop_l, vsop_b, vsop_r)[source]¶

This function computes the position of a celestial body at a given epoch when its VSOP87 periodic term tables are provided.

Parameters:	epoch (`Epoch`) – Epoch to compute the position, given as an `Epoch` object vsop_l (list) – Table of VSOP87 terms for the heliocentric longitude vsop_b (list) – Table of VSOP87 terms for the heliocentric latitude vsop_r (list) – Table of VSOP87 terms for the radius vector
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

Curveffiting¶

Class to get the best fit of a curve to a set of (x, y) points.

class pymeeus.CurveFitting.CurveFitting(*args)[source]¶

Class CurveFitting deals with finding the function (linear, cuadratic, etc) that best fit a given set of points.

The constructor takes pairs of (x, y) values from the table of interest. These pairs of values can be given as a sequence of int/floats, tuples, lists or Angles. It is also possible to provide a CurveFitting object to the constructor in order to get a copy.

Note

When using Angles, be careful with the 360-to-0 discontinuity.

If a sequence of int, floats or Angles is given, the values in the odd positions are considered to belong to the ‘x’ set, while the values in the even positions belong to the ‘y’ set. If only one tuple or list is provided, it is assumed that it is the ‘y’ set, and the ‘x’ set is build from 0 onwards with steps of length 1.

Please keep in mind that a minimum of two data pairs are needed in order to carry out any fitting. If only one value pair is provided, a ValueError exception will be raised.

__init__(*args)[source]¶

CurveFitting constructor.

This takes pairs of (x, y) values from the table of interest. These pairs of values can be given as a sequence of int/floats, tuples, lists or Angles. It is also possible to provide a CurveFitting object to the constructor in order to get a copy.

Note

When using Angles, be careful with the 360-to-0 discontinuity

If a sequence of int, floats or Angles is given, the values in the odd positions are considered to belong to the ‘x’ set, while the values in the even positions belong to the ‘y’ set. If only one tuple or list is provided, it is assumed that it is the ‘y’ set, and the ‘x’ set is build from 0 onwards with steps of length 1.

Please keep in mind that a minimum of two data pairs are needed in order to carry out any interpolation. If only one value pair is provided, a ValueError exception will be raised.

Parameters:	args (int, float, list, tuple, `Angle`, `CurveFitting`) – Input tabular values, or another CurveFitting object.
Returns:	CurveFitting object.
Return type:	`CurveFitting`
Raises:	ValueError if not enough input data pairs are provided.
Raises:	TypeError if input values are of wrong type.

>>> i = CurveFitting([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> print(i._x)
[5, 3, 6, 1, 2, 4]
>>> print(i._y)
[10, 6, 12, 2, 4, 8]
>>> j = CurveFitting([3, -8, 1, 12, 2, 5, 8])
>>> print(j._x)
[0, 1, 2, 3, 4, 5, 6]
>>> print(j._y)
[3, -8, 1, 12, 2, 5, 8]
>>> k = CurveFitting(3, -8, 1, 12, 2, 5, 8)
>>> print(k._x)
[3, 1, 2]
>>> print(k._y)
[-8, 12, 5]
>>> m = CurveFitting(k)
>>> print(m._x)
[3, 1, 2]
>>> print(m._y)
[-8, 12, 5]

__len__()[source]¶

This method returns the number of value pairs internally stored in this object.

Returns:	Number of value pairs internally stored
Return type:	int

>>> i = CurveFitting([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> len(i)
6

__repr__()[source]¶

Method providing the ‘official’ string representation of the object. It provides a valid expression that could be used to recreate the object.

Returns:	As string with a valid expression to recreate the object
Return type:	string

>>> i = CurveFitting([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> repr(i)
'CurveFitting([5, 3, 6, 1, 2, 4], [10, 6, 12, 2, 4, 8])'

__str__()[source]¶

Method used when trying to print the object.

Returns:	Internal tabular values as strings.
Return type:	string

>>> i = CurveFitting([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> print(i)
X: [5, 3, 6, 1, 2, 4]
Y: [10, 6, 12, 2, 4, 8]

__weakref__¶: list of weak references to the object (if defined)

correlation_coeff()[source]¶

This method returns the coefficient of correlation, as a float.

Returns:	Coefficient of correlation.
Return type:	float

>>> cf = CurveFitting([73.0, 38.0, 35.0, 42.0, 78.0, 68.0, 74.0, 42.0,
...                    52.0, 54.0, 39.0, 61.0, 42.0, 49.0, 50.0, 62.0,
...                    44.0, 39.0, 43.0, 54.0, 44.0, 37.0],
...                   [90.4, 125.3, 161.8, 143.4, 52.5, 50.8, 71.5,
...                    152.8, 131.3, 98.5, 144.8, 78.1, 89.5, 63.9,
...                    112.1, 82.0, 119.8, 161.2, 208.4, 111.6, 167.1,
...                    162.1])
>>> r = cf.correlation_coeff()
>>> print(round(r, 3))
-0.767

general_fitting(f0, f1=<function <lambda>>, f2=<function <lambda>>)[source]¶

This method returns a tuple with the ‘a’, ‘b’, ‘c’ coefficients of the general equation ‘y = a*f0(x) + b*f1(x) + c*f2(x)’ that best fits the table data, using the least squares approach.

Parameters:	f1, f2 (f0,) – Functions used to build the general equation.
Returns:	‘a’, ‘b’, ‘c’ coefficients of best general equation fit.
Return type:	tuple
Raises:	ZeroDivisionError if input functions are null or input data leads to a division by zero

>>> cf4 = CurveFitting([3, 20, 34, 50, 75, 88, 111, 129, 143, 160, 183,
...                     200, 218, 230, 248, 269, 290, 303, 320, 344],
...                    [0.0433, 0.2532, 0.3386, 0.3560, 0.4983, 0.7577,
...                     1.4585, 1.8628, 1.8264, 1.2431, -0.2043,
...                     -1.2431, -1.8422, -1.8726, -1.4889, -0.8372,
...                     -0.4377, -0.3640, -0.3508, -0.2126])
>>> def sin1(x): return sin(radians(x))
>>> def sin2(x): return sin(radians(2.0*x))
>>> def sin3(x): return sin(radians(3.0*x))
>>> a, b, c = cf4.general_fitting(sin1, sin2, sin3)
>>> print("a = {}; b = {}; c = {}".format(round(a, 2), round(b, 2),
...                                       round(c, 2)))
a = 1.2; b = -0.77; c = 0.39

>>> cf5 = CurveFitting([0, 1.2, 1.4, 1.7, 2.1, 2.2])
>>> a, b, c = cf5.general_fitting(sqrt)
>>> print("a = {}; b = {}; c = {}".format(round(a, 3), round(b, 3),
...                                       round(c, 3)))
a = 1.016; b = 0.0; c = 0.0

linear_fitting()[source]¶

This method returns a tuple with the ‘a’, ‘b’ coefficients of the linear equation ‘y = a*x + b’ that best fits the table data, using the least squares approach.

Returns:	‘a’, ‘b’ coefficients of best linear equation fit.
Return type:	tuple
Raises:	ZeroDivisionError if input data leads to a division by zero

>>> cf = CurveFitting([73.0, 38.0, 35.0, 42.0, 78.0, 68.0, 74.0, 42.0,
...                    52.0, 54.0, 39.0, 61.0, 42.0, 49.0, 50.0, 62.0,
...                    44.0, 39.0, 43.0, 54.0, 44.0, 37.0],
...                   [90.4, 125.3, 161.8, 143.4, 52.5, 50.8, 71.5,
...                    152.8, 131.3, 98.5, 144.8, 78.1, 89.5, 63.9,
...                    112.1, 82.0, 119.8, 161.2, 208.4, 111.6, 167.1,
...                    162.1])
>>> a, b = cf.linear_fitting()
>>> print("a = {}       b = {}".format(round(a, 2), round(b, 2)))
a = -2.49       b = 244.18

quadratic_fitting()[source]¶

This method returns a tuple with the ‘a’, ‘b’, ‘c’ coefficients of the quadratic equation ‘y = a*x*x + b*x + c’ that best fits the table data, using the least squares approach.

Returns:	‘a’, ‘b’, ‘c’ coefficients of best quadratic equation fit.
Return type:	tuple
Raises:	ZeroDivisionError if input data leads to a division by zero

>>> cf2 = CurveFitting([-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5,
...                     2.0, 2.5,3.0],
...                    [-9.372, -3.821, 0.291, 3.730, 5.822, 8.324,
...                     9.083, 6.957, 7.006, 0.365, -1.722])
>>> a, b, c = cf2.quadratic_fitting()
>>> print("a = {}; b = {}; c = {}".format(round(a, 2), round(b, 2),
...                                       round(c, 2)))
a = -2.22; b = 3.76; c = 6.64

set(*args)[source]¶

Method used to define the value pairs of CurveFitting object.

This takes pairs of (x, y) values from the table of interest. These pairs of values can be given as a sequence of int/floats, tuples, lists, or Angles. It is also possible to provide a CurveFitting object to this method in order to get a copy.

Note

When using Angles, be careful with the 360-to-0 discontinuity

If a sequence of int, floats or Angles is given, the values in the odd positions are considered to belong to the ‘x’ set, while the values in the even positions belong to the ‘y’ set. If only one tuple or list is provided, it is assumed that it is the ‘y’ set, and the ‘x’ set is build from 0 onwards with steps of length 1.

Please keep in mind that a minimum of two data pairs are needed in order to carry out any interpolation. If only one value is provided, a ValueError exception will be raised.

Parameters:	args (int, float, list, tuple, `Angle`) – Input tabular values, or another CurveFitting object.
Returns:	None.
Return type:	None
Raises:	ValueError if not enough input data pairs are provided.
Raises:	TypeError if input values are of wrong type.

>>> i = CurveFitting()
>>> i.set([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> print(i._x)
[5, 3, 6, 1, 2, 4]
>>> print(i._y)
[10, 6, 12, 2, 4, 8]
>>> j = CurveFitting()
>>> j.set([3, -8, 1, 12, 2, 5, 8])
>>> print(j._x)
[0, 1, 2, 3, 4, 5, 6]
>>> print(j._y)
[3, -8, 1, 12, 2, 5, 8]
>>> k = CurveFitting(3, -8, 1, 12, 2, 5, 8)
>>> print(k._x)
[3, 1, 2]
>>> print(k._y)
[-8, 12, 5]

Epoch¶

Class to handle time.

pymeeus.Epoch.DAY2HOURS = 24.0¶: Number of hours per day

pymeeus.Epoch.DAY2MIN = 1440.0¶: Number of minutes per day

pymeeus.Epoch.DAY2SEC = 86400.0¶: Number of seconds per day

class pymeeus.Epoch.Epoch(*args, **kwargs)[source]¶

Class Epoch deals with the tasks related to time handling.

The constructor takes either a single JDE value, another Epoch object, or a series of values representing year, month, day, hours, minutes, seconds. This series of values are by default supposed to be in Terrestial Time (TT).

This is not necesarily the truth, though. For instance, the time of a current observation is tipically in UTC time (civil time), not in TT, and there is some offset between those two time references.

When a UTC time is provided, the parameter utc=True must be given. Then, the input is converted to International Atomic Time (TAI) using an internal table of leap seconds, and from there, it is converted to (and stored as) Terrestrial Time (TT).

Given that leap seconds are added or subtracted in a rather irregular basis, it is not possible to predict them in advance, and the internal leap seconds table will become outdated at some point in time. To counter this, you have two options:

Download an updated version of this Pymeeus package.
Use the argument leap_seconds in the constructor or set() method to provide the correct number of leap seconds (w.r.t. TAI) to be applied.

Note

Providing the leap_seconds argument will automatically set the argument utc to True.

For instance, if at some time in the future the TAI-UTC difference is 43 seconds, you should set leap_seconds=43 if you don’t have an updated version of this class.

In order to know which is the most updated leap second value stored in this class, you may use the get_last_leap_second() method.

Note

The current version of UTC was implemented in January 1st, 1972. Therefore, for dates before that date the correction is NOT carried out, even if the utc argument is set to True, and it is supposed that the input data is already in TT scale.

Note

For conversions between TT and Universal Time (UT), please use the method tt2ut().

Note

Internally, time values are stored as a Julian Ephemeris Day (JDE), based on the uniform scale of Dynamical Time, or more specifically, Terrestial Time (TT) (itself the redefinition of Terrestrial Dynamical Time, TDT).

Note

The UTC-TT conversion is composed of three corrections:

TT-TAI, comprising 32.184 s,
TAI-UTC(1972), 10 s, and
UTC(1972)-UTC(target)

item c. is the corresponding amount of leap seconds to the target Epoch. When you do, for instance, leap_seconds=43, you modify the c. part.

Note

Given that this class stores the epoch as JDE, if the JDE value is in the order of millions of days then, for a computer with 15-digit accuracy, the final time resolution is about 10 milliseconds. That is considered enough for most applications of this class.

__add__(b)[source]¶

This method defines the addition between an Epoch and some days.

Parameters:	b (int, float) – Value to be added, in days.
Returns:	A new Epoch object.
Return type:	`Epoch`
Raises:	TypeError if operand is of wrong type.

>>> a = Epoch(1991, 7, 11)
>>> b = a + 10000
>>> y, m, d = b.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
2018/11/26.0

__call__()[source]¶

Method used when Epoch is called only with parenthesis.

Returns:	The internal value of the Julian Ephemeris Day.
Return type:	float

>>> a = Epoch(-122, 1, 1.0)
>>> print(a())
1676497.5

__eq__(b)[source]¶

This method defines the ‘is equal’ operator between Epochs.

Note

For the comparison, the base.TOL value is used.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Epoch(2007, 5, 20.0)
>>> b = Epoch(2007, 5, 20.000001)
>>> a == b
False
>>> a = Epoch(2004, 10, 15.7)
>>> b = Epoch(a)
>>> a == b
True
>>> a = Epoch(2434923.85)
>>> a == 2434923.85
True

__float__()[source]¶

This method returns the internal JDE value as a float.

Returns:	Internal JDE value as a float.
Return type:	float

>>> a = Epoch(2434923.85)
>>> float(a)
2434923.85

__ge__(b)[source]¶

This method defines ‘is equal or greater’ operator between Epochs.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Epoch(2004, 10, 15.71)
>>> b = Epoch(2004, 10, 15.7)
>>> a >= b
True

__gt__(b)[source]¶

This method defines the ‘is greater than’ operator between Epochs.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Epoch(2004, 10, 15.71)
>>> b = Epoch(2004, 10, 15.7)
>>> a > b
True
>>> a = Epoch(-207, 11, 5.2)
>>> b = Epoch(-207, 11, 5.2)
>>> a > b
False

__hash__()[source]¶

Method used to create hash from an Epoch object.

This method allows Epoch objects to be used as the key in a dictionary.

__iadd__(b)[source]¶

This method defines the accumulative addition to this Epoch.

Parameters:	b (int, float) – Value to be added, in days.
Returns:	This Epoch.
Return type:	`Epoch`
Raises:	TypeError if operand is of wrong type.

>>> a = Epoch(2003, 12, 31.0)
>>> a += 32.5
>>> y, m, d = a.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
2004/2/1.5

__init__(*args, **kwargs)[source]¶

Epoch constructor.

This constructor takes either a single JDE value, another Epoch object, or a series of values representing year, month, day, hours, minutes, seconds. This series of values are by default supposed to be in Terrestial Time (TT).

It is also possible that the year, month, etc. arguments be provided in a tuple or list. Moreover, it is also possible provide date or datetime objects for initialization.

The month value can be provided as an integer (1 = January, 2 = February, etc), or it can be provided with short (Jan, Feb,…) or long (January, February,…) names. Also, hours, minutes, seconds can be provided separately, or as decimals of the day value.

When a UTC time is provided, the parameter utc=True must be given. Then, the input is converted to International Atomic Time (TAI) using an internal table of leap seconds, and from there, it is converted to (and stored as) Terrestrial Time (TT). If utc is not provided, it is supposed that the input data is already in TT scale.

If a value is provided with the leap_seconds argument, then that value will be used for the UTC->TAI conversion, and the internal leap seconds table will be bypassed.

Parameters:	args (int, float, `Epoch`, tuple, list, date, datetime) – Either JDE, Epoch, date, datetime or year, month, day, hours, minutes, seconds values, by themselves or inside a tuple or list utc (bool) – Whether the provided epoch is a civil time (UTC) leap_seconds (int, float) – This is the value to be used in the UTC->TAI conversion, instead of taking it from internal leap seconds table.
Returns:	Epoch object.
Return type:	`Epoch`
Raises:	ValueError if input values are in the wrong range.
Raises:	TypeError if input values are of wrong type.

>>> e = Epoch(1987, 6, 19.5)
>>> print(e)
2446966.0

__int__()[source]¶

This method returns the internal JDE value as an int.

Returns:	Internal JDE value as an int.
Return type:	int

>>> a = Epoch(2434923.85)
>>> int(a)
2434923

__isub__(b)[source]¶

This method defines the accumulative subtraction to this Epoch.

Parameters:	b (int, float) – Value to be subtracted, in days.
Returns:	This Epoch.
Return type:	`Epoch`
Raises:	TypeError if operand is of wrong type.

>>> a = Epoch(2001, 12, 31.0)
>>> a -= 2*365
>>> y, m, d = a.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
2000/1/1.0

__le__(b)[source]¶

This method defines ‘is equal or less’ operator between Epochs.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Epoch(2004, 10, 15.71)
>>> b = Epoch(2004, 10, 15.7)
>>> a <= b
False
>>> a = Epoch(-207, 11, 5.2)
>>> b = Epoch(-207, 11, 5.2)
>>> a <= b
True

__lt__(b)[source]¶

This method defines the ‘is less than’ operator between Epochs.

Returns:	A boolean.
Return type:	bool
Raises:	TypeError if input values are of wrong type.

>>> a = Epoch(2004, 10, 15.7)
>>> b = Epoch(2004, 10, 15.7)
>>> a < b
False

__ne__(b)[source]¶

This method defines the ‘is not equal’ operator between Epochs.

Note

For the comparison, the base.TOL value is used.

Returns:	A boolean.
Return type:	bool

>>> a = Epoch(2007, 5, 20.0)
>>> b = Epoch(2007, 5, 20.000001)
>>> a != b
True
>>> a = Epoch(2004, 10, 15.7)
>>> b = Epoch(a)
>>> a != b
False
>>> a = Epoch(2434923.85)
>>> a != 2434923.85
False

__radd__(b)[source]¶

This method defines the addition to a Epoch by the right

Parameters:	b (int, float) – Value to be added, in days.
Returns:	A new Epoch object.
Return type:	`Epoch`
Raises:	TypeError if operand is of wrong type.

>>> a = Epoch(2004, 2, 27.8)
>>> b = 2.2 + a
>>> y, m, d = b.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
2004/3/1.0

__repr__()[source]¶

Method providing the ‘official’ string representation of the object. It provides a valid expression that could be used to recreate the object.

Returns:	As string with a valid expression to recreate the object
Return type:	string

>>> e = Epoch(1987, 6, 19.5)
>>> repr(e)
'Epoch(2446966.0)'

__str__()[source]¶

Method used when trying to print the object.

Returns:	Internal JDE value as a string.
Return type:	string

>>> e = Epoch(1987, 6, 19.5)
>>> print(e)
2446966.0

__sub__(b)[source]¶

This method defines the subtraction between Epochs or between an Epoch and a given number of days.

Parameters:	b (py:class:Epoch, int, float) – Value to be subtracted, either an Epoch or days.
Returns:	A new Epoch object if parameter ‘b’ is in days, or the difference between provided Epochs, in days.
Return type:	`Epoch`, float
Raises:	TypeError if operand is of wrong type.

>>> a = Epoch(1986, 2, 9.0)
>>> print(round(a(), 2))
2446470.5
>>> b = Epoch(1910, 4, 20.0)
>>> print(round(b(), 2))
2418781.5
>>> c = a - b
>>> print(round(c, 2))
27689.0
>>> a = Epoch(2003, 12, 31.0)
>>> b = a - 365.5
>>> y, m, d = b.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
2002/12/30.5

__weakref__¶: list of weak references to the object (if defined)

apparent_sidereal_time(true_obliquity, nutation_longitude)[source]¶

Method to compute the _apparent_ sidereal time at Greenwich for the epoch stored in this object. It represents the Greenwich hour angle of the true vernal equinox.

Note

If you require the result as an angle, you should convert the result from this method to hours with decimals (with DAY2HOURS), and then multiply by 15 deg/hr. Alternatively, you can convert the result to hours with decimals, and feed this value to an Angle object, setting ra=True, and making use of Angle facilities for further handling.

Parameters:	true_obliquity (int, float, `Angle`) – The true obliquity of the ecliptic as an int, float or `Angle`, in degrees. You can use the method Earth.true_obliquity() to find it. nutation_longitude (int, float, `Angle`) – The nutation in longitude as an int, float or `Angle`, in degrees. You can use method Earth.nutation_longitude() to find it.
Returns:	Apparent sidereal time, in days
Return type:	float
Raises:	TypeError if input value is of wrong type.

>>> e = Epoch(1987, 4, 10)
>>> round(e.apparent_sidereal_time(23.44357, (-3.788)/3600.0), 8)
0.54914508

static check_input_date(*args, **kwargs)[source]¶

Method to check that the input is a proper date.

This method returns an Epoch object, and the leap_seconds argument then controls the way the UTC->TT conversion is handled for that new object. If leap_seconds argument is set to a value different than zero, then that value will be used for the UTC->TAI conversion, and the internal leap seconds table will be bypassed. On the other hand, if it is set to zero, then the UTC to TT correction is disabled, and it is supposed that the input data is already in TT scale.

Parameters:	args (int, float, `Epoch`, datetime, date, tuple, list) – Either Epoch, date, datetime or year, month, day values, by themselves or inside a tuple or list leap_seconds (int, float) – If different from zero, this is the value to be used in the UTC->TAI conversion. If equals to zero, conversion is disabled. If not given, UTC to TT conversion is carried out (default).
Returns:	Epoch object corresponding to the input date
Return type:	`Epoch`
Raises:	ValueError if input values are in the wrong range.
Raises:	TypeError if input values are of wrong type.

dow(as_string=False)[source]¶

Method to return the day of week corresponding to this Epoch.

By default, this method returns an integer value: 0 for Sunday, 1 for Monday, etc. However, when as_string=True is passed, the names of the days are returned.

Parameters:	as_string (bool) – Whether result will be given as a integer or as a string. False by default.
Returns:	Day of the week, as a integer or as a string.
Return type:	int, str

>>> e = Epoch(1954, 'June', 30)
>>> e.dow()
3
>>> e = Epoch(2018, 'Feb', 14.9)
>>> e.dow(as_string=True)
'Wednesday'
>>> e = Epoch(2018, 'Feb', 15)
>>> e.dow(as_string=True)
'Thursday'
>>> e = Epoch(2018, 'Feb', 15.99)
>>> e.dow(as_string=True)
'Thursday'
>>> e.set(2018, 'Jul', 15.4)
>>> e.dow(as_string=True)
'Sunday'
>>> e.set(2018, 'Jul', 15.9)
>>> e.dow(as_string=True)
'Sunday'

doy()[source]¶

This method returns the Day Of Year (DOY) for the current Epoch object.

Returns:	Day Of Year (DOY).
Return type:	float

>>> e = Epoch(1999, 1, 29)
>>> round(e.doy(), 1)
29.0
>>> e = Epoch(2017, 12, 31.7)
>>> round(e.doy(), 1)
365.7
>>> e = Epoch(2012, 3, 3.1)
>>> round(e.doy(), 1)
63.1
>>> e = Epoch(-400, 2, 29.9)
>>> round(e.doy(), 1)
60.9

static doy2date(year, doy)[source]¶

This method takes a year and a Day Of Year values, and returns the corresponding date.

Parameters:	year (int, float) – Year, in four digits format doy (int, float) – Day of Year number
Returns:	Year, month, day.
Return type:	tuple
Raises:	ValueError if either input year or doy values are invalid.

>>> t = Epoch.doy2date(1999, 29)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
1999/1/29.0
>>> t = Epoch.doy2date(2017, 365.7)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
2017/12/31.7
>>> t = Epoch.doy2date(2012, 63.1)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
2012/3/3.1
>>> t = Epoch.doy2date(-1004, 60)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
-1004/2/29.0
>>> t = Epoch.doy2date(0, 60)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
0/2/29.0
>>> t = Epoch.doy2date(1, 60)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
1/3/1.0
>>> t = Epoch.doy2date(-1, 60)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
-1/3/1.0
>>> t = Epoch.doy2date(-2, 60)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
-2/3/1.0
>>> t = Epoch.doy2date(-3, 60)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
-3/3/1.0
>>> t = Epoch.doy2date(-4, 60)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
-4/2/29.0
>>> t = Epoch.doy2date(-5, 60)
>>> print("{}/{}/{}".format(t[0], t[1], round(t[2], 1)))
-5/3/1.0

static easter(year)[source]¶

Method to return the Easter day for given year.

Note

This method is valid for both Gregorian and Julian years.

Parameters:	year (int) – Year
Returns:	Easter month and day, as a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> Epoch.easter(1991)
(3, 31)
>>> Epoch.easter(1818)
(3, 22)
>>> Epoch.easter(1943)
(4, 25)
>>> Epoch.easter(2000)
(4, 23)
>>> Epoch.easter(1954)
(4, 18)
>>> Epoch.easter(179)
(4, 12)
>>> Epoch.easter(1243)
(4, 12)

get_date(**kwargs)[source]¶

This method converts the internal JDE value back to a date.

Use utc=True to enable the TT to UTC conversion mechanism, or provide a non zero value to leap_seconds to apply a specific leap seconds value.

It is also possible to retrieve a local time with the parameter local=True. In such case, the method utc2local() is called to compute the LocalTime-UTC difference. This implies that the parameter utc=True is automatically set.

Note

Please bear in mind that, in order for the method utc2local() to work, your operative system must be correctly configured, with the right time and corresponding time zone.

Parameters:	utc (bool) – Whether the TT to UTC conversion mechanism will be enabled leap_seconds (int, float) – Optional value for leap seconds. local – Whether the retrieved epoch is converted to local time.
Returns:	Year, month, day in a tuple
Return type:	tuple

>>> e = Epoch(2436116.31)
>>> y, m, d = e.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
1957/10/4.81
>>> e = Epoch(1988, 1, 27)
>>> y, m, d = e.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
1988/1/27.0
>>> e = Epoch(1842713.0)
>>> y, m, d = e.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
333/1/27.5
>>> e = Epoch(1507900.13)
>>> y, m, d = e.get_date()
>>> print("{}/{}/{}".format(y, m, round(d, 2)))
-584/5/28.63

static get_doy(yyyy, mm, dd)[source]¶

This method returns the Day Of Year (DOY) for the given date.

Parameters:	yyyy (int, float) – Year, in four digits format mm (int, float) – Month, in numeric format (1 = January, 2 = February, etc) dd (int, float) – Day, in numeric format
Returns:	Day Of Year (DOY).
Return type:	float
Raises:	ValueError if input values correspond to a wrong date.

>>> Epoch.get_doy(1999, 1, 29)
29.0
>>> Epoch.get_doy(1978, 11, 14)
318.0
>>> Epoch.get_doy(2017, 12, 31.7)
365.7
>>> Epoch.get_doy(2012, 3, 3.1)
63.1
>>> Epoch.get_doy(-400, 2, 29.9)
60.9

get_full_date(**kwargs)[source]¶

This method converts the internal JDE value back to a full date.

Use utc=True to enable the TT to UTC conversion mechanism, or provide a non zero value to leap_seconds to apply a specific leap seconds value.

It is also possible to retrieve a local time with the parameter local=True. In such case, the method utc2local() is called to compute the LocalTime-UTC difference. This implies that the parameter utc=True is automatically set.

Note

Please bear in mind that, in order for the method utc2local() to work, your operative system must be correctly configured, with the right time and corresponding time zone.

Parameters:	utc (bool) – Whether the TT to UTC conversion mechanism will be enabled leap_seconds (int, float) – Optional value for leap seconds. local – Whether the retrieved epoch is converted to local time.
Returns:	Year, month, day, hours, minutes, seconds in a tuple
Return type:	tuple

>>> e = Epoch(2436116.31)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 1)))
1957/10/4 19:26:24.0
>>> e = Epoch(1988, 1, 27)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 1)))
1988/1/27 0:0:0.0
>>> e = Epoch(1842713.0)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 1)))
333/1/27 12:0:0.0
>>> e = Epoch(1507900.13)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 1)))
-584/5/28 15:7:12.0

static get_last_leap_second()[source]¶

Method to get the date and value of the last leap second added to the table

Returns:	Tuple with year, month, day, leap second value.
Return type:	tuple

static get_month(month, as_string=False)[source]¶

Method to get the month as a integer in the [1, 12] range, or as a full name.

Parameters:	month (int, float, str) – Month, in numeric, short name or long name format as_string (bool) – Whether the output will be numeric, or a long name.
Returns:	Month as integer in the [1, 12] range, or as a long name.
Return type:	int, str
Raises:	ValueError if input month value is invalid.

>>> Epoch.get_month(4.0)
4
>>> Epoch.get_month('Oct')
10
>>> Epoch.get_month('FEB')
2
>>> Epoch.get_month('August')
8
>>> Epoch.get_month('august')
8
>>> Epoch.get_month('NOVEMBER')
11
>>> Epoch.get_month(9.0, as_string=True)
'September'
>>> Epoch.get_month('Feb', as_string=True)
'February'
>>> Epoch.get_month('March', as_string=True)
'March'

static gregorian2moslem(year, month, day)[source]¶

Method to convert a date in the Gregorian (or Julian) calendar to the Moslen calendar.

Parameters:	year (int) – Year month (int) – Month day (int) – Day
Returns:	Date in Moslem calendar: year, month and day, as a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> Epoch.gregorian2moslem(1991, 8, 13)
(1412, 2, 2)

static is_julian(year, month, day)[source]¶

This method returns True if given date is in the Julian calendar.

Parameters:	year – Year month – Month day (int) – Day
Returns:	Whether the provided date belongs to Julian calendar or not.
Return type:	bool

>>> Epoch.is_julian(1997, 5, 27.1)
False
>>> Epoch.is_julian(1397, 7, 7.0)
True

static is_leap(year)[source]¶

Method to check if a given year is a leap year.

Parameters:	year (int, float) – Year to be checked.
Returns:	Whether or not year is a leap year.
Return type:	bool
Raises:	ValueError if input year value is invalid.

>>> Epoch.is_leap(2003)
False
>>> Epoch.is_leap(2012)
True
>>> Epoch.is_leap(1900)
False
>>> Epoch.is_leap(-1000)
True
>>> Epoch.is_leap(1000)
True

jde()[source]¶

Method to return the internal value of the Julian Ephemeris Day.

Returns:	The internal value of the Julian Ephemeris Day.
Return type:	float

>>> a = Epoch(-1000, 2, 29.0)
>>> print(a.jde())
1355866.5

static jewish_pesach(year)[source]¶

Method to return the Jewish Easter (Pesach) day for given year.

Note

This method is valid for both Gregorian and Julian years.

Parameters:	year (int) – Year
Returns:	Jewish Easter (Pesach) month and day, as a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> Epoch.jewish_pesach(1990)
(4, 10)

julian()[source]¶

This method returns True if this Epoch object holds a date in the Julian calendar.

Returns:	Whether this Epoch object holds a date belonging to Julian calendar or not.
Return type:	bool

>>> e = Epoch(1997, 5, 27.1)
>>> e.julian()
False
>>> e = Epoch(1397, 7, 7.0)
>>> e.julian()
True

leap()[source]¶

This method checks if the current Epoch object holds a leap year.

Returns:	Whether or the year in this Epoch object is a leap year.
Return type:	bool

>>> e = Epoch(2003, 1, 1)
>>> e.leap()
False
>>> e = Epoch(2012, 1, 1)
>>> e.leap()
True
>>> e = Epoch(1900, 1, 1)
>>> e.leap()
False
>>> e = Epoch(-1000, 1, 1)
>>> e.leap()
True
>>> e = Epoch(1000, 1, 1)
>>> e.leap()
True

static leap_seconds(year, month)[source]¶

Returns the leap seconds accumulated for the given year and month.

Parameters:	year (int) – Year month (int) – Month, in numeric format ([1:12] range)
Returns:	Leap seconds accumulated for given year and month.
Return type:	int

>>> Epoch.leap_seconds(1972, 4)
0
>>> Epoch.leap_seconds(1972, 6)
0
>>> Epoch.leap_seconds(1972, 7)
1
>>> Epoch.leap_seconds(1983, 6)
11
>>> Epoch.leap_seconds(1983, 7)
12
>>> Epoch.leap_seconds(1985, 8)
13
>>> Epoch.leap_seconds(2016, 11)
26
>>> Epoch.leap_seconds(2017, 1)
27
>>> Epoch.leap_seconds(2018, 7)
27

mean_sidereal_time()[source]¶

Method to compute the _mean_ sidereal time at Greenwich for the epoch stored in this object. It represents the Greenwich hour angle of the mean vernal equinox.

Note

If you require the result as an angle, you should convert the result from this method to hours with decimals (with DAY2HOURS), and then multiply by 15 deg/hr. Alternatively, you can convert the result to hours with decimals, and feed this value to an Angle object, setting ra=True, and making use of Angle facilities for further handling.

Returns:	Mean sidereal time, in days
Return type:	float

>>> e = Epoch(1987, 4, 10)
>>> round(e.mean_sidereal_time(), 9)
0.549147764
>>> e = Epoch(1987, 4, 10, 19, 21, 0.0)
>>> round(e.mean_sidereal_time(), 9)
0.357605204

mjd()[source]¶

This method returns the Modified Julian Day (MJD).

Returns:	Modified Julian Day (MJD).
Return type:	float

>>> e = Epoch(1858, 'NOVEMBER', 17)
>>> e.mjd()
0.0

static moslem2gregorian(year, month, day)[source]¶

Method to convert a date in the Moslen calendar to the Gregorian (or Julian) calendar.

Note

This method is valid for both Gregorian and Julian years.

Parameters:	year (int) – Year month (int) – Month day (int) – Day
Returns:	Date in Gregorian (Julian) calendar: year, month and day, as a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> Epoch.moslem2gregorian(1421, 1, 1)
(2000, 4, 6)

rise_set(latitude, longitude, altitude=0.0)[source]¶

This method computes the times of rising and setting of the Sun.

Note

The algorithm used is the one explained in the article “Sunrise equation” of the Wikipedia at: https://en.wikipedia.org/wiki/Sunrise_equation

Note

This algorithm is only valid outside the artic and antartic circles (+/- 66d 33’). For latitudes higher than +66d 33’ and smaller than -66d 33’ this method returns a ValueError exception

Note

The results are given in UTC time.

Parameters:	latitude (`Angle`) – Latitude of the observer, as an Angle object. Positive to the North longitude (`Angle`) – Longitude of the observer, as an Angle object. Positive to the East altitude (int, float) – Altitude of the observer, as meters above sea level
Returns:	Two `Epoch` objects representing rising time and setting time, in a tuple
Return type:	tuple
Raises:	TypeError if input values are of wrong type.
Raises:	ValueError if latitude outside the +/- 66d 33’ range.

>>> e = Epoch(2019, 4, 2)
>>> latitude = Angle(48, 8, 0)
>>> longitude = Angle(11, 34, 0)
>>> altitude = 520.0
>>> rising, setting = e.rise_set(latitude, longitude, altitude)
>>> y, m, d, h, mi, s = rising.get_full_date()
>>> print("{}:{}".format(h, mi))
4:48
>>> y, m, d, h, mi, s = setting.get_full_date()
>>> print("{}:{}".format(h, mi))
17:48

set(*args, **kwargs)[source]¶

Method used to set the value of this object.

This method takes either a single JDE value, or a series of values representing year, month, day, hours, minutes, seconds. This series of values are by default supposed to be in Terrestial Time (TT).

It is also possible to provide another Epoch object as input for the set() method, or the year, month, etc arguments can be provided in a tuple or list. Moreover, it is also possible provide date or datetime objects for initialization.

The month value can be provided as an integer (1 = January, 2 = February, etc), or it can be provided as short (Jan, Feb, …) or long (January, February, …) names. Also, hours, minutes, seconds can be provided separately, or as decimals of the day value.

When a UTC time is provided, the parameter utc=True must be given. Then, the input is converted to International Atomic Time (TAI) using an internal table of leap seconds, and from there, it is converted to (and stored as) Terrestrial Time (TT). If utc is not provided, it is supposed that the input data is already in TT scale.

If a value is provided with the leap_seconds argument, then that value will be used for the UTC->TAI conversion, and the internal leap seconds table will be bypassed.

It is also possible to provide a local time with the parameter local=True. In such case, the method utc2local() is called to compute the LocalTime-UTC difference. This implies that the parameter utc=True is automatically set.

Note

The UTC to TT correction is only carried out for dates after January 1st, 1972.

Note

Please bear in mind that, in order for the method utc2local() to work, your operative system must be correctly configured, with the right time and corresponding time zone.

Parameters:	args (int, float, `Epoch`, tuple, list, date, datetime) – Either JDE, Epoch, date, datetime or year, month, day, hours, minutes, seconds values, by themselves or inside a tuple or list. utc (bool) – Whether the provided epoch is a civil time (UTC). leap_seconds (int, float) – This is the value to be used in the UTC->TAI conversion, instead of taking it from internal leap seconds table. local – Whether the provided epoch is a local time.
Returns:	None.
Return type:	None
Raises:	ValueError if input values are in the wrong range.
Raises:	TypeError if input values are of wrong type.

>>> e = Epoch()
>>> e.set(1987, 6, 19.5)
>>> print(e)
2446966.0
>>> e.set(1977, 'Apr', 26.4)
>>> print(e)
2443259.9
>>> e.set(1957, 'October', 4.81)
>>> print(e)
2436116.31
>>> e.set(333, 'Jan', 27, 12)
>>> print(e)
1842713.0
>>> e.set(1900, 'Jan', 1)
>>> print(e)
2415020.5
>>> e.set(-1001, 'august', 17.9)
>>> print(e)
1355671.4
>>> e.set(-4712, 1, 1.5)
>>> print(e)
0.0
>>> e.set((1600, 12, 31))
>>> print(e)
2305812.5
>>> e.set([1988, 'JUN', 19, 12])
>>> print(e)
2447332.0
>>> d = datetime.date(2000, 1, 1)
>>> e.set(d)
>>> print(e)
2451544.5
>>> e.set(837, 'Apr', 10, 7, 12)
>>> print(e)
2026871.8
>>> d = datetime.datetime(837, 4, 10, 7, 12, 0, 0)
>>> e.set(d)
>>> print(e)
2026871.8
>>> e = Epoch(JDE2000, utc=True)
>>> print(round((e - JDE2000) * DAY2SEC, 3))
64.184
>>> e = Epoch(2451545.0, utc=True)
>>> print(round((e - JDE2000) * DAY2SEC, 3))
64.184
>>> e = Epoch(JDE2000, local=True)
>>> print(round((e - JDE2000) * DAY2SEC - Epoch.utc2local(), 3))
64.184
>>> e = Epoch(JDE2000, local=True, leap_seconds=35.0)
>>> print(round((e - JDE2000) * DAY2SEC - Epoch.utc2local(), 3))
77.184

static tt2ut(year, month)[source]¶

This method provides an approximation of the difference, in seconds, between Terrestrial Time and Universal Time, denoted DeltaT, where: DeltaT = TT - UT.

Here we depart from Meeus book and use the polynomial expressions from:

https://eclipse.gsfc.nasa.gov/LEcat5/deltatpoly.html

Which are regarded as more elaborate and precise than Meeus’.

Please note that, by definition, the UTC time used internally in this Epoch class by default is kept within 0.9 seconds from UT. Therefore, UTC is in itself a quite good approximation to UT, arguably better than some of the results provided by this method.

Parameters:	year (int, float) – Year we want to compute DeltaT for. month (int, float) – Month we want to compute DeltaT for.
Returns:	DeltaT, in seconds
Return type:	float

>>> round(Epoch.tt2ut(1642, 1), 1)
62.1
>>> round(Epoch.tt2ut(1680, 1), 1)
15.3
>>> round(Epoch.tt2ut(1700, 1), 1)
8.8
>>> round(Epoch.tt2ut(1726, 1), 1)
10.9
>>> round(Epoch.tt2ut(1750, 1), 1)
13.4
>>> round(Epoch.tt2ut(1774, 1), 1)
16.7
>>> round(Epoch.tt2ut(1800, 1), 1)
13.7
>>> round(Epoch.tt2ut(1820, 1), 1)
11.9
>>> round(Epoch.tt2ut(1890, 1), 1)
-6.1
>>> round(Epoch.tt2ut(1928, 2), 1)
24.2
>>> round(Epoch.tt2ut(1977, 2), 1)
47.7
>>> round(Epoch.tt2ut(1998, 1), 1)
63.0
>>> round(Epoch.tt2ut(2015, 7), 1)
69.3

static utc2local()[source]¶

Method to return the difference between UTC and local time.

This method provides you the seconds that you have to add or subtract to UTC time to convert to your local time. The correct way to apply this method is according to the expression:

utc2local() = LocalTime - UTC

Therefore, if for example you are located in a place whose corresponding time zone is UTC+2, then this method will yield 7200 (seconds in two hours) and you must then apply this expression:

LocalTime = UTC + utclocal() = UTC + 7200

Note

Please bear in mind that, in order for this method to work, your operative system must be correctly configured, with the right time and corresponding time zone.

Note

Remember that class Epoch internally stores time as Terrestrial Time (TT), not UTC. In order to get the UTC time you must use a method like get_date() or get_full_date() and pass the parameter utc=True.

Returns:	Difference in seconds between local and UTC time, in that order.
Return type:	float

year()[source]¶

This method returns the contents of this object as a year with decimals.

Returns:	Year with decimals.
Return type:	float

>>> e = Epoch(1993, 'October', 1)
>>> print(round(e.year(), 4))
1993.7479

pymeeus.Epoch.JDE2000 = Epoch(2451545.0)¶: Standard epoch for January 1st, 2000 at 12h corresponding to JDE2451545.0

pymeeus.Epoch.LEAP_TABLE = {1972.5: 1, 1973.0: 2, 1974.0: 3, 1975.0: 4, 1976.0: 5, 1977.0: 6, 1978.0: 7, 1979.0: 8, 1980.0: 9, 1981.5: 10, 1982.5: 11, 1983.5: 12, 1985.5: 13, 1988.0: 14, 1990.0: 15, 1991.0: 16, 1992.5: 17, 1993.5: 18, 1994.5: 19, 1996.0: 20, 1997.5: 21, 1999.0: 22, 2006.0: 23, 2009.0: 24, 2012.5: 25, 2015.5: 26, 2017.0: 27}¶: This table represents the point in time FROM WHERE the given number of leap seconds is valid. Given that leap seconds are (so far) always added at June 30th or December 31st, a leap second added in 1997/06/30 is represented here as ‘1997.5’, while a leap second added in 2005/12/31 appears here as ‘2006.0’.

Interpolation¶

Class to find intermediate values from those given in a table.

class pymeeus.Interpolation.Interpolation(*args)[source]¶

Class Interpolation deals with finding intermediate values to those given in a table.

Besides the basic interpolation, this class also provides methods to find the extrema (maximum or minimum value) in a given interval (if present), and it also has methods to find roots (the values of the argument ‘x’ for which the function ‘y’ becomes zero).

Please note that it seems that Meeus uses the Bessel interpolation method (Chapter 3). However, this class uses the Newton interpolation method because it is (arguably) more flexible regarding the number of entries provided in the interpolation table.

The constructor takes pairs of (x, y) values from the table of interest. These pairs of values can be given as a sequence of int/floats, tuples, lists or Angles. It is also possible to provide an Interpolation object to the constructor in order to get a copy.

Note

When using Angles, be careful with the 360-to-0 discontinuity.

If a sequence of int, floats or Angles is given, the values in the odd positions are considered to belong to the ‘x’ set, while the values in the even positions belong to the ‘y’ set. If only one tuple or list is provided, it is assumed that it is the ‘y’ set, and the ‘x’ set is build from 0 onwards with steps of length 1.

Please keep in mind that a minimum of two data pairs are needed in order to carry out any interpolation. If only one value pair is provided, a ValueError exception will be raised.

__call__(x)[source]¶

Method to interpolate the function at a given ‘x’.

Parameters:	x (int, float, `Angle`) – Point where the interpolation will be carried out.
Returns:	Resulting value of the interpolation.
Return type:	float
Raises:	ValueError if input value is outside of interpolation range.
Raises:	TypeError if input value is of wrong type.

>>> i = Interpolation([7, 8, 9], [0.884226, 0.877366, 0.870531])
>>> y = round(i(8.18125), 6)
>>> print(y)
0.876125

__init__(*args)[source]¶

Interpolation constructor.

This takes pairs of (x, y) values from the table of interest. These pairs of values can be given as a sequence of int/floats, tuples, lists or Angles. It is also possible to provide an Interpolation object to the constructor in order to get a copy.

Note

When using Angles, be careful with the 360-to-0 discontinuity

If a sequence of int, floats or Angles is given, the values in the odd positions are considered to belong to the ‘x’ set, while the values in the even positions belong to the ‘y’ set. If only one tuple or list is provided, it is assumed that it is the ‘y’ set, and the ‘x’ set is build from 0 onwards with steps of length 1.

Please keep in mind that a minimum of two data pairs are needed in order to carry out any interpolation. If only one value pair is provided, a ValueError exception will be raised.

Parameters:	args (int, float, list, tuple, `Angle`, `Interpolation`) – Input tabular values, or another Interpolation object.
Returns:	Interpolation object.
Return type:	`Interpolation`
Raises:	ValueError if not enough input data pairs are provided.
Raises:	TypeError if input values are of wrong type.

>>> i = Interpolation([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> print(i._x)
[1, 2, 3, 4, 5, 6]
>>> print(i._y)
[2, 4, 6, 8, 10, 12]
>>> j = Interpolation([3, -8, 1, 12, 2, 5, 8])
>>> print(j._x)
[0, 1, 2, 3, 4, 5, 6]
>>> print(j._y)
[3, -8, 1, 12, 2, 5, 8]
>>> k = Interpolation(3, -8, 1, 12, 2, 5, 8)
>>> print(k._x)
[1, 2, 3]
>>> print(k._y)
[12, 5, -8]

__len__()[source]¶

This method returns the number of interpolation points (x, y, pairs) stored in this Interpolation object.

Returns:	Number of interpolation points internally stored
Return type:	int

>>> i = Interpolation([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> len(i)
6

__repr__()[source]¶

Method providing the ‘official’ string representation of the object. It provides a valid expression that could be used to recreate the object.

Returns:	As string with a valid expression to recreate the object
Return type:	string

>>> i = Interpolation([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> repr(i)
'Interpolation([1, 2, 3, 4, 5, 6], [2, 4, 6, 8, 10, 12])'

__str__()[source]¶

Method used when trying to print the object.

Returns:	Internal tabular values as strings.
Return type:	string

>>> i = Interpolation([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> print(i)
X: [1, 2, 3, 4, 5, 6]
Y: [2, 4, 6, 8, 10, 12]

__weakref__¶: list of weak references to the object (if defined)

derivative(x)[source]¶

Method to compute the derivative from interpolation polynomial.

Parameters:	x (int, float, `Angle`) – Point where the interpolation derivative will be carried out.
Returns:	Resulting value of the interpolation derivative.
Return type:	float
Raises:	ValueError if input value is outside of interpolation range.
Raises:	TypeError if input value is of wrong type.

>>> m = Interpolation([-1.0, 0.0, 1.0], [-2.0, 3.0, 2.0])
>>> m.derivative(-1.0)
8.0
>>> m.derivative(0.5)
-1.0

get_tolerance()[source]¶

Gets the internal tolerance value used for comparisons.

Note

The default tolerance value is TOL.

Returns:	Internal tolerance.
Return type:	float

minmax(xl=0, xh=0, max_iter=1000)[source]¶

Method to find the minimum or maximum inside the [xl, xh] range.

Finding the minimum or maximum (extremum) of a function within a given interval is akin to find the root of its derivative. Therefore, this method creates an interpolation object for the derivative function, and calls the root method of that object. See root() method for more details.

If values xl, xh are not given, the limits of the interpolation table values will be used.

Note

This method returns a ValueError exception if the corresponding derivatives yl’ = f’(xl) and yh’ = f’(xh) values have the same sign. In that case, the method assumes there is no extremum in the [xl, xh] interval.

Note

If any of the xl, xh values is beyond the limits given by the interpolation values, its value will be set to the corresponding limit.

Note

If xl == xh (and not zero), a ValueError exception is raised.

Note

If the method doesn’t converge within max_iter ierations, then a ValueError exception is raised.

Parameters:	xl (int, float, `Angle`) – Lower limit of interval where a extremum will be looked for. xh (int, float, `Angle`) – Higher limit of interval where extremum will be looked for. max_iter (int) – Maximum number of iterations allowed.
Returns:	Extremum of interpolated function within [xl, xh] interval.
Return type:	int, float, `Angle`
Raises:	ValueError if yl = f(xl), yh = f(xh) have same sign.
Raises:	ValueError if xl == xh.
Raises:	ValueError if maximum number of iterations is exceeded.
Raises:	TypeError if input value is of wrong type.

>>> m = Interpolation([-1.0, 0.0, 1.0], [-2.0, 3.0, 2.0])
>>> round(m.minmax(), 8)
0.33333333

root(xl=0, xh=0, max_iter=1000)[source]¶

Method to find the root inside the [xl, xh] range.

This method applies, in principle, the Newton method to find the root; however, if conditions are such that Newton method may not bei properly behaving or converging, then it switches to the linear Interpolation method.

If values xl, xh are not given, the limits of the interpolation table values will be used.

Note

This method returns a ValueError exception if the corresponding yl = f(xl) and yh = f(xh) values have the same sign. In that case, the method assumes there is no root in the [xl, xh] interval.

Note

If any of the xl, xh values is beyond the limits given by the interpolation values, its value will be set to the corresponding limit.

Note

If xl == xh (and not zero), a ValueError exception is raised.

Note

If the method doesn’t converge within max_iter ierations, then a ValueError exception is raised.

Parameters:	xl (int, float, `Angle`) – Lower limit of interval where the root will be looked for. xh (int, float, `Angle`) – Higher limit of interval where the root will be looked for. max_iter (int) – Maximum number of iterations allowed.
Returns:	Root of the interpolated function within [xl, xh] interval.
Return type:	int, float, `Angle`
Raises:	ValueError if yl = f(xl), yh = f(xh) have same sign.
Raises:	ValueError if xl == xh.
Raises:	ValueError if maximum number of iterations is exceeded.
Raises:	TypeError if input value is of wrong type.

>>> m = Interpolation([-1.0, 0.0, 1.0], [-2.0, 3.0, 2.0])
>>> round(m.root(), 8)
-0.72075922

set(*args)[source]¶

Method used to define the value pairs of Interpolation object.

This takes pairs of (x, y) values from the table of interest. These pairs of values can be given as a sequence of int/floats, tuples, lists, or Angles. It is also possible to provide an Interpolation object to this method in order to get a copy.

Note

When using Angles, be careful with the 360-to-0 discontinuity

If a sequence of int, floats or Angles is given, the values in the odd positions are considered to belong to the ‘x’ set, while the values in the even positions belong to the ‘y’ set. If only one tuple or list is provided, it is assumed that it is the ‘y’ set, and the ‘x’ set is build from 0 onwards with steps of length 1.

Please keep in mind that a minimum of two data pairs are needed in order to carry out any interpolation. If only one value pair is provided, a ValueError exception will be raised.

Parameters:	args (int, float, list, tuple, `Angle`) – Input tabular values, or another Interpolation object.
Returns:	None.
Return type:	None
Raises:	ValueError if not enough input data pairs are provided.
Raises:	TypeError if input values are of wrong type.

>>> i = Interpolation()
>>> i.set([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])
>>> print(i._x)
[1, 2, 3, 4, 5, 6]
>>> print(i._y)
[2, 4, 6, 8, 10, 12]
>>> j = Interpolation()
>>> j.set([3, -8, 1, 12, 2, 5, 8])
>>> print(j._x)
[0, 1, 2, 3, 4, 5, 6]
>>> print(j._y)
[3, -8, 1, 12, 2, 5, 8]
>>> k = Interpolation(3, -8, 1, 12, 2, 5, 8)
>>> print(k._x)
[1, 2, 3]
>>> print(k._y)
[12, 5, -8]
>>> m = Interpolation(k)
>>> print(m._x)
[1, 2, 3]
>>> print(m._y)
[12, 5, -8]

set_tolerance(tol)[source]¶

Changes the internal tolerance value used for comparisons.

Parameters:	tol (int, float) – New tolerance value.
Returns:	None
Return type:	None

Bodies¶

Earth¶

Class to model Earth’s globe.

class pymeeus.Earth.Earth(ellipsoid=Ellipsoid(6378137.0, 0.00335281066475, 7.292115e-05))[source]¶

Class Earth models the figure of the Earth surface and, with the help of a configurable reference ellipsoid, provides a set of handy method to compute different parameters, like the distance between two points on the surface.

Please note that here we depart a little bit from Meeus’ book because the Earth class uses the World Geodetic System 1984 (WGS84) as the default reference ellipsoid, instead of the International Astronomical Union 1974, which Meeus uses. This change is done because WGS84 is regarded as more modern.

__init__(ellipsoid=Ellipsoid(6378137.0, 0.00335281066475, 7.292115e-05))[source]¶

Earth constructor.

It takes a reference ellipsoid as input. If not provided, the ellipsoid used is the WGS84 by default.

Parameters:	ellipsoid – Reference ellipsoid to be used. WGS84 by default.
Returns:	Earth object.
Return type:	`Earth`
Raises:	TypeError if input value is of wrong type.

__repr__()[source]¶

Method providing the ‘official’ string representation of the object. It provides a valid expression that could be used to recreate the object.

Returns:	As string with a valid expression to recreate the object
Return type:	string

__str__()[source]¶

Method used when trying to print the Earth object. It essentially returns the corresponting ‘__str__()’ method from the reference ellipsoid being used.

Returns:	Semi-major equatorial radius, flattening and angular velocity of the current reference ellipsoid, as a string.
Return type:	string

>>> e = Earth()
>>> s = str(e)
>>> v = s.split(':')
>>> print(v[0] + '|' + str(round(float(v[1]), 14)) + '|' + v[2] )
6378137.0|0.00335281066475|7.292115e-05

__weakref__¶: list of weak references to the object (if defined)

static apparent_heliocentric_position(epoch, nutation=True)[source]¶

This method computes the apparent heliocentric position of the Earth for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Earth position, as an Epoch object nutation (bool) – Whether the nutation correction will be applied
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> lon, lat, r = Earth.apparent_heliocentric_position(epoch)
>>> print(round(lon.to_positive(), 6))
19.905986
>>> print(lat.dms_str(n_dec=3))
-0.721''
>>> print(round(r, 8))
0.99760852

distance(lon1, lat1, lon2, lat2)[source]¶

This method computes the distance between two points on the Earth’s surface using the method from H. Andoyer.

Note

We will consider that positions ‘East’ and ‘South’ are negative.

Parameters:	lon1 (int, float, `Angle`) – Longitude of the first point, in degrees lat1 (int, float, `Angle`) – Geodetical or geographical latitude of the first point, in degrees lon2 (int, float, `Angle`) – Longitude of the second point, in degrees lat2 (int, float, `Angle`) – Geodetical or geographical latitude of the second point, in degrees
Returns:	Tuple with distance between the two points along Earth’s surface, and approximate error, in meters
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> e = Earth(ellipsoid=IAU76)
>>> lon1 = Angle(-2, 20, 14.0)
>>> lat1 = Angle(48, 50, 11.0)
>>> lon2 = Angle(77, 3, 56.0)
>>> lat2 = Angle(38, 55, 17.0)
>>> dist, error = e.distance(lon1, lat1, lon2, lat2)
>>> round(dist, 0)
6181628.0
>>> error
69.0
>>> lon1 = Angle(-2.09)
>>> lat1 = Angle(41.3)
>>> lon2 = Angle(73.99)
>>> lat2 = Angle(40.75)
>>> dist, error = e.distance(lon1, lat1, lon2, lat2)
>>> round(dist, 0)
6176760.0
>>> error
69.0

static geometric_heliocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of the Earth for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Earth position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> l, b, r = Earth.geometric_heliocentric_position(epoch, tofk5=False)
>>> print(round(l.to_positive(), 6))
19.907297
>>> print(b.dms_str(n_dec=3))
-0.744''
>>> print(round(r, 8))
0.99760852

static geometric_heliocentric_position_j2000(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of the Earth for a given epoch, using the VSOP87 theory, referred to the equinox J2000.0.

Parameters:	epoch (`Epoch`) – Epoch to compute Earth position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

linear_velocity(latitude)[source]¶

Method to compute the linear velocity of a point at latitude, due to the rotation of the Earth.

Parameters:	latitude (int, float, `Angle`) – Geodetical or geographical latitude of the observer, in degrees
Returns:	Linear velocity of a point at latitude, in meters per second
Return type:	float
Raises:	TypeError if input value is of wrong type.

>>> e = Earth(ellipsoid=IAU76)
>>> round(e.linear_velocity(42.0), 2)
346.16

static orbital_elements_j2000(epoch)[source]¶

This method computes the orbital elements of Earth for the standard equinox J2000.0 for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Earth.orbital_elements_j2000(epoch)
>>> print(round(l, 6))
271.801199
>>> print(round(a, 8))
1.00000102
>>> print(round(e, 7))
0.0166811
>>> print(round(i, 6))
0.008544
>>> print(round(ome, 5))
174.71534
>>> print(round(arg, 6))
-71.566717

static orbital_elements_mean_equinox(epoch)[source]¶

This method computes the orbital elements of Earth for the mean equinox of the date for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Earth.orbital_elements_mean_equinox(epoch)
>>> print(round(l, 6))
272.716028
>>> print(round(a, 8))
1.00000102
>>> print(round(e, 7))
0.0166811
>>> print(round(i, 6))
0.0
>>> print(round(ome, 5))
174.71534
>>> print(round(arg, 6))
-70.651889

static parallax_correction(right_ascension, declination, latitude, distance, hour_angle, height=0.0)[source]¶

This function computes the parallaxes in right ascension and declination in order to obtain the topocentric values.

Parameters:	right_ascension (`Angle`) – Geocentric right ascension, as an `Angle` object declination (`Angle`) – Geocentric declination, as an `Angle` object latitude (`Angle`) – Latitude of the observation point distance (float) – Distance from the celestial object to the Earth, in Astronomical Units hour_angle (`Angle`) – Geocentric hour angle of the celestial object, as an `Angle` heigth – Height of observation point above sea level, in meters
Returns:	Tuple containing the topocentric right ascension and declination
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> right_ascension = Angle(22, 38, 7.25, ra=True)
>>> declination = Angle(-15, 46, 15.9)
>>> latitude = Angle(33, 21, 22)
>>> distance = 0.37276
>>> hour_angle = Angle(288.7958)
>>> topo_ra, topo_dec = Earth.parallax_correction(right_ascension,                                                           declination,                                                           latitude, distance,                                                           hour_angle)
>>> print(topo_ra.ra_str(n_dec=2))
22h 38' 8.54''
>>> print(topo_dec.dms_str(n_dec=1))
-15d 46' 30.0''

static parallax_ecliptical(longitude, latitude, semidiameter, obs_lat, obliquity, sidereal_time, distance, height=0.0)[source]¶

This function computes the topocentric coordinates of a celestial body (Moon or planet) directly from its geocentric values in ecliptical coordinates.

Parameters:	longitude (`Angle`) – Geocentric ecliptical longitude as an `Angle` latitude (`Angle`) – Geocentric ecliptical latitude as an `Angle` semidiameter (`Angle`) – Geocentric semidiameter as an `Angle` obs_lat (`Angle`) – Latitude of the observation point obliquity (`Angle`) – Obliquity of the eliptic, as an `Angle` sidereal_time (`Angle`) – Local sidereal time, as an `Angle` distance (float) – Distance from the celestial object to the Earth, in Astronomical Units heigth – Height of observation point above sea level, in meters
Returns:	Tuple containing the topocentric longitude, latitude and semidiameter
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> longitude = Angle(181, 46, 22.5)
>>> latitude = Angle(2, 17, 26.2)
>>> semidiameter = Angle(0, 16, 15.5)
>>> obs_lat = Angle(50, 5, 7.8)
>>> obliquity = Angle(23, 28, 0.8)
>>> sidereal_time = Angle(209, 46, 7.9)
>>> distance = 0.0024650163
>>> topo_lon, topo_lat, topo_diam =                 Earth.parallax_ecliptical(longitude, latitude, semidiameter,                                           obs_lat, obliquity, sidereal_time,                                           distance)
>>> print(topo_lon.dms_str(n_dec=1))
181d 48' 5.0''
>>> print(topo_lat.dms_str(n_dec=1))
1d 29' 7.1''
>>> print(topo_diam.dms_str(n_dec=1))
16' 25.5''

static passage_nodes(epoch, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of Earth, nearest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch closest to the node passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2019, 1, 1)
>>> time, r = Earth.passage_nodes(epoch)
>>> year, month, day = time.get_date()
>>> print(year)
2019
>>> print(month)
3
>>> print(round(day, 1))
15.0
>>> print(round(r, 4))
0.9945

static perihelion_aphelion(epoch, perihelion=True)[source]¶

This method computes the time of Perihelion (or Aphelion) closer to a given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired Perihelion (or Aphelion) peihelion – If True, the epoch of the closest Perihelion is computed, if False, the epoch of the closest Aphelion is found.
Returns:	The epoch of the desired Perihelion (or Aphelion)
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1989, 11, 20.0)
>>> e = Earth.perihelion_aphelion(epoch)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
1990
>>> print(m)
1
>>> print(d)
4
>>> print(h)
17
>>> epoch = Epoch(2000, 4, 1.0)
>>> e = Earth.perihelion_aphelion(epoch, perihelion=False)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2000
>>> print(m)
7
>>> print(d)
3
>>> print(h)
23
>>> print(mi)
51
>>> epoch = Epoch(2003, 3, 10.0)
>>> e = Earth.perihelion_aphelion(epoch)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2003
>>> print(m)
1
>>> print(d)
4
>>> print(h)
5
>>> print(mi)
1
>>> epoch = Epoch(2009, 5, 1.0)
>>> e = Earth.perihelion_aphelion(epoch, perihelion=False)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2009
>>> print(m)
7
>>> print(d)
4
>>> print(h)
1
>>> print(mi)
41

rho(latitude)[source]¶

Method to compute the rho term, which is the observer distance to the center of the Earth, when the observer is at sea level. In this case, the Earth’s equatorial radius is taken as unity.

Parameters:	latitude (int, float, `Angle`) – Geodetical or geographical latitude of the observer, in degrees
Returns:	Rho: Distance to the center of the Earth from sea level. It is a ratio with respect to Earth equatorial radius.
Return type:	float
Raises:	TypeError if input value is of wrong type.

>>> e = Earth(ellipsoid=IAU76)
>>> round(e.rho(0.0), 1)
1.0

rho_cosphi(latitude, height)[source]¶

Method to compute the rho*cos(phi’) term, needed in the calculation of diurnal parallaxes, eclipses and occulatitions.

Parameters:	latitude (int, float, `Angle`) – Geodetical or geographical latitude of the observer, in degrees height (int, float) – Height of the observer above the sea level, in meters
Returns:	rho*cos(phi’) term
Return type:	float
Raises:	TypeError if input value is of wrong type.

>>> lat = Angle(33, 21, 22.0)
>>> e = Earth(ellipsoid=IAU76)
>>> round(e.rho_cosphi(lat, 1706), 6)
0.836339

rho_sinphi(latitude, height)[source]¶

Method to compute the rho*sin(phi’) term, needed in the calculation of diurnal parallaxes, eclipses and occulatitions.

Parameters:	latitude (int, float, `Angle`) – Geodetical or geographical latitude of the observer, in degrees height (int, float) – Height of the observer above the sea level, in meters
Returns:	rho*sin(phi’) term
Return type:	float
Raises:	TypeError if input value is of wrong type.

>>> lat = Angle(33, 21, 22.0)
>>> e = Earth(ellipsoid=IAU76)
>>> round(e.rho_sinphi(lat, 1706), 6)
0.546861

rm(latitude)[source]¶

Method to compute the radius of curvature of the Earth’s meridian at the given latitude.

Parameters:	latitude (int, float, `Angle`) – Geodetical or geographical latitude of the observer, in degrees
Returns:	Radius of curvature of the Earth’s meridian at the given latitude, in meters
Return type:	float
Raises:	TypeError if input value is of wrong type.

>>> e = Earth(ellipsoid=IAU76)
>>> round(e.rm(42.0), 1)
6364033.3

rp(latitude)[source]¶

Method to compute the radius of the parallel circle at the given latitude.

Parameters:	latitude (int, float, `Angle`) – Geodetical or geographical latitude of the observer, in degrees
Returns:	Radius of the parallel circle at given latitude, in meters
Return type:	float
Raises:	TypeError if input value is of wrong type.

>>> e = Earth(ellipsoid=IAU76)
>>> round(e.rp(42.0), 1)
4747001.2

set(ellipsoid)[source]¶

Method used to define an Earth object.

It takes a reference ellipsoid as input. If not provided, the ellipsoid used is the WGS84 by default.

Parameters:	ellipsoid – Reference ellipsoid to be used. WGS84 by default.
Returns:	None
Return type:	None
Raises:	TypeError if input value is of wrong type.

class pymeeus.Earth.Ellipsoid(a, f, omega)[source]¶

Class Ellipsoid is useful to encapsulate the most important parameters of a given reference ellipsoid.

__init__(a, f, omega)[source]¶

Ellipsoid constructor.

Parameters:	a (float) – Semi-major or equatorial radius, in meters f (float) – Flattening omega (float) – Angular velocity of the Earth, in rad/s

__repr__()[source]¶

Method providing the ‘official’ string representation of the object. It provides a valid expression that could be used to recreate the object.

Returns:	As string with a valid expression to recreate the object
Return type:	string

>>> a = Ellipsoid(6378140.0, 0.0033528132, 7.292e-5)
>>> repr(a)
'Ellipsoid(6378140.0, 0.0033528132, 7.292e-05)'

__str__()[source]¶

Method used when trying to print the object.

Returns:	Semi-major equatorial radius, flattening and angular velocity as a string.
Return type:	string

>>> a = Ellipsoid(6378140.0, 0.0033528132, 7.292e-5)
>>> print(a)
6378140.0:0.0033528132:7.292e-05

__weakref__¶: list of weak references to the object (if defined)

b()[source]¶

Method to return the semi-minor radius.

Returns:	Semi-minor radius, in meters
Return type:	float

>>> a = Ellipsoid(6378140.0, 0.0033528132, 7.292e-5)
>>> round(a.b(), 3)
6356755.288

e()[source]¶

Method to return the eccentricity of the Earth’s meridian.

Returns:	Eccentricity of the Earth’s meridian
Return type:	float

>>> a = Ellipsoid(6378140.0, 0.0033528132, 7.292e-5)
>>> round(a.e(), 8)
0.08181922

pymeeus.Earth.IAU76 = Ellipsoid(6378140.0, 0.0033528131779, 7.292114992e-05)¶: Reference ellipsoid defined by the International Astronomic Union in 1976

pymeeus.Earth.ORBITAL_ELEM = [[100.466457, 36000.7698278, 0.00030322, 2e-08], [1.000001018, 0.0, 0.0, 0.0], [0.01670863, -4.2037e-05, -1.267e-07, 1.4e-10], [0.0, 0.0, 0.0, 0.0], [174.873176, -0.2410908, 4.262e-05, 1e-09], [102.937348, 1.7195366, 0.00045688, -1.8e-08]]¶: This table contains the parameters to compute Earth’s orbital elements for the mean equinox of date. Based in Table 31.A, page 212

pymeeus.Earth.ORBITAL_ELEM_J2000 = [[100.466457, 35999.3728565, -5.68e-06, -1e-09], [0.0, 0.0130548, -9.31e-06, -3.4e-08], [174.873176, -0.2410908, 4.262e-05, 1e-09], [102.937348, 0.3225654, 0.00014799, -3.9e-08]]¶: This table contains the parameters to compute Earth’s orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 214

pymeeus.Earth.VSOP87_B = [[[279.62, 3.19870156017, 84334.66158130829], [101.643, 5.42248619256, 5507.5532386674], [80.445, 3.88013204458, 5223.6939198022], [43.806, 3.70444689758, 2352.8661537718], [31.933, 4.00026369781, 1577.3435424478], [22.724, 3.9847383156, 1047.7473117547], [16.392, 3.56456119782, 5856.4776591154], [18.141, 4.98367470263, 6283.0758499914], [14.443, 3.70275614914, 9437.762934887], [14.304, 3.41117857525, 10213.285546211], [11.246, 4.8282069053, 14143.4952424306], [10.9, 2.08574562327, 6812.766815086], [9.714, 3.47303947752, 4694.0029547076], [10.367, 4.05663927946, 71092.88135493269], [8.775, 4.44016515669, 5753.3848848968], [8.366, 4.9925151218, 7084.8967811152], [6.921, 4.32559054073, 6275.9623029906], [9.145, 1.14182646613, 6620.8901131878], [7.194, 3.60193205752, 529.6909650946], [7.698, 5.55425745881, 167621.5758508619], [5.285, 2.48446991566, 4705.7323075436], [5.208, 6.24992674537, 18073.7049386502], [4.529, 2.33827747356, 6309.3741697912], [5.579, 4.41023653738, 7860.4193924392], [4.743, 0.70995680136, 5884.9268465832], [4.301, 1.10255777773, 6681.2248533996], [3.849, 1.82229412531, 5486.777843175], [4.093, 5.11700141207, 13367.9726311066], [3.681, 0.43793170356, 3154.6870848956], [3.42, 5.42034800952, 6069.7767545534], [3.617, 6.04641937526, 3930.2096962196], [3.67, 4.58210192227, 12194.0329146209], [2.918, 1.95463881126, 10977.078804699], [2.797, 5.61259275048, 11790.6290886588], [2.502, 0.60499729367, 6496.3749454294], [2.319, 5.01648216014, 1059.3819301892], [2.684, 1.39470396488, 22003.9146348698], [2.428, 3.24183056052, 78051.5857313169], [2.12, 4.30691000285, 5643.1785636774], [2.257, 3.15557225618, 90617.7374312997], [1.813, 3.75574218285, 3340.6124266998], [2.226, 2.79699346659, 12036.4607348882], [1.888, 0.86991545823, 8635.9420037632], [1.517, 1.95852055701, 398.1490034082], [1.581, 3.19976230948, 5088.6288397668], [1.421, 6.25530883827, 2544.3144198834], [1.595, 0.25619915135, 17298.1823273262], [1.391, 4.69964175561, 7058.5984613154], [1.478, 2.81808207569, 25934.1243310894], [1.481, 3.65823554806, 11506.7697697936], [1.693, 4.95689385293, 156475.2902479957], [1.183, 1.29343061246, 775.522611324], [1.114, 2.37889311846, 3738.761430108], [0.994, 4.30088900425, 9225.539273283], [0.924, 3.06451026812, 4164.311989613], [0.867, 0.55606931068, 8429.2412664666], [0.988, 5.97286104208, 7079.3738568078], [0.824, 1.50984806173, 10447.3878396044], [0.915, 0.12635654592, 11015.1064773348], [0.742, 1.99159139281, 26087.9031415742], [1.039, 3.14159265359, 0.0], [0.85, 4.24120016095, 29864.334027309], [0.755, 2.8963187332, 4732.0306273434], [0.714, 1.37548118603, 2146.1654164752], [0.708, 1.91406542362, 8031.0922630584], [0.746, 0.57893808616, 796.2980068164], [0.802, 5.1233913723, 2942.4634232916], [0.751, 1.67479850166, 21228.3920235458], [0.602, 4.09976538826, 64809.80550494129], [0.594, 3.49580704962, 16496.3613962024], [0.592, 4.59481504319, 4690.4798363586], [0.53, 5.739792952, 8827.3902698748], [0.503, 5.66433137112, 33794.5437235286], [0.483, 1.57106522411, 801.8209311238], [0.438, 0.06707733767, 3128.3887650958], [0.423, 2.86944595927, 12566.1516999828], [0.504, 3.2620766916, 7632.9432596502], [0.552, 1.02926440457, 239762.20451754928], [0.427, 3.6743437821, 213.299095438], [0.404, 1.46193297142, 15720.8387848784], [0.503, 4.85802444134, 6290.1893969922], [0.417, 0.81920713533, 5216.5803728014], [0.365, 0.01002966162, 12168.0026965746], [0.363, 1.28376436579, 6206.8097787158], [0.353, 4.7005913311, 7234.794256242], [0.415, 0.96862624175, 4136.9104335162], [0.387, 3.09145061418, 25158.6017197654], [0.373, 2.65119262792, 7342.4577801806], [0.361, 2.97762937739, 9623.6882766912], [0.418, 3.75759994446, 5230.807466803], [0.396, 1.22507712354, 6438.4962494256], [0.322, 1.21162178805, 8662.240323563], [0.284, 5.64170320068, 1589.0728952838], [0.379, 1.72248432748, 14945.3161735544], [0.32, 3.94161159962, 7330.8231617461], [0.313, 5.47602376446, 1194.4470102246], [0.292, 1.38971327603, 11769.8536931664], [0.305, 0.80429352049, 37724.7534197482], [0.257, 5.81382809757, 426.598190876], [0.265, 6.10358507671, 6836.6452528338], [0.25, 4.56452895547, 7477.522860216], [0.266, 2.62926282354, 7238.6755916], [0.263, 6.22089501237, 6133.5126528568], [0.306, 2.79682380531, 1748.016413067], [0.236, 2.46093023714, 11371.7046897582], [0.316, 1.62662805006, 250908.4901204155], [0.216, 3.68721275185, 5849.3641121146], [0.23, 0.36165162947, 5863.5912061162], [0.233, 5.03509933858, 20426.571092422], [0.2, 5.86073159059, 4535.0594369244], [0.277, 4.65400292395, 82239.1669577989], [0.209, 3.72323200804, 10973.55568635], [0.199, 5.05186622555, 5429.8794682394], [0.256, 2.4092327977, 19651.048481098], [0.21, 4.50691909144, 29088.811415985], [0.181, 6.00294783127, 4292.3308329504], [0.249, 0.12900984422, 154379.7956244863], [0.209, 3.87759458598, 17789.845619785], [0.225, 3.18339652605, 18875.525869774], [0.191, 4.53897489299, 18477.1087646123], [0.172, 2.09694183014, 13095.8426650774], [0.182, 3.161079435, 16730.4636895958], [0.188, 2.22746128596, 41654.9631159678], [0.164, 5.18686275017, 5481.2549188676], [0.16, 2.49298855159, 12592.4500197826], [0.155, 1.5959543823, 10021.8372800994], [0.135, 0.21349051064, 10988.808157535], [0.178, 3.8037517797, 23581.2581773176], [0.123, 1.66800739151, 15110.4661198662], [0.122, 2.72678272244, 18849.2275499742], [0.126, 1.1767551291, 14919.0178537546], [0.142, 3.95053441332, 337.8142631964], [0.116, 6.06340906229, 6709.6740408674], [0.137, 3.52143246757, 12139.5535091068], [0.136, 2.92179113542, 32217.2001810808], [0.11, 3.51203379263, 18052.9295431578], [0.147, 4.63371971408, 22805.7355659936], [0.108, 5.45280814878, 7.1135470008], [0.148, 0.65447253687, 95480.9471841745], [0.119, 5.92110458985, 33019.0211122046], [0.11, 5.34824206306, 639.897286314], [0.106, 3.71081682629, 14314.1681130498], [0.139, 6.17607198418, 24356.7807886416], [0.118, 5.5973871267, 161338.5000008705], [0.117, 3.6506527164, 45585.1728121874], [0.127, 4.74596574209, 49515.382508407], [0.12, 1.04211499785, 6915.8595893046], [0.12, 5.60638811846, 5650.2921106782], [0.115, 3.10668213289, 14712.317116458], [0.099, 0.69018940049, 12779.4507954208], [0.097, 1.07908724794, 9917.6968745098], [0.093, 2.62295197319, 17260.1546546904], [0.099, 4.45774681732, 4933.2084403326], [0.123, 1.37488922089, 28286.9904848612], [0.121, 5.19767249813, 27511.4678735372], [0.105, 0.87192267806, 77375.95720492408], [0.087, 3.9363781295, 17654.7805397496], [0.122, 2.2395606868, 83997.09113559539], [0.087, 4.18201600952, 22779.4372461938], [0.104, 4.59580877295, 1349.8674096588], [0.102, 2.83545248411, 12352.8526045448], [0.102, 3.97386522171, 10818.1352869158], [0.101, 4.32892825857, 36147.4098773004], [0.094, 5.00001709261, 150192.2143980043], [0.077, 3.97199369296, 1592.5960136328], [0.1, 6.07733097102, 26735.9452622132], [0.086, 5.2602963825, 28313.288804661], [0.093, 4.31900620254, 44809.6502008634], [0.076, 6.22743405935, 13521.7514415914], [0.072, 1.55820597747, 6256.7775301916], [0.082, 4.95202664555, 10575.4066829418], [0.082, 1.69647647075, 1990.745017041], [0.075, 2.29836095644, 3634.6210245184], [0.075, 2.66367876557, 16200.7727245012], [0.087, 0.26630214764, 31441.6775697568], [0.077, 2.25530954137, 5235.3285382367], [0.076, 1.09869730846, 12903.9659631792], [0.058, 4.28246138307, 12559.038152982], [0.064, 5.51112830114, 173904.65170085328], [0.056, 2.60133794851, 73188.3759784421], [0.055, 5.81483150022, 143233.51002162008], [0.054, 3.38482031504, 323049.11878710287], [0.039, 3.28500401343, 71768.50988132549], [0.039, 3.1123991069, 96900.81328129109]], [[9.03, 3.8972906189, 5507.5532386674], [6.177, 1.73038850355, 5223.6939198022], [3.8, 5.24404145734, 2352.8661537718], [2.834, 2.4734503745, 1577.3435424478], [1.817, 0.41874743765, 6283.0758499914], [1.499, 1.83320979291, 5856.4776591154], [1.466, 5.69401926017, 5753.3848848968], [1.301, 2.18890066314, 9437.762934887], [1.233, 4.95222451476, 10213.285546211], [1.021, 0.12866660208, 7860.4193924392], [0.982, 0.09005453285, 14143.4952424306], [0.865, 1.73949953555, 3930.2096962196], [0.581, 2.26949174067, 5884.9268465832], [0.524, 5.65662503159, 529.6909650946], [0.473, 6.22750969242, 6309.3741697912], [0.451, 1.53288619213, 18073.7049386502], [0.364, 3.61614477374, 13367.9726311066], [0.372, 3.2247072132, 6275.9623029906], [0.268, 2.34341267879, 11790.6290886588], [0.322, 0.94084045832, 6069.7767545534], [0.232, 0.26781182579, 7058.5984613154], [0.216, 6.05952221329, 10977.078804699], [0.232, 2.93325646109, 22003.9146348698], [0.204, 3.86264841382, 6496.3749454294], [0.202, 2.81892511133, 15720.8387848784], [0.185, 4.93512381859, 12036.4607348882], [0.22, 3.99305643742, 6812.766815086], [0.166, 1.74970002999, 11506.7697697936], [0.212, 1.57166285369, 4694.0029547076], [0.157, 1.08259734788, 5643.1785636774], [0.154, 5.99434678412, 5486.777843175], [0.144, 5.23285656085, 78051.5857313169], [0.144, 1.16454655948, 90617.7374312997], [0.137, 2.67760436027, 6290.1893969922], [0.18, 2.06509026215, 7084.8967811152], [0.121, 5.90212574947, 9225.539273283], [0.15, 2.00175038718, 5230.807466803], [0.149, 5.06157254516, 17298.1823273262], [0.118, 5.39979058038, 3340.6124266998], [0.161, 3.32421999691, 6283.3196674749], [0.121, 4.36722193162, 19651.048481098], [0.116, 5.83462858507, 4705.7323075436], [0.128, 4.35489873365, 25934.1243310894], [0.143, 0.0, 0.0], [0.109, 2.52157834166, 6438.4962494256], [0.099, 2.70727488041, 5216.5803728014], [0.103, 0.93782340879, 8827.3902698748], [0.082, 4.2921468039, 8635.9420037632], [0.079, 2.24085737326, 1059.3819301892], [0.097, 5.50959692365, 29864.334027309], [0.072, 0.21891639822, 21228.3920235458], [0.071, 2.86755026812, 6681.2248533996], [0.074, 2.20184828895, 37724.7534197482], [0.063, 4.45586625948, 7079.3738568078], [0.061, 0.63918772258, 33794.5437235286], [0.047, 2.09070235724, 3128.3887650958], [0.047, 3.325438433, 26087.9031415742], [0.049, 1.60680905005, 6702.5604938666], [0.057, 0.11215813438, 29088.811415985], [0.056, 5.47982934911, 775.522611324], [0.05, 1.89396788463, 12139.5535091068], [0.047, 2.9721490724, 20426.571092422], [0.041, 5.5532939489, 11015.1064773348], [0.041, 5.91861144924, 23581.2581773176], [0.045, 4.95273290181, 5863.5912061162], [0.05, 3.62740835096, 41654.9631159678], [0.037, 6.09033460601, 64809.80550494129], [0.037, 5.86153655431, 12566.1516999828], [0.046, 1.65798680284, 25158.6017197654], [0.038, 2.00673650251, 426.598190876], [0.036, 6.24373396652, 6283.14316029419], [0.036, 0.40465162918, 6283.0085396886], [0.032, 6.03707103538, 2942.4634232916], [0.041, 4.86809570283, 1592.5960136328], [0.028, 4.38359423735, 7632.9432596502], [0.028, 6.03334294232, 17789.845619785], [0.026, 3.88971333608, 5331.3574437408], [0.026, 5.94932724051, 16496.3613962024], [0.031, 1.44666331503, 16730.4636895958], [0.026, 6.26376705837, 23543.23050468179], [0.033, 0.93797239147, 213.299095438], [0.026, 3.71858432944, 13095.8426650774], [0.027, 0.60565274405, 10988.808157535], [0.023, 4.4438898555, 18849.2275499742], [0.028, 1.53862289477, 6279.4854213396], [0.028, 1.96831814872, 6286.6662786432], [0.028, 5.78094918529, 15110.4661198662], [0.026, 2.48165809843, 5729.506447149], [0.02, 3.85655029499, 9623.6882766912], [0.021, 5.83006047147, 7234.794256242], [0.021, 0.69628570421, 398.1490034082], [0.022, 5.02222806555, 6127.6554505572], [0.02, 3.4761126529, 6148.010769956], [0.02, 0.90769829044, 5481.2549188676], [0.02, 0.03081589303, 6418.1409300268], [0.02, 3.74220084927, 1589.0728952838], [0.021, 4.00149269576, 3154.6870848956], [0.018, 1.58348238359, 2118.7638603784], [0.019, 0.85407021371, 14712.317116458]], [[1.662, 1.62703209173, 84334.66158130829], [0.492, 2.41382223971, 1047.7473117547], [0.344, 2.24353004539, 5507.5532386674], [0.258, 6.00906896311, 5223.6939198022], [0.131, 0.9544734524, 6283.0758499914], [0.086, 1.67530247303, 7860.4193924392], [0.09, 0.97606804452, 1577.3435424478], [0.09, 0.37899871725, 2352.8661537718], [0.089, 6.25807507963, 10213.285546211], [0.075, 0.84213523741, 167621.5758508619], [0.052, 1.70501566089, 14143.4952424306], [0.057, 6.15295833679, 12194.0329146209], [0.051, 1.2761601674, 5753.3848848968], [0.051, 5.37229738682, 6812.766815086], [0.034, 1.73672994279, 7058.5984613154], [0.038, 2.77761031485, 10988.808157535], [0.046, 3.38617099014, 156475.2902479957], [0.021, 1.95248349228, 8827.3902698748], [0.018, 3.33419222028, 8429.2412664666], [0.019, 4.32945160287, 17789.845619785], [0.017, 0.66191210656, 6283.0085396886], [0.018, 3.74885333072, 11769.8536931664], [0.017, 4.23058370776, 10977.078804699], [0.017, 1.78116162721, 5486.777843175], [0.021, 1.36972913918, 12036.4607348882], [0.017, 2.79601092529, 796.2980068164], [0.015, 0.4308784885, 11790.6290886588], [0.017, 1.35132152761, 78051.5857313169], [0.015, 1.17032155085, 213.299095438], [0.018, 2.85221514199, 5088.6288397668], [0.017, 0.21780913672, 6283.14316029419], [0.013, 1.21201504386, 25132.3033999656], [0.012, 1.12953712197, 90617.7374312997], [0.012, 5.13714452592, 7079.3738568078], [0.013, 3.79842135217, 4933.2084403326], [0.012, 4.89407978213, 3738.761430108], [0.015, 6.05682328852, 398.1490034082], [0.014, 4.81029291856, 4694.0029547076], [0.011, 0.61684523405, 3128.3887650958], [0.011, 5.328765385, 6040.3472460174], [0.014, 5.27227350286, 4535.0594369244], [0.011, 2.39292099451, 5331.3574437408], [0.01, 4.4529653271, 6525.8044539654], [0.014, 4.66400985037, 8031.0922630584], [0.01, 3.22472385926, 9437.762934887], [0.011, 3.80913404437, 801.8209311238], [0.01, 5.15032130575, 11371.7046897582], [0.013, 0.98720797401, 5729.506447149], [0.009, 5.94191743597, 7632.9432596502]], [[0.011, 0.23877262399, 7860.4193924392], [0.009, 1.16069982609, 5507.5532386674], [0.008, 1.65357552925, 5884.9268465832], [0.008, 2.86720038197, 7058.5984613154], [0.007, 3.04818741666, 5486.777843175], [0.007, 2.59437103785, 529.6909650946], [0.008, 4.02863090524, 6256.7775301916], [0.008, 2.42003508927, 5753.3848848968], [0.006, 0.84181087594, 6275.9623029906], [0.006, 5.40160929468, 1577.3435424478], [0.007, 2.73399865247, 6309.3741697912]], [[0.004, 0.79662198849, 6438.4962494256], [0.005, 0.84308705203, 1047.7473117547], [0.005, 0.05711572303, 84334.66158130829], [0.003, 3.46779895686, 6279.5527316424], [0.003, 2.89822201212, 6127.6554505572]]]¶: This table contains Earth’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in page 420.

pymeeus.Earth.VSOP87_B_J2000 = [[[280.0, 3.199, 84334.662], [102.0, 5.422, 5507.553], [80.0, 3.88, 5223.69], [44.0, 3.7, 2352.87], [32.0, 4.0, 1577.34]], [[227778.0, 3.413766, 6283.07585], [3806.0, 3.3706, 12566.1517], [3620.0, 0.0, 0.0], [72.0, 3.33, 18849.23], [8.0, 3.89, 5507.55], [8.0, 1.79, 5223.69], [6.0, 5.2, 2352.87]], [[9721.0, 5.1519, 6283.07585], [233.0, 3.1416, 0.0], [134.0, 0.644, 12566.152], [7.0, 1.07, 18849.23]], [[276.0, 0.595, 6283.076], [17.0, 3.14, 0.0], [4.0, 0.12, 12566.15]], [[6.0, 2.27, 6283.08], [1.0, 0.0, 0.0]]]¶: This table contains Earth’s most important periodic terms from the planetary theory VSOP87 for the heliocentric latitude, referred to the equinox J2000.0. In Meeus’ book these values can be found in page 420 and page 173.

pymeeus.Earth.VSOP87_L = [[[175347045.673, 0.0, 0.0], [3341656.456, 4.66925680417, 6283.0758499914], [34894.275, 4.62610241759, 12566.1516999828], [3417.571, 2.82886579606, 3.523118349], [3497.056, 2.74411800971, 5753.3848848968], [3135.896, 3.62767041758, 77713.7714681205], [2676.218, 4.41808351397, 7860.4193924392], [2342.687, 6.13516237631, 3930.2096962196], [1273.166, 2.03709655772, 529.6909650946], [1324.292, 0.74246356352, 11506.7697697936], [901.855, 2.04505443513, 26.2983197998], [1199.167, 1.10962944315, 1577.3435424478], [857.223, 3.50849156957, 398.1490034082], [779.786, 1.17882652114, 5223.6939198022], [990.25, 5.23268129594, 5884.9268465832], [753.141, 2.53339053818, 5507.5532386674], [505.264, 4.58292563052, 18849.2275499742], [492.379, 4.20506639861, 775.522611324], [356.655, 2.91954116867, 0.0673103028], [284.125, 1.89869034186, 796.2980068164], [242.81, 0.34481140906, 5486.777843175], [317.087, 5.84901952218, 11790.6290886588], [271.039, 0.31488607649, 10977.078804699], [206.16, 4.80646606059, 2544.3144198834], [205.385, 1.86947813692, 5573.1428014331], [202.261, 2.45767795458, 6069.7767545534], [126.184, 1.0830263021, 20.7753954924], [155.516, 0.83306073807, 213.299095438], [115.132, 0.64544911683, 0.9803210682], [102.851, 0.63599846727, 4694.0029547076], [101.724, 4.26679821365, 7.1135470008], [99.206, 6.20992940258, 2146.1654164752], [132.212, 3.41118275555, 2942.4634232916], [97.607, 0.6810127227, 155.4203994342], [85.128, 1.29870743025, 6275.9623029906], [74.651, 1.75508916159, 5088.6288397668], [101.895, 0.97569221824, 15720.8387848784], [84.711, 3.67080093025, 71430.69561812909], [73.547, 4.67926565481, 801.8209311238], [73.874, 3.50319443167, 3154.6870848956], [78.756, 3.03698313141, 12036.4607348882], [79.637, 1.807913307, 17260.1546546904], [85.803, 5.98322631256, 161000.6857376741], [56.963, 2.78430398043, 6286.5989683404], [61.148, 1.81839811024, 7084.8967811152], [69.627, 0.83297596966, 9437.762934887], [56.116, 4.38694880779, 14143.4952424306], [62.449, 3.97763880587, 8827.3902698748], [51.145, 0.28306864501, 5856.4776591154], [55.577, 3.47006009062, 6279.5527316424], [41.036, 5.36817351402, 8429.2412664666], [51.605, 1.33282746983, 1748.016413067], [51.992, 0.18914945834, 12139.5535091068], [49.0, 0.48735065033, 1194.4470102246], [39.2, 6.16832995016, 10447.3878396044], [35.566, 1.77597314691, 6812.766815086], [36.77, 6.04133859347, 10213.285546211], [36.596, 2.56955238628, 1059.3819301892], [33.291, 0.59309499459, 17789.845619785], [35.954, 1.70876111898, 2352.8661537718], [40.938, 2.39850881707, 19651.048481098], [30.047, 2.73975123935, 1349.8674096588], [30.412, 0.44294464135, 83996.84731811189], [23.663, 0.48473567763, 8031.0922630584], [23.574, 2.06527720049, 3340.6124266998], [21.089, 4.14825464101, 951.7184062506], [24.738, 0.21484762138, 3.5904286518], [25.352, 3.16470953405, 4690.4798363586], [22.82, 5.22197888032, 4705.7323075436], [21.419, 1.42563735525, 16730.4636895958], [21.891, 5.55594302562, 553.5694028424], [17.481, 4.56052900359, 135.0650800354], [19.925, 5.22208471269, 12168.0026965746], [19.86, 5.77470167653, 6309.3741697912], [20.3, 0.37133792946, 283.8593188652], [14.421, 4.19315332546, 242.728603974], [16.225, 5.98837722564, 11769.8536931664], [15.077, 4.19567181073, 6256.7775301916], [19.124, 3.82219996949, 23581.2581773176], [18.888, 5.38626880969, 149854.4001348079], [14.346, 3.72355084422, 38.0276726358], [17.898, 2.21490735647, 13367.9726311066], [12.054, 2.62229588349, 955.5997416086], [11.287, 0.17739328092, 4164.311989613], [13.971, 4.40138139996, 6681.2248533996], [13.621, 1.88934471407, 7632.9432596502], [12.503, 1.13052412208, 5.5229243074], [10.498, 5.35909518669, 1592.5960136328], [9.803, 0.99947478995, 11371.7046897582], [9.22, 4.57138609781, 4292.3308329504], [10.327, 6.19982566125, 6438.4962494256], [12.003, 1.003514567, 632.7837393132], [10.827, 0.32734520222, 103.0927742186], [8.356, 4.53902685948, 25132.3033999656], 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3.11002782516, 4136.9104335162], [0.032, 5.52273255667, 5753.3848848968], [0.035, 3.7969999668, 143571.32428481648], [0.026, 1.50634201907, 154717.6098876827], [0.03, 3.53519084118, 6284.0561710596], [0.023, 4.41808025967, 5884.9268465832], [0.025, 1.38477355808, 65147.6197681377], [0.023, 3.49782549797, 7477.522860216], [0.019, 3.14329413716, 6496.3749454294], [0.019, 2.20135125199, 18073.7049386502], [0.019, 4.95020255309, 3930.2096962196], [0.019, 0.57998702747, 31415.379249957], [0.021, 1.75474323399, 12139.5535091068], [0.019, 3.92233070499, 19651.048481098], [0.014, 0.98131213224, 12559.038152982], [0.019, 4.93309333729, 2942.4634232916], [0.016, 5.55997534558, 8827.3902698748], [0.013, 1.68808165516, 4535.0594369244], [0.013, 0.33982116161, 4933.2084403326], [0.012, 1.85426309994, 5856.4776591154], [0.01, 4.82763996845, 13095.8426650774], [0.011, 5.38005490571, 11790.6290886588], [0.01, 1.40815507226, 10988.808157535], [0.011, 3.05005267431, 17260.1546546904], [0.01, 4.93364992366, 12352.8526045448]], [[289.226, 5.84384198723, 6283.0758499914], [34.955, 0.0, 0.0], [16.819, 5.48766912348, 12566.1516999828], [2.962, 5.19577265202, 155.4203994342], [1.288, 4.72200252235, 3.523118349], [0.635, 5.96925937141, 242.728603974], [0.714, 5.30045809128, 18849.2275499742], [0.402, 3.78682982419, 553.5694028424], [0.072, 4.2976812618, 6286.5989683404], [0.067, 0.90721687647, 6127.6554505572], [0.036, 5.24029648014, 6438.4962494256], [0.024, 5.16003960716, 25132.3033999656], [0.023, 3.01921570335, 6309.3741697912], [0.017, 5.82863573502, 6525.8044539654], [0.017, 3.6777286393, 71430.69561812909], [0.009, 4.58467294499, 1577.3435424478], [0.008, 1.40626662824, 11856.2186514245], [0.008, 5.07561257196, 6256.7775301916], [0.007, 2.82473374405, 83996.84731811189], [0.005, 2.71488713339, 10977.078804699], [0.005, 3.76879847273, 12036.4607348882], [0.005, 4.28412873331, 6275.9623029906]], [[114.084, 3.14159265359, 0.0], [7.717, 4.13446589358, 6283.0758499914], [0.765, 3.83803776214, 12566.1516999828], [0.42, 0.41925861858, 155.4203994342], [0.04, 3.5984758584, 18849.2275499742], [0.041, 3.14398414077, 3.523118349], [0.035, 5.00298940826, 5573.1428014331], [0.013, 0.48794833701, 77713.7714681205], [0.01, 5.6480176635, 6127.6554505572], [0.008, 2.84160570605, 161000.6857376741], [0.002, 0.54912904658, 6438.4962494256]], [[0.878, 3.14159265359, 0.0], [0.172, 2.7657906951, 6283.0758499914], [0.05, 2.01353298182, 155.4203994342], [0.028, 2.21496423926, 12566.1516999828], [0.005, 1.75600058765, 18849.2275499742]]]¶: This table contains Earth’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 418-420.

pymeeus.Earth.VSOP87_L_J2000 = [[[175347046.0, 0.0, 0.0], [3341656.0, 4.6692568, 6283.07585], [34894.0, 4.6261, 12556.1517], [3497.0, 2.7441, 5753.3849], [3418.0, 2.8289, 3.5231], [3136.0, 3.6277, 77713.7715], [2676.0, 4.4181, 7860.4194], [2343.0, 6.1352, 3930.2097], [1324.0, 0.7425, 11506.7698], [1273.0, 2.0371, 529.691], [1199.0, 1.1096, 1577.3435], [990.0, 5.233, 5884.927], [902.0, 2.045, 26.298], [857.0, 3.508, 398.149], [780.0, 1.179, 5223.694], [753.0, 2.533, 5507.553], [505.0, 4.583, 18849.228], [492.0, 4.205, 775.523], [357.0, 2.92, 0.067], [317.0, 5.849, 11790.629], [284.0, 1.899, 796.298], [271.0, 0.315, 10977.079], [243.0, 0.345, 5486.778], [206.0, 4.806, 2544.314], [205.0, 1.869, 5573.143], [202.0, 2.458, 6069.777], [156.0, 0.833, 213.299], [132.0, 3.411, 2942.463], [126.0, 1.083, 20.775], [115.0, 0.645, 0.98], [103.0, 0.636, 4694.003], [102.0, 0.976, 15720.839], [102.0, 4.267, 7.114], [99.0, 6.21, 2146.17], [98.0, 0.68, 155.42], [86.0, 5.98, 161000.69], [85.0, 1.3, 6275.96], [85.0, 3.67, 71430.7], [80.0, 1.81, 17260.15], [79.0, 3.04, 12036.46], [75.0, 1.76, 5088.63], [74.0, 3.5, 3154.69], [74.0, 4.68, 801.82], [70.0, 0.83, 9437.76], [62.0, 3.98, 8827.39], [61.0, 1.82, 7084.9], [57.0, 2.78, 6286.6], [56.0, 4.39, 14143.5], [56.0, 3.47, 6279.55], [52.0, 0.19, 12139.55], [52.0, 1.33, 1748.02], [51.0, 0.28, 5856.48], [49.0, 0.49, 1194.45], [41.0, 5.37, 8429.24], [41.0, 2.4, 19651.05], [39.0, 6.17, 10447.39], [37.0, 6.04, 10213.29], [37.0, 2.57, 1059.38], [36.0, 1.71, 2352.87], [36.0, 1.78, 6812.77], [33.0, 0.59, 17789.85], [30.0, 0.44, 83996.85], [30.0, 2.74, 1349.87], [25.0, 3.16, 4690.48]], [[628307584999.0, 0.0, 0.0], [206059.0, 2.678235, 6283.07585], [4303.0, 2.6351, 12566.1517], [425.0, 1.59, 3.523], [119.0, 5.796, 26.298], [109.0, 2.966, 1577.344], [93.0, 2.59, 18849.23], [72.0, 1.14, 529.69], [68.0, 1.87, 398.15], [67.0, 4.41, 5507.55], [59.0, 2.89, 5223.69], [56.0, 2.17, 155.42], [45.0, 0.4, 796.3], [36.0, 0.47, 775.52], [29.0, 2.65, 7.11], [21.0, 5.34, 0.98], [19.0, 1.85, 5486.78], [19.0, 4.97, 213.3], [17.0, 2.99, 6275.96], [16.0, 0.03, 2544.31], [16.0, 1.43, 2146.17], [15.0, 1.21, 10977.08], [12.0, 2.83, 1748.02], [12.0, 3.26, 5088.63], [12.0, 5.27, 1194.45], [12.0, 2.08, 4694.0], [11.0, 0.77, 553.57], [10.0, 1.3, 6286.6], [10.0, 4.24, 1349.87], [9.0, 2.7, 242.73], [9.0, 5.64, 951.72], [8.0, 5.3, 2352.87], [6.0, 2.65, 9437.76], [6.0, 4.67, 4690.48]], [[8722.0, 1.0725, 6283.0758], [991.0, 3.1416, 0.0], [295.0, 0.437, 12566.152], [27.0, 0.05, 3.52], [16.0, 5.19, 26.3], [16.0, 3.69, 155.42], [9.0, 0.3, 18849.23], [9.0, 2.06, 77713.77], [7.0, 0.83, 775.52], [5.0, 4.66, 1577.34], [4.0, 1.03, 7.11], [4.0, 3.44, 5573.14], [3.0, 5.14, 796.3], [3.0, 6.05, 5507.55], [3.0, 1.19, 242.73], [3.0, 6.12, 529.69], [3.0, 0.3, 398.15], [3.0, 2.28, 553.57], [2.0, 4.38, 5223.69], [2.0, 3.75, 0.98]], [[289.0, 5.842, 6283.076], [21.0, 6.05, 12556.15], [3.0, 5.2, 155.42], [3.0, 3.14, 0.0], [1.0, 4.72, 3.52], [1.0, 5.97, 242.73], [1.0, 5.54, 18849.23]], [[8.0, 4.14, 6283.08], [1.0, 3.28, 12566.15]]]¶: This table contains Earth’s most important periodic terms from the planetary theory VSOP87 for the heliocentric longitude, referred to the equinox J2000.0. In Meeus’ book these values can be found in pages 418-420 and page 173.

pymeeus.Earth.VSOP87_R = [[[100013988.799, 0.0, 0.0], [1670699.626, 3.09846350771, 6283.0758499914], [13956.023, 3.0552460962, 12566.1516999828], [3083.72, 5.19846674381, 77713.7714681205], [1628.461, 1.17387749012, 5753.3848848968], [1575.568, 2.84685245825, 7860.4193924392], [924.799, 5.45292234084, 11506.7697697936], [542.444, 4.56409149777, 3930.2096962196], [472.11, 3.66100022149, 5884.9268465832], [328.78, 5.89983646482, 5223.6939198022], [345.983, 0.96368617687, 5507.5532386674], [306.784, 0.29867139512, 5573.1428014331], [174.844, 3.01193636534, 18849.2275499742], [243.189, 4.27349536153, 11790.6290886588], [211.829, 5.84714540314, 1577.3435424478], [185.752, 5.02194447178, 10977.078804699], [109.835, 5.05510636285, 5486.777843175], [98.316, 0.88681311277, 6069.7767545534], [86.499, 5.68959778254, 15720.8387848784], [85.825, 1.27083733351, 161000.6857376741], [62.916, 0.92177108832, 529.6909650946], [57.056, 2.01374292014, 83996.84731811189], [64.903, 0.27250613787, 17260.1546546904], [49.384, 3.24501240359, 2544.3144198834], [55.736, 5.24159798933, 71430.69561812909], [42.515, 6.01110242003, 6275.9623029906], [46.963, 2.57805070386, 775.522611324], [38.968, 5.36071738169, 4694.0029547076], [44.661, 5.53715807302, 9437.762934887], [35.66, 1.67468058995, 12036.4607348882], [31.921, 0.18368229781, 5088.6288397668], [31.846, 1.77775642085, 398.1490034082], [33.193, 0.24370300098, 7084.8967811152], [38.245, 2.39255343974, 8827.3902698748], [28.464, 1.21344868176, 6286.5989683404], [37.49, 0.82952922332, 19651.048481098], [36.957, 4.90107591914, 12139.5535091068], [34.537, 1.84270693282, 2942.4634232916], [26.275, 4.58896850401, 10447.3878396044], [24.596, 3.78660875483, 8429.2412664666], [23.587, 0.26866117066, 796.2980068164], [27.793, 1.89934330904, 6279.5527316424], [23.927, 4.99598548138, 5856.4776591154], [20.349, 4.65267995431, 2146.1654164752], [23.287, 2.80783650928, 14143.4952424306], [22.103, 1.95004702988, 3154.6870848956], [19.506, 5.38227371393, 2352.8661537718], [17.958, 0.19871379385, 6812.766815086], [17.174, 4.43315560735, 10213.285546211], [16.19, 5.23160507859, 17789.845619785], [17.314, 6.15200787916, 16730.4636895958], [13.814, 5.18962074032, 8031.0922630584], [18.833, 0.67306674027, 149854.4001348079], [18.331, 2.25348733734, 23581.2581773176], [13.641, 3.68516118804, 4705.7323075436], [13.139, 0.65289581324, 13367.9726311066], [10.414, 4.33285688538, 11769.8536931664], [9.978, 4.20126336355, 6309.3741697912], [10.169, 1.59390681369, 4690.4798363586], [7.564, 2.6256059739, 6256.7775301916], [9.661, 3.6758679122, 27511.4678735372], [6.743, 0.56270332741, 3340.6124266998], [8.743, 6.06359123461, 1748.016413067], [7.786, 3.67371235637, 12168.0026965746], [6.633, 5.66149277792, 11371.7046897582], [7.712, 0.31242577789, 7632.9432596502], [6.592, 3.13576266188, 801.8209311238], [7.46, 5.64757188143, 11926.2544136688], [6.933, 2.923845864, 6681.2248533996], [6.802, 1.4232980642, 23013.5395395872], [6.115, 5.13393615454, 1194.4470102246], [6.477, 2.64986648492, 19804.8272915828], [5.233, 4.62434053374, 6438.4962494256], [6.147, 3.02863936662, 233141.3144043615], [4.608, 1.72194702724, 7234.794256242], [4.221, 1.55697533729, 7238.6755916], [5.314, 2.40716580847, 11499.6562227928], [5.128, 5.3239896569, 11513.8833167944], [4.77, 0.25554312006, 11856.2186514245], [5.519, 2.09089154502, 17298.1823273262], [5.625, 4.34052903053, 90955.5516944961], [4.578, 4.4656964157, 5746.271337896], [3.788, 4.9072938351, 4164.311989613], [5.337, 5.09957905104, 31441.6775697568], [3.967, 1.20054555174, 1349.8674096588], [4.008, 3.03007204392, 1059.3819301892], [3.476, 0.7608027703, 10973.55568635], [4.232, 1.05485713117, 5760.4984318976], [4.582, 3.76570026763, 6386.16862421], [3.335, 3.13829943354, 6836.6452528338], [3.418, 3.00072390334, 4292.3308329504], [3.598, 5.70718084323, 5643.1785636774], [3.237, 4.16448773994, 9917.6968745098], [4.154, 2.59941292162, 7058.5984613154], [3.362, 4.54577697964, 4732.0306273434], [2.978, 1.3056126882, 6283.14316029419], [2.765, 0.51311975679, 26.2983197998], [2.802, 5.66263240521, 8635.9420037632], [2.927, 5.73787481548, 16200.7727245012], [3.164, 1.69140262657, 11015.1064773348], [2.598, 2.96244118586, 25132.3033999656], [3.519, 3.62639325753, 244287.60000722768], [2.676, 4.2072570085, 18073.7049386502], [2.978, 1.74971565805, 6283.0085396886], [2.287, 1.06975704977, 14314.1681130498], [2.863, 5.92838131397, 14712.317116458], [3.071, 0.23793217002, 35371.8872659764], [2.656, 0.8995930178, 12352.8526045448], [2.415, 2.79975176257, 709.9330485583], [2.814, 3.51488206882, 21228.3920235458], [1.977, 2.6135829755, 951.7184062506], [2.548, 2.47684686575, 6208.2942514241], [1.999, 0.5609038816, 7079.3738568078], [2.305, 1.05376461628, 22483.84857449259], [1.855, 2.86090681163, 5216.5803728014], [2.157, 1.31396741861, 154717.6098876827], [1.97, 4.36929875289, 167283.7615876655], [1.635, 5.85571606764, 10984.1923516998], [1.754, 2.14452408833, 6290.1893969922], [2.154, 6.03828341543, 10873.9860304804], [1.714, 3.70157691113, 1592.5960136328], [1.541, 6.21598380732, 23543.23050468179], [1.611, 1.99824499377, 10969.9652576982], [1.712, 1.34295663542, 3128.3887650958], [1.642, 5.55026665339, 6496.3749454294], [1.502, 5.43948825854, 155.4203994342], [1.827, 5.91227480261, 3738.761430108], [1.726, 2.16764983583, 10575.4066829418], [1.532, 5.3568310707, 13521.7514415914], [1.829, 1.66006148731, 39302.096962196], [1.605, 1.90928637633, 6133.5126528568], [1.282, 2.46014880418, 13916.0191096416], [1.211, 4.4136063155, 3894.1818295422], [1.394, 1.77801929354, 9225.539273283], [1.571, 4.95512957592, 25158.6017197654], [1.205, 1.19212540615, 3.523118349], [1.132, 2.69830084955, 6040.3472460174], [1.504, 5.77002730341, 18209.33026366019], [1.393, 1.62621805428, 5120.6011455836], [1.077, 2.93931554233, 17256.6315363414], [1.232, 0.71655165307, 143571.32428481648], [1.087, 0.99769687939, 955.5997416086], [1.068, 5.28472576231, 65147.6197681377], [0.98, 5.10949204607, 6172.869528772], [1.169, 3.11664290862, 14945.3161735544], [1.202, 4.02992510402, 553.5694028424], [0.979, 2.00000879212, 15110.4661198662], [0.962, 4.023807714, 6282.0955289232], [0.999, 3.6264300279, 6262.300454499], [1.03, 5.84989900289, 213.299095438], [1.014, 2.84221578218, 8662.240323563], [1.185, 1.51330541132, 17654.7805397496], [0.967, 2.67081017562, 5650.2921106782], [1.222, 2.65423784904, 88860.05707098669], [0.981, 2.36370360283, 6206.8097787158], [1.033, 0.13874927606, 11712.9553182308], [1.103, 3.08477302937, 43232.3066584156], [0.781, 2.53372735932, 16496.3613962024], [1.019, 3.04569392376, 6037.244203762], [0.795, 5.80662989111, 5230.807466803], [0.813, 3.57710279439, 10177.2576795336], [0.962, 5.31470594766, 6284.0561710596], [0.721, 5.96264301567, 12559.038152982], [0.966, 2.74714939953, 6244.9428143536], [0.921, 0.10155275926, 29088.811415985], [0.692, 3.89764447548, 1589.0728952838], [0.719, 5.91791450402, 4136.9104335162], [0.772, 4.05505682353, 6127.6554505572], [0.712, 5.49291532439, 22003.9146348698], [0.672, 1.60700490811, 11087.2851259184], [0.69, 4.50539825563, 426.598190876], [0.854, 3.26104981596, 20426.571092422], [0.656, 4.3241018294, 16858.4825329332], [0.84, 2.59572585222, 28766.924424484], [0.692, 0.61650089011, 11403.676995575], [0.7, 3.40901167143, 7.1135470008], [0.726, 0.04243053594, 5481.2549188676], [0.557, 4.78317696534, 20199.094959633], [0.649, 1.04027912958, 6062.6632075526], [0.633, 5.70229959167, 45892.73043315699], [0.592, 6.11836729658, 9623.6882766912], [0.523, 3.62840021266, 5333.9002410216], [0.604, 5.57734696185, 10344.2950653858], [0.496, 2.21023499449, 1990.745017041], [0.691, 1.96071732602, 12416.5885028482], [0.64, 1.59074172032, 18319.5365848796], [0.625, 3.82362791378, 13517.8701062334], [0.663, 5.08444996779, 283.8593188652], [0.475, 1.17025894287, 12569.6748183318], [0.664, 4.50029469969, 47162.5163546352], [0.569, 0.16310365162, 17267.26820169119], [0.568, 3.86100969474, 6076.8903015542], [0.539, 4.83282276086, 18422.62935909819], [0.466, 0.75872342878, 7342.4577801806], [0.541, 3.07212190507, 226858.23855437007], [0.458, 0.26774483096, 4590.910180489], [0.61, 1.53597051291, 33019.0211122046], [0.617, 2.62356328726, 11190.377900137], [0.548, 4.55798855791, 18875.525869774], [0.633, 4.60110281228, 66567.48586525429], [0.596, 5.78202396722, 632.7837393132], [0.533, 5.01786882904, 12132.439962106], [0.603, 5.38458554802, 316428.22867391503], [0.469, 0.59168241917, 21954.15760939799], [0.548, 3.50613163558, 17253.04110768959], [0.502, 0.98804327589, 11609.8625440122], [0.568, 1.98497313089, 7668.6374249425], [0.482, 1.62141803864, 12146.6670561076], [0.391, 3.68718382989, 18052.9295431578], [0.457, 3.7720573734, 156137.47598479927], [0.401, 5.28260651958, 15671.0817594066], [0.469, 1.80963184268, 12562.6285816338], [0.508, 3.36399024699, 20597.2439630412], [0.45, 5.6605429925, 10454.5013866052], [0.375, 4.98534633105, 9779.1086761254], 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6283.14316029419], [0.02, 6.10644140189, 167283.7615876655], [0.015, 4.09775914607, 11712.9553182308], [0.016, 5.717699407, 17298.1823273262], [0.016, 3.28894009404, 5884.9268465832], [0.015, 5.64785377164, 12559.038152982], [0.016, 4.4345208093, 6283.0085396886], [0.014, 2.31721603062, 5481.2549188676], [0.014, 4.43479032305, 13517.8701062334], [0.014, 4.73209312936, 7342.4577801806], [0.012, 0.64705975463, 18073.7049386502], [0.011, 1.514433322, 16200.7727245012], [0.011, 0.88708889185, 21228.3920235458], [0.014, 4.50116508534, 640.8776073822], [0.011, 4.64339996198, 11790.6290886588], [0.011, 1.31064298246, 4164.311989613], [0.009, 3.02238989305, 23543.23050468179], [0.009, 2.04999402381, 22003.9146348698], [0.009, 4.91488110218, 213.299095438]], [[144.595, 4.27319435148, 6283.0758499914], [6.729, 3.91697608662, 12566.1516999828], [0.774, 0.0, 0.0], [0.247, 3.73019298781, 18849.2275499742], [0.036, 2.8008140905, 6286.5989683404], [0.033, 5.62216602775, 6127.6554505572], [0.019, 3.71292621802, 6438.4962494256], [0.016, 4.26011484232, 6525.8044539654], [0.016, 3.50416887054, 6256.7775301916], [0.014, 3.62127621114, 25132.3033999656], [0.011, 4.39200958819, 4705.7323075436], [0.011, 5.22327127059, 6040.3472460174], [0.01, 4.28045254647, 83996.84731811189], [0.009, 1.56864096494, 5507.5532386674], [0.011, 1.37795688024, 6309.3741697912], [0.01, 5.19937959068, 71430.69561812909], [0.009, 0.4727519993, 6279.5527316424], [0.009, 0.74642756529, 5729.506447149], [0.007, 2.9737489156, 775.522611324], [0.007, 3.28615691021, 7058.5984613154], [0.007, 2.19184402142, 6812.766815086], [0.005, 3.15419034438, 529.6909650946], [0.006, 4.54725567047, 1059.3819301892], [0.005, 1.51104406936, 7079.3738568078], [0.007, 2.98052059053, 6681.2248533996], [0.005, 2.30961231391, 12036.4607348882], [0.005, 3.71102966917, 6290.1893969922]], [[3.858, 2.56384387339, 6283.0758499914], [0.306, 2.2676950123, 12566.1516999828], [0.053, 3.44031471924, 5573.1428014331], [0.015, 2.04794573436, 18849.2275499742], [0.013, 2.05688873673, 77713.7714681205], [0.007, 4.4121885448, 161000.6857376741], [0.005, 5.26154653107, 6438.4962494256], [0.005, 4.07695126049, 6127.6554505572], [0.006, 3.81514213664, 149854.4001348079], [0.003, 1.28175749811, 6286.5989683404]], [[0.086, 1.21579741687, 6283.0758499914], [0.012, 0.65617264033, 12566.1516999828], [0.001, 0.38068797142, 18849.2275499742]]]¶: This table contains Earth’s periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 420-421.

pymeeus.Earth.WGS84 = Ellipsoid(6378137.0, 0.00335281066475, 7.292115e-05)¶: Reference ellipsoid World Geodetic System 1984, a modern ellipsoid used by the GPS system, and the standard in many applications

JupiterMoons¶

Class to model JupiterMoons.

class pymeeus.JupiterMoons.JupiterMoons[source]¶

Class JupiterMoons models the four galilean satellites of Jupiter. With: 1: Io 2: Europa 3: Ganymede 4: Callisto The algorithm used can be found in chapter 44 (high accuracy method) of Meeus’ book Astronomic Algorithms

__weakref__¶: list of weak references to the object (if defined)

static apparent_rectangular_coordinates(epoch, X, Y, Z, OMEGA, psi, i, lambda_0, beta_0, D=0, isFictional=False)[source]¶

This method computes the apparent rectangular coordinates of a Jupiter satellite for given coordinates.

Parameters:	epoch (`Epoch`) – Epoch to compute satellite position, as an Epoch object X (float) – X-coordinate of the satellite in Jupiter radii Y (float) – Y-coordinate of the satellite in Jupiter radii Z (float) – Z-coordinate of the satellite in Jupiter radii OMEGA (float) – longitude of the node of Jupiter psi (float) – longitude of the node of Jupiter i (float) – inclination on the plane of the ecliptic beta_0 (float) – Jupiter’s geocentric latitude lambda_0 (float) – Jupiter’s geocentric longitude D (float) – parameter calculated by the fifth fictional satellite (fictional satellite has to be calculated first, in order to calculate the coordinates of the remaining “true” satellites) isFictional (bool) – Whether or not the satellite is the fictional satellite No. 5
Returns:	A tuple with the apparent rectangular coordinates of the satellite or parameter D if isFictional=True
Return type:	tuple, float
Raises:	TypeError if input values are wrong type

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> psi_corrected = 317.1058009213959
>>> i_ecliptic_jupiter = 1.3036541530886598
>>> lambda_0 = -2.9355662143229146
>>> beta_0 = 0.021667777174910842
>>> OMEGA_ascending_node_jupiter= 100.39249942976576
>>> X_1 =  5.60361395790844
>>> Y_1 = 1.9398758261880644
>>> Z_1 = 0.00449258104769796
>>> X_5 =  0
>>> Y_5 = 0
>>> Z_5 = 1
>>> d = JupiterMoons.apparent_rectangular_coordinates(         utc_1992_12_16_00_00_00, X_5, Y_5, Z_5,         OMEGA_ascending_node_jupiter, psi_corrected, i_ecliptic_jupiter,         lambda_0, beta_0, isFictional=True)
>>> print(round(d, 10))
-0.0320562469
>>> io = JupiterMoons.apparent_rectangular_coordinates(         utc_1992_12_16_00_00_00, X_1, Y_1, Z_1,         OMEGA_ascending_node_jupiter, psi_corrected, i_ecliptic_jupiter,         lambda_0, beta_0, d)
>>> print(io)
(-3.4489935969836503, 0.21361563816963675, -4.818966623735296)

static calculate_delta(epoch)[source]¶

This method calculates the distance between Earth and Jupiter (DELTA) for a given epoch by iteration.

Parameters:	epoch (`Epoch`) – Epoch the distance should be calculated for.
Returns:	Distance Earth-Jupiter in AU and light-time delay from Earth-Jupiter, together with Coordinates of Jupiter
Return type:	tuple
Raises:	TypeError if input values are wrong type

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> delta, tau, l, b, r = JupiterMoons.calculate_delta(         utc_1992_12_16_00_00_00)
>>> print(round(delta, 10))
5.6611211815
>>> print(round(tau, 10))
0.032695909

static check_coordinates(X, Y)[source]¶

This method checks if the given coordinates correspond with a satellite phenomena. It returns if the satellite with the given coordinates is hidden behind Jupiter or directly in front.

Parameters:	X (float) – X-coordinate of the satellite in Jupiter’s radii Y (float) – Y-coordinate of the satellite in Jupiter’s radii
Returns:	Perspective distance to Jupiter’s center in Jupiter’s radii
Return type:	float

Calculation of the perspective distance of the planet Io to the center of Jupiter for December 16 at 0h UTC as seen from the Earth

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> result_matrix =         JupiterMoons.rectangular_positions_jovian_equatorial(         utc_1992_12_16_00_00_00, solar=False)
>>> io_radius_to_center_of_jupiter_earth =         JupiterMoons.check_coordinates(result_matrix[0][0], result_matrix[         0][1])
>>> print(round(io_radius_to_center_of_jupiter_earth, 10))
3.4577572706

static check_eclipse(X_0=0, Y_0=0, Z_0=0, epoch=None, i_sat=None)[source]¶

This method checks if the given coordinates or Epoch correspond with a satellite being in eclipse.

Parameters:	X_0 (float) – X-coordinate of the satellite in Jupiter’s radii observed from the sun Y_0 (float) – Y-coordinate of the satellite in Jupiter’s radii observed from the sun Z_0 (float) – Z-coordinate of the satellite in Jupiter’s radii observed from the sun epoch (`Epoch`) – Epoch that should be checked i_sat (int) – Index of the satellite (only for given Epoch)
Returns:	perspective distance to center of Jupiter in Jupiter radii as seen from the Sun (value of perspective distance is negative when the satellite is closer to the Sun than Jupiter, otherwise positive)
Return type:	float
Raises:	TypeError if input values are wrong type

Calculation of the Perspective distance of the planet Io squareroot(X_0^2 + Y_0^2) to the center of Jupiter for December 16 at 0h UTC as seen from the Sun

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> result_matrix =         JupiterMoons.rectangular_positions_jovian_equatorial(         utc_1992_12_16_00_00_00, solar=True)
>>> io_radius_to_center_of_jupiter_sunview =         JupiterMoons.check_eclipse(result_matrix[0][0], result_matrix[0][1])
>>> print(round(io_radius_to_center_of_jupiter_sunview, 10))
-2.5533012642

static check_occultation(X=0, Y=0, Z=0, epoch=None, i_sat=None)[source]¶

This method checks if the given coordinates or Epoch correspond with a satellite being in occultation.

Parameters:	X (float) – X-coordinate of the satellite in Jupiter’s radii Y (float) – Y-coordinate of the satellite in Jupiter’s radii Z (float) – Z-coordinate of the satellite in Jupiter’s radii epoch (`Epoch`) – Epoch that should be checked i_sat (int) – Index of the satellite (only for given Epoch)
Returns:	Perspective distance to center of Jupiter in Jupiter radii as seen from the Earth (value of perspective distance is negative when the satellite is closer to the Earth than Jupiter, otherwise positive)
Return type:	float
Raises:	TypeError if input values are wrong type

Calculation of the perspective distance of the planet Io squareroot(X^2 + Y^2) to the center of Jupiter for December 16 at 0h UTC as seen from the Earth

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> result_matrix =         JupiterMoons.rectangular_positions_jovian_equatorial(         utc_1992_12_16_00_00_00, solar=False)
>>> io_distance_to_center_of_jupiter_earthview =         JupiterMoons.check_occultation(result_matrix[0][0], result_matrix[         0][1])
>>> print(round(io_distance_to_center_of_jupiter_earthview, 10))
-3.4577572706

static check_phenomena(epoch, check_all=True, i_sat=0)[source]¶

This method returns the perspective distance to any phenomena of all satellites for the given epoch.

Parameters:	epoch (:py:class:'Epoch') – Epoch the calculations should be made for check_all (bool) – Whether all satellites should be checked i_sat (int) – Which satellite should be checked
Returns:	Distance to the satellite being ecclipsed, occulted in penumbra
Return type:	tuple
Raises:	TypeError if input values are wrong type

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> result_matrix = JupiterMoons.check_phenomena(         utc_1992_12_16_00_00_00)
>>> print(result_matrix[0])
[-3.457757270630766, -2.553301264153796, 0.0]
>>> print(result_matrix[1])
[-7.44770945299594, -8.33419997337025, 0.0]
>>> print(result_matrix[2])
[-1.3572840767173413, -3.817302564886177, 0.0]
>>> print(result_matrix[3])
[-7.157430454898491, -11.373611474420906, 0.0]

>>> io_ecc_start_2021_02_12_14_19_14 = Epoch(2021, 2, 12.5966898148148)
>>> result_matrix = JupiterMoons.check_phenomena(         io_ecc_start_2021_02_12_14_19_14)
>>> print([round(result_matrix[0][0], 10), round(result_matrix[0][         1], 10), round(result_matrix[0][2], 10)])
[1.192605868, 0.8560277162, 0.0]
>>> print([round(result_matrix[1][0], 10), round(result_matrix[1][         1], 10), round(result_matrix[1][2], 10)])
[-8.7397202369, -8.8930940921, 0.0]
>>> print([round(result_matrix[2][0], 10), round(result_matrix[2][         1], 10), round(result_matrix[2][2], 10)])
[14.0691219925, 13.8323491768, 0.0]
>>> print([round(result_matrix[3][0], 10), round(result_matrix[3][         1], 10), round(result_matrix[3][2], 10)])
[-2.9341209156, -3.9904598153, 0.0]

static correct_rectangular_positions(R, i_sat, DELTA, X_coordinate, Y_coordinate=0, Z_coordinate=0)[source]¶

This method corrects the given rectangular coordinates of a Jupiter satellite in order to obtain higher accuracy by considering differential light-time and the perspective effect.

Parameters:	R (float) – Radius vector of the satellite i_sat (int) – Number of the satellite DELTA (float) – Distance Observer-Jupiter in AU X_coordinate (float, tuple, list) – Uncorrected X-coordinate of the satellite in Jupiter’s radii or tuple for all coordinates Y_coordinate (float) – Uncorrected Y-coordinate of the satellite in Jupiter’s radii Z_coordinate (float) – Uncorrected Z-coordinate of the satellite in Jupiter’s radii
Returns:	A tuple with the corrected rectangular coordinates (X, Y, Z) of the satellite in Jupiter’s radii
Return type:	tuple
Raises:	TypeError if input values are wrong type

Calculate corrected rectangular Coordinates X, Y and Z in Jupiter’s radii for Io (1)

>>> R = 5.929892730360271
>>> i_sat = 1
>>> DELTA = 5.6611211815432645
>>> X_coordinate = -3.4489935969836503
>>> Y_coordinate = 0.21361563816963675
>>> Z_coordinate = -4.818966623735296
>>> io = JupiterMoons.correct_rectangular_positions(R, i_sat, DELTA,         X_coordinate, Y_coordinate, Z_coordinate)
>>> print(io)
(-3.450168811390241, 0.21370246960509387, -4.818966623735296)

static is_phenomena(epoch)[source]¶

This method checks if the given coordinates correspond with any satellite phenomena. It returns the type of phenomena for all satellites.

Parameters:	epoch (:py:class:'Epoch') – Epoch the calculations should be made for
Returns:	Result matrix for the four Galilean satellites Row 0: Io Column 0: Occultation Row 1: Europa Column 1: Eclipse Row 2: Ganymede Column 2: No use Row 3: Callisto
Return type:	tuple
Raises:	TypeError if input values are wrong type

Calculation of result matrix for December 16 at 0h UTC

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> result_matrix = JupiterMoons.is_phenomena(utc_1992_12_16_00_00_00)
>>> print(result_matrix[0])
[False, False, False]
>>> print(result_matrix[1])
[False, False, False]
>>> print(result_matrix[2])
[False, False, False]
>>> print(result_matrix[3])
[False, False, False]
>>> io_ecc_start_2021_02_12_14_19_14 = Epoch(2021, 2, 12.5966898148148)
>>> result_matrix = JupiterMoons.is_phenomena(         io_ecc_start_2021_02_12_14_19_14)
>>> print(result_matrix[0])
[False, True, False]
>>> print(result_matrix[1])
[False, False, False]
>>> print(result_matrix[2])
[False, False, False]
>>> print(result_matrix[3])
[False, False, False]

static jupiter_system_angles(epoch)[source]¶

This method computes the ascending node of Jupiter as well as the node of the equator of Jupiter on the ecliptic (psi).

Parameters:	epoch (`Epoch`) – Epoch to compute satellites’ positions, as an Epoch object
Returns:	Two float values with the ascending node of Jupiter and psi
Return type:	(float, float)

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> psi_corrected, OMEGA_ascending_node_jupiter =         JupiterMoons.jupiter_system_angles(utc_1992_12_16_00_00_00)
>>> print(round(psi_corrected, 9))
317.105800921
>>> print(round(OMEGA_ascending_node_jupiter, 9))
100.39249943

static rectangular_positions_jovian_equatorial(epoch, tofk5=True, solar=False, do_correction=True)[source]¶

This method computes the rectangular geocentric position of Jupiter’s satellites for a given epoch, using the E5-theory.

Parameters:	epoch (`Epoch`) – Epoch to compute satellites’ positions, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not solar (bool) – Whether the satellites’ positions are observed from the Earth (False) or the Sun (True) do_correction (bool) – Whether the satellites’ positions is corrected for the effect of different light-time and perspective effect or not
Returns:	Four tuple with the rectangular X, Y and Z coordinates in Jupiter radii (float), where Z is positive if the satellite is more distant than Jupiter.
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
>>> io, europa, ganymede, callisto =         JupiterMoons.rectangular_positions_jovian_equatorial(         utc_1992_12_16_00_00_00)
>>> print(io)
(-3.450168811390241, 0.21370246960509387, -4.818966623735296)
>>> print(europa)
(7.441869121153001, 0.27524463479625677, -5.747104399729193)
>>> print(ganymede)
(1.201111684800708, 0.5899903274317162, -14.940581367576527)
>>> print(callisto)
(7.072022641384802, 1.0289678450543338, -25.224420329457175)

Jupiter¶

Class to model Jupiter planet.

class pymeeus.Jupiter.Jupiter[source]¶

Class Jupiter models that planet.

__weakref__¶: list of weak references to the object (if defined)

static apparent_heliocentric_position(epoch)[source]¶

This method computes the apparent heliocentric position of planet Jupiter for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Jupiter position, as an Epoch object
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

static conjunction(epoch)[source]¶

This method computes the time of the conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired conjunction
Returns:	The time when the conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> conj = Jupiter.conjunction(epoch)
>>> y, m, d = conj.get_date()
>>> print(y)
1993
>>> print(m)
10
>>> print(round(d, 4))
18.3341

static geocentric_position(epoch)[source]¶

This method computes the geocentric position of Jupiter (right ascension and declination) for the given epoch, as well as the elongation angle.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension, the declination and the elongation angle as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 20.0)
>>> ra, dec, elon = Jupiter.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
12h 47' 9.6''
>>> print(dec.dms_str(n_dec=1))
-3d 41' 55.3''
>>> print(elon.dms_str(n_dec=1))
76d 2' 26.0''

static geometric_heliocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of planet Jupiter for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Jupiter position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2018, 10, 27.0)
>>> l, b, r = Jupiter.geometric_heliocentric_position(epoch)
>>> print(round(l.to_positive(), 4))
241.5873
>>> print(round(b, 4))
0.8216
>>> print(round(r, 5))
5.36848

static magnitude(sun_dist, earth_dist)[source]¶

This function computes the approximate magnitude of Jupiter.

Parameters:	sun_dist (float) – Distance from Jupiter to Sun, in Astronomical Units earth_dist (float) – Distance Jupiter to Earth, in Astronomical Units
Returns:	Jupiter’s magnitude
Return type:	float
Raises:	TypeError if input values are of wrong type.

static opposition(epoch)[source]¶

This method computes the time of the opposition closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	The time when the opposition happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(-6, 9, 1.0)
>>> oppo = Jupiter.opposition(epoch)
>>> y, m, d = oppo.get_date()
>>> print(y)
-6
>>> print(m)
9
>>> print(round(d, 4))
15.2865

static orbital_elements_j2000(epoch)[source]¶

This method computes the orbital elements of Jupiter for the standard equinox J2000.0 for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Jupiter.orbital_elements_j2000(epoch)
>>> print(round(l, 6))
221.518802
>>> print(round(a, 8))
5.20260333
>>> print(round(e, 7))
0.0486046
>>> print(round(i, 6))
1.30198
>>> print(round(ome, 5))
100.58051
>>> print(round(arg, 6))
-86.107875

static orbital_elements_mean_equinox(epoch)[source]¶

This method computes the orbital elements of Jupiter for the mean equinox of the date for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Jupiter.orbital_elements_mean_equinox(epoch)
>>> print(round(l, 6))
222.433723
>>> print(round(a, 8))
5.20260333
>>> print(round(e, 7))
0.0486046
>>> print(round(i, 6))
1.29967
>>> print(round(ome, 5))
101.13309
>>> print(round(arg, 6))
-85.745532

static passage_nodes(epoch, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of Jupiter, nearest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch closest to the node passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2019, 1, 1)
>>> time, r = Jupiter.passage_nodes(epoch)
>>> year, month, day = time.get_date()
>>> print(year)
2025
>>> print(month)
9
>>> print(round(day, 1))
15.6
>>> print(round(r, 4))
5.1729

static perihelion_aphelion(epoch, perihelion=True)[source]¶

This method computes the time of Perihelion (or Aphelion) closer to a given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired Perihelion (or Aphelion) peihelion – If True, the epoch of the closest Perihelion is computed, if False, the epoch of the closest Aphelion is found.
Returns:	The epoch of the desired Perihelion (or Aphelion)
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2019, 2, 23.0)
>>> e = Jupiter.perihelion_aphelion(epoch)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2023
>>> print(m)
1
>>> print(d)
20
>>> print(h)
11
>>> epoch = Epoch(1981, 6, 1.0)
>>> e = Jupiter.perihelion_aphelion(epoch, perihelion=False)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
1981
>>> print(m)
7
>>> print(d)
28
>>> print(h)
6

static station_longitude_1(epoch)[source]¶

This method computes the time of the 1st station in longitude (i.e. when the planet is stationary and begins to move westward - retrograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	Time when the 1st station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2018, 11, 1.0)
>>> sta1 = Jupiter.station_longitude_1(epoch)
>>> y, m, d = sta1.get_date()
>>> print(y)
2018
>>> print(m)
3
>>> print(round(d, 4))
9.1288

static station_longitude_2(epoch)[source]¶

This method computes the time of the 2nd station in longitude (i.e. when the planet is stationary and begins to move eastward - prograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	Time when the 1st station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2018, 11, 1.0)
>>> sta2 = Jupiter.station_longitude_2(epoch)
>>> y, m, d = sta2.get_date()
>>> print(y)
2018
>>> print(m)
7
>>> print(round(d, 4))
10.6679

pymeeus.Jupiter.ORBITAL_ELEM = [[34.351519, 3036.3027748, 0.0002233, 3.7e-08], [5.202603209, 1.913e-07, 0.0, 0.0], [0.04849793, 0.000163225, -4.714e-07, -2.01e-09], [1.303267, -0.0054965, 4.66e-06, -2e-09], [100.464407, 1.0209774, 0.00040315, 4.04e-07], [14.331207, 1.6126352, 0.00103042, -4.464e-06]]¶: This table contains the parameters to compute Jupiter’s orbital elements for the mean equinox of date. Based in Table 31.A, page 213

pymeeus.Jupiter.ORBITAL_ELEM_J2000 = [[34.351519, 3034.9056606, -8.501e-05, 1.6e-08], [1.303267, -0.0019877, 3.32e-05, 9.7e-08], [100.464407, 0.1767232, 0.000907, -7.272e-06], [14.331207, 0.2155209, 0.00072211, -4.485e-06]]¶: This table contains the parameters to compute Jupiter’s orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 215

pymeeus.Jupiter.VSOP87_B = [[[2268615.703, 3.55852606718, 529.6909650946], [109971.634, 3.90809347389, 1059.3819301892], [110090.358, 0.0, 0.0], [8101.427, 3.60509573368, 522.5774180938], [6043.996, 4.25883108794, 1589.0728952838], [6437.782, 0.30627121409, 536.8045120954], [1106.88, 2.98534421928, 1162.4747044078], [941.651, 2.93619072405, 1052.2683831884], [894.088, 1.75447429921, 7.1135470008], [767.28, 2.1547359406, 632.7837393132], [944.328, 1.67522288396, 426.598190876], [684.22, 3.67808770098, 213.299095438], [629.223, 0.64343282328, 1066.49547719], [835.861, 5.17881973234, 103.0927742186], [531.67, 2.70305954352, 110.2063212194], [558.524, 0.01354830508, 846.0828347512], [464.449, 1.17337249185, 949.1756089698], [431.072, 2.60825000494, 419.4846438752], [351.433, 4.61062990714, 2118.7638603784], [123.148, 3.34968181384, 1692.1656695024], [115.038, 5.04892295442, 316.3918696566], [132.16, 4.7781699067, 742.9900605326], [103.402, 2.31878999565, 1478.8665740644], [116.379, 1.38688232033, 323.5054166574], [102.42, 3.15293785436, 1581.959348283], [103.762, 3.7010383811, 515.463871093], [78.65, 3.98318653238, 1265.5674786264], [69.935, 2.56006216424, 956.2891559706], [55.597, 0.37500753017, 1375.7737998458], [51.986, 0.99007119033, 1596.1864422846], [55.194, 0.40176412035, 525.7588118315], [63.456, 4.50073574333, 735.8765135318], [49.691, 0.18649893085, 543.9180590962], [48.831, 3.57260550671, 533.6231183577], [28.353, 1.53532744749, 625.6701923124], [29.209, 5.43145863011, 206.1855484372], [23.255, 5.95197992848, 838.9692877504], [22.841, 6.19262787685, 532.8723588323], [23.202, 4.06473368575, 526.5095713569], [24.436, 6.10947656959, 1169.5882514086], [21.116, 4.96322972735, 2648.454825473], [17.879, 3.08704395969, 1795.258443721], [16.234, 4.83515727869, 1368.660252845], [21.314, 2.69476951059, 1045.1548361876], [15.74, 1.15130330106, 942.062061969], [17.325, 1.61550009206, 14.2270940016], [13.396, 2.30539585502, 853.196381752], [11.904, 3.09811974536, 2111.6503133776], [11.734, 2.83006431723, 2008.557539159], [11.291, 0.98957560201, 433.7117378768], [11.83, 4.76527836803, 309.2783226558], [10.702, 3.70181397065, 2221.856634597], [10.815, 5.81958878617, 1272.6810256272], [13.505, 3.2812697576, 1155.361157407], [10.179, 2.58691128827, 117.3198682202], [10.632, 5.23487936086, 95.9792272178], [8.771, 0.40456546655, 220.4126424388], [7.439, 2.94638292086, 412.3710968744], [6.151, 2.69100382247, 380.12776796], [5.028, 0.72750312028, 1055.4497769261], [4.939, 0.73756716762, 1905.4647649404], [5.421, 4.08612438558, 1685.0521225016], [5.936, 4.32059910537, 1063.3140834523], [4.737, 4.0930301685, 527.2432845398], [4.01, 0.51530008355, 1073.6090241908], [4.709, 1.84067645204, 984.6003316219], [3.974, 1.33608029246, 2125.8774073792], [3.762, 3.58647039394, 529.7391492044], [4.731, 6.16377350841, 532.1386456494], [4.666, 5.88762905802, 639.897286314], [3.763, 0.38865925413, 529.6427809848], [3.409, 4.05398247269, 1898.3512179396], [3.457, 3.43865563497, 1485.9801210652], [4.229, 2.23767157901, 74.7815985673], [3.091, 0.16470256025, 1699.2792165032], [2.975, 0.72268908074, 530.6541729411], [3.162, 1.2504841642, 330.6189636582], [2.727, 4.37679213321, 149.5631971346], [2.837, 0.05987107395, 1439.5096981492], [2.983, 3.2525120722, 528.7277572481], [2.232, 0.26149880534, 1062.5633239269], [2.464, 1.1691330442, 453.424893819], [2.596, 3.30510149086, 2324.9494088156], [1.988, 2.85269577619, 1574.8458012822], [2.527, 5.9445820295, 909.8187330546], [2.269, 1.30379329597, 3.9321532631], [1.742, 4.49909767044, 1258.4539316256], [1.714, 4.12945878208, 2001.4439921582], [2.029, 3.97938086639, 1056.2005364515], [1.667, 0.36037092553, 10213.285546211], [1.579, 6.11640144795, 1802.3719907218], [1.393, 3.69324470827, 2214.7430875962], [1.604, 1.98841031703, 38.1330356378], [1.325, 1.74025919863, 529.1697002328], [1.451, 2.39804501178, 2428.0421830342], [1.594, 2.07556780757, 1021.2488945514], [1.32, 1.33770977126, 618.5566453116], [1.346, 3.2759149254, 2641.3412784722], [1.23, 0.1955272822, 305.3461693927], [1.223, 2.86681556337, 1382.8873468466], [1.324, 2.23549334986, 530.2122299564], [1.056, 3.80579750957, 76.2660712756], [1.05, 4.68011652614, 1788.1448967202], [1.226, 5.34003255221, 3178.1457905676], [1.009, 3.19608028376, 2538.2485042536], [1.266, 3.04704446731, 604.4725636619], [0.954, 3.86932544808, 728.762966531], [1.124, 1.5956036748, 3.1813937377], [0.978, 0.25223689838, 983.1158589136], [0.948, 0.21552742733, 750.1036075334], [0.946, 3.9392774812, 508.3503240922], [0.92, 1.14672086939, 963.4027029714], [0.817, 5.93809619876, 831.8557407496], [0.77, 2.96062737592, 526.7702037878], [1.017, 5.55711112145, 199.0720014364], [0.761, 1.38163787157, 532.6117264014], [0.726, 3.98337964395, 2317.8358618148], [0.862, 0.87975657414, 490.3340891794], [0.868, 3.44331872364, 569.0478410098], [0.711, 4.11107052823, 2751.5475996916], [0.708, 0.33555577415, 528.9402055692], [0.708, 4.00539820601, 530.44172462], [0.656, 4.39568451439, 519.3960243561], [0.801, 4.03984430862, 1364.7280995819], [0.679, 1.18645749024, 525.4981794006], [0.645, 5.10510349996, 1361.5467058442], [0.668, 3.15607509055, 533.8837507886], [0.663, 0.73722024843, 5223.6939198022], [0.663, 1.57092786811, 6283.0758499914], [0.543, 0.26376529935, 227.5261894396], [0.525, 6.22318693939, 539.9859058331], [0.513, 4.98337900151, 302.164775655], [0.544, 2.22227019273, 2744.4340526908], [0.532, 2.62425372687, 99.1606209555], [0.602, 1.56074089013, 454.9093665273], [0.518, 0.26343805959, 551.031606097], [0.516, 1.09376390349, 934.9485149682], [0.659, 0.62560671589, 1512.8068240082], [0.524, 0.64710955846, 524.0618908021], [0.516, 3.69478866795, 535.3200393871], [0.491, 3.63039940597, 2531.1349572528], [0.57, 0.61976758791, 540.7366653585], [0.496, 2.19398015038, 1514.2912967165], [0.532, 0.20040217534, 525.0250986486], [0.493, 0.39160693598, 224.3447957019], [0.449, 0.62392433691, 529.5309064002], [0.449, 3.71676131146, 529.851023789], [0.45, 5.02467015031, 1048.3362299253], [0.428, 5.4480466029, 11.0457002639], [0.499, 4.13924061941, 534.3568315406], [0.528, 1.76471074936, 524.2743391232], [0.454, 4.53321742354, 1056.9342496344], [0.52, 2.57406093768, 535.107591066], [0.398, 1.40345870113, 960.2213092337], [0.457, 4.17708652827, 2104.5367663768], [0.505, 5.36536256321, 1057.8974574809], [0.535, 4.80455380313, 1593.0050485469], [0.415, 0.96548127237, 2435.155730035], [0.519, 0.54543519483, 1061.829610744], [0.359, 4.02704454075, 1059.430114299], [0.356, 2.66818105522, 835.0371344873], [0.443, 5.27513700376, 1.4844727083], [0.358, 5.94423960514, 440.8252848776], [0.471, 6.05791940453, 1471.7530270636], [0.386, 2.15984900214, 9153.9036160218], [0.424, 2.7092967003, 1038.0412891868], [0.359, 0.82922836987, 1059.3337460794], [0.31, 0.88102053266, 529.9034134157], [0.31, 3.45966511571, 529.4785167735], [0.3, 3.70331799503, 2634.2277314714], [0.292, 2.63594456361, 415.5524906121], [0.279, 1.60669121578, 643.8294395771], [0.291, 5.8313407182, 1148.2476104062], [0.37, 5.71572992274, 531.1754378029], [0.268, 5.39275891813, 1891.2376709388], [0.275, 3.34108666036, 518.6452648307], [0.269, 1.06051406954, 1585.1407420207], [0.306, 2.5028901737, 511.5317178299], [0.295, 1.84394223501, 547.8502123593], [0.254, 2.98312992496, 1134.1635287565], [0.289, 1.86070918711, 21.3406410024], [0.265, 4.93075479744, 679.2541622292], [0.25, 0.42860925124, 1969.2006632438], [0.308, 2.67237933272, 2957.7331481288], [0.313, 4.88085697819, 528.2064923863], [0.222, 4.78828764413, 514.7131115676], [0.221, 4.32763468981, 1677.9385755008], [0.217, 3.46278526461, 2950.619601128], [0.216, 0.5220766798, 2228.9701815978], [0.214, 5.83569926578, 544.6688186216], [0.283, 2.8870971609, 35.4247226521], [0.272, 1.65708415457, 3060.8259223474], [0.234, 1.68821537711, 2655.5683724738], [0.205, 3.3618688829, 2847.5268269094], [0.264, 3.62722625694, 2420.9286360334], [0.191, 4.26821147044, 430.5303441391], [0.179, 3.91470663005, 3340.6124266998], [0.18, 0.04531671003, 387.2413149608], [0.241, 4.03927631611, 494.2662424425], [0.176, 4.26298906325, 672.1406152284], [0.187, 2.72587420586, 299.1263942692], [0.234, 1.3447482745, 173.9422195228], [0.171, 0.85473611718, 1603.2999892854], [0.224, 0.33130232434, 565.1156877467], [0.2, 1.27632489123, 39.3568759152], [0.17, 4.96479470273, 1464.6394800628], [0.211, 1.00937080256, 523.5406259403], [0.21, 3.75793720248, 2854.6403739102], [0.162, 5.87784787295, 3480.3105662226], [0.163, 4.62850343495, 2015.6710861598], [0.191, 3.3315928375, 535.8413042489], [0.151, 1.17096741034, 1060.3451380357], [0.16, 1.81852636004, 312.4597163935], [0.158, 2.59595816107, 529.4303326637], [0.158, 1.7447274873, 529.9515975255], [0.173, 3.62399350412, 230.5645708254], [0.142, 0.70435921398, 522.529233984], [0.144, 5.3576312243, 107.0249274817], [0.144, 6.13954848857, 1158.5425511447], [0.178, 0.27566275049, 3906.9087570986], [0.126, 5.14832919826, 2207.6295405954], [0.126, 3.41994798109, 2.4476805548], [0.127, 0.39825164051, 70.8494453042], [0.123, 4.77865550523, 2524.021410252], [0.123, 0.46184813516, 647.0108333148], [0.144, 3.60261852727, 1058.4187223427], [0.158, 3.76231915252, 92.0470739547], [0.119, 4.08266911415, 1585.8915015461], [0.125, 2.35496721797, 3163.918696566], [0.122, 3.21027426317, 3377.217792004], [0.121, 3.39770381916, 18.1592472647], [0.131, 1.67926417552, 1289.9465010146], [0.115, 2.35735471566, 1550.939859646], [0.126, 2.40833814513, 106.2741679563], [0.131, 1.37610474529, 1023.9572075371], [0.121, 1.60252617273, 10.2949407385], [0.121, 0.61420823557, 1592.2542890215], [0.135, 3.60177675518, 124.433415221], [0.137, 2.41724947062, 3274.1250177854], [0.129, 0.09702914345, 2332.0629558164], [0.093, 4.88949890397, 1098.7388061044], [0.106, 5.18592950792, 2281.2304965106], [0.114, 2.96523316419, 1166.4068576709], [0.092, 1.65166124027, 860.3099287528], [0.102, 3.64093193142, 3171.0322435668], [0.103, 1.63066232967, 1894.4190646765], [0.08, 0.38766601876, 4694.0029547076], [0.074, 3.86865238736, 3067.9394693482], [0.095, 1.66362447044, 1151.4290041439]], [[177351.787, 5.70166488486, 529.6909650946], [3230.171, 5.7794161934, 1059.3819301892], [3081.364, 5.47464296527, 522.5774180938], [2211.914, 4.73477480209, 536.8045120954], [1694.232, 3.14159265359, 0.0], [346.445, 4.74595174109, 1052.2683831884], [234.264, 5.18856099929, 1066.49547719], [196.154, 6.18554286642, 7.1135470008], [150.468, 3.92721226087, 1589.0728952838], [114.128, 3.4389727183, 632.7837393132], [96.667, 2.9142630409, 949.1756089698], [76.599, 2.50522188662, 103.0927742186], [81.671, 5.07666097497, 1162.4747044078], [76.572, 0.61288981445, 419.4846438752], [73.875, 5.49958292155, 515.463871093], [49.915, 3.94799616572, 735.8765135318], [60.544, 5.44740084359, 213.299095438], [36.561, 4.69828392839, 543.9180590962], [46.032, 0.53850360901, 110.2063212194], [45.123, 1.89516645239, 846.0828347512], [36.019, 6.10952578764, 316.3918696566], [31.975, 4.92452714629, 1581.959348283], [21.015, 5.6295773141, 1596.1864422846], [23.156, 5.84829490183, 323.5054166574], [24.719, 3.94107395247, 2118.7638603784], [17.274, 5.65310656429, 533.6231183577], [16.521, 5.89840100621, 526.5095713569], [16.698, 5.66663034948, 1265.5674786264], [15.815, 4.43314786393, 1045.1548361876], [13.398, 4.30179033605, 532.8723588323], [11.744, 1.80990486955, 956.2891559706], [11.925, 4.30094564154, 525.7588118315], [9.514, 2.02589667166, 206.1855484372], [10.542, 6.15533910933, 14.2270940016], [8.414, 3.9291045034, 1478.8665740644], [8.099, 4.20152809071, 1169.5882514086], [7.712, 2.99160389601, 942.062061969], [8.825, 1.55897920307, 426.598190876], [8.884, 4.87430124264, 1155.361157407], [7.793, 3.84684930196, 625.6701923124], [5.646, 3.40915964493, 639.897286314], [4.615, 0.83374662294, 117.3198682202], [4.02, 5.50502127885, 433.7117378768], [3.704, 0.90226777963, 95.9792272178], [3.859, 0.69640284662, 853.196381752], [3.091, 5.09115860882, 1073.6090241908], [3.36, 5.10133284081, 1692.1656695024], [2.892, 4.9041891666, 220.4126424388], [2.772, 5.09066125724, 2111.6503133776], [2.425, 3.74438653232, 742.9900605326], [2.558, 5.46955948791, 1795.258443721], [2.466, 4.2227835543, 2648.454825473], [1.968, 0.57192251841, 309.2783226558], [1.794, 4.60765219417, 1272.6810256272], [1.822, 1.98842964323, 1375.7737998458], [1.703, 6.12660562937, 2125.8774073792], [2.011, 5.00936865256, 412.3710968744], [1.645, 0.08830372958, 1063.3140834523], [1.875, 5.81006158403, 330.6189636582], [1.741, 4.58650290431, 1574.8458012822], [1.529, 5.81660291389, 1258.4539316256], [1.916, 0.85150399517, 1368.660252845], [1.614, 4.36839107221, 728.762966531], [1.51, 2.79374165455, 1485.9801210652], [1.333, 4.84260898693, 1062.5633239269], [1.359, 5.16511980864, 838.9692877504], [1.165, 5.66275740881, 508.3503240922], [1.092, 4.68797557406, 1699.2792165032], [1.438, 5.78105679279, 1056.2005364515], [1.083, 3.99886917926, 1471.7530270636], [1.002, 4.79949608524, 1055.4497769261], [0.749, 6.1440086203, 519.3960243561], [0.657, 5.63765568876, 1898.3512179396], [0.702, 5.04126574492, 1685.0521225016], [0.607, 3.15707515246, 618.5566453116], [0.587, 1.37658820775, 199.0720014364], [0.552, 4.8065772945, 551.031606097], [0.494, 4.43417307482, 539.9859058331], [0.517, 0.05161181997, 3.1813937377], [0.469, 3.81715950042, 2008.557539159], [0.415, 1.34693184108, 1382.8873468466], [0.382, 4.86764073919, 227.5261894396], [0.473, 1.72405831407, 532.1386456494], [0.458, 4.44604993015, 1038.0412891868], [0.376, 2.23190744786, 529.6427809848], [0.451, 3.75869883836, 984.6003316219], [0.376, 5.42971857629, 529.7391492044], [0.389, 1.92698506631, 525.0250986486], [0.364, 3.35456685746, 2221.856634597], [0.476, 5.93625415892, 527.2432845398], [0.383, 6.12255867339, 149.5631971346], [0.301, 4.09378934049, 440.8252848776], [0.31, 5.58150418981, 2428.0421830342], [0.282, 4.85996662231, 1788.1448967202], [0.298, 5.09589374634, 528.7277572481], [0.34, 4.5653707022, 750.1036075334], [0.272, 2.3534696034, 534.3568315406], [0.36, 3.91050161665, 74.7815985673], [0.299, 1.43093538841, 909.8187330546], [0.297, 2.56584512211, 530.6541729411], [0.235, 4.81644489422, 535.107591066], [0.306, 0.68420442848, 380.12776796], [0.236, 4.63162956792, 526.7702037878], [0.27, 0.18549916939, 21.3406410024], [0.288, 4.26655874393, 1802.3719907218], [0.196, 5.35950443033, 2214.7430875962], [0.19, 4.5461519326, 2104.5367663768], [0.193, 4.35426216497, 511.5317178299], [0.178, 4.51895208036, 3178.1457905676], [0.194, 0.57050756837, 1361.5467058442], [0.2, 1.48040474749, 302.164775655], [0.168, 5.40141749419, 524.2743391232], [0.152, 0.68077486546, 1905.4647649404], [0.149, 1.06678990744, 831.8557407496], [0.182, 3.62401009613, 38.1330356378], [0.176, 5.64331384323, 963.4027029714], [0.184, 4.48850356629, 604.4725636619], [0.133, 5.45026366125, 2641.3412784722], [0.143, 2.21577268292, 1439.5096981492], [0.13, 4.88155705493, 2531.1349572528], [0.129, 6.15206333598, 547.8502123593], [0.133, 5.43193972385, 1603.2999892854], [0.133, 3.49297492409, 529.1697002328], [0.132, 3.98820790955, 530.2122299564], [0.118, 5.38352943814, 1891.2376709388], [0.133, 5.65694269884, 76.2660712756], [0.145, 2.94976686191, 454.9093665273], [0.115, 3.29206553804, 3.9321532631], [0.102, 4.48856749557, 2001.4439921582], [0.106, 6.08434275898, 10.2949407385], [0.093, 5.8473777184, 2324.9494088156], [0.101, 0.15815934254, 2655.5683724738], [0.115, 3.59221021604, 2015.6710861598], [0.103, 4.70399583323, 305.3461693927], [0.084, 0.44180206332, 1593.0050485469], [0.092, 2.44863388631, 490.3340891794], [0.087, 6.23817512863, 6283.0758499914], [0.095, 3.30154605532, 2317.8358618148], [0.072, 1.90578907085, 528.9402055692], [0.072, 5.57619428876, 530.44172462], [0.078, 5.97323507836, 1585.8915015461]], [[8094.051, 1.46322843658, 529.6909650946], [742.415, 0.95691639003, 522.5774180938], [813.244, 3.14159265359, 0.0], [398.951, 2.89888666447, 536.8045120954], [342.226, 1.44683789727, 1059.3819301892], [73.948, 0.40724675866, 1052.2683831884], [46.151, 3.48036895772, 1066.49547719], [29.314, 0.99088831805, 515.463871093], [29.717, 1.92504171329, 1589.0728952838], [22.753, 4.27124052435, 7.1135470008], [13.916, 2.92242387338, 543.9180590962], [12.067, 5.22168932482, 632.7837393132], [10.703, 4.88024222475, 949.1756089698], [6.078, 6.21089108431, 1045.1548361876], [5.935, 0.52977760072, 1581.959348283], [5.037, 1.43444929374, 526.5095713569], [4.564, 0.91811732585, 1162.4747044078], [4.547, 4.01953745202, 1596.1864422846], [5.098, 6.03169795231, 735.8765135318], [3.593, 4.54080164408, 110.2063212194], [3.443, 1.38618954572, 533.6231183577], [3.277, 4.39650286553, 14.2270940016], [3.407, 0.42275631534, 419.4846438752], [2.904, 2.06041641723, 316.3918696566], [2.541, 3.98323842017, 323.5054166574], [3.113, 2.48079280193, 2118.7638603784], [3.061, 2.39880866911, 532.8723588323], [2.155, 4.7799063714, 942.062061969], [2.143, 3.88727338786, 426.598190876], [2.252, 0.3719643412, 1155.361157407], [2.019, 3.89985000464, 846.0828347512], [1.857, 1.19658907851, 103.0927742186], [1.683, 1.42264195434, 1265.5674786264], [2.313, 0.87671613055, 213.299095438], [1.443, 2.38565505909, 1169.5882514086], [1.823, 5.80106463776, 625.6701923124], [1.728, 2.24114678267, 525.7588118315], [1.198, 0.03252059731, 956.2891559706], [1.138, 3.46420904745, 1073.6090241908], [1.086, 5.352791467, 117.3198682202], [0.84, 2.89946334223, 95.9792272178], [0.746, 5.53017890231, 1478.8665740644], [0.944, 4.055870535, 206.1855484372], [0.758, 3.74770617289, 433.7117378768], [0.673, 1.26396626349, 508.3503240922], [0.889, 6.07878453176, 728.762966531], [0.6, 1.82954494089, 639.897286314], [0.589, 1.23625943417, 1258.4539316256], [0.619, 0.67923057477, 838.9692877504], [0.566, 5.36336098734, 742.9900605326], [0.648, 5.32990375008, 853.196381752], [0.553, 3.15511946637, 220.4126424388], [0.432, 1.03719283016, 1692.1656695024], [0.435, 1.65056479007, 519.3960243561], [0.43, 1.41830384501, 412.3710968744], [0.431, 2.20986254651, 1368.660252845], [0.415, 4.35372561905, 330.6189636582], [0.438, 0.1655227729, 1574.8458012822], [0.312, 4.50639455819, 2125.8774073792], [0.28, 3.01441283033, 551.031606097], [0.309, 0.67399908949, 2111.6503133776], [0.301, 3.06868080871, 1062.5633239269], [0.236, 1.946968422, 1485.9801210652], [0.235, 3.41850395941, 199.0720014364], [0.246, 2.61803442505, 309.2783226558], [0.238, 2.56643737684, 539.9859058331], [0.248, 2.96997778167, 2648.454825473], [0.209, 5.82481690851, 1471.7530270636], [0.205, 1.20202002469, 1056.2005364515], [0.188, 0.97113663101, 1685.0521225016], [0.137, 2.91203499563, 1699.2792165032], [0.131, 1.79274504072, 1063.3140834523], [0.161, 1.05926568614, 1795.258443721], [0.112, 2.62660288825, 440.8252848776], [0.11, 3.56263668146, 227.5261894396], [0.114, 6.13907482464, 1038.0412891868], [0.103, 4.6428710104, 3.1813937377], [0.123, 4.81268110532, 21.3406410024], [0.102, 4.2760382797, 1375.7737998458], [0.089, 1.22926014128, 1898.3512179396], [0.08, 0.62129648755, 831.8557407496]], [[251.624, 3.38087923084, 529.6909650946], [121.738, 2.733118372, 522.5774180938], [48.694, 1.03689996685, 536.8045120954], [10.988, 2.31463561347, 1052.2683831884], [8.067, 2.76729757621, 515.463871093], [6.205, 1.7811582737, 1066.49547719], [7.287, 4.25268318975, 1059.3819301892], [3.627, 1.13028917221, 543.9180590962], [2.798, 3.14159265359, 0.0], [1.898, 2.28934054087, 7.1135470008], [1.643, 1.77507208483, 1045.1548361876], [0.945, 0.45261136388, 632.7837393132], [0.758, 0.30577920142, 949.1756089698], [0.731, 2.63748223583, 14.2270940016], [0.876, 0.32927768725, 1589.0728952838], [0.678, 2.36909615348, 1581.959348283], [0.623, 2.480562136, 1596.1864422846], [0.736, 1.52532370632, 735.8765135318], [0.499, 3.67985494258, 419.4846438752], [0.454, 0.26977404624, 942.062061969], [0.453, 3.18232334886, 526.5095713569], [0.409, 2.88147337106, 110.2063212194], [0.347, 5.7624428587, 103.0927742186], [0.31, 2.98017326384, 508.3503240922], [0.321, 4.40642025933, 532.8723588323], [0.3, 1.66936571536, 625.6701923124], [0.295, 1.75924202728, 1073.6090241908], [0.282, 3.11087801399, 533.6231183577], [0.263, 0.55255030187, 426.598190876], [0.208, 2.17540496886, 1155.361157407], [0.183, 4.34670868038, 525.7588118315], [0.18, 6.07777744541, 639.897286314], [0.159, 2.60843864402, 1162.4747044078], [0.117, 4.70141431381, 95.9792272178], [0.107, 5.48942805114, 433.7117378768], [0.105, 3.75192101775, 316.3918696566], [0.13, 1.37897716939, 323.5054166574], [0.094, 3.05797832024, 1265.5674786264], [0.114, 3.75170981478, 117.3198682202], [0.095, 0.54905691533, 1169.5882514086], [0.088, 3.26874502411, 213.299095438], [0.098, 2.00704668688, 1574.8458012822]], [[15.05, 4.52956999637, 522.5774180938], [5.37, 4.47427159142, 529.6909650946], [4.456, 5.43908581047, 536.8045120954], [3.422, 0.0, 0.0], [1.833, 4.51807036227, 515.463871093], [1.322, 4.20117611581, 1052.2683831884], [0.755, 5.59451554966, 543.9180590962], [0.512, 0.05803177475, 1066.49547719], [0.282, 3.66807771223, 1059.3819301892], [0.147, 3.56490986181, 1045.1548361876], [0.142, 5.69936472988, 7.1135470008], [0.112, 1.16718383135, 14.2270940016]], [[1.445, 0.09198554072, 522.5774180938], [0.368, 0.00874408003, 515.463871093], [0.304, 3.27902945138, 536.8045120954], [0.129, 0.33959775247, 529.6909650946], [0.095, 1.29305954542, 543.9180590962]]]¶: This table contains Jupiter’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 430-432.

pymeeus.Jupiter.VSOP87_L = [[[59954691.495, 0.0, 0.0], [9695898.711, 5.06191793105, 529.6909650946], [573610.145, 1.44406205976, 7.1135470008], [306389.18, 5.41734729976, 1059.3819301892], [97178.28, 4.14264708819, 632.7837393132], [72903.096, 3.64042909255, 522.5774180938], [64263.986, 3.41145185203, 103.0927742186], [39806.051, 2.29376744855, 419.4846438752], [38857.78, 1.2723172486, 316.3918696566], [27964.622, 1.78454589485, 536.8045120954], [13589.738, 5.7748103159, 1589.0728952838], [8246.362, 3.58227961655, 206.1855484372], [8768.686, 3.63000324417, 949.1756089698], [7368.057, 5.08101125612, 735.8765135318], [6263.171, 0.02497643742, 213.299095438], [6114.05, 4.51319531666, 1162.4747044078], [4905.419, 1.32084631684, 110.2063212194], [5305.283, 1.30671236848, 14.2270940016], [5305.457, 4.18625053495, 1052.2683831884], [4647.249, 4.69958109497, 3.9321532631], [3045.009, 4.31675960318, 426.598190876], [2610.001, 1.5666759485, 846.0828347512], [2028.191, 1.06376547379, 3.1813937377], [1764.768, 2.14148077766, 1066.49547719], [1722.983, 3.88036008872, 1265.5674786264], [1920.959, 0.97168928755, 639.897286314], [1633.217, 3.58201089758, 515.463871093], [1431.997, 4.29683690269, 625.6701923124], [973.278, 4.09764957065, 95.9792272178], [884.439, 2.43701426123, 412.3710968744], [732.875, 6.08534113239, 838.9692877504], [731.072, 3.80591233956, 1581.959348283], [691.928, 6.13368222939, 2118.7638603784], [709.19, 1.29272573658, 742.9900605326], [614.464, 4.10853496756, 1478.8665740644], [495.224, 3.75567461379, 323.5054166574], [581.902, 4.53967717552, 309.2783226558], [375.657, 4.70299124833, 1368.660252845], [389.864, 4.89716105852, 1692.1656695024], [341.006, 5.71452525783, 533.6231183577], [330.458, 4.74049819491, 0.0481841098], [440.854, 2.95818460943, 454.9093665273], [417.266, 1.03554430161, 2.4476805548], [244.17, 5.220208789, 728.762966531], [261.54, 1.87652461032, 0.9632078465], [256.568, 3.72410724159, 199.0720014364], [261.009, 0.82047246448, 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[31.813, 4.8582998665, 522.5774180938], [14.742, 4.29061635784, 515.463871093], [8.899, 0.71478520741, 1059.3819301892], [4.957, 1.29502259434, 543.9180590962], [4.484, 2.31715516627, 1066.49547719], [4.251, 0.48326797501, 21.3406410024], [3.1, 3.00245542678, 412.3710968744], [2.055, 0.39858940218, 639.897286314], [1.762, 4.90536207307, 625.6701923124], [1.902, 4.25925620271, 199.0720014364], [1.695, 4.26147580803, 206.1855484372], [1.375, 5.25546955667, 1052.2683831884], [1.203, 4.71614633845, 95.9792272178], [1.086, 1.28604571172, 1589.0728952838], [0.982, 4.77990073662, 1045.1548361876], [0.935, 6.05847062188, 88.865680217], [0.916, 5.77537499431, 728.762966531], [0.89, 4.55299189579, 426.598190876], [0.784, 3.4016156795, 419.4846438752], [0.768, 3.54672049322, 103.0927742186], [0.67, 0.522233077, 110.2063212194], [0.415, 5.22809480633, 302.164775655], [0.393, 6.24184621807, 956.2891559706], [0.381, 5.2546696604, 309.2783226558], [0.421, 0.59561318533, 117.3198682202], [0.346, 4.78348312106, 508.3503240922], [0.319, 3.47979828725, 323.5054166574], [0.331, 2.95893485883, 1596.1864422846], [0.295, 4.32713459459, 942.062061969], [0.319, 0.47990052824, 831.8557407496], [0.251, 1.79898001222, 1073.6090241908], [0.212, 0.43917684084, 220.4126424388], [0.188, 1.12654974776, 1169.5882514086], [0.188, 2.16135407548, 1361.5467058442], [0.18, 3.43266428069, 1148.2476104062], [0.164, 1.92864127211, 2118.7638603784], [0.157, 3.02963907392, 1272.6810256272], [0.093, 5.60436000012, 1581.959348283], [0.085, 5.023172562, 1155.361157407], [0.075, 3.13198879608, 632.7837393132]], [[49.577, 5.25658966184, 7.1135470008], [15.761, 5.25126837478, 14.2270940016], [4.343, 0.01461869263, 536.8045120954], [1.526, 1.09739911439, 522.5774180938], [0.728, 5.85949047619, 543.9180590962], [0.694, 0.87382487754, 515.463871093], [0.845, 3.14159265359, 0.0], [0.456, 0.81521692852, 1066.49547719], [0.293, 5.62909357048, 1059.3819301892], [0.09, 0.2117811971, 529.6909650946]]]¶: This table contains Jupiter’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 426-430.

pymeeus.Jupiter.VSOP87_R = [[[520887429.471, 0.0, 0.0], [25209327.02, 3.49108640015, 529.6909650946], [610599.902, 3.84115365602, 1059.3819301892], [282029.465, 2.57419879933, 632.7837393132], [187647.391, 2.07590380082, 522.5774180938], [86792.941, 0.71001090609, 419.4846438752], [72062.869, 0.21465694745, 536.8045120954], [65517.227, 5.97995850843, 316.3918696566], [29134.62, 1.6775924371, 103.0927742186], [30135.275, 2.16132058449, 949.1756089698], [23453.209, 3.54023147303, 735.8765135318], [22283.71, 4.19362773546, 1589.0728952838], [23947.34, 0.27457854894, 7.1135470008], [13032.6, 2.96043055741, 1162.4747044078], [9703.346, 1.90669572402, 206.1855484372], [12749.004, 2.71550102862, 1052.2683831884], [9161.431, 4.41352618935, 213.299095438], [7894.539, 2.47907551404, 426.598190876], [7057.978, 2.18184753111, 1265.5674786264], [6137.755, 6.26417542514, 846.0828347512], [5477.093, 5.65729325169, 639.897286314], [3502.519, 0.56531297394, 1066.49547719], [4136.89, 2.72219979684, 625.6701923124], [4170.012, 2.01605033912, 515.463871093], [2499.966, 4.55182055941, 838.9692877504], [2616.955, 2.00993967129, 1581.959348283], [1911.876, 0.85621927419, 412.3710968744], [2127.644, 6.1275146175, 742.9900605326], [1610.549, 3.08867789275, 1368.660252845], [1479.484, 2.68026191372, 1478.8665740644], [1230.708, 1.89042979701, 323.5054166574], [1216.81, 1.80171561024, 110.2063212194], [961.072, 4.54876989805, 2118.7638603784], [885.708, 4.14785948471, 533.6231183577], [776.7, 3.6769695469, 728.762966531], [998.579, 2.8720894011, 309.2783226558], [1014.959, 1.38673237666, 454.9093665273], [727.162, 3.98824686402, 1155.361157407], [655.289, 2.79065604219, 1685.0521225016], [821.465, 1.59342534396, 1898.3512179396], [620.798, 4.82284338962, 956.2891559706], [653.981, 3.38150775269, 1692.1656695024], [812.036, 5.94091899141, 909.8187330546], [562.12, 0.08095987241, 543.9180590962], [542.221, 0.28360266386, 525.7588118315], [457.859, 0.1272269451, 1375.7737998458], [614.784, 2.27624915604, 942.062061969], [435.805, 2.60272129748, 95.9792272178], [496.066, 5.53005947761, 380.12776796], [469.965, 2.81896276101, 1795.258443721], [445.003, 0.14623567024, 14.2270940016], [290.869, 3.89339143564, 1471.7530270636], [276.627, 2.52238450687, 2001.4439921582], [275.084, 2.98863518924, 526.5095713569], [293.875, 2.04938438861, 199.0720014364], [290.985, 6.03131226226, 1169.5882514086], [338.342, 2.79873192583, 1045.1548361876], [257.482, 6.13395478303, 532.8723588323], [319.013, 1.34803130803, 2214.7430875962], [309.352, 5.36855804945, 1272.6810256272], [345.804, 1.56404293688, 491.5579294568], [303.364, 1.15407454372, 5753.3848848968], [192.325, 0.91996333387, 1596.1864422846], [215.398, 2.63572815848, 2111.6503133776], [200.738, 2.37259566683, 1258.4539316256], [239.036, 3.57397189838, 835.0371344873], [197.073, 5.92859096863, 453.424893819], [139.44, 3.63960322318, 1788.1448967202], [191.373, 6.2825131187, 983.1158589136], [176.551, 2.57669991654, 9683.5945811164], [123.567, 2.26158186345, 2317.8358618148], [128.176, 4.6658590767, 831.8557407496], [112.43, 0.85604150812, 433.7117378768], [128.817, 1.10567106595, 2531.1349572528], [99.39, 4.50312054049, 518.6452648307], [93.87, 2.7255387999, 853.196381752], [106.481, 5.8146222229, 220.4126424388], [120.188, 2.95156363556, 3.9321532631], [104.002, 2.22221906187, 74.7815985673], [81.655, 3.23481337678, 1361.5467058442], [112.513, 4.86216964016, 528.2064923863], [79.539, 0.8854224683, 430.5303441391], [85.801, 2.11458386763, 1574.8458012822], [85.685, 2.33823884827, 2428.0421830342], [68.311, 3.35727048905, 2104.5367663768], [69.57, 3.04164697156, 302.164775655], [69.775, 3.22402404312, 305.3461693927], [69.57, 0.20494979941, 532.1386456494], [56.991, 2.00204191909, 2634.2277314714], [77.062, 2.09816000231, 508.3503240922], [56.716, 3.91743976711, 2221.856634597], [58.325, 5.72360355252, 628.8515860501], [52.485, 4.02485010492, 527.2432845398], [63.645, 1.09973563964, 1364.7280995819], [53.607, 0.87425992614, 2847.5268269094], [59.598, 0.95822471775, 494.2662424425], [57.96, 3.45779497978, 2008.557539159], [41.512, 3.51955526735, 529.7391492044], [44.666, 1.62313786651, 984.6003316219], [44.883, 4.90091959557, 2648.454825473], [53.206, 1.19800364308, 760.25553592], [44.393, 4.42623747662, 1063.3140834523], [37.566, 2.93021095213, 1677.9385755008], [41.516, 0.32174409278, 529.6427809848], [42.855, 0.03093594081, 1439.5096981492], [45.963, 2.54342106514, 636.7158925763], [40.181, 4.39381642864, 1148.2476104062], [38.77, 4.31675565025, 149.5631971346], [40.348, 2.10140891053, 2744.4340526908], [48.851, 5.60297777544, 2810.9214616052], [37.085, 5.07828164301, 1905.4647649404], [43.875, 1.24536971083, 621.7380390493], [34.005, 3.09360167248, 2420.9286360334], [36.782, 0.84232174637, 530.6541729411], [31.139, 5.35811251334, 1485.9801210652], [39.295, 4.70800489067, 569.0478410098], [39.7, 2.46163878814, 355.7487455718], [31.527, 6.19284070863, 3.1813937377], [28.399, 2.48456666067, 519.3960243561], [32.432, 2.73281750275, 604.4725636619], [27.119, 3.92341697086, 2324.9494088156], [26.753, 1.74975198417, 2950.619601128], [28.986, 1.83535862643, 1891.2376709388], [26.493, 0.60380196895, 1055.4497769261], [33.525, 0.76068430639, 643.8294395771], [26.568, 1.03594610835, 405.2575498736], [25.534, 3.46320665375, 458.8415197904], [24.421, 0.8818183693, 423.4167971383], [32.949, 3.18597137308, 528.7277572481], [22.456, 0.43129919683, 1073.6090241908], [21.599, 1.41820425091, 540.7366653585], [25.673, 0.5235819476, 511.5317178299], [21.115, 3.08023522766, 629.6023455755], [22.713, 0.65234613144, 3163.918696566], [19.189, 5.16589014963, 635.9651330509], [26.042, 1.33629471285, 330.6189636582], [18.263, 3.59973446951, 746.9222137957], [18.21, 2.66819439927, 1994.3304451574], [19.724, 4.13552133321, 1464.6394800628], [19.48, 1.85656428109, 3060.8259223474], [23.927, 4.99826361784, 1289.9465010146], [21.886, 5.91718683551, 1802.3719907218], [17.482, 2.82161612542, 2737.32050569], [16.608, 5.67394889755, 408.4389436113], [22.892, 5.26731352093, 672.1406152284], [18.349, 1.89869734949, 1021.2488945514], [19.123, 3.65882402977, 415.5524906121], [15.735, 3.34772676006, 1056.2005364515], [16.373, 0.18094878053, 1699.2792165032], [18.899, 3.69120638874, 88.865680217], [18.655, 1.97327300097, 38.1330356378], [15.542, 3.8220488101, 721.6494195302], [16.78, 1.90976657921, 217.2312487011], [15.313, 1.05907174619, 114.1384744825], [15.19, 1.32317039042, 117.3198682202], [15.08, 3.74469077216, 2641.3412784722], [19.836, 2.73184571324, 39.3568759152], [14.708, 1.67270454473, 529.1697002328], [14.036, 3.54305270022, 142.4496501338], [12.931, 1.48829749349, 3267.0114707846], [14.924, 1.3254608594, 490.3340891794], [14.753, 4.64530618027, 6283.0758499914], [14.672, 0.80451954754, 5223.6939198022], [12.085, 3.67072510553, 750.1036075334], [11.954, 2.97127390765, 505.3119427064], [14.65, 2.1679293025, 530.2122299564], [11.869, 1.66551754962, 2207.6295405954], [12.273, 0.20690014405, 1062.5633239269], [11.46, 1.11906683214, 561.934294009], [11.083, 3.22049096074, 535.107591066], [11.567, 5.22625628971, 524.0618908021], [11.161, 3.82945634036, 76.2660712756], [10.918, 1.27796962818, 2125.8774073792], [12.685, 3.96848605476, 2538.2485042536], [11.23, 3.23092119889, 422.6660376129], [12.645, 0.7367042858, 908.3342603463], [11.33, 5.56127247007, 531.1754378029], [9.509, 5.00507284204, 597.3590166611], [10.291, 3.84159025239, 1781.0313497194], [10.762, 4.91380719453, 525.0250986486], [11.786, 5.11863653538, 685.4739373527], [11.98, 1.72470898635, 911.3032057629], [8.937, 2.40338241992, 2310.722314814], [9.253, 2.57670338148, 3053.7123753466], [9.488, 2.95089828501, 1382.8873468466], [9.889, 0.43758517388, 3480.3105662226], [8.781, 3.66562388594, 739.8086667949], [8.664, 2.70398612383, 526.7702037878], [9.505, 1.61249870019, 3377.217792004], [11.54, 1.59520481029, 1474.6737883704], [9.533, 0.35468711552, 1512.8068240082], [9.98, 4.80984684596, 558.0021407459], [9.014, 1.21458362718, 416.3032501375], [7.969, 0.08480602718, 528.9402055692], [8.668, 5.29060005706, 945.2434557067], [7.851, 1.46751861875, 963.4027029714], [8.611, 1.13232641062, 532.6117264014], [7.838, 6.26933498027, 647.0108333148], [7.581, 2.90608705954, 533.8837507886], [8.583, 6.06634530166, 10213.285546211], [10.198, 2.48743123636, 1819.6374661092], [8.536, 2.2270070179, 9153.9036160218], [9.759, 6.15593336218, 593.426863398], [7.968, 3.75535355212, 530.44172462], [7.142, 3.58836120327, 2957.7331481288], [7.122, 0.11970048938, 224.3447957019], [8.731, 0.7530291397, 960.2213092337], [7.063, 2.1679303769, 724.8308132679], [7.263, 2.29499675875, 520.129737539], [6.418, 1.25058991868, 3583.4033404412], [8.27, 1.24806288317, 495.7507151508], [6.483, 4.7456777264, 202.2533951741], [7.197, 3.84169279666, 618.5566453116], [8.146, 0.73147060302, 230.5645708254], [6.165, 5.50124418381, 11.0457002639], [7.946, 2.07754951174, 953.1077622329], [7.675, 0.92400307662, 525.4981794006], [6.21, 1.45641362115, 483.2205421786], [7.359, 0.31355650764, 378.6432952517], [6.707, 2.92071167098, 1038.0412891868], [7.143, 0.18218134889, 731.9443602687], [7.309, 6.27084533477, 21.3406410024], [6.135, 2.67651237303, 312.4597163935], [5.558, 3.83419160288, 534.3568315406], [5.344, 5.25294750019, 1048.3362299253], [7.504, 0.74281415471, 457.617679513], [5.335, 6.23059924424, 551.031606097], [5.613, 1.51210605952, 524.2743391232], [5.284, 2.18579185671, 280.9671470045], [5.475, 5.95864753605, 539.9859058331], [5.056, 0.37387972537, 529.5309064002], [6.202, 5.53813122743, 2.4476805548], [5.49, 5.97692444199, 227.5261894396], [6.266, 0.76632858238, 938.1299087059], [5.75, 2.13496323512, 191.9584544356], [5.218, 4.69335266854, 560.7104537316], [5.48, 5.21157595558, 1057.8974574809], [5.738, 0.34249718209, 535.9107402181], [4.816, 1.51326236835, 2524.021410252], [5.056, 3.46671669992, 529.851023789], [4.71, 2.2781383055, 3370.1042450032], [5.228, 3.61776977584, 2097.423219376], [4.878, 1.39829798223, 3693.6096616606], [5.727, 4.80120381106, 598.8434893694], [5.707, 3.94177950323, 2854.6403739102], [4.988, 4.87244187719, 1.4844727083], [5.424, 3.53268613904, 456.3938392356], [4.288, 4.84438067847, 70.8494453042], [5.944, 3.79180483544, 25558.2121764796], [4.195, 2.09136830994, 2627.1141844706], [4.582, 5.61707254513, 2435.155730035], [4.268, 6.20250525415, 775.233389447], [4.521, 0.20049967962, 92.0470739547], [5.405, 4.66492781581, 833.552661779], [5.607, 3.30226645638, 535.3200393871], [4.171, 3.14873010832, 944.9828232758], [4.108, 5.84489743779, 440.8252848776], [4.367, 4.68363584557, 327.4375699205], [4.033, 3.30883782817, 3274.1250177854], [4.292, 0.20604269202, 3796.7024358792], [4.27, 0.98941708997, 387.2413149608], [4.259, 3.21120589971, 696.5196376166], [4.673, 1.96606729969, 107.0249274817], [4.031, 4.62854606236, 2751.5475996916], [5.115, 2.66416451377, 1215.1649024473], [4.181, 4.74527698816, 988.532484885], [4.374, 1.50010561403, 1894.4190646765], [3.803, 3.59911687954, 437.6438911399], [3.761, 3.96903199782, 732.6951197941], [3.62, 1.57847427805, 381.6122406683], [3.49, 0.63097592112, 529.9034134157], [4.019, 2.5766416572, 916.9322800554], [4.133, 4.78417930217, 824.7421937488], [4.411, 3.13179382423, 630.3360587584], [4.099, 3.63702212253, 810.6581120991], [3.704, 6.17243801274, 537.7677199419], [4.124, 2.14248285449, 210.1177017003], [3.49, 3.20962050417, 529.4785167735], [3.281, 1.53106243317, 547.8502123593], [3.554, 6.03787799174, 739.0579072695], [4.101, 6.00406226999, 902.7051860538], [3.267, 3.49354065789, 1166.4068576709], [3.286, 2.5596687053, 945.9942152321], [4.041, 4.78735413707, 850.0149880143], [4.304, 0.11406117717, 1744.8558675419], [4.043, 5.204170936, 635.231419868], [3.115, 4.61986265585, 952.3570027075], [3.016, 0.95126220905, 3899.7952100978], [3.017, 2.59699501992, 632.831923423], [3.219, 1.83594791142, 18.1592472647], [3.203, 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1.04290245896, 1589.0728952838], [30.023, 4.63236245032, 426.598190876], [33.531, 0.84784977903, 206.1855484372], [20.804, 2.50071243814, 728.762966531], [14.466, 0.96040197071, 508.3503240922], [12.969, 1.5023378855, 1045.1548361876], [11.654, 3.55513510121, 323.5054166574], [12.319, 2.60952614503, 735.8765135318], [15.023, 0.89136998434, 199.0720014364], [11.16, 1.79041437555, 309.2783226558], [10.554, 6.27845112678, 956.2891559706], [9.812, 6.26016859519, 103.0927742186], [9.301, 3.45126812476, 838.9692877504], [6.672, 1.87004905364, 302.164775655], [7.442, 1.28047007623, 742.9900605326], [7.178, 0.92022189637, 942.062061969], [5.577, 1.37980792905, 95.9792272178], [6.834, 3.45228722967, 831.8557407496], [4.632, 2.82934545414, 1596.1864422846], [3.969, 1.21290005054, 1169.5882514086], [3.869, 5.99495313698, 213.299095438], [3.551, 6.10714791535, 405.2575498736], [2.943, 2.32831075458, 1155.361157407], [2.442, 1.86965213405, 532.8723588323], [2.41, 0.42627205128, 220.4126424388], [2.289, 1.94941487274, 1073.6090241908], [2.274, 0.09211517505, 632.7837393132], [2.189, 1.58907745204, 2118.7638603784], [2.387, 5.97080671477, 1162.4747044078], [2.104, 1.06751462671, 21.3406410024], [2.128, 1.51119399925, 1258.4539316256], [2.491, 0.35125020737, 1272.6810256272], [2.006, 5.9448738836, 110.2063212194], [1.98, 2.54989377864, 88.865680217], [2.04, 2.16463966964, 433.7117378768], [1.955, 2.70341589777, 721.6494195302], [1.67, 4.46255717328, 853.196381752], [1.91, 2.25964760758, 1361.5467058442], [1.71, 1.98372066321, 525.7588118315], [1.52, 0.11641358425, 949.1756089698], [2.003, 3.16520599208, 1148.2476104062], [1.71, 2.70850417287, 330.6189636582], [1.629, 0.47376028854, 526.5095713569], [1.229, 3.01987279595, 963.4027029714], [1.671, 0.44352103086, 533.6231183577], [1.207, 1.15774089269, 1574.8458012822], [1.146, 2.54505851138, 846.0828347512], [1.355, 1.17462112647, 1038.0412891868], [1.001, 2.70272799283, 519.3960243561], [1.372, 0.67467128629, 551.031606097], [0.983, 4.17198081351, 2627.1141844706], [1.084, 1.07011164067, 227.5261894396], [0.892, 2.92543286761, 1368.660252845], [0.823, 4.86559196955, 611.4430983108], [1.136, 1.78981738432, 1581.959348283], [0.897, 4.9107363027, 1670.8250285], [0.908, 3.6880404733, 824.7421937488], [0.789, 3.2338089325, 2125.8774073792], [0.771, 2.39070707004, 2317.8358618148], [0.891, 0.59692950778, 539.9859058331], [0.876, 4.52127091462, 750.1036075334], [0.802, 0.20759322884, 1141.1340634054], [0.85, 0.94145487094, 191.9584544356], [0.762, 2.25149516048, 2538.2485042536], [0.694, 0.67080348659, 440.8252848776], [0.741, 5.79934203525, 1485.9801210652], [0.643, 2.48127580335, 1265.5674786264], [0.575, 6.13756590872, 1279.794572628], [0.636, 5.51001645505, 2413.8150890326], [0.636, 4.40777238491, 1382.8873468466], [0.555, 2.18233983981, 1062.5633239269], [0.564, 1.92775967119, 2634.2277314714], [0.531, 2.04824376019, 295.0512286542], [0.541, 2.32424368689, 1471.7530270636], [0.697, 2.27179476322, 1699.2792165032], [0.546, 1.9577490573, 1677.9385755008], [0.465, 4.35550844067, 1692.1656695024], [0.508, 2.50298248836, 2207.6295405954], [0.496, 5.77087043616, 1478.8665740644], [0.44, 5.98661963879, 934.9485149682], [0.424, 2.80194129521, 81.7521332162], [0.406, 3.93940190897, 316.3918696566], [0.506, 0.18719982992, 10.2949407385]], [[128.628, 0.08419309557, 536.8045120954], [113.458, 4.24858855779, 529.6909650946], [82.65, 3.29754909408, 522.5774180938], [37.883, 2.73326611144, 515.463871093], [26.694, 5.69142588558, 7.1135470008], [17.65, 5.40012536918, 1059.3819301892], [12.612, 6.01560416057, 543.9180590962], [9.287, 0.76813946494, 1066.49547719], [8.107, 5.68228065707, 14.2270940016], [6.271, 5.12286932534, 639.897286314], [6.978, 1.42751292055, 412.3710968744], [5.377, 3.33501947275, 625.6701923124], [2.911, 3.40334805052, 1052.2683831884], [2.593, 4.16090412984, 728.762966531], [2.562, 2.89802035072, 426.598190876], [2.268, 6.22195938856, 1589.0728952838], [2.114, 3.11758855774, 1045.1548361876], [1.673, 2.81399290364, 206.1855484372], [1.805, 2.60030006919, 199.0720014364], [1.823, 1.89432426038, 419.4846438752], [1.522, 1.33432648232, 1596.1864422846], [1.697, 0.0, 0.0], [1.039, 4.41904942302, 956.2891559706], [1.161, 5.16181311538, 831.8557407496], [0.916, 3.17245716108, 508.3503240922], [0.87, 5.793878135, 1169.5882514086], [0.916, 1.87129662931, 1148.2476104062], [0.955, 0.66801367802, 1361.5467058442], [0.788, 1.47515450553, 1272.6810256272], [0.966, 5.47457968043, 220.4126424388], [0.788, 2.42252866885, 117.3198682202], [0.712, 0.4965589703, 1073.6090241908], [0.656, 3.53022740783, 302.164775655], [0.681, 2.8450717434, 191.9584544356], [0.771, 2.19893222018, 942.062061969], [0.765, 5.311472577, 551.031606097], [0.667, 3.72432305249, 88.865680217], [0.534, 1.83172084748, 647.0108333148], [0.553, 0.85896003802, 330.6189636582], [0.543, 5.26057584439, 21.3406410024], [0.584, 3.82243061802, 618.5566453116], [0.512, 4.44485521707, 110.2063212194], [0.612, 1.59320941864, 3.1813937377], [0.631, 1.83863158533, 10.2949407385], [0.491, 1.52912023181, 405.2575498736], [0.521, 0.24011424451, 433.7117378768]], [[11.188, 4.75249399945, 536.8045120954], [4.255, 5.9151622917, 522.5774180938], [2.079, 5.56781555864, 515.463871093], [1.908, 4.29659647286, 543.9180590962], [1.875, 3.69357495838, 7.1135470008], [1.59, 5.49312796166, 1066.49547719], [1.612, 4.13222808529, 1059.3819301892], [1.24, 3.77981722506, 14.2270940016], [1.033, 4.50671820436, 529.6909650946]]]¶: This table contains Jupiter’s periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 432-434.

Mars¶

Class to model Mars planet.

class pymeeus.Mars.Mars[source]¶

Class Mars models that planet.

__weakref__¶: list of weak references to the object (if defined)

static apparent_heliocentric_position(epoch)[source]¶

This method computes the apparent heliocentric position of planet Mars for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Mars position, as an Epoch object
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

static conjunction(epoch)[source]¶

This method computes the time of the conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired conjunction
Returns:	The time when the conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> conj = Mars.conjunction(epoch)
>>> y, m, d = conj.get_date()
>>> print(y)
1993
>>> print(m)
12
>>> print(round(d, 4))
27.0898

static geocentric_position(epoch)[source]¶

This method computes the geocentric position of Mars (right ascension and declination) for the given epoch, as well as the elongation angle.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension, the declination and the elongation angle as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 20.0)
>>> ra, dec, elon = Mars.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
7h 48' 35.4''
>>> print(dec.dms_str(n_dec=1))
24d 35' 33.9''
>>> print(elon.dms_str(n_dec=1))
153d 35' 1.6''

static geometric_heliocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of planet Mars for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Mars position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2018, 10, 27.0)
>>> l, b, r = Mars.geometric_heliocentric_position(epoch)
>>> print(round(l.to_positive(), 4))
2.0015
>>> print(round(b, 4))
-1.3683
>>> print(round(r, 5))
1.39306

static magnitude(sun_dist, earth_dist, phase_angle)[source]¶

This function computes the approximate magnitude of Mars.

Parameters:	sun_dist (float) – Distance from Mars to the Sun, in Astronomical Units earth_dist (float) – Distance from Mars to Earth, in Astronomical Units phase_angle (float, `Angle`) – Mars phase angle
Returns:	Mars’ magnitude
Return type:	float
Raises:	TypeError if input values are of wrong type.

static opposition(epoch)[source]¶

This method computes the time of the opposition closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	The time when the opposition happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2729, 10, 1.0)
>>> oppo = Mars.opposition(epoch)
>>> y, m, d = oppo.get_date()
>>> print(y)
2729
>>> print(m)
9
>>> print(round(d, 4))
9.1412

static orbital_elements_j2000(epoch)[source]¶

This method computes the orbital elements of Mars for the standard equinox J2000.0 for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Mars.orbital_elements_j2000(epoch)
>>> print(round(l, 6))
287.94027
>>> print(round(a, 8))
1.52367934
>>> print(round(e, 7))
0.0934599
>>> print(round(i, 6))
1.844381
>>> print(round(ome, 5))
49.36464
>>> print(round(arg, 6))
286.98617

static orbital_elements_mean_equinox(epoch)[source]¶

This method computes the orbital elements of Mars for the mean equinox of the date for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Mars.orbital_elements_mean_equinox(epoch)
>>> print(round(l, 6))
288.855211
>>> print(round(a, 8))
1.52367934
>>> print(round(e, 7))
0.0934599
>>> print(round(i, 6))
1.849338
>>> print(round(ome, 5))
50.06365
>>> print(round(arg, 6))
287.202108

static passage_nodes(epoch, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of Mars, nearest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch closest to the node passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2019, 1, 1)
>>> time, r = Mars.passage_nodes(epoch)
>>> year, month, day = time.get_date()
>>> print(year)
2019
>>> print(month)
1
>>> print(round(day, 1))
15.2
>>> print(round(r, 4))
1.4709

static perihelion_aphelion(epoch, perihelion=True)[source]¶

This method computes the time of Perihelion (or Aphelion) closer to a given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired Perihelion (or Aphelion) peihelion – If True, the epoch of the closest Perihelion is computed, if False, the epoch of the closest Aphelion is found.
Returns:	The epoch of the desired Perihelion (or Aphelion)
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2019, 2, 23.0)
>>> e = Mars.perihelion_aphelion(epoch)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2018
>>> print(m)
9
>>> print(d)
16
>>> print(h)
12
>>> epoch = Epoch(2032, 1, 1.0)
>>> e = Mars.perihelion_aphelion(epoch, perihelion=False)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2032
>>> print(m)
10
>>> print(d)
24
>>> print(h)
22

static station_longitude_1(epoch)[source]¶

This method computes the time of the 1st station in longitude (i.e. when the planet is stationary and begins to move westward - retrograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	Time when the 1st station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1997, 3, 1.0)
>>> sta1 = Mars.station_longitude_1(epoch)
>>> y, m, d = sta1.get_date()
>>> print(y)
1997
>>> print(m)
2
>>> print(round(d, 4))
6.033

static station_longitude_2(epoch)[source]¶

This method computes the time of the 2nd station in longitude (i.e. when the planet is stationary and begins to move eastward - prograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	Time when the 2nd station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1997, 3, 1.0)
>>> sta2 = Mars.station_longitude_2(epoch)
>>> y, m, d = sta2.get_date()
>>> print(y)
1997
>>> print(m)
4
>>> print(round(d, 4))
27.7553

pymeeus.Mars.ORBITAL_ELEM = [[355.433, 19141.6964471, 0.00031052, 1.6e-08], [1.523679342, 0.0, 0.0, 0.0], [0.09340065, 9.0484e-05, -8.06e-08, -2.5e-10], [1.849726, -0.0006011, 1.276e-05, -7e-09], [49.558093, 0.7720959, 1.557e-05, 2.267e-06], [336.060234, 1.8410449, 0.00013477, 5.36e-07]]¶: This table contains the parameters to compute Mars’ orbital elements for the mean equinox of date. Based in Table 31.A, page 212

pymeeus.Mars.ORBITAL_ELEM_J2000 = [[355.433, 19140.2993039, 2.62e-06, -3e-09], [1.849726, -0.0081477, -2.255e-05, -2.9e-08], [49.558093, -0.295025, -0.00064048, -1.964e-06], [336.060234, 0.4439016, -0.00017313, 5.18e-07]]¶: This table contains the parameters to compute Mars’ orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 214

pymeeus.Mars.VSOP87_B = [[[3197134.986, 3.76832042432, 3340.6124266998], [298033.234, 4.10616996243, 6681.2248533996], [289104.742, 0.0, 0.0], [31365.538, 4.44651052853, 10021.8372800994], [3484.1, 4.78812547889, 13362.4497067992], [442.999, 5.65233015876, 3337.0893083508], [443.401, 5.02642620491, 3344.1355450488], [399.109, 5.130568147, 16703.062133499], [292.506, 3.79290644595, 2281.2304965106], [181.982, 6.13648011704, 6151.533888305], [163.159, 4.26399626634, 529.6909650946], [159.678, 2.23194610246, 1059.3819301892], [139.323, 2.41796344238, 8962.4553499102], [149.297, 2.16501209917, 5621.8429232104], [142.686, 1.1821501611, 3340.5951730476], [142.685, 3.2129218082, 3340.629680352], [82.544, 5.36667872319, 6684.7479717486], [73.64, 5.09187524843, 398.1490034082], [72.66, 5.53775710437, 6283.0758499914], [86.377, 5.74429648412, 3738.761430108], [83.276, 5.98866315739, 6677.7017350506], [60.116, 3.67960808826, 796.2980068164], [63.111, 0.73049113369, 5884.9268465832], [62.338, 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3340.6124266998], [4986.799, 3.14159265359, 0.0], [302.141, 5.55871276021, 6681.2248533996], [25.767, 1.89662673499, 13362.4497067992], [21.452, 0.91749968618, 10021.8372800994], [11.82, 2.242407387, 3337.0893083508], [7.985, 2.24892866611, 16703.062133499], [2.96, 5.89425825808, 3496.032826134], [2.445, 5.18770525274, 5621.8429232104], [1.428, 1.2523814058, 2281.2304965106], [1.779, 2.5875996852, 20043.6745601988], [1.501, 3.18533003542, 3532.0606928114], [1.259, 4.80695172904, 3185.1920272656], [1.029, 2.35029907056, 6677.7017350506], [1.109, 3.80982317372, 5884.9268465832], [0.928, 0.29719160927, 3344.1355450488], [0.86, 3.1169831893, 6151.533888305], [0.853, 1.33003321402, 529.6909650946], [0.883, 5.18681316017, 5486.777843175], [1.1, 1.82962075794, 2942.4634232916], [0.815, 3.40910567373, 2544.3144198834], [0.702, 5.49076132554, 8962.4553499102], [0.644, 0.31400306761, 5088.6288397668], [0.745, 4.30248377111, 3340.5951730476], [0.586, 4.48071459693, 3894.1818295422], [0.681, 5.40951261308, 2810.9214616052], [0.745, 0.05006463495, 3340.629680352], [0.542, 4.68895461946, 3339.6321056316], [0.497, 4.65665197749, 5092.1519581158], [0.46, 1.0667034361, 3097.88382272579], [0.423, 3.08034568777, 4292.3308329504], [0.381, 2.2493311519, 6283.0758499914], [0.429, 1.70668733378, 4690.4798363586], [0.339, 2.92195933142, 23384.2869868986], [0.31, 5.63324402105, 3723.508958923], [0.287, 1.98619629868, 398.1490034082], [0.298, 2.58990636818, 6684.7479717486], [0.323, 5.1523014439, 6525.8044539654], [0.247, 2.36923535528, 3340.545116397], [0.247, 1.96071951597, 3340.6797370026], [0.223, 4.75562842441, 2146.1654164752], [0.24, 3.53331653789, 3583.3410306738], [0.238, 4.69268478666, 6254.6266625236], [0.198, 6.21548965568, 2787.0430238574], [0.202, 4.25439775032, 3333.498879699], [0.158, 1.89216973387, 6674.1113063988], [0.185, 2.99350590687, 10025.3603984484], [0.177, 1.41002572971, 6438.4962494256], [0.138, 1.23172102079, 3127.3133312618], [0.153, 5.93528616595, 6836.6452528338], [0.106, 5.35414400998, 3738.761430108], [0.102, 3.67040445208, 1059.3819301892], [0.136, 1.06039656714, 2388.8940204492], [0.093, 5.04434801864, 155.4203994342], [0.093, 2.7081739491, 8955.3418029094], [0.108, 6.13917250674, 1748.016413067], [0.116, 5.9773948353, 2914.0142358238], [0.099, 1.50221319099, 1751.539531416], [0.092, 2.68470906437, 1990.745017041], [0.077, 4.01966292109, 1592.5960136328], [0.076, 4.49660101731, 4562.4609930212], [0.076, 5.67354102576, 6041.3275670856], [0.078, 0.79220883728, 2288.3440435114], [0.074, 5.84795427965, 3341.592747768], [0.08, 5.48556941416, 8432.7643848156], [0.065, 2.11705931744, 10018.3141617504], [0.066, 5.08073805943, 2957.7158944766], [0.059, 3.79695096075, 6923.9534573736], [0.061, 4.51116100631, 4933.2084403326], [0.059, 0.27765984344, 6127.6554505572], [0.052, 5.84885902302, 4535.0594369244], [0.051, 1.45495904398, 7477.522860216], [0.054, 1.50278239577, 6489.776587288], [0.056, 5.23212313891, 9866.4168806652], [0.057, 4.73711357046, 2274.1169495098], [0.051, 2.48654805044, 6467.9257579616], [0.047, 3.25899585469, 26724.8994135984], [0.048, 1.12255878532, 1349.8674096588], [0.048, 2.87122326236, 242.728603974], [0.045, 1.61358524585, 5729.506447149], [0.043, 2.2433473149, 7.1135470008], [0.044, 4.61065704096, 11773.3768115154], [0.042, 3.32038536161, 13365.9728251482], [0.047, 5.10736479739, 8969.568896911], [0.044, 4.64194663616, 4399.994356889], [0.039, 5.45779680648, 8031.0922630584], [0.044, 5.2140954242, 4136.9104335162], [0.045, 4.2620606379, 796.2980068164], [0.048, 3.04342098072, 5331.3574437408], [0.042, 2.12882700518, 213.299095438], [0.038, 2.49365585516, 951.7184062506], [0.037, 3.44816851943, 3325.3599555148], [0.038, 3.1398138598, 3149.1641605882], [0.039, 3.6498494618, 553.5694028424], [0.036, 0.54858124109, 7632.9432596502], [0.032, 4.29985057106, 3355.8648978848], [0.037, 6.26349600634, 6682.2051744678], [0.037, 3.83188032092, 3870.3033917944], [0.031, 4.94959395405, 6680.2445323314], [0.035, 6.14813219827, 3360.96774609859], [0.029, 0.54115054572, 6681.2921637024], [0.035, 1.49899662719, 3320.257107301], [0.033, 0.19183878029, 3347.7259737006], [0.036, 3.28055425527, 1589.0728952838], [0.031, 5.66015228291, 1066.49547719], [0.028, 4.1817845554, 7210.9158184942], [0.025, 1.21627749818, 9492.1463150048], [0.029, 2.02283118033, 7234.794256242], [0.024, 2.96777681837, 7064.1213856228], [0.024, 2.16823248221, 10419.9862835076], [0.022, 4.67496434823, 1194.4470102246], [0.022, 5.01797528499, 9595.2390892234], [0.028, 4.19843924702, 8429.2412664666], [0.023, 6.2119501005, 7740.6067835888], [0.025, 1.45100528915, 6872.6731195112], [0.024, 1.93342510858, 2699.7348193176], [0.028, 5.8700989815, 1052.2683831884], [0.025, 2.49707182259, 2118.7638603784], [0.02, 0.15682384975, 3205.5473466644], [0.019, 3.82087005916, 12303.06777661], [0.02, 6.26534330674, 7875.6718636242], [0.019, 0.99282533197, 522.5774180938], [0.018, 4.11668144665, 426.598190876], [0.02, 3.28531167827, 9225.539273283], [0.019, 5.72169822815, 11371.7046897582], [0.018, 4.76871968197, 382.8965322232], [0.017, 0.96519535826, 191.4482661116], [0.017, 0.86184324841, 10973.55568635], [0.018, 6.22706341047, 9381.9399937854], [0.015, 0.54135050421, 640.8776073822]], [[606.506, 1.98050633529, 3340.6124266998], [42.611, 0.0, 0.0], [13.652, 1.795882288, 6681.2248533996], [2.73, 3.45377082121, 10021.8372800994], [0.929, 3.75226159072, 3337.0893083508], [0.607, 0.10618486408, 13362.4497067992], [0.617, 1.14471772765, 3496.032826134], [0.479, 0.70504966293, 16703.062133499], [0.185, 3.28778562029, 3185.1920272656], [0.169, 0.29980532608, 5621.8429232104], [0.158, 1.09025317222, 20043.6745601988], [0.123, 2.55664973413, 3097.88382272579], [0.126, 4.74517022983, 3532.0606928114], [0.093, 6.03607685759, 3894.1818295422], [0.099, 4.92257049901, 2544.3144198834], [0.106, 3.41315845439, 2942.4634232916], [0.062, 6.257389499, 3339.6321056316], [0.058, 4.64867983361, 4292.3308329504], [0.05, 3.75358626972, 6677.7017350506], [0.05, 1.89408668049, 5088.6288397668], [0.058, 3.65295480755, 6525.8044539654], [0.045, 0.38295906298, 5486.777843175], [0.052, 3.25535694335, 4690.4798363586], [0.044, 1.89342588822, 3583.3410306738], [0.037, 0.04871920725, 2146.1654164752], [0.039, 0.91787211117, 3723.508958923], [0.051, 5.24063514394, 5884.9268465832], [0.036, 2.79633276263, 529.6909650946], [0.032, 1.44641701752, 23384.2869868986], [0.024, 5.41362501363, 3340.5951730476], [0.023, 4.70639896508, 2787.0430238574], [0.021, 1.12581894224, 3340.629680352], [0.021, 0.51744721589, 8962.4553499102], [0.018, 0.36433365806, 155.4203994342], [0.018, 5.53170427847, 3333.498879699], [0.019, 4.34401282378, 6151.533888305], [0.021, 2.9476550159, 398.1490034082], [0.02, 5.96181389299, 5092.1519581158], [0.017, 0.80891817293, 1194.4470102246], [0.011, 0.43817745316, 1059.3819301892], [0.014, 1.06654064248, 6836.6452528338]], [[11.334, 3.45724352586, 3340.6124266998], [13.369, 0.0, 0.0], [0.744, 0.50445805257, 6681.2248533996], [0.148, 1.05056602649, 10021.8372800994], [0.102, 2.66185835593, 3496.032826134], [0.053, 5.27888218929, 3337.0893083508], [0.022, 4.09971603267, 3097.88382272579], [0.023, 1.7242234213, 3185.1920272656], [0.013, 5.41704779112, 16703.062133499], [0.013, 2.06957065662, 13362.4497067992], [0.01, 0.959895956, 3894.1818295422]], [[0.457, 4.86794125358, 3340.6124266998], [0.053, 5.30547050586, 6681.2248533996], [0.012, 5.75114070583, 10021.8372800994], [0.013, 4.17736925293, 3496.032826134], [0.007, 0.0, 0.0]]]¶: This table contains Mars’ periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 424-425.

pymeeus.Mars.VSOP87_L = [[[620347711.583, 0.0, 0.0], [18656368.1, 5.05037100303, 3340.6124266998], [1108216.792, 5.40099836958, 6681.2248533996], [91798.394, 5.75478745111, 10021.8372800994], [27744.987, 5.97049512942, 3.523118349], [10610.23, 2.93958524973, 2281.2304965106], [12315.897, 0.84956081238, 2810.9214616052], [8926.772, 4.15697845939, 0.0172536522], [8715.688, 6.11005159792, 13362.4497067992], [6797.552, 0.36462243626, 398.1490034082], [7774.867, 3.33968655074, 5621.8429232104], [3575.079, 1.66186540141, 2544.3144198834], [4161.101, 0.2281497533, 2942.4634232916], [3075.25, 0.85696597082, 191.4482661116], [2628.122, 0.6480614357, 3337.0893083508], [2937.543, 6.07893711408, 0.0673103028], [2389.42, 5.03896401349, 796.2980068164], [2579.842, 0.02996706197, 3344.1355450488], [1528.14, 1.14979306228, 6151.533888305], [1798.808, 0.65634026844, 529.6909650946], [1264.356, 3.62275092231, 5092.1519581158], [1286.232, 3.06795924626, 2146.1654164752], [1546.408, 2.91579633392, 1751.539531416], [1024.907, 3.69334293555, 8962.4553499102], [891.567, 0.1829389909, 16703.062133499], [858.76, 2.40093704204, 2914.0142358238], [832.718, 2.46418591282, 3340.5951730476], [832.724, 4.49495753458, 3340.629680352], [712.899, 3.66336014788, 1059.3819301892], [748.724, 3.82248399468, 155.4203994342], [723.863, 0.67497565801, 3738.761430108], [635.557, 2.92182704275, 8432.7643848156], [655.163, 0.48864075176, 3127.3133312618], [550.472, 3.81001205408, 0.9803210682], [552.746, 4.47478863016, 1748.016413067], [425.972, 0.55365138172, 6283.0758499914], [415.132, 0.49662314774, 213.299095438], [472.164, 3.6254781941, 1194.4470102246], [306.552, 0.38052862973, 6684.7479717486], [312.141, 0.99853322843, 6677.7017350506], [293.199, 4.22131277914, 20.7753954924], [302.377, 4.48618150321, 3532.0606928114], [274.028, 0.54222141841, 3340.545116397], [281.073, 5.88163372945, 1349.8674096588], [231.185, 1.28240685294, 3870.3033917944], [283.6, 5.76885494123, 3149.1641605882], [236.114, 5.75504515576, 3333.498879699], [274.035, 0.13372501211, 3340.6797370026], [299.396, 2.78323705697, 6254.6266625236], [204.161, 2.82133266185, 1221.8485663214], [238.857, 5.37155471672, 4136.9104335162], [188.639, 1.49103016486, 9492.1463150048], [221.225, 3.50466672203, 382.8965322232], [179.196, 1.00561112574, 951.7184062506], [172.11, 0.43943041719, 5486.777843175], [193.126, 3.35715137745, 3.5904286518], [144.305, 1.41874193418, 135.0650800354], [160.011, 3.94854735192, 4562.4609930212], [174.068, 2.41360332576, 553.5694028424], [130.993, 4.04491720264, 12303.06777661], [138.245, 4.30145176915, 7.1135470008], [128.062, 1.80665643332, 5088.6288397668], [139.897, 3.32592516164, 2700.7151403858], [128.102, 2.20806651008, 1592.5960136328], [116.945, 3.12805282207, 7903.073419721], [110.375, 1.05195079687, 242.728603974], [113.486, 3.70070798123, 1589.0728952838], [100.09, 3.24343740861, 11773.3768115154], [95.592, 0.53954181149, 20043.6745601988], [98.947, 4.8455829474, 6681.2421070518], [104.541, 0.78535382076, 8827.3902698748], [84.187, 3.9897072073, 4399.994356889], [86.931, 2.20186740523, 11243.6858464208], [71.437, 2.80307550016, 3185.1920272656], [72.091, 5.84672102525, 5884.9268465832], [73.476, 2.18428012567, 8429.2412664666], [98.946, 2.81481140371, 6681.2075997474], [68.414, 2.73834914412, 2288.3440435114], [86.751, 1.02092221563, 7079.3738568078], [65.32, 2.68118597578, 28.4491874678], [83.749, 3.2025613099, 4690.4798363586], [75.034, 0.76643418252, 6467.9257579616], [68.984, 3.76399731788, 6041.3275670856], [66.706, 0.73630620766, 3723.508958923], [63.314, 4.5277147047, 426.598190876], [61.683, 6.16831509419, 2274.1169495098], [52.256, 0.89941531307, 9623.6882766912], [55.488, 4.6062546702, 4292.3308329504], [51.332, 4.14823636534, 3341.592747768], [56.629, 5.06250410206, 15.252471185], [63.376, 0.91296240798, 3553.9115221378], [45.829, 0.78784235062, 1990.745017041], [48.542, 3.95670418719, 4535.0594369244], [41.223, 6.02019329922, 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5.88548900893, 19513.9835951042], [0.03, 4.38908125588, 12295.9542296092], [0.029, 6.25276686131, 17924.9106998204], [0.025, 4.89000247975, 30065.5118402982], [0.026, 2.43119321236, 6518.7582172674], [0.028, 3.97032105354, 4407.1079038898], [0.025, 1.88631752737, 4379.6390374902], [0.028, 3.06196380356, 6247.5131155228], [0.029, 5.98057498931, 6680.2445323314], [0.032, 4.65556810177, 6701.5801727984], [0.026, 0.76018385758, 3863.1898447936], [0.023, 0.31107548102, 8827.3902698748], [0.022, 1.11553014451, 6684.8152820514], [0.02, 4.81193192299, 15508.6151232744], [0.019, 5.11656455993, 12012.5822971404], [0.026, 1.07254469525, 8425.6508378148], [0.021, 2.68436255141, 6261.7402095244], [0.022, 2.15631095909, 3335.0895023924], [0.017, 0.03684125273, 3877.4169387952], [0.017, 2.84467149903, 3344.4937620578], [0.02, 5.41519706836, 3205.5473466644]], [[1482.423, 0.44434694876, 3340.6124266998], [662.095, 0.88469178686, 6681.2248533996], [188.268, 1.28799982497, 10021.8372800994], [41.474, 1.64850786997, 13362.4497067992], [22.661, 2.05267665262, 155.4203994342], [25.994, 0.0, 0.0], [8.024, 1.99858757687, 16703.062133499], [10.454, 1.58006906385, 3.523118349], [4.9, 2.82452457966, 242.728603974], [3.782, 2.01914272515, 3344.1355450488], [3.176, 4.59144897927, 3185.1920272656], [3.134, 0.65044714325, 553.5694028424], [1.684, 5.53835848782, 951.7184062506], [1.511, 5.71795850828, 191.4482661116], [1.448, 0.45869142895, 796.2980068164], [1.442, 2.34368495577, 20043.6745601988], [1.302, 5.36284013048, 0.9803210682], [1.169, 4.14601161433, 1349.8674096588], [1.133, 2.38180830662, 6684.7479717486], [1.037, 1.76892750558, 382.8965322232], [0.894, 5.33688328934, 1194.4470102246], [0.807, 2.74798886181, 1748.016413067], [0.64, 6.10665147849, 3496.032826134], [0.558, 1.8521234236, 398.1490034082], [0.567, 5.85922384979, 7.1135470008], [0.647, 3.17645475605, 3583.3410306738], [0.452, 5.98109989317, 2787.0430238574], [0.508, 1.01139298015, 3149.1641605882], [0.519, 4.93376176788, 6525.8044539654], [0.405, 1.27295444059, 2281.2304965106], [0.399, 2.32888685659, 3738.761430108], [0.472, 0.84411483892, 4136.9104335162], [0.333, 5.42704539231, 1059.3819301892], [0.313, 3.70599897858, 3097.88382272579], [0.281, 5.74581724084, 3341.592747768], [0.287, 2.69304799864, 10025.3603984484], [0.237, 5.68519881994, 4535.0594369244], [0.215, 2.49494803822, 1990.745017041], [0.26, 2.67996877129, 23384.2869868986], [0.269, 1.10390153866, 2388.8940204492], [0.186, 4.86408411823, 2957.7158944766], [0.176, 3.86618588087, 1592.5960136328], [0.177, 0.85374318134, 3894.1818295422], [0.191, 4.58805692093, 3337.0893083508], [0.185, 5.8089931655, 4399.994356889], [0.143, 1.15592612974, 7477.522860216], [0.173, 2.70622920014, 7079.3738568078], [0.146, 1.37569151302, 6489.776587288], [0.121, 3.24333934982, 9492.1463150048], [0.116, 5.7163981953, 12303.06777661], [0.111, 2.69566947038, 6151.533888305], [0.103, 3.53587741373, 6923.9534573736], [0.097, 5.23712981002, 8962.4553499102], [0.089, 0.0595132422, 6127.6554505572], [0.093, 5.29818155587, 9866.4168806652], [0.084, 5.68544044325, 4292.3308329504], [0.091, 6.16223680965, 6682.2051744678], [0.081, 4.3272333062, 4933.2084403326], [0.077, 5.6318982417, 1589.0728952838], [0.102, 6.26937663026, 3347.7259737006], [0.076, 3.05724276091, 13365.9728251482], [0.063, 3.12770931753, 10419.9862835076], [0.062, 1.45647168723, 7234.794256242], [0.069, 4.82266605869, 6677.7017350506], [0.064, 1.73713448446, 3870.3033917944], [0.058, 4.85723088492, 5486.777843175], [0.058, 6.0984951679, 7875.6718636242], [0.063, 4.14716431158, 3340.5951730476], [0.065, 6.17932665318, 3340.629680352], [0.057, 1.48905992502, 5729.506447149], [0.06, 6.08710604997, 7740.6067835888], [0.056, 1.96253427165, 3553.9115221378], [0.059, 0.89874385953, 3339.6321056316], [0.053, 3.04776975379, 26724.8994135984], [0.061, 3.63390789623, 12832.7587417046], [0.059, 0.2298974604, 6681.2421070518], [0.053, 3.97996929188, 5092.1519581158], [0.059, 4.48255178273, 6681.2075997474], [0.045, 3.88055745121, 10264.5658840734], [0.058, 4.85371375265, 5621.8429232104], [0.049, 4.03623343126, 16173.3711684044], [0.061, 2.90942510134, 5331.3574437408], [0.043, 1.8099071834, 9830.3890139878], [0.044, 4.10828944542, 4690.4798363586], [0.044, 3.49332765553, 13760.5987102074], [0.048, 1.81207342615, 3723.508958923], [0.041, 1.59906754314, 10818.1352869158], [0.041, 2.05797417369, 6681.2921637024], [0.038, 3.38619280993, 16706.585251848], [0.041, 2.53336616026, 6681.1575430968], [0.041, 1.79026132252, 3340.545116397], [0.05, 4.7032478409, 8273.8208670324], [0.037, 3.37898394417, 30065.5118402982], [0.039, 2.65566008587, 6894.5239488376], [0.035, 2.58475025674, 9623.6882766912], [0.037, 6.07442797404, 4929.6853219836], [0.042, 3.04642561189, 12964.300703391], [0.037, 5.23784488646, 6298.3283211764], [0.043, 6.17835198533, 15643.6802033098], [0.034, 2.44806511268, 7210.9158184942], [0.038, 0.26295105909, 10022.8176011676], [0.028, 0.74281728305, 10021.8545337516], [0.028, 4.99536302577, 10021.8200264472], [0.034, 5.04118058624, 8969.568896911], [0.025, 2.59849002421, 8955.3418029094], [0.03, 1.14354290938, 2146.1654164752], [0.027, 0.26980783576, 18984.2926300096], [0.025, 0.39810308575, 6688.3384004004], [0.024, 5.64558695441, 13207.029307365], [0.027, 6.24398988438, 5088.6288397668], [0.024, 0.25455613132, 11081.2192102886], [0.023, 0.53673547304, 3333.498879699], [0.024, 1.80155161992, 6674.1113063988], [0.029, 4.3995350792, 10018.3141617504], [0.023, 2.91040060956, 10551.528245194], [0.022, 1.20895727897, 529.6909650946], [0.026, 0.08254324904, 7632.9432596502], [0.026, 1.81668969835, 6283.0758499914], [0.021, 0.67232850824, 6836.6452528338], [0.017, 5.19518401928, 2914.0142358238], [0.017, 5.16488977775, 8031.0922630584], [0.021, 3.62973082412, 5884.9268465832], [0.018, 5.06870872024, 2544.3144198834], [0.014, 2.50522181917, 7064.1213856228], [0.017, 0.4183559895, 9468.267877257], [0.011, 2.21363101654, 6438.4962494256], [0.013, 1.38424462832, 3340.6797370026], [0.012, 3.28248484262, 8671.9698704406], [0.014, 0.23027665815, 3767.2106175758]], [[113.969, 3.14159265359, 0.0], [28.725, 5.63662412043, 6681.2248533996], [24.447, 5.13868481454, 3340.6124266998], [11.187, 6.03161074431, 10021.8372800994], [3.19, 3.56267988299, 155.4203994342], [3.252, 0.13228350651, 13362.4497067992], [0.787, 0.49340783377, 16703.062133499], [0.776, 1.31734531594, 242.728603974], [0.494, 3.06356214498, 3185.1920272656], [0.374, 2.15785846355, 553.5694028424], [0.331, 6.23159792887, 3.523118349], [0.197, 0.44350153983, 3344.1355450488], [0.181, 0.81531283571, 20043.6745601988], [0.168, 3.73509781785, 3496.032826134], [0.086, 0.79259553758, 6684.7479717486], [0.115, 1.66898531261, 3583.3410306738], [0.092, 3.40530361815, 6525.8044539654], [0.064, 4.47443580658, 2787.0430238574], [0.045, 5.17216217058, 3097.88382272579], [0.041, 1.21875027733, 23384.2869868986], [0.036, 5.53975653407, 3149.1641605882], [0.039, 5.40966345885, 4136.9104335162], [0.028, 4.85378781404, 6127.6554505572], [0.027, 0.42811091036, 3738.761430108], [0.032, 1.22337093927, 10025.3603984484], [0.025, 1.56348878811, 26724.8994135984], [0.029, 2.40178626971, 3894.1818295422], [0.023, 4.16371253947, 3341.592747768], [0.022, 5.98429191667, 7477.522860216], [0.019, 2.03228980685, 6923.9534573736], [0.02, 3.79275573433, 9866.4168806652], [0.021, 4.27532111122, 6836.6452528338], [0.015, 1.56315497374, 13365.9728251482], [0.011, 1.33261955275, 2281.2304965106], [0.009, 6.15390464542, 6489.776587288], [0.007, 4.23894194106, 4535.0594369244]], [[0.71, 4.04089996521, 6681.2248533996], [0.868, 3.14159265359, 0.0], [0.51, 4.49214901625, 10021.8372800994], [0.357, 5.07435505061, 155.4203994342], [0.223, 3.51351884241, 3340.6124266998], [0.199, 4.85313666795, 13362.4497067992], [0.1, 6.09089356066, 242.728603974], [0.069, 5.19017483537, 16703.062133499], [0.064, 1.55783055571, 3185.1920272656], [0.035, 3.68246171643, 553.5694028424], [0.025, 5.22079788019, 3496.032826134], [0.024, 0.17861991485, 3583.3410306738], [0.019, 5.5918471446, 20043.6745601988], [0.014, 5.95565787085, 23384.2869868986], [0.012, 1.93859256739, 6525.8044539654]]]¶: This table contains Mars’ periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 421-424.

pymeeus.Mars.VSOP87_R = [[[153033488.276, 0.0, 0.0], [14184953.153, 3.47971283519, 3340.6124266998], [660776.357, 3.81783442097, 6681.2248533996], [46179.117, 4.15595316284, 10021.8372800994], [8109.738, 5.55958460165, 2810.9214616052], [7485.315, 1.77238998069, 5621.8429232104], [5523.193, 1.3643631888, 2281.2304965106], [3825.16, 4.49407182408, 13362.4497067992], [2306.539, 0.09081742493, 2544.3144198834], [1999.399, 5.36059605227, 3337.0893083508], [2484.385, 4.92545577893, 2942.4634232916], [1960.198, 4.74249386323, 3344.1355450488], [1167.115, 2.11261501155, 5092.1519581158], [1102.828, 5.0090826416, 398.1490034082], [899.077, 4.40790433994, 529.6909650946], [992.252, 5.83862401067, 6151.533888305], [807.348, 2.10216647104, 1059.3819301892], [797.91, 3.44839026172, 796.2980068164], [740.98, 1.49906336892, 2146.1654164752], [692.34, 2.13378814785, 8962.4553499102], [633.144, 0.89353285018, 3340.5951730476], [725.583, 1.24516913473, 8432.7643848156], [633.14, 2.92430448169, 3340.629680352], [574.352, 0.82896196337, 2914.0142358238], [526.187, 5.38292276228, 3738.761430108], [629.976, 1.28738135858, 1751.539531416], [472.776, 5.19850457873, 3127.3133312618], [348.095, 4.83219198908, 16703.062133499], [283.702, 2.90692294913, 3532.0606928114], [279.552, 5.25749247548, 6283.0758499914], [233.827, 5.10546492529, 5486.777843175], [219.428, 5.58340248784, 191.4482661116], [269.891, 3.76394728622, 5884.9268465832], [208.333, 5.25476080773, 3340.545116397], [275.224, 2.90818883832, 1748.016413067], [275.501, 1.21767967781, 6254.6266625236], [239.133, 2.03669896238, 1194.4470102246], [223.19, 4.19861593779, 3149.1641605882], [182.686, 5.08062683355, 6684.7479717486], [186.213, 5.69871555748, 6677.7017350506], [175.995, 5.95341786369, 3870.3033917944], [178.613, 4.18423025538, 3333.498879699], [208.336, 4.84626442122, 3340.6797370026], [228.128, 3.2552902062, 6872.6731195112], [144.286, 0.21296012258, 5088.6288397668], [163.534, 3.79889068111, 4136.9104335162], [133.12, 1.5391010671, 7903.073419721], [141.759, 2.47790321309, 4562.4609930212], [114.941, 4.31745088059, 1349.8674096588], [118.781, 2.12178071222, 1589.0728952838], [102.096, 6.18138550087, 9492.1463150048], [128.555, 5.49883294915, 8827.3902698748], [111.538, 0.55339169625, 11243.6858464208], [82.498, 1.6222704459, 11773.3768115154], [83.212, 0.61553380568, 8429.2412664666], [84.47, 0.6227459311, 1592.5960136328], [86.659, 1.74988330093, 2700.7151403858], [71.826, 2.47489899385, 12303.06777661], [85.312, 1.61621097912, 4690.4798363586], [63.641, 2.67334126661, 426.598190876], [68.599, 2.40197828418, 4399.994356889], [58.559, 4.72052787516, 213.299095438], [62.015, 1.10065866221, 1221.8485663214], [66.509, 2.21307705185, 6041.3275670856], [55.811, 1.23288325946, 3185.1920272656], [54.989, 5.72691385306, 951.7184062506], [52.418, 3.02366828926, 4292.3308329504], [55.686, 5.44686699242, 3723.508958923], [58.959, 3.26242666052, 6681.2421070518], [44.629, 2.0147364039, 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[1.532, 4.84374321619, 13362.4497067992], [0.375, 1.50962233608, 3185.1920272656], [0.339, 5.20684967613, 16703.062133499], [0.151, 5.16472931648, 3344.1355450488], [0.125, 2.19233532803, 3496.032826134], [0.148, 0.0, 0.0], [0.087, 0.10275067375, 3583.3410306738], [0.07, 5.55303619365, 20043.6745601988], [0.048, 2.91797786512, 2787.0430238574], [0.049, 5.5703851483, 6684.7479717486], [0.057, 1.86867280743, 6525.8044539654], [0.034, 3.63370917313, 3097.88382272579], [0.021, 2.30677637991, 1059.3819301892], [0.021, 4.03682122005, 4136.9104335162], [0.018, 4.1743875589, 3149.1641605882], [0.013, 1.97805475719, 155.4203994342], [0.017, 5.9485641469, 10025.3603984484], [0.012, 0.51097435253, 6923.9534573736], [0.012, 3.27945299261, 6127.6554505572], [0.013, 2.22938285522, 9866.4168806652], [0.016, 2.64206771227, 4399.994356889], [0.013, 2.75813969974, 6836.6452528338], [0.013, 4.51988349676, 6489.776587288], [0.014, 0.76591550754, 3894.1818295422], [0.011, 5.47381242145, 3738.761430108]], [[0.476, 2.47617204701, 6681.2248533996], [0.268, 2.91510547706, 10021.8372800994], [0.115, 1.76888962311, 3340.6124266998], [0.096, 3.31378377179, 13362.4497067992], [0.048, 6.27457828287, 3185.1920272656], [0.027, 3.69380877066, 16703.062133499], [0.013, 3.72005807206, 3496.032826134], [0.012, 4.88179002689, 3583.3410306738], [0.012, 3.14159265359, 0.0]]]¶: This table contains Mars’ periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 425-426.

Mercury¶

Class to model Mercury planet.

class pymeeus.Mercury.Mercury[source]¶

Class Mercury models that planet.

__weakref__¶: list of weak references to the object (if defined)

static apparent_heliocentric_position(epoch)[source]¶

This method computes the apparent heliocentric position of planet Mercury for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Mercury position, as an Epoch object
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

static eastern_elongation(epoch)[source]¶

This method computes the time of the eastern elongation closest to the given epoch, as well as the corresponding maximum elongation angle.

Parameters:	epoch (`Epoch`) – Epoch close to the desired eastern elongation
Returns:	A tuple with the time when the eastern elongation happens, as an Epoch, and the maximum elongation angle, as an Angle
Return type:	tuple
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1990, 8, 1.0)
>>> time, elongation = Mercury.eastern_elongation(epoch)
>>> y, m, d = time.get_date()
>>> print(y)
1990
>>> print(m)
8
>>> print(round(d, 4))
11.8514
>>> print(round(elongation, 4))
27.4201

static geocentric_position(epoch)[source]¶

This method computes the geocentric position of Mercury (right ascension and declination) for the given epoch, as well as the elongation angle.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension, the declination and the elongation angle as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 20.0)
>>> ra, dec, elon = Mercury.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
16h 33' 59.3''
>>> print(dec.dms_str(n_dec=1))
-20d 53' 31.6''
>>> print(elon.dms_str(n_dec=1))
18d 24' 29.8''

static geometric_heliocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of planet Mercury for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Mercury position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2018, 10, 27.0)
>>> l, b, r = Mercury.geometric_heliocentric_position(epoch)
>>> print(round(l.to_positive(), 4))
287.4887
>>> print(round(b, 4))
-6.0086
>>> print(round(r, 5))
0.45113

static inferior_conjunction(epoch)[source]¶

This method computes the time of the inferior conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired inferior conjunction
Returns:	The time when the inferior conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> conjunction = Mercury.inferior_conjunction(epoch)
>>> y, m, d = conjunction.get_date()
>>> print(y)
1993
>>> print(m)
11
>>> print(round(d, 4))
6.1449
>>> epoch = Epoch(1631, 10, 1.0)
>>> conjunction = Mercury.inferior_conjunction(epoch)
>>> y, m, d = conjunction.get_date()
>>> print(y)
1631
>>> print(m)
11
>>> print(round(d, 3))
7.306

static magnitude(sun_dist, earth_dist, phase_angle)[source]¶

This function computes the approximate magnitude of Mercury.

Parameters:	sun_dist (float) – Distance from Mercury to Sun, in Astronomical Units earth_dist (float) – Distance Mercury to Earth, in Astronomical Units phase_angle (float, `Angle`) – Mercury phase angle
Returns:	Mercury’s magnitude
Return type:	float
Raises:	TypeError if input values are of wrong type.

static orbital_elements_j2000(epoch)[source]¶

This method computes the orbital elements of Mercury for the standard equinox J2000.0 for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Mercury.orbital_elements_j2000(epoch)
>>> print(round(l, 6))
202.579453
>>> print(round(a, 8))
0.38709831
>>> print(round(e, 7))
0.2056451
>>> print(round(i, 6))
7.001089
>>> print(round(ome, 5))
48.24873
>>> print(round(arg, 6))
29.311401

static orbital_elements_mean_equinox(epoch)[source]¶

This method computes the orbital elements of Mercury for the mean equinox of the date for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Mercury.orbital_elements_mean_equinox(epoch)
>>> print(round(l, 6))
203.494701
>>> print(round(a, 8))
0.38709831
>>> print(round(e, 7))
0.2056451
>>> print(round(i, 6))
7.006171
>>> print(round(ome, 5))
49.10765
>>> print(round(arg, 6))
29.367732

static passage_nodes(epoch, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of Mercury, nearest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch closest to the node passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2019, 1, 1)
>>> time, r = Mercury.passage_nodes(epoch)
>>> year, month, day = time.get_date()
>>> print(year)
2018
>>> print(month)
11
>>> print(round(day, 1))
24.7
>>> print(round(r, 4))
0.3143

static perihelion_aphelion(epoch, perihelion=True)[source]¶

This method computes the time of Perihelion (or Aphelion) closer to a given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired Perihelion (or Aphelion) peihelion – If True, the epoch of the closest Perihelion is computed, if False, the epoch of the closest Aphelion is found.
Returns:	The epoch of the desired Perihelion (or Aphelion)
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2000, 1, 1.0)
>>> e = Mercury.perihelion_aphelion(epoch)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2000
>>> print(m)
2
>>> print(d)
15
>>> print(h)
18
>>> epoch = Epoch(2000, 3, 1.0)
>>> e = Mercury.perihelion_aphelion(epoch, perihelion=False)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2000
>>> print(m)
3
>>> print(d)
30
>>> print(h)
17

static station_longitude_1(epoch)[source]¶

This method computes the time of the 1st station in longitude (i.e. when the planet is stationary and begins to move westward - retrograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired inferior conjunction
Returns:	Time when the 1st statin in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> sta1 = Mercury.station_longitude_1(epoch)
>>> y, m, d = sta1.get_date()
>>> print(y)
1993
>>> print(m)
10
>>> print(round(d, 4))
25.9358

static station_longitude_2(epoch)[source]¶

This method computes the time of the 2nd station in longitude (i.e. when the planet is stationary and begins to move eastward - prograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired inferior conjunction
Returns:	Time when the 2nd station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> sta2 = Mercury.station_longitude_2(epoch)
>>> y, m, d = sta2.get_date()
>>> print(y)
1993
>>> print(m)
11
>>> print(round(d, 4))
15.0724

static superior_conjunction(epoch)[source]¶

This method computes the time of the superior conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired superior conjunction
Returns:	The time when the superior conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> conjunction = Mercury.superior_conjunction(epoch)
>>> y, m, d = conjunction.get_date()
>>> print(y)
1993
>>> print(m)
8
>>> print(round(d, 4))
29.3301

static western_elongation(epoch)[source]¶

This method computes the time of the western elongation closest to the given epoch, as well as the corresponding maximum elongation angle.

Parameters:	epoch (`Epoch`) – Epoch close to the desired western elongation
Returns:	A tuple with the time when the western elongation happens, as an Epoch, and the maximum elongation angle, as an Angle
Return type:	tuple
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 11, 1.0)
>>> time, elongation = Mercury.western_elongation(epoch)
>>> y, m, d = time.get_date()
>>> print(y)
1993
>>> print(m)
11
>>> print(round(d, 4))
22.6386
>>> print(round(elongation, 4))
19.7506

pymeeus.Mercury.ORBITAL_ELEM = [[252.250906, 149474.0722491, 0.0003035, 1.8e-08], [0.38709831, 0.0, 0.0, 0.0], [0.20563175, 2.0407e-05, -2.83e-08, -1.8e-10], [7.004986, 0.0018215, -1.81e-05, 5.6e-08], [48.330893, 1.1861883, 0.00017542, 2.15e-07], [77.456119, 1.5564776, 0.00029544, 9e-09]]¶: This table contains the parameters to compute Mercury’s orbital elements for the mean equinox of date. Based in Table 31.A, page 212

pymeeus.Mercury.ORBITAL_ELEM_J2000 = [[252.250906, 149472.6746358, -5.36e-06, 2e-09], [7.004986, -0.0059516, 8e-07, 4.3e-08], [48.330893, -0.1254227, -8.833e-05, -2e-07], [77.456119, 0.1588643, -1.342e-05, -7e-09]]¶: This table contains the parameters to compute Mercury’s orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 214

pymeeus.Mercury.VSOP87_B = [[[11737528.962, 1.98357498767, 26087.9031415742], [2388076.996, 5.03738959685, 52175.8062831484], [1222839.532, 3.14159265359, 0.0], [543251.81, 1.79644363963, 78263.70942472259], [129778.77, 4.83232503961, 104351.61256629678], [31866.927, 1.58088495667, 130439.51570787099], [7963.301, 4.60972126348, 156527.41884944518], [2014.189, 1.35324164694, 182615.3219910194], [513.953, 4.37835409309, 208703.2251325936], [207.674, 4.91772564073, 27197.2816936676], [208.584, 2.02020294153, 24978.5245894808], [132.013, 1.11908492283, 234791.12827416777], [100.454, 5.65684734206, 20426.571092422], [121.395, 1.81271752059, 53285.1848352418], [91.566, 2.28163128692, 25028.521211385], [99.214, 0.09391887097, 51116.4243529592], [94.574, 1.24184909234, 31749.2351907264], [78.785, 4.4072588, 57837.1383323006], [77.747, 0.52557061749, 1059.3819301892], [84.264, 5.08510388314, 51066.427731055], [49.948, 3.49752993688, 5661.3320491522], [46.454, 3.23739270829, 77204.32749453338], [44.767, 4.87849816734, 79373.087976816], [40.766, 2.46558332165, 46514.4742339962], [37.378, 4.45768797944, 4551.9534970588], [34.082, 4.14209210575, 260879.03141574195], [35.911, 1.09057317869, 1109.3785520934], [31.953, 1.18516389747, 83925.0414738748], [30.954, 3.5032802721, 21535.9496445154], [31.808, 2.41474588439, 47623.8527860896], [28.691, 1.84828614269, 77154.33087262919], [25.765, 2.77593370583, 27043.5028831828], [25.199, 3.5906226646, 27147.28507176339], [20.244, 3.06833797229, 51646.11531805379], [18.591, 5.5842727444, 73711.75592766379], [16.971, 0.02791276551, 103292.23063610759], [20.099, 4.06593040301, 25132.3033999656], [17.002, 6.13739392193, 41962.5207369374], [14.984, 1.64717994813, 105460.99111839019], [14.186, 0.33074185469, 10213.285546211], [15.577, 6.07693643204, 53131.406024757], [15.795, 3.79629547258, 529.6909650946], [14.011, 5.52786452723, 72602.37737557039], [12.309, 3.16626298867, 14765.2390432698], [11.261, 0.11326534696, 13521.7514415914], [12.448, 4.05109331029, 39609.6545831656], [13.044, 3.48016433624, 37410.5672398786], [11.042, 4.23192662377, 110012.94461544899], [11.152, 0.5565846182, 63498.47038145279], [10.717, 1.53686240986, 25661.3049506982], [10.213, 2.87881017166, 12566.1516999828], [9.924, 0.94824584866, 65697.55772473979], [11.047, 5.79741510309, 51749.20809227239], [10.46, 5.82962163777, 50586.73338786459], [9.774, 1.6627176265, 24498.8302462904], [9.412, 1.82364886045, 15874.6175953632], [12.866, 4.81650804018, 30639.856638633], [9.123, 4.8860439411, 103242.2340142034], [9.011, 1.04262074744, 426.598190876], [8.735, 3.04132552652, 68050.42387851159], [8.491, 1.05130498445, 1589.0728952838], [8.835, 0.88128343813, 286966.9345573162], [8.823, 5.81343662067, 11322.6640983044], [8.196, 0.84015129448, 51220.20654153979], [7.64, 3.43584035231, 36301.18868778519], [9.175, 6.16059719071, 77734.01845962799], [8.897, 0.39691733779, 53235.18821333759], [7.253, 2.37179970828, 99799.65906923798], [7.104, 5.73557153523, 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3.14407932401, 51962.5071877104], [0.006, 5.13879522396, 76044.9523205358], [0.007, 2.06447590782, 124778.18365871879], [0.006, 1.61443478349, 3442.5749449654], [0.005, 2.013597281, 27676.976036858], [0.007, 5.49596353318, 23754.70674870219], [0.006, 0.7144527205, 27147.28507176339], [0.005, 5.5815366372, 68050.42387851159], [0.006, 4.72387999717, 52290.24557183361], [0.005, 1.65565332809, 80482.46652890938], [0.005, 4.67655631038, 181505.94343892598], [0.005, 5.35730277715, 12566.1516999828], [0.005, 2.24953488234, 52061.36699446317], [0.004, 1.93085125724, 52156.1365222486], [0.006, 5.23350745087, 129380.13377768178], [0.005, 6.27908880897, 66941.04532641819], [0.006, 0.86124028508, 52195.47604404819], [0.004, 0.23100295125, 89586.37352302698], [0.005, 4.23944686799, 103396.01282468818], [0.004, 0.29398633221, 51109.31080595839], [0.005, 0.38451698643, 955.5997416086], [0.005, 5.7581296517, 104881.30353139139], [0.005, 3.78671774559, 52171.9249477904], [0.004, 5.32828531774, 25448.00585526019], [0.005, 3.20016226905, 426.598190876], [0.005, 3.86489045633, 9103.9069941176], [0.004, 0.62543334322, 11322.6640983044], [0.004, 6.07508617692, 2118.7638603784], [0.004, 2.43638238426, 74821.13447975718], [0.004, 5.29735530203, 52179.6876185064], [0.003, 3.19112063073, 13521.7514415914], [0.004, 5.09850398144, 77837.11123384659], [0.004, 1.81128771043, 102762.53967101299], [0.004, 2.92685125689, 639.897286314], [0.004, 4.37998582536, 52182.9198301492], [0.004, 0.905624683, 77623.81213840858], [0.004, 3.40653389629, 105410.99449648599], [0.004, 5.56139758889, 26301.2022370122], [0.003, 3.84817111044, 2218.7571041868], [0.004, 3.40685398189, 52389.1053785864], [0.003, 3.64497587259, 64607.84893354619], [0.003, 4.90907522926, 99799.65906923798], [0.003, 0.33551297325, 13655.8604911764], [0.003, 4.38738526963, 108903.56606335558], [0.003, 2.7199654281, 38654.05484155699], [0.003, 5.2163956913, 150866.08680029298], [0.003, 5.79669813464, 131548.89425996438], [0.003, 4.24481036893, 136100.84775702318], [0.003, 4.74285528385, 53764.8791784322], [0.003, 5.12230469501, 32370.9789915656], [0.003, 4.9197929909, 18849.2275499742], [0.003, 1.21305125502, 105307.21230790539], [0.002, 2.6174140723, 71492.99882347698], [0.002, 5.29927753161, 64741.95798313119], [0.002, 2.59250638038, 155468.036919256], [0.002, 5.60083003625, 26514.5013324502], [0.003, 4.14284031224, 85034.42002596818], [0.002, 3.04958336441, 93028.94846799239], [0.003, 6.16142033746, 78477.00852016058], [0.003, 4.49779099, 76144.94556434419], [0.002, 2.74114399659, 130969.20667296558], [0.002, 6.18606575651, 51123.53789995999], [0.003, 1.68659744145, 28421.0995344462], [0.002, 1.39891525246, 81706.28436968799], [0.002, 1.36487993355, 79852.78232000639], [0.002, 0.86961235954, 129483.91596626239], [0.002, 0.16105302705, 25021.4076643842], [0.002, 0.20969999844, 19804.8272915828], [0.002, 0.99320491085, 162188.75089859738], [0.002, 3.94039306142, 391318.54712361295], [0.002, 1.80783236802, 23969.1392811958], [0.002, 5.1209935622, 51322.60990139639], [0.002, 2.80140642811, 62389.09182935939], [0.002, 2.60965333457, 52602.4044740244], [0.002, 2.72257468038, 7994.5284420242], [0.002, 5.84222491614, 103821.92160120218], [0.002, 6.0451587872, 25035.6347583858], [0.001, 2.34997564292, 85502.38501632259], [0.001, 0.17409967102, 131498.89763806018], [0.001, 3.11496551907, 25934.1243310894], [0.002, 5.12532278095, 78267.59076008058], [0.001, 5.49175475674, 104331.94280539699], [0.001, 0.44808516059, 78050.41032928458], [0.002, 1.92656028088, 102132.85546210999], [0.002, 1.93198050022, 853.196381752], [0.001, 2.42972717822, 94138.32702008578], [0.002, 5.13550087739, 6770.7106012456], [0.001, 4.6544150087, 78259.82808936459], [0.001, 3.9968745549, 188276.6540401716], [0.001, 4.3906969233, 104371.28232719658], [0.001, 1.64083661387, 78283.37918562238], [0.001, 1.50558734911, 78378.1487134078], [0.002, 6.27597485867, 91785.46086631398], [0.001, 4.00786092689, 78244.0396638228], [0.001, 1.70741856509, 28206.6670019526], [0.001, 5.31900403501, 78149.27013603736], [0.001, 1.40205887414, 44937.1306915484], [0.001, 1.03752853909, 104347.73123093879], [0.001, 0.66520680194, 419.4846438752], [0.001, 2.46437105095, 104355.49390165479], [0.001, 0.81835364761, 35191.8101356918], [0.001, 0.3378067084, 88476.99497093359], [0.001, 6.0834623193, 70269.18098269838], [0.001, 1.86810476133, 125887.56221081219], [0.001, 2.1046897466, 25619.9381512198], [0.001, 3.99494842792, 71025.0338331226], [0.001, 0.57714874874, 71980.63357473118], [0.001, 5.66312942513, 97580.90196505119], [0.001, 3.06688416266, 129909.82474277639], [0.001, 5.66709359706, 433.7117378768], [0.001, 0.91634007063, 103925.01437542078], [0.001, 6.02359186416, 181555.94006083018], [0.001, 1.44980129997, 26727.8004278882], [0.001, 1.4544292039, 207593.8465805002], [0.001, 1.12941846578, 60055.89543648739], [0.001, 1.16964771504, 134991.4692049298]], [[235.423, 0.35387524604, 26087.9031415742], [160.537, 0.0, 0.0], [18.904, 4.36275460261, 52175.8062831484], [6.376, 2.50715381439, 78263.70942472259], [4.58, 6.14257817571, 104351.61256629678], [3.061, 3.12497552681, 130439.51570787099], [1.732, 6.26642412058, 156527.41884944518], [0.857, 3.07673166705, 182615.3219910194], [0.384, 6.14815319932, 208703.2251325936], [0.159, 2.9243737832, 234791.12827416777], [0.062, 5.97292432097, 260879.03141574195], [0.054, 3.31612529961, 27197.2816936676], [0.023, 2.74287679452, 286966.9345573162], [0.017, 0.77798463435, 24978.5245894808], [0.012, 6.24116133415, 53285.1848352418], [0.008, 5.79551081392, 313054.83769889036], [0.006, 5.81555175073, 31749.2351907264], [0.004, 4.67212195693, 51116.4243529592], [0.004, 3.36833718999, 77154.33087262919], [0.003, 0.26522113546, 103242.2340142034], [0.003, 0.79794705473, 27043.5028831828], [0.004, 0.82832725003, 21535.9496445154], [0.002, 2.57068697582, 339142.7408404646], [0.002, 1.8230939704, 1109.3785520934], [0.002, 0.74986942696, 47623.8527860896], [0.002, 3.61692083154, 50586.73338786459], [0.002, 4.70766241226, 51066.427731055], [0.001, 3.3368854909, 129330.13715577759], [0.001, 2.64641192612, 57837.1383323006], [0.001, 4.93247555987, 25558.2121764796], [0.002, 5.00344393497, 20426.571092422], [0.002, 4.80841415889, 25028.521211385], [0.002, 5.85039338716, 1059.3819301892], [0.001, 1.85116662745, 37410.5672398786], [0.002, 3.97317842825, 25132.3033999656], [0.001, 2.144942913, 79373.087976816], [0.001, 2.53223701416, 46514.4742339962], [0.001, 5.8492163243, 5661.3320491522], [0.001, 3.81958470276, 53131.406024757], [0.001, 1.05410669937, 1589.0728952838], [0.001, 1.39436798574, 51646.11531805379], [0.001, 0.36358097174, 22645.32819660879], [0.001, 2.7660997227, 28306.66024576099], [0.001, 3.67689055309, 4551.9534970588], [0.001, 5.39398540975, 32858.61374281979], [0.001, 3.7357912717, 24498.8302462904], [0.001, 0.60289607514, 26068.2333806744], [0.001, 5.78167324673, 26107.57290247399], [0.001, 3.81913361795, 26091.7844769322], [0.001, 5.64455351094, 365230.6439820388], [0.001, 4.87981157471, 10213.285546211], [0.001, 2.44816565966, 26084.0218062162], [0.0, 3.45284067711, 105460.99111839019]], [[4.276, 1.74579932115, 26087.9031415742], [1.023, 3.14159265359, 0.0], [0.425, 4.03419509143, 52175.8062831484], [0.257, 0.20643590425, 78263.70942472259], [0.116, 3.75237354024, 104351.61256629678], [0.073, 1.18210375402, 130439.51570787099], [0.051, 4.54581086194, 156527.41884944518], [0.031, 1.44226942756, 182615.3219910194], [0.016, 4.56372679459, 208703.2251325936], [0.008, 1.34684622635, 234791.12827416777], [0.003, 4.40269422669, 260879.03141574195], [0.001, 4.69096566687, 27197.2816936676], [0.001, 1.16082903756, 286966.9345573162], [0.0, 1.44918128781, 24978.5245894808], [0.0, 4.1804627869, 313054.83769889036]], [[0.106, 3.94555784256, 26087.9031415742], [0.075, 3.14159265359, 0.0], [0.022, 1.30514874546, 52175.8062831484], [0.007, 4.99717136857, 78263.70942472259], [0.004, 2.05662545278, 104351.61256629678], [0.002, 5.43162058632, 130439.51570787099], [0.001, 2.68052517331, 156527.41884944518], [0.001, 6.00841870284, 182615.3219910194], [0.0, 2.85941734701, 208703.2251325936], [0.0, 5.96962108614, 234791.12827416777]]]¶: This table contains Mercury’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 415-416.

pymeeus.Mercury.VSOP87_L = [[[440250710.144, 0.0, 0.0], [40989414.976, 1.48302034194, 26087.9031415742], [5046294.199, 4.4778548954, 52175.8062831484], [855346.843, 1.16520322351, 78263.70942472259], [165590.362, 4.11969163181, 104351.61256629678], [34561.897, 0.77930765817, 130439.51570787099], [7583.476, 3.7134840051, 156527.41884944518], [3559.74, 1.51202669419, 1109.3785520934], [1726.012, 0.35832239908, 182615.3219910194], [1803.463, 4.1033317841, 5661.3320491522], [1364.682, 4.59918318745, 27197.2816936676], [1589.923, 2.99510417815, 25028.521211385], [1017.332, 0.8803143904, 31749.2351907264], [714.182, 1.54144865265, 24978.5245894808], [643.759, 5.30266110787, 21535.9496445154], [404.2, 3.28228847025, 208703.2251325936], [352.441, 5.24156297101, 20426.571092422], [343.313, 5.76531885335, 955.5997416086], [339.214, 5.86327765, 25558.2121764796], [451.137, 6.04989275289, 51116.4243529592], [325.335, 1.3367433478, 53285.1848352418], [259.587, 0.98732428184, 4551.9534970588], [345.212, 2.79211901539, 15874.6175953632], [272.947, 2.49451163975, 529.6909650946], [234.83, 0.266721189, 11322.6640983044], [238.793, 0.11343953378, 1059.3819301892], [264.336, 3.91705094013, 57837.1383323006], [216.645, 0.65987207348, 13521.7514415914], [183.359, 2.62878670784, 27043.5028831828], [175.965, 4.53636829858, 51066.427731055], [181.629, 2.43413502466, 25661.3049506982], [208.995, 2.09178234008, 47623.8527860896], [172.643, 2.45200164173, 24498.8302462904], [142.316, 3.36003948842, 37410.5672398786], [137.942, 0.29098447849, 10213.285546211], [118.233, 2.78149786369, 77204.32749453338], [96.86, 6.2039820274, 234791.12827416777], [125.219, 3.72079804425, 39609.6545831656], [86.819, 2.64219349385, 51646.11531805379], [86.723, 1.9595304265, 46514.4742339962], [88.329, 5.41338795963, 26617.5941066688], [106.422, 4.20572116254, 19804.8272915828], [89.987, 5.85243631094, 41962.5207369374], [84.971, 4.33100364958, 79373.087976816], [69.247, 4.19446437496, 19.66976089979], [63.463, 3.14700877722, 7238.6755916], [68.493, 0.63424819267, 83925.0414738748], [69.729, 3.57201709671, 25132.3033999656], [59.481, 2.74692752, 16983.9961474566], [64.83, 0.0476292581, 33326.5787331742], [55.376, 4.05312663019, 30639.856638633], [54.442, 3.14331542453, 27147.28507176339], [47.56, 5.49722099211, 3.881335358], [49.567, 3.98985863874, 6770.7106012456], [56.531, 5.11920557675, 73711.75592766379], [41.764, 5.64185159566, 53131.406024757], [51.458, 5.47786463494, 50586.73338786459], [44.744, 1.22366857463, 77154.33087262919], [41.882, 5.19309298528, 6283.0758499914], [38.045, 2.43117327523, 12566.1516999828], [35.627, 0.81390126585, 32858.61374281979], [48.007, 5.49260554912, 51749.20809227239], [35.392, 3.36964859355, 36301.18868778519], [33.951, 2.78618091049, 14765.2390432698], [30.56, 5.84045074182, 43071.8992890308], [35.964, 1.4238083863, 2218.7571041868], [34.044, 0.47470299167, 65697.55772473979], [30.8, 5.77017310191, 103292.23063610759], [28.496, 0.65048992658, 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26084.0218062162], [0.001, 1.75551273491, 26095.016688575], [0.001, 2.60061927559, 12566.1516999828], [0.001, 2.69537843932, 52290.24557183361], [0.001, 4.87760109673, 25874.6040461362], [0.001, 2.82170820105, 26091.7844769322], [0.001, 0.20736391933, 52061.36699446317], [0.001, 3.82330817663, 68050.42387851159], [0.001, 4.42501613287, 22645.32819660879], [0.001, 6.01128227925, 52168.69273614759], [0.001, 0.28595674172, 11322.6640983044], [0.001, 5.86280830053, 26202.34243025941], [0.001, 3.42109354799, 25973.46385288896], [0.001, 0.300500653, 41962.5207369374], [0.001, 2.73414310032, 25035.6347583858], [0.001, 1.72346523407, 51962.5071877104], [0.001, 4.77297529316, 78477.00852016058], [0.001, 5.84491832575, 50057.04242277], [0.001, 3.2207229087, 25021.4076643842], [0.001, 4.74388045372, 26514.5013324502], [0.001, 4.31760630574, 69159.80243060499], [0.001, 0.65529515413, 426.598190876], [0.0, 1.96307435492, 1059.3819301892], [0.001, 5.58694327783, 79852.78232000639], [0.0, 3.84680432548, 26080.78959457339], [0.001, 6.07111696987, 102762.53967101299], [0.0, 0.59142287644, 73711.75592766379], [0.001, 5.84438975852, 105307.21230790539], [0.001, 2.82906439989, 24498.8302462904]], [[114.078, 3.14159265359, 0.0], [3.247, 2.02848007619, 26087.9031415742], [1.914, 1.41731803758, 78263.70942472259], [1.727, 4.50137643801, 52175.8062831484], [1.237, 4.49970181057, 104351.61256629678], [0.645, 1.26591776986, 130439.51570787099], [0.298, 4.30600984981, 156527.41884944518], [0.128, 1.05702505039, 182615.3219910194], [0.051, 4.08566191934, 208703.2251325936], [0.02, 0.82130394681, 234791.12827416777], [0.008, 3.82199615635, 260879.03141574195], [0.003, 0.54824060574, 286966.9345573162], [0.002, 3.52134138964, 313054.83769889036], [0.001, 1.6743511854, 24978.5245894808], [0.002, 4.60718203612, 51066.427731055], [0.001, 1.11396201203, 129330.13715577759], [0.001, 4.35040700322, 103242.2340142034], [0.001, 4.17434865844, 79373.087976816], [0.001, 1.20388630609, 77154.33087262919], [0.001, 4.8467708474, 27197.2816936676], [0.001, 0.29913093643, 339142.7408404646], [0.0, 4.11587125815, 155418.04029735178], [0.0, 0.73293870976, 46514.4742339962], [0.0, 3.68822427283, 21535.9496445154], [0.0, 3.83816920045, 20426.571092422], [0.0, 4.73086078962, 1059.3819301892], [0.0, 0.40939402954, 24498.8302462904], [0.0, 3.30678220146, 365230.6439820388]], [[0.877, 3.14159265359, 0.0], [0.059, 3.37513289692, 52175.8062831484], [0.042, 0.02433099382, 78263.70942472259], [0.043, 4.65071406046, 26087.9031415742], [0.031, 2.98112204944, 104351.61256629678], [0.019, 6.00992467582, 130439.51570787099], [0.01, 2.7569979192, 156527.41884944518], [0.005, 5.79082359724, 182615.3219910194], [0.002, 2.5143544823, 208703.2251325936], [0.001, 5.49411470932, 234791.12827416777], [0.001, 2.23688504668, 260879.03141574195], [0.001, 5.18718589133, 286966.9345573162], [0.0, 2.11315496491, 313054.83769889036]]]¶: This table contains Mercury’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 414-415.

pymeeus.Mercury.VSOP87_R = [[[39528271.652, 0.0, 0.0], [7834131.817, 6.19233722599, 26087.9031415742], [795525.557, 2.95989690096, 52175.8062831484], [121281.763, 6.01064153805, 78263.70942472259], [21921.969, 2.77820093975, 104351.61256629678], [4354.065, 5.82894543257, 130439.51570787099], [918.228, 2.59650562598, 156527.41884944518], [260.033, 3.02817753482, 27197.2816936676], [289.955, 1.42441936951, 25028.521211385], [201.855, 5.6472504035, 182615.3219910194], [201.499, 5.59227724202, 31749.2351907264], [141.98, 6.25264202645, 24978.5245894808], [100.144, 3.73435608689, 21535.9496445154], [77.561, 3.66972526976, 20426.571092422], [63.277, 4.29905918105, 25558.2121764796], [62.951, 4.76588899933, 1059.3819301892], [66.754, 2.52520309182, 5661.3320491522], [75.5, 4.47428642962, 51116.4243529592], [48.266, 6.06824478778, 53285.1848352418], [45.748, 2.41480951648, 208703.2251325936], [35.224, 1.05917802674, 27043.5028831828], [40.815, 2.35882016415, 57837.1383323006], [44.234, 1.21957314874, 15874.6175953632], [33.873, 0.86381554651, 25661.3049506982], [37.203, 0.5173382147, 47623.8527860896], [30.092, 1.79500530627, 37410.5672398786], [28.417, 3.02063625668, 51066.427731055], [30.903, 0.88366335532, 24498.8302462904], [26.105, 2.15021963174, 39609.6545831656], [18.699, 4.96496008403, 11322.6640983044], [21.27, 5.36857139841, 13521.7514415914], [19.422, 4.98378647655, 10213.285546211], [16.941, 3.88765393402, 26617.5941066688], [15.109, 0.44510589948, 46514.4742339962], [17.087, 1.24077764194, 77204.32749453338], [13.94, 1.62573946865, 27147.28507176339], [13.382, 1.07657890477, 51646.11531805379], [15.012, 4.28173463507, 41962.5207369374], [13.977, 4.77056848793, 33326.5787331742], [12.794, 6.06437138766, 1109.3785520934], [13.938, 1.99984876578, 25132.3033999656], [16.297, 2.63293587817, 19804.8272915828], [11.933, 2.36500939134, 4551.9534970588], [10.612, 5.46555460932, 234791.12827416777], [12.754, 2.07613721222, 529.6909650946], [12.069, 2.84997619452, 79373.087976816], [9.069, 1.21263611811, 14765.2390432698], [9.491, 0.83697007534, 12566.1516999828], [9.379, 5.41195286503, 83925.0414738748], [7.499, 2.44636811119, 30639.856638633], [7.463, 5.53233943198, 32858.61374281979], [7.215, 1.17101960691, 16983.9961474566], [8.492, 3.56622930635, 73711.75592766379], [7.109, 5.32625264624, 426.598190876], [6.863, 1.82313992707, 36301.18868778519], [6.56, 4.27818149213, 43071.8992890308], [8.853, 3.87737694074, 50586.73338786459], [6.885, 5.3110852663, 1589.0728952838], [5.943, 4.06893157254, 53131.406024757], [7.653, 3.91505031889, 51749.20809227239], [6.46, 6.07127449283, 77154.33087262919], [5.415, 5.20028107807, 65697.55772473979], [5.186, 3.56743214904, 6283.0758499914], [4.09, 3.6759365871, 22645.32819660879], [4.075, 4.29142111073, 103292.23063610759], [4.428, 5.69109527379, 45892.73043315699], [3.426, 0.64911736234, 52705.49724824299], [3.354, 3.49345105494, 72602.37737557039], [4.12, 1.65386255382, 25448.00585526019], [3.261, 1.03195890028, 68050.42387851159], [3.278, 3.13863529552, 25874.6040461362], [3.054, 4.12578324522, 77734.01845962799], [3.614, 2.95861599353, 28306.66024576099], [2.82, 4.62628436074, 18849.2275499742], [2.746, 4.67880529205, 53235.18821333759], [2.713, 3.5687491398, 26068.2333806744], [2.507, 2.23312085627, 260879.03141574195], [3.044, 5.90613310181, 105460.99111839019], [2.755, 5.67587087632, 26107.57290247399], [2.267, 2.17941340037, 110012.94461544899], [2.274, 0.50173671332, 32370.9789915656], [2.741, 4.75427920262, 63498.47038145279], [2.792, 5.05148206806, 51220.20654153979], [2.557, 2.95114696617, 26080.78959457339], [2.135, 0.84621449019, 76674.63652943878], [1.971, 0.4740749431, 48733.23133818299], [1.965, 1.37420902653, 7238.6755916], [2.526, 0.34535290139, 23969.1392811958], [2.022, 0.33828477567, 99799.65906923798], [1.872, 3.93046425037, 38654.05484155699], [1.873, 0.69327393635, 26091.7844769322], [1.945, 2.37383894182, 6770.7106012456], [1.863, 2.26591720786, 26084.0218062162], [1.75, 5.50651903992, 56727.7597802072], [1.971, 3.77455887616, 19317.1925403286], [1.672, 2.15275897581, 26514.5013324502], [1.703, 4.28965990785, 40853.142184844], [1.657, 4.98021773372, 9103.9069941176], [2.084, 2.19427123968, 26301.2022370122], [1.818, 4.20870869718, 25938.3399444396], [1.78, 0.91702846577, 79219.3091663312], [1.525, 2.15720744047, 27676.976036858], [1.533, 1.54503054415, 955.5997416086], [1.614, 1.27565104562, 25035.6347583858], [1.743, 0.68269002122, 77837.11123384659], [1.389, 4.85102048256, 62389.09182935939], [1.436, 0.05742216761, 23869.1460373874], [1.563, 5.84795233948, 26095.016688575], [1.64, 4.66281337712, 25021.4076643842], [1.552, 2.83910580545, 103242.2340142034], [1.249, 1.97225274067, 91785.46086631398], [1.16, 1.73113341823, 38519.945791972], [1.005, 1.0589831486, 129380.13377768178], [1.041, 2.78036322434, 213.299095438], [1.105, 4.55512109515, 26011.6370702986], [0.979, 2.43870419667, 71980.63357473118], [0.853, 0.56334001923, 20760.4270331914], [0.874, 4.53679970936, 639.897286314], [0.783, 3.69782403005, 78793.40038981718], [0.772, 4.08714349549, 94138.32702008578], [0.797, 0.26237215917, 98690.28051714458], [0.965, 4.70784179954, 51535.90899683439], [0.952, 2.53604676495, 29530.4780865396], [0.773, 2.67893222158, 131548.89425996438], [0.734, 0.8924366949, 103821.92160120218], [0.668, 3.73113394924, 46848.3301747656], [0.828, 2.88184821542, 19406.6782881746], [0.622, 1.05579535631, 59414.4818747484], [0.676, 4.5035759936, 25934.1243310894], [0.602, 5.28399057704, 286966.9345573162], [0.619, 1.44855037685, 79323.09135491178], [0.691, 6.19733262608, 51962.5071877104], [0.563, 5.22994296186, 136100.84775702318], [0.782, 3.25257254691, 50057.04242277], [0.551, 0.33644229836, 52156.1365222486], [0.705, 4.7370537192, 26241.681952059], [0.577, 5.85654734429, 2118.7638603784], [0.517, 3.93195379429, 102762.53967101299], [0.635, 4.87720838965, 12432.0426503978], [0.635, 1.81903685898, 77308.10968311399], [0.543, 1.40877130839, 44937.1306915484], [0.556, 5.44324390449, 85034.42002596818], [0.496, 3.39004132624, 125887.56221081219], [0.546, 4.78005317483, 22747.2907148744], [0.52, 6.00172746972, 52168.69273614759], [0.463, 3.67549187956, 25668.418497699], [0.452, 3.53009000682, 58458.88213313979], [0.474, 5.9433051204, 54394.56338733519], [0.559, 2.44355554028, 52195.47604404819], [0.418, 3.73369494476, 103925.01437542078], [0.397, 0.6901460802, 64741.95798313119], [0.511, 5.88243734524, 45494.58142974879], [0.412, 2.26754404446, 82815.66292178139], [0.425, 5.24511927496, 52389.1053785864], [0.482, 0.98165400059, 25654.19140369739], [0.38, 3.74413132157, 52179.6876185064], [0.359, 5.50961077408, 26555.8681319286], [0.375, 1.97920050124, 149.5631971346], [0.385, 5.8899194879, 129330.13715577759], [0.348, 4.91064019959, 1052.2683831884], [0.417, 4.32916053867, 51123.53789995999], [0.425, 1.43228996047, 51109.31080595839], [0.464, 3.98651628073, 105307.21230790539], [0.329, 5.20636278221, 52602.4044740244], [0.354, 1.04383146533, 66941.04532641819], [0.324, 1.62975761811, 88476.99497093359], [0.378, 5.31676765847, 52171.9249477904], [0.44, 1.46381183715, 89586.37352302698], [0.355, 1.52732265492, 1066.49547719], [0.302, 5.21113786435, 53764.8791784322], [0.301, 5.02488829134, 117873.36400788819], [0.351, 2.0043334173, 24395.7374720718], [0.289, 6.19161799918, 26094.53170047421], [0.286, 3.77170690304, 7.1135470008], [0.284, 3.10942616808, 49957.0491789616], [0.385, 2.47184729453, 25234.70675982219], [0.382, 0.97683830518, 52026.2430860138], [0.271, 5.74710334372, 25455.119402261], [0.375, 0.55187466535, 45405.0956819028], [0.267, 2.86410554301, 80482.46652890938], [0.304, 6.00190488272, 2218.7571041868], [0.259, 1.82921035765, 24505.94379329119], [0.251, 4.10979574073, 155468.036919256], [0.289, 3.05373259745, 26081.27458267419], [0.242, 5.05501089774, 42153.969003049], [0.244, 2.1562166805, 3442.5749449654], [0.315, 2.61613386386, 52182.9198301492], [0.256, 3.71117332461, 25619.9381512198], [0.232, 2.21576531181, 16066.0658614748], [0.233, 5.49033130906, 98068.53671630539], [0.236, 4.08612620637, 26013.1215430069], [0.225, 0.4444933881, 76144.94556434419], [0.242, 3.29851820234, 6681.2248533996], [0.227, 1.63236736805, 76.2660712756], [0.268, 1.09037236588, 18093.37469954999], [0.255, 4.20635818545, 44181.27784112419], [0.205, 1.00234895959, 26190.99591579279], [0.228, 1.46077047931, 77623.81213840858], [0.197, 1.36693360783, 26727.8004278882], [0.259, 1.28296039354, 13655.8604911764], [0.197, 5.7330758343, 157636.79740153858], [0.196, 0.06606260924, 40565.2543247742], [0.187, 0.86187283408, 120226.23016165999], [0.262, 4.14850032709, 24491.71669928959], [0.244, 4.14399890076, 95247.70557217918], [0.232, 5.45165695316, 33967.99229491319], [0.206, 5.59196913034, 55618.3812281138], [0.203, 1.64252462466, 58946.51688439399], [0.187, 0.46291089182, 104881.30353139139], [0.192, 1.827611583, 25551.09862947879], [0.2, 1.81376901527, 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25551.09862947879], [0.001, 4.51335885923, 12546.481939083], [0.001, 3.0701219029, 104138.31347085879], [0.001, 4.25105785914, 65831.6667743248], [0.001, 0.26591217377, 89586.37352302698], [0.001, 3.25491459524, 46848.3301747656], [0.001, 5.00501348827, 51639.00177105299], [0.0, 5.50334834007, 124156.43985787958], [0.0, 4.83101429854, 76.2660712756], [0.0, 4.34030366776, 80482.46652890938], [0.001, 3.55161443787, 116917.76426627958], [0.0, 4.96302997387, 26521.614879451], [0.001, 0.12176395772, 6681.2248533996], [0.001, 0.06932929209, 48847.6706268682], [0.001, 0.08695556653, 26507.38778544939], [0.0, 1.50601747714, 632.7837393132], [0.0, 3.68453733674, 25035.6347583858], [0.0, 2.10077795936, 86143.79857806159], [0.0, 2.61352703953, 53399.624123927], [0.0, 0.84481589887, 25881.717593137], [0.0, 3.95854005744, 95247.70557217918], [0.001, 4.47600641528, 97112.93697469679], [0.001, 0.40812953108, 77844.22478084739], [0.0, 4.89912517892, 25984.8103673556], [0.0, 5.16843610094, 52698.38370124219], [0.0, 3.99449121788, 131395.11544947958], [0.0, 5.65446547429, 80596.9058175946]], [[32.676, 1.67971635359, 26087.9031415742], [24.166, 4.63403168997, 52175.8062831484], [12.133, 1.38983781545, 78263.70942472259], [5.14, 4.4391538693, 104351.61256629678], [1.981, 1.20733880274, 130439.51570787099], [1.46, 3.14159265359, 0.0], [0.719, 4.25913631362, 156527.41884944518], [0.25, 1.02794425848, 182615.3219910194], [0.084, 4.08008618813, 208703.2251325936], [0.028, 0.85051616237, 234791.12827416777], [0.023, 1.60029451014, 24978.5245894808], [0.011, 4.58074493182, 27197.2816936676], [0.012, 4.6587866039, 51066.427731055], [0.009, 3.90622332667, 260879.03141574195], [0.005, 1.43247327249, 77154.33087262919], [0.004, 1.37106678794, 53285.1848352418], [0.003, 0.67217979275, 286966.9345573162], [0.002, 5.31135121818, 20426.571092422], [0.002, 1.54812921017, 1109.3785520934], [0.002, 4.48222146022, 103242.2340142034], [0.002, 0.20057520755, 1059.3819301892], [0.002, 4.21865197056, 25132.3033999656], [0.001, 2.11997334018, 46514.4742339962], [0.001, 3.24267833975, 27147.28507176339], [0.001, 1.33238537892, 1589.0728952838], [0.001, 5.14405498793, 72602.37737557039], [0.001, 6.14595652541, 26617.5941066688], [0.001, 3.90494988936, 4551.9534970588], [0.001, 2.85134614467, 14765.2390432698], [0.001, 3.74339959279, 313054.83769889036], [0.001, 2.82234410581, 52705.49724824299], [0.001, 1.70132625273, 23869.1460373874], [0.001, 1.4410244446, 27043.5028831828], [0.001, 1.2535842334, 129330.13715577759], [0.001, 0.32054002273, 9103.9069941176], [0.001, 0.90273607509, 51220.20654153979], [0.001, 4.28935605425, 27676.976036858], [0.001, 4.41206223998, 79373.087976816], [0.001, 3.23518929162, 26301.2022370122], [0.001, 2.25667255668, 21535.9496445154], [0.001, 4.70414203873, 50586.73338786459], [0.001, 1.88608259832, 98690.28051714458], [0.001, 5.90420893056, 40853.142184844], [0.001, 4.46892853356, 53131.406024757], [0.001, 3.83684899837, 77308.10968311399], [0.001, 1.31974603498, 54394.56338733519], [0.001, 0.74728939443, 32858.61374281979], [0.001, 6.27673970277, 53235.18821333759], [0.001, 4.1481457861, 29530.4780865396], [0.001, 0.87302135775, 83925.0414738748], [0.001, 5.20339566427, 26068.2333806744], [0.001, 2.88631079597, 79323.09135491178], [0.001, 5.39853672849, 19317.1925403286], [0.001, 5.73721596586, 78793.40038981718], [0.0, 1.85241611022, 25973.46385288896], [0.0, 4.31134434847, 26202.34243025941], [0.0, 4.7430520578, 49957.0491789616], [0.001, 3.87802524387, 82815.66292178139], [0.0, 4.28937691652, 155418.04029735178], [0.001, 3.35625397389, 37410.5672398786], [0.0, 2.00313301846, 22645.32819660879], [0.0, 2.49236829957, 25028.521211385], [0.0, 0.01947787727, 25558.2121764796], [0.001, 4.13825019881, 3442.5749449654], [0.0, 2.81832388305, 25874.6040461362], [0.0, 2.53416253042, 12566.1516999828], [0.001, 4.46169102731, 28306.66024576099], [0.001, 2.16986592809, 45405.0956819028], [0.0, 2.27621739263, 26091.7844769322], [0.0, 1.95034764539, 52156.1365222486], [0.0, 4.03054078678, 30639.856638633], [0.0, 5.76722418743, 43071.8992890308], [0.0, 4.12739704581, 26107.57290247399], [0.0, 1.37713138665, 31749.2351907264], [0.001, 3.54716896556, 51646.11531805379], [0.0, 3.3916159585, 36301.18868778519], [0.0, 4.34410863436, 57837.1383323006], [0.0, 5.6208296069, 639.897286314], [0.0, 3.16268062826, 529.6909650946], [0.0, 0.77186843357, 26084.0218062162], [0.0, 3.78612072314, 110012.94461544899], [0.0, 5.50453358586, 51116.4243529592], [0.0, 0.0470913142, 52389.1053785864], [0.0, 3.93991894858, 6770.7106012456], [0.0, 1.41961180167, 76674.63652943878], [0.0, 2.61258742182, 66941.04532641819], [0.0, 0.70544164813, 426.598190876], [0.0, 0.91883551986, 55618.3812281138], [0.0, 5.30612883135, 52179.6876185064], [0.0, 2.00604152518, 2118.7638603784], [0.0, 1.08640210817, 56727.7597802072], [0.0, 3.92743267498, 58946.51688439399], [0.0, 5.52202827702, 38654.05484155699], [0.0, 1.57363600114, 24498.8302462904], [0.0, 3.49522546958, 5661.3320491522], [0.0, 0.63375195581, 51535.90899683439], [0.0, 1.34305878069, 105460.99111839019], [0.0, 0.44820055565, 339142.7408404646], [0.0, 1.39951362362, 2218.7571041868], [0.0, 0.63311095079, 18849.2275499742], [0.0, 0.82120127326, 52195.47604404819], [0.0, 2.78211363987, 47623.8527860896], [0.0, 0.94752126944, 53764.8791784322], [0.0, 3.75988526858, 52171.9249477904], [0.0, 0.4435143591, 136100.84775702318], [0.0, 1.44824925227, 76044.9523205358], [0.0, 4.83985082201, 78244.0396638228], [0.0, 0.04522486479, 62389.09182935939], [0.0, 6.28144869714, 213.299095438], [0.0, 5.17875557687, 955.5997416086], [0.0, 4.62674052433, 52168.69273614759], [0.0, 2.19907823066, 77204.32749453338], [0.0, 3.40216597806, 26514.5013324502], [0.0, 2.89630853457, 13655.8604911764], [0.0, 3.92639394174, 50057.04242277], [0.0, 0.27991454535, 11322.6640983044], [0.0, 5.63968341805, 103292.23063610759], [0.0, 1.86829495425, 48733.23133818299], [0.0, 5.77443179008, 105410.99449648599]], [[0.394, 0.3673540384, 26087.9031415742], [0.387, 3.18568771507, 52175.8062831484], [0.27, 6.16983616444, 78263.70942472259], [0.149, 2.91591904641, 104351.61256629678], [0.071, 5.95867889641, 130439.51570787099], [0.031, 2.72386700044, 156527.41884944518], [0.017, 0.0, 0.0], [0.012, 5.77775334056, 182615.3219910194], [0.005, 2.54506442647, 208703.2251325936], [0.002, 5.59978443851, 234791.12827416777], [0.001, 2.33785160871, 260879.03141574195], [0.0, 5.37943916893, 286966.9345573162], [0.0, 3.09643524101, 51066.427731055], [0.0, 0.0934285875, 24978.5245894808], [0.0, 6.02837115675, 53285.1848352418], [0.0, 6.13600004668, 77154.33087262919], [0.0, 2.93310416266, 27197.2816936676], [0.0, 2.89897922039, 103242.2340142034]], [[0.006, 3.98900269603, 26087.9031415742], [0.006, 1.55248278782, 52175.8062831484], [0.005, 4.65461721116, 78263.70942472259], [0.004, 1.40238366492, 104351.61256629678], [0.002, 4.44231488663, 130439.51570787099], [0.001, 3.14159265359, 0.0], [0.001, 1.21278177664, 156527.41884944518], [0.001, 4.2614902634, 182615.3219910194], [0.0, 0.99831133595, 208703.2251325936], [0.0, 4.0026706421, 234791.12827416777]]]¶: This table contains Mercury’s periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in page 416.

Minor¶

Class to model minor celestial objetcs.

class pymeeus.Minor.Minor(q, e, i, omega, w, t)[source]¶

Class Minor models minor celestial bodies.

__init__(q, e, i, omega, w, t)[source]¶

Minor constructor.

The Minor object is initialized with this constructor, setting the orbital values and computing some internal parameters. This constructor is build upon the ‘set()’ method.

Parameters:	q (float) – Perihelion distance, in Astronomical Units e (float) – Eccentricity of the orbit i (`Angle`) – Inclination of the orbit, as an Angle object omega (`Angle`) – Longitude of the ascending node, as an Angle object w (`Angle`) – Argument of the perihelion, as an Angle object t (`Epoch`) – Epoch of passage by perihelion, as an Epoch object
Raises:	TypeError if input value is of wrong type.

__weakref__¶: list of weak references to the object (if defined)

geocentric_position(epoch)[source]¶

This method computes the geocentric position of a minor celestial body (right ascension and declination) for the given epoch, and referred to the standard equinox J2000.0. Additionally, it also computes the elongation angle to the Sun.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension, the declination and the elongation angle to the Sun, as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> a = 2.2091404
>>> e = 0.8502196
>>> q = a * (1.0 - e)
>>> i = Angle(11.94524)
>>> omega = Angle(334.75006)
>>> w = Angle(186.23352)
>>> t = Epoch(1990, 10, 28.54502)
>>> minor = Minor(q, e, i, omega, w, t)
>>> epoch = Epoch(1990, 10, 6.0)
>>> ra, dec, p = minor.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
10h 34' 13.7''
>>> print(dec.dms_str(n_dec=0))
19d 9' 32.0''
>>> print(round(p, 2))
40.51
>>> t = Epoch(1998, 4, 14.4358)
>>> q = 1.487469
>>> e = 1.0
>>> i = Angle(0.0)
>>> omega = Angle(0.0)
>>> w = Angle(0.0)
>>> minor = Minor(q, e, i, omega, w, t)
>>> epoch = Epoch(1998, 8, 5.0)
>>> ra, dec, p = minor.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
5h 45' 34.5''
>>> print(dec.dms_str(n_dec=0))
23d 23' 53.0''
>>> print(round(p, 2))
45.73

heliocentric_ecliptical_position(epoch)[source]¶

This method computes the heliocentric position of a minor celestial body, providing the result in ecliptical coordinates.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing longitude and latitude, as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> a = 2.2091404
>>> e = 0.8502196
>>> q = a * (1.0 - e)
>>> i = Angle(11.94524)
>>> omega = Angle(334.75006)
>>> w = Angle(186.23352)
>>> t = Epoch(1990, 10, 28.54502)
>>> epoch = Epoch(1990, 10, 6.0)
>>> minor = Minor(q, e, i, omega, w, t)
>>> lon, lat = minor.heliocentric_ecliptical_position(epoch)
>>> print(lon.dms_str(n_dec=1))
66d 51' 57.8''
>>> print(lat.dms_str(n_dec=1))
11d 56' 14.4''

set(q, e, i, omega, w, t)[source]¶

Method used to set the orbital values and set some internal parameters.

Parameters:	q (float) – Perihelion distance, in Astronomical Units e (float) – Eccentricity of the orbit i (`Angle`) – Inclination of the orbit, as an Angle object omega (`Angle`) – Longitude of the ascending node, as an Angle object w (`Angle`) – Argument of the perihelion, as an Angle object t (`Epoch`) – Epoch of passage by perihelion, as an Epoch object
Raises:	TypeError if input value is of wrong type.

Moon¶

Module holding functions to handle coordinates.

class pymeeus.Moon.Moon[source]¶

Class Moon models Earth’s satellite.

__weakref__¶: list of weak references to the object (if defined)

static apparent_ecliptical_pos(epoch)[source]¶

This method computes the apparent geocentric ecliptical position (longitude, latitude) of the Moon for a given instant, referred to the mean equinox of the date, as well as the Moon-Earth distance in kilometers and the equatorial horizontal parallax.

Parameters:	epoch (`Epoch`) – Instant to compute the Moon’s position, as an py:class:Epoch object.
Returns:	Tuple containing: Apparent geocentric longitude of the center of the Moon, as an py:class:Epoch object. Apparent geocentric latitude of the center of the Moon, as an py:class:Epoch object. Distance in kilometers between the centers of Earth and Moon, in kilometers (float) Equatorial horizontal parallax of the Moon, as an py:class:Epoch object.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 4, 12.0)
>>> Lambda, Beta, Delta, ppi = Moon.apparent_ecliptical_pos(epoch)
>>> print(round(Lambda, 5))
133.16726
>>> print(round(Beta, 6))
-3.229126
>>> print(round(Delta, 1))
368409.7
>>> print(round(ppi, 5))
0.99199

static apparent_equatorial_pos(epoch)[source]¶

This method computes the apparent equatorial position (right ascension, declination) of the Moon for a given instant, referred to the mean equinox of the date, as well as the Moon-Earth distance in kilometers and the equatorial horizontal parallax.

Parameters:	epoch (`Epoch`) – Instant to compute the Moon’s position, as an py:class:Epoch object.
Returns:	Tuple containing: Apparent right ascension of the center of the Moon, as an py:class:Epoch object. Apparent declination of the center of the Moon, as an py:class:Epoch object. Distance in kilometers between the centers of Earth and Moon, in kilometers (float) Equatorial horizontal parallax of the Moon, as an py:class:Epoch object.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 4, 12.0)
>>> ra, dec, Delta, ppi = Moon.apparent_equatorial_pos(epoch)
>>> print(round(ra, 6))
134.688469
>>> print(round(dec, 6))
13.768367
>>> print(round(Delta, 1))
368409.7
>>> print(round(ppi, 5))
0.99199

static geocentric_ecliptical_pos(epoch)[source]¶

This method computes the geocentric ecliptical position (longitude, latitude) of the Moon for a given instant, referred to the mean equinox of the date, as well as the Moon-Earth distance in kilometers and the equatorial horizontal parallax.

Parameters:	epoch (`Epoch`) – Instant to compute the Moon’s position, as an py:class:Epoch object.
Returns:	Tuple containing: Geocentric longitude of the center of the Moon, as an py:class:Epoch object. Geocentric latitude of the center of the Moon, as an py:class:Epoch object. Distance in kilometers between the centers of Earth and Moon, in kilometers (float) Equatorial horizontal parallax of the Moon, as an py:class:Epoch object.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 4, 12.0)
>>> Lambda, Beta, Delta, ppi = Moon.geocentric_ecliptical_pos(epoch)
>>> print(round(Lambda, 6))
133.162655
>>> print(round(Beta, 6))
-3.229126
>>> print(round(Delta, 1))
368409.7
>>> print(round(ppi, 5))
0.99199

static illuminated_fraction_disk(epoch)[source]¶

This method computes the approximate illuminated fraction ‘k’ of the disk of the Moon. The method used has a relatively low accuracy, but it is enough to the 2nd decimal place.

Parameters:	epoch (`Epoch`) – Instant to compute the Moon’s illuminated fraction of the disk, as a py:class:Epoch object.
Returns:	The approximate illuminated fraction of the Moon’s disk.
Return type:	float
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 4, 12.0)
>>> k = Moon.illuminated_fraction_disk(epoch)
>>> print(round(k, 2))
0.68

static longitude_mean_ascending_node(epoch)[source]¶

This method computes the longitude of the mean ascending node of the Moon in degrees, for a given instant, measured from the mean equinox of the date.

Parameters:	epoch (`Epoch`) – Instant to compute the Moon’s mean ascending node, as an py:class:Epoch object.
Returns:	The longitude of the mean ascending node.
Return type:	py:class:Angle
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1913, 5, 27.0)
>>> Omega = Moon.longitude_mean_ascending_node(epoch)
>>> print(round(Omega, 1))
0.0
>>> epoch = Epoch(2043, 9, 10.0)
>>> Omega = Moon.longitude_mean_ascending_node(epoch)
>>> print(round(Omega, 1))
0.0
>>> epoch = Epoch(1959, 12, 7.0)
>>> Omega = Moon.longitude_mean_ascending_node(epoch)
>>> print(round(Omega, 1))
180.0
>>> epoch = Epoch(2108, 11, 3.0)
>>> Omega = Moon.longitude_mean_ascending_node(epoch)
>>> print(round(Omega, 1))
180.0

static longitude_mean_perigee(epoch)[source]¶

This method computes the longitude of the mean perigee of the lunar orbitn in degrees, for a given instant, measured from the mean equinox of the date.

Parameters:	epoch (`Epoch`) – Instant to compute the Moon’s mean perigee, as an py:class:Epoch object.
Returns:	The longitude of the mean perigee.
Return type:	py:class:Angle
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(2021, 3, 5.0)
>>> Pi = Moon.longitude_mean_perigee(epoch)
>>> print(round(Pi, 5))
224.89194

static longitude_true_ascending_node(epoch)[source]¶

This method computes the longitude of the true ascending node of the Moon in degrees, for a given instant, measured from the mean equinox of the date.

Parameters:	epoch (`Epoch`) – Instant to compute the Moon’s true ascending node, as an py:class:Epoch object.
Returns:	The longitude of the true ascending node.
Return type:	py:class:Angle
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1913, 5, 27.0)
>>> Omega = Moon.longitude_true_ascending_node(epoch)
>>> print(round(Omega, 4))
0.8763

static moon_librations(epoch)[source]¶

This method computes the librations in longitude and latitude of the moon. There are several librations: The optical librations, that are the apparent oscillations in the hemisphere that the Moon turns towards the Earth, due to variations in the geometric position of the Earth relative to the lunar surface during the course of the orbital motion of the Moon. These variations allow to observe about 59% of the surface of the Moon from the Earth.

There is also the physical libration of the Moon, i.e., the libration due to the actual rotational motion of the Moon about its mean rotation. The physical libration is much smaller than the optical libration, and can never be larger than 0.04 degree in both longitude and latitude.

Finally, there is the total libration, which is the sum of the two librations mentioned above

Parameters:	epoch – Epoch we want to compute the Moon’s librations.
Returns:	A tuple containing the optical libration in longitude and in latitude, the physical libration also in longitude and latitude, and the total librations which is the sum of the previouse ones, all of them as `Angle` objects.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 4, 12.0)
>>> lopt, bopt, lphys, bphys, ltot, btot = Moon.moon_librations(epoch)
>>> print(round(lopt, 3))
-1.206
>>> print(round(bopt, 3))
4.194
>>> print(round(lphys, 3))
-0.025
>>> print(round(bphys, 3))
0.006
>>> print(round(ltot, 2))
-1.23
>>> print(round(btot, 3))
4.2

static moon_maximum_declination(epoch, target='northern')[source]¶

This method computes the approximate times when the Moon reaches its maximum declination (either ‘northern’ or ‘southern’), as well as the values of these extreme declinations. The resulting times will be expressed in the uniform time scale of Dynamical Time (TT).

Parameters:	epoch – Approximate epoch we want to compute the Moon’s maximum declination. target (str) – Either ‘northern’ or ‘southern’, depending on the maximum declination being looked for. It is ‘northern’ by default.
Returns:	A tuple containing the instant of time when the maximum declination happens, as a `Epoch` object, and the angle value of such declination, as a `Angle` object.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1988, 12, 15.0)
>>> epo, dec = Moon.moon_maximum_declination(epoch)
>>> y, m, d, h, mi, s = epo.get_full_date()
>>> print("{}/{}/{} {}:0{}".format(y, m, d, h, mi))
1988/12/22 20:01
>>> print("{}".format(dec.dms_str(n_dec=0)))
28d 9' 22.0''
>>> epoch = Epoch(2049, 4, 15.0)
>>> epo, dec = Moon.moon_maximum_declination(epoch, target='southern')
>>> y, m, d, h, mi, s = epo.get_full_date()
>>> print("{}/{}/{} {}:{}".format(y, m, d, h, mi))
2049/4/21 14:0
>>> print("{}".format(dec.dms_str(n_dec=0)))
-22d 8' 18.0''
>>> epoch = Epoch(-4, 3, 15.0)
>>> epo, dec = Moon.moon_maximum_declination(epoch, target='northern')
>>> y, m, d, h, mi, s = epo.get_full_date()
>>> print("{}/{}/{} {}h".format(y, m, d, h))
-4/3/16 15h
>>> print("{}".format(dec.dms_str(n_dec=0)))
28d 58' 26.0''

static moon_passage_nodes(epoch, target='ascending')[source]¶

This method computes the approximate times when the center of the Moon passes through the ascending or descending node of its orbit. The resulting times will be expressed in the uniform time scale of Dynamical Time (TT).

Parameters:	epoch – Approximate epoch we want to compute the Moon’s passage through the ascending or descending node. target (str) – Either ‘ascending’ or ‘descending’. It is ‘ascending’ by default.
Returns:	The instant of time when the Moon passes thhrough the ascending or descending node.
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.
Raises:	ValueError if ‘target’ value is invalid.

>>> epoch = Epoch(1987, 5, 15.0)
>>> passage = Moon.moon_passage_nodes(epoch, target="ascending")
>>> y, m, d, h, mi, s = passage.get_full_date()
>>> mi += s/60.0
>>> print("{}/{}/{} {}:{}".format(y, m, d, h, round(mi)))
1987/5/23 6:26

static moon_perigee_apogee(epoch, target='perigee')[source]¶

This method computes the approximate times when the distance between the Earth and the Moon is a minimum (perigee) or a maximum (apogee). The resulting times will be expressed in the uniform time scale of Dynamical Time (TT).

Parameters:	epoch – Approximate epoch we want to compute the Moon’s perigee or apogee for. target (str) – Either ‘perigee’ or ‘apogee’. It’s ‘perigee’ by default.
Returns:	A tuple containing the instant of time when the perigee or apogee happens, as a `Epoch` object, and the Moon’s corresponding equatorial horizontal parallax, as a `Angle` object.
Return type:	tuple
Raises:	TypeError if input values are of wrong type.
Raises:	ValueError if ‘target’ value is invalid.

>>> epoch = Epoch(1988, 10, 1.0)
>>> apogee, parallax = Moon.moon_perigee_apogee(epoch, target="apogee")
>>> y, m, d, h, mi, s = apogee.get_full_date()
>>> print("{}/{}/{} {}:{}".format(y, m, d, h, mi))
1988/10/7 20:30
>>> print("{}".format(parallax.dms_str(n_dec=3)))
54' 0.679''

static moon_phase(epoch, target='new')[source]¶

This method computes the time of the phase of the moon closest to the provided epoch. The resulting time is expressed in the uniform time scale of Dynamical Time (TT).

Parameters:	epoch – Approximate epoch we want to compute the Moon phase for. target (str) – Corresponding phase. It can be “new” (New Moon), “first” (First Quarter), “full” (Full Moon) and “last” (Last Quarter). It is ‘new’ by default.
Returns:	The instant of time when the provided phase happens.
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.
Raises:	ValueError if ‘target’ value is invalid.

>>> epoch = Epoch(1977, 2, 15.0)
>>> new_moon = Moon.moon_phase(epoch, target="new")
>>> y, m, d, h, mi, s = new_moon.get_full_date()
>>> print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 0)))
1977/2/18 3:37:42.0
>>> epoch = Epoch(2044, 1, 1.0)
>>> new_moon = Moon.moon_phase(epoch, target="last")
>>> y, m, d, h, mi, s = new_moon.get_full_date()
>>> print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s)))
2044/1/21 23:48:17

static moon_position_angle_axis(epoch)[source]¶

This method computes the position angle of the Moon’s axis of rotation. The effect of the physical libration is taken into account.

Parameters:	epoch – Epoch we want to compute the position angle of the Moon’s axis of rotation.
Returns:	The position angle of the Moon’s axis of rotation, as a `Angle` object.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 4, 12.0)
>>> p = Moon.moon_position_angle_axis(epoch)
>>> print(round(p, 2))
15.08

static position_bright_limb(epoch)[source]¶

This method computes the position angle of the Moon’s bright limb, i.e., the position angle of the midpoint of the illuminated limb, reckoned eastward from the North Point of the disk (not from the axis of rotation of the lunar globe).

Parameters:	epoch (`Epoch`) – Instant to compute the position angle of the Moon’s bright limb, as a py:class:Epoch object.
Returns:	The position angle of the Moon’s bright limb.
Return type:	`Angle`
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 4, 12.0)
>>> xi = Moon.position_bright_limb(epoch)
>>> print(round(xi, 1))
285.0

pymeeus.Moon.PERIODIC_TERMS_B_TABLE = [[0, 0, 0, 1, 5128122.0], [0, 0, 1, 1, 280602.0], [0, 0, 1, -1, 277693.0], [2, 0, 0, -1, 173237.0], [2, 0, -1, 1, 55413.0], [2, 0, -1, -1, 46271.0], [2, 0, 0, 1, 32573.0], [0, 0, 2, 1, 17198.0], [2, 0, 1, -1, 9266.0], [0, 0, 2, -1, 8822.0], [2, -1, 0, -1, 8216.0], [2, 0, -2, -1, 4324.0], [2, 0, 1, 1, 4200.0], [2, 1, 0, -1, -3359.0], [2, -1, -1, 1, 2463.0], [2, -1, 0, 1, 2211.0], [2, -1, -1, -1, 2065.0], [0, 1, -1, -1, -1870.0], [4, 0, -1, -1, 1828.0], [0, 1, 0, 1, -1794.0], [0, 0, 0, 3, -1749.0], [0, 1, -1, 1, -1565.0], [1, 0, 0, 1, -1491.0], [0, 1, 1, 1, -1475.0], [0, 1, 1, -1, -1410.0], [0, 1, 0, -1, -1344.0], [1, 0, 0, -1, -1335.0], [0, 0, 3, 1, 1107.0], [4, 0, 0, -1, 1021.0], [4, 0, -1, 1, 833.0], [0, 0, 1, -3, 777.0], [4, 0, -2, 1, 671.0], [2, 0, 0, -3, 607.0], [2, 0, 2, -1, 596.0], [2, -1, 1, -1, 491.0], [2, 0, -2, 1, -451.0], [0, 0, 3, -1, 439.0], [2, 0, 2, 1, 422.0], [2, 0, -3, -1, 421.0], [2, 1, -1, 1, -366.0], [2, 1, 0, 1, -351.0], [4, 0, 0, 1, 331.0], [2, -1, 1, 1, 315.0], [2, -2, 0, -1, 302.0], [0, 0, 1, 3, -283.0], [2, 1, 1, -1, -229.0], [1, 1, 0, -1, 223.0], [1, 1, 0, 1, 223.0], [0, 1, -2, -1, -220.0], [2, 1, -1, -1, -220.0], [1, 0, 1, 1, -185.0], [2, -1, -2, -1, 181.0], [0, 1, 2, 1, -177.0], [4, 0, -2, -1, 176.0], [4, -1, -1, -1, 166.0], [1, 0, 1, -1, -164.0], [4, 0, 1, -1, 132.0], [1, 0, -1, -1, -119.0], [4, -1, 0, -1, 115.0], [2, -2, 0, 1, 107.0]]¶: This table contains the periodic terms for the latitude of the Moon Sigmab. Units are 0.000001 degree. In Meeus’ book this is Table 47.B and can be found in page 341.

pymeeus.Moon.PERIODIC_TERMS_LR_TABLE = [[0, 0, 1, 0, 6288774.0, -20905355.0], [2, 0, -1, 0, 1274027.0, -3699111.0], [2, 0, 0, 0, 658314.0, -2955968.0], [0, 0, 2, 0, 213618.0, -569925.0], [0, 1, 0, 0, -185116.0, 48888.0], [0, 0, 0, 2, -114332.0, -3149.0], [2, 0, -2, 0, 58793.0, 246158.0], [2, -1, -1, 0, 57066.0, -152138.0], [2, 0, 1, 0, 53322.0, -170733.0], [2, -1, 0, 0, 45758.0, -204586.0], [0, 1, -1, 0, -40923.0, -129620.0], [1, 0, 0, 0, -34720.0, 108743.0], [0, 1, 1, 0, -30383.0, 104755.0], [2, 0, 0, -2, 15327.0, 10321.0], [0, 0, 1, 2, -12528.0, 0.0], [0, 0, 1, -2, 10980.0, 79661.0], [4, 0, -1, 0, 10675.0, -34782.0], [0, 0, 3, 0, 10034.0, -23210.0], [4, 0, -2, 0, 8548.0, -21636.0], [2, 1, -1, 0, -7888.0, 24208.0], [2, 1, 0, 0, -6766.0, 30824.0], [1, 0, -1, 0, -5163.0, -8379.0], [1, 1, 0, 0, 4987.0, -16675.0], [2, -1, 1, 0, 4036.0, -12831.0], [2, 0, 2, 0, 3994.0, -10445.0], [4, 0, 0, 0, 3861.0, -11650.0], [2, 0, -3, 0, 3665.0, 14403.0], [0, 1, -2, 0, -2689.0, -7003.0], [2, 0, -1, 2, -2602.0, 0.0], [2, -1, -2, 0, 2390.0, 10056.0], [1, 0, 1, 0, -2348.0, 6322.0], [2, -2, 0, 0, 2236.0, -9884.0], [0, 1, 2, 0, -2120.0, 5751.0], [0, 2, 0, 0, -2069.0, 0.0], [2, -2, -1, 0, 2048.0, -4950.0], [2, 0, 1, -2, -1773.0, 4130.0], [2, 0, 0, 2, -1595.0, 0.0], [4, -1, -1, 0, 1215.0, -3958.0], [0, 0, 2, 2, -1110.0, 0.0], [3, 0, -1, 0, -892.0, 3258.0], [2, 1, 1, 0, -810.0, 2616.0], [4, -1, -2, 0, 759.0, -1897.0], [0, 2, -1, 0, -713.0, -2117.0], [2, 2, -1, 0, -700.0, 2354.0], [2, 1, -2, 0, 691.0, 0.0], [2, -1, 0, -2, 596.0, 0.0], [4, 0, 1, 0, 549.0, -1423.0], [0, 0, 4, 0, 537.0, -1117.0], [4, -1, 0, 0, 520.0, -1571.0], [1, 0, -2, 0, -487.0, -1739.0], [2, 1, 0, -2, -399.0, 0.0], [0, 0, 2, -2, -381.0, -4421.0], [1, 1, 1, 0, 351.0, 0.0], [3, 0, -2, 0, -340.0, 0.0], [4, 0, -3, 0, 330.0, 0.0], [2, -1, 2, 0, 327.0, 0.0], [0, 2, 1, 0, -323.0, 1165.0], [1, 1, -1, 0, 299.0, 0.0], [2, 0, 3, 0, 294.0, 0.0], [2, 0, -1, -2, 0.0, 8752.0]]¶: This table contains the periodic terms for the longitude (Sigmal) and distance (Sigmar) of the Moon. Units are 0.000001 degree for Sigmal, and 0.001 kilometer for Sigmar. In Meeus’ book this is Table 47.A and can be found in pages 339-340.

Neptune¶

Class to model Neptune planet.

class pymeeus.Neptune.Neptune[source]¶

Class Neptune models that planet.

__weakref__¶: list of weak references to the object (if defined)

static apparent_heliocentric_position(epoch)[source]¶

This method computes the apparent heliocentric position of planet Neptune for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Neptune position, as an Epoch object
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

static conjunction(epoch)[source]¶

This method computes the time of the conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired conjunction
Returns:	The time when the conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> conj = Neptune.conjunction(epoch)
>>> y, m, d = conj.get_date()
>>> print(y)
1994
>>> print(m)
1
>>> print(round(d, 4))
11.3057

static geocentric_position(epoch)[source]¶

This method computes the geocentric position of Neptune (right ascension and declination) for the given epoch, as well as the elongation angle.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension, the declination and the elongation angle as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 20.0)
>>> ra, dec, elon = Neptune.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
19h 17' 14.5''
>>> print(dec.dms_str(n_dec=1))
-21d 34' 15.1''
>>> print(elon.dms_str(n_dec=1))
19d 44' 59.6''

static geometric_heliocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of planet Neptune for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Neptune position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2018, 10, 27.0)
>>> l, b, r = Neptune.geometric_heliocentric_position(epoch)
>>> print(round(l.to_positive(), 4))
345.3776
>>> print(round(b, 4))
-0.9735
>>> print(round(r, 5))
29.93966

static magnitude(sun_dist, earth_dist)[source]¶

This function computes the approximate magnitude of Neptune.

Parameters:	sun_dist (float) – Distance from Neptune to Sun, in Astronomical Units earth_dist (float) – Distance Neptune to Earth, in Astronomical Units
Returns:	Neptune’s magnitude
Return type:	float
Raises:	TypeError if input values are of wrong type.

static opposition(epoch)[source]¶

This method computes the time of the opposition closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	The time when the opposition happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1846, 8, 1)
>>> oppo = Neptune.opposition(epoch)
>>> y, m, d = oppo.get_date()
>>> print(y)
1846
>>> print(m)
8
>>> print(round(d, 4))
20.1623

static orbital_elements_j2000(epoch)[source]¶

This method computes the orbital elements of Neptune for the standard equinox J2000.0 for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Neptune.orbital_elements_j2000(epoch)
>>> print(round(l, 6))
87.407029
>>> print(round(a, 8))
30.11038676
>>> print(round(e, 7))
0.0094597
>>> print(round(i, 6))
1.770101
>>> print(round(ome, 5))
131.74402
>>> print(round(arg, 6))
-83.6046

static orbital_elements_mean_equinox(epoch)[source]¶

This method computes the orbital elements of Neptune for the mean equinox of the date for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Neptune.orbital_elements_mean_equinox(epoch)
>>> print(round(l, 6))
88.321947
>>> print(round(a, 8))
30.11038676
>>> print(round(e, 7))
0.0094597
>>> print(round(i, 6))
1.763855
>>> print(round(ome, 5))
132.46986
>>> print(round(arg, 6))
-83.415521

pymeeus.Neptune.ORBITAL_ELEM = [[304.348665, 219.8833092, 0.00030882, 1.8e-08], [30.110386869, -1.663e-07, 6.9e-10, 0.0], [0.00945575, 6.033e-06, 0.0, -5e-11], [1.769953, -0.0093082, -7.08e-06, 2.7e-08], [131.748057, 1.1022039, 0.00025952, -6.37e-07], [48.120276, 1.4262957, 0.00038434, 2e-08]]¶: This table contains the parameters to compute Neptune’s orbital elements for the mean equinox of date. Based in Table 31.A, page 213

pymeeus.Neptune.ORBITAL_ELEM_J2000 = [[304.348665, 218.4862002, 5.9e-07, -2e-09], [1.769953, 0.0002256, 2.3e-07, 0.0], [131.748057, -0.0061651, -2.19e-06, -7.8e-08], [48.120276, 0.0291866, 7.61e-05, 0.0]]¶: This table contains the parameters to compute Neptune’s orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 215

pymeeus.Neptune.VSOP87_B = [[[3088622.933, 1.44104372626, 38.1330356378], [27780.087, 5.91271882843, 76.2660712756], [27623.609, 0.0, 0.0], [15355.49, 2.52123799481, 36.6485629295], [15448.133, 3.50877080888, 39.6175083461], [1999.919, 1.50998669505, 74.7815985673], [1967.54, 4.37778195768, 1.4844727083], [1015.137, 3.21561035875, 35.1640902212], [605.767, 2.80246601405, 73.297125859], [594.878, 2.12892708114, 41.1019810544], [588.805, 3.18655882497, 2.9689454166], [401.83, 4.16883287237, 114.3991069134], [254.333, 3.27120499438, 453.424893819], [261.647, 3.76722704749, 213.299095438], [279.964, 1.68165309699, 77.7505439839], [205.59, 4.25652348864, 529.6909650946], [140.455, 3.52969556376, 137.0330241624], [98.53, 4.16774829927, 33.6796175129], [51.257, 1.95121181203, 4.4534181249], [67.971, 4.66970781659, 71.8126531507], [41.931, 5.41783694467, 111.4301614968], [41.822, 5.94832001477, 112.9146342051], [30.637, 0.93620571932, 42.5864537627], [11.084, 5.88898793049, 108.4612160802], [9.62, 0.03944255108, 70.3281804424], [9.664, 0.22455797403, 79.2350166922], [9.728, 5.30069593532, 32.1951448046], [7.386, 3.00684933642, 426.598190876], [7.087, 0.12535040656, 109.9456887885], [6.021, 6.20514068152, 115.8835796217], [6.169, 3.62098109648, 983.1158589136], [4.777, 0.75210194972, 5.9378908332], [6.391, 5.8464610106, 148.0787244263], [6.251, 2.41678769385, 152.5321425512], [4.539, 5.581820987, 175.1660598002], [5.006, 4.60815664851, 1059.3819301892], [4.289, 4.19647392821, 47.6942631934], [5.795, 5.07516716087, 415.2918581812], [4.749, 2.51605725604, 37.611770776], [4.119, 1.72779509865, 28.5718080822], [4.076, 6.00252170354, 145.1097790097], [4.429, 5.65995321659, 98.8999885246], [3.95, 2.74104636753, 350.3321196004], [4.091, 1.61787956945, 39.0962434843], [4.131, 4.40682554313, 37.1698277913], [4.71, 3.50929350767, 38.6543004996], [4.44, 4.78977105547, 38.084851528], [4.433, 1.23386935925, 38.1812197476], [3.762, 4.83940791709, 491.5579294568], [2.606, 1.20956732792, 451.9404211107], [2.537, 2.18628045751, 454.9093665273], [2.328, 5.19779918719, 72.0732855816], [2.502, 0.8598790435, 106.9767433719], [2.342, 0.81387240947, 4.192785694], [1.981, 0.46617960831, 184.7272873558], [1.963, 6.01909114576, 44.070926471], [2.18, 0.70099749844, 206.1855484372], [1.811, 0.40456996647, 40.5807161926], [1.814, 3.64699555185, 220.4126424388], [1.705, 6.13551142362, 181.7583419392], [1.855, 5.61635630213, 35.685355083], [1.595, 2.97147156093, 37.8724032069], [1.785, 2.42154818096, 388.4651552382], [1.595, 3.05266110075, 38.3936680687], [1.437, 1.48678704605, 135.5485514541], [1.387, 2.46149266117, 138.5174968707], [1.366, 1.52026779665, 68.8437077341], [1.575, 3.58964541604, 38.0211610532], [1.297, 5.06156596196, 33.9402499438], [1.487, 0.20211121607, 30.0562807905], [1.504, 5.80298577327, 46.2097904851], [1.192, 0.87275514483, 42.3258213318], [1.569, 2.43405967107, 38.2449102224], [1.207, 1.84658687853, 251.4321310758], [1.015, 0.53439848924, 129.9194771616], [0.999, 2.47463873948, 312.1990839626], [0.99, 3.41514319052, 144.1465711632], [0.963, 4.31733242907, 151.0476698429], [1.02, 0.98226686775, 143.6253063014], [0.941, 1.02993053785, 221.3758502853], [0.938, 2.43648356625, 567.8240007324], [1.111, 0.65175024456, 146.594251718], [0.777, 0.00175975222, 218.4069048687], [0.895, 0.2512386962, 30.7106720963], [0.795, 5.80519741659, 149.5631971346], [0.737, 3.40060492866, 446.3113468182], [0.719, 1.43795191278, 8.0767548473], [0.72, 0.0065100755, 460.5384408198], [0.766, 4.03399506246, 522.5774180938], [0.666, 1.39457824982, 84.3428261229], [0.584, 1.01405548136, 536.8045120954], [0.596, 0.62390100715, 35.212274331], [0.598, 5.39946724188, 41.0537969446], [0.475, 5.80072248338, 7.4223635415], [0.51, 1.3447857974, 258.0244132148], [0.458, 5.25325523118, 80.7194894005], [0.421, 3.24496387889, 416.7763308895], [0.446, 1.19167306357, 180.2738692309], [0.471, 0.92632922375, 44.7253177768], [0.387, 1.68488418788, 183.2428146475], [0.375, 0.15223869165, 255.0554677982], [0.354, 4.21526988674, 0.9632078465], [0.379, 2.16947487177, 105.4922706636], [0.341, 4.7919405168, 110.2063212194], [0.427, 5.15774894584, 31.5407534988], [0.302, 3.4570630628, 100.3844612329], [0.298, 2.26790695187, 639.897286314], [0.279, 0.25689162963, 39.5056337615], [0.32, 3.58085653166, 45.2465826386], [0.269, 5.72024180826, 36.7604375141], [0.247, 0.61040148804, 186.2117600641], [0.245, 0.64173616273, 419.4846438752], [0.235, 0.73189197665, 10213.285546211], [0.232, 0.37399822852, 490.0734567485], [0.23, 5.76570492457, 12.5301729722], [0.24, 4.13447692727, 0.5212648618], [0.279, 1.62614865256, 294.6729761443], [0.238, 2.1852891655, 219.891377577], [0.262, 3.08384135298, 6.592282139], [0.217, 2.93214905312, 27.0873353739], [0.217, 4.69210602828, 406.1031376411], [0.219, 1.3521271256, 216.9224321604], [0.2, 2.35215465744, 605.9570363702], [0.232, 3.92583619589, 1512.8068240082], [0.223, 5.52392277606, 187.6962327724], [0.19, 0.29169556516, 291.7040307277], [0.236, 3.12464145036, 563.6312150384], [0.193, 0.53675942386, 60.7669528868], [0.215, 3.78391259001, 103.0927742186], [0.172, 5.63262770743, 7.1135470008], [0.164, 4.14700645532, 77.2292791221], [0.162, 0.72021213236, 11.0457002639], [0.16, 4.23490438166, 487.3651437628], [0.191, 0.37651439206, 31.019488637], [0.157, 1.02419759383, 6283.0758499914], [0.157, 4.42530429545, 6206.8097787158], [0.178, 6.24797160202, 316.3918696566], [0.161, 5.65988283675, 343.2185725996], [0.153, 5.58405022784, 252.0865223816], [0.189, 4.8079103997, 641.1211265914], [0.166, 5.50438043692, 662.531203563], [0.146, 5.08949604858, 286.596221297], [0.145, 2.13015521881, 2042.4977891028], [0.156, 2.19452173251, 274.0660483248], [0.148, 4.85696640135, 442.7517005706], [0.187, 4.96121139073, 1589.0728952838], [0.155, 2.28260574227, 142.1408335931], [0.134, 1.29277093566, 456.3938392356], [0.126, 5.59769497652, 179.3588454942], [0.146, 2.53359213478, 256.5399405065], [0.14, 1.57962199954, 75.7448064138], [0.123, 0.05442220184, 944.9828232758], [0.122, 1.90676379802, 418.2608035978], [0.154, 1.86865302773, 331.3215390738], [0.144, 5.52229258454, 14.0146456805], [0.138, 2.80728175526, 82.8583534146], [0.107, 0.66995358132, 190.665178189], [0.114, 1.4889498028, 253.5709950899], [0.11, 5.32587573069, 240.125798381], [0.105, 0.65548440578, 173.6815870919], [0.102, 2.58735617801, 450.4559484024], [0.098, 0.44044795266, 328.3525936572], [0.101, 4.71267656829, 117.36805233], [0.094, 0.54938580474, 293.188503436], [0.095, 2.17636214523, 101.8689339412], [0.093, 0.63687810471, 377.1588225434], [0.091, 5.84828809934, 10137.0194749354], [0.089, 1.02830167997, 1021.2488945514], [0.094, 1.79320597168, 493.0424021651], [0.08, 1.58140274465, 69.1525242748], [0.075, 0.23453373368, 63.7358983034], [0.071, 1.5196198969, 488.5889840402]], [[227279.214, 3.8079308987, 38.1330356378], [1803.12, 1.97576485377, 76.2660712756], [1385.733, 4.82555548018, 36.6485629295], [1433.3, 3.14159265359, 0.0], [1073.298, 6.08054240712, 39.6175083461], [147.903, 3.85766231348, 74.7815985673], [136.448, 0.47764957338, 1.4844727083], [70.285, 6.18782052139, 35.1640902212], [51.899, 5.05221791891, 73.297125859], [37.273, 4.89476629246, 41.1019810544], [42.568, 0.30721737205, 114.3991069134], [37.104, 5.75999349109, 2.9689454166], [26.399, 5.21566335936, 213.299095438], [16.949, 4.26463671859, 77.7505439839], [18.747, 0.90426522185, 453.424893819], [12.951, 6.17709713139, 529.6909650946], [10.502, 1.20336443465, 137.0330241624], [4.416, 1.25478204684, 111.4301614968], [4.383, 6.14147099615, 71.8126531507], [3.694, 0.94837702528, 33.6796175129], [2.957, 4.7753287121, 4.4534181249], [2.698, 1.92435531119, 112.9146342051], [1.989, 3.96637567224, 42.5864537627], [1.15, 4.30568700024, 37.611770776], [0.871, 4.81775882249, 152.5321425512], [0.944, 2.2177777205, 109.9456887885], [0.936, 1.1705498394, 148.0787244263], [0.925, 2.40329074, 206.1855484372], [0.69, 1.57381082857, 38.6543004996], [0.624, 2.79466003645, 79.2350166922], [0.726, 4.13829519132, 28.5718080822], [0.64, 2.46161252327, 115.8835796217], [0.531, 2.969915305, 98.8999885246], [0.537, 1.95986772922, 220.4126424388], [0.539, 2.06690307827, 40.5807161926], [0.716, 0.5578184701, 350.3321196004], [0.563, 1.84072805158, 983.1158589136], [0.533, 1.3478767794, 47.6942631934], [0.566, 1.80111775954, 175.1660598002], [0.449, 1.62191691011, 144.1465711632], [0.371, 2.74239666472, 415.2918581812], [0.381, 6.11910193382, 426.598190876], [0.366, 2.3975258536, 129.9194771616], [0.456, 3.19611413854, 108.4612160802], [0.327, 3.62341506247, 38.1812197476], [0.328, 0.89613145346, 38.084851528], [0.341, 3.8726546907, 35.685355083], [0.331, 4.48858774501, 460.5384408198], [0.414, 1.03543720726, 70.3281804424], [0.31, 0.51297445145, 37.1698277913], [0.287, 2.183516518, 491.5579294568], [0.274, 6.11504724934, 522.5774180938], [0.281, 3.81657117512, 5.9378908332], [0.298, 4.00532631258, 39.0962434843], [0.265, 5.26569823181, 446.3113468182], [0.319, 1.34097217817, 184.7272873558], [0.203, 6.02944475303, 149.5631971346], [0.205, 5.5393573202, 536.8045120954], [0.226, 6.17710997862, 454.9093665273], [0.186, 3.24302117645, 4.192785694], [0.179, 4.91458426239, 451.9404211107], [0.198, 2.3077585288, 146.594251718], [0.166, 1.16793600058, 72.0732855816], [0.147, 2.10574339673, 44.070926471], [0.123, 1.98250467171, 46.2097904851], [0.159, 3.46955908364, 145.1097790097], [0.116, 5.8897111359, 38.0211610532], [0.115, 4.73412534395, 38.2449102224], [0.125, 3.42713474801, 251.4321310758], [0.128, 1.51108932026, 221.3758502853], [0.127, 0.17176461812, 138.5174968707], [0.124, 5.85160407534, 1059.3819301892], [0.091, 2.38273591235, 30.0562807905], [0.118, 5.27114846878, 37.8724032069], [0.117, 5.35267669439, 38.3936680687], [0.099, 5.19920708255, 135.5485514541], [0.114, 4.37452353441, 388.4651552382], [0.093, 4.64183693718, 106.9767433719], [0.084, 1.35269684746, 33.9402499438], [0.111, 3.5622646377, 181.7583419392], [0.082, 3.18401661435, 42.3258213318], [0.084, 5.51669920239, 8.0767548473]], [[9690.766, 5.57123750291, 38.1330356378], [78.815, 3.62705474219, 76.2660712756], [71.523, 0.4547668858, 36.6485629295], [58.646, 3.14159265359, 0.0], [29.915, 1.60671721861, 39.6175083461], [6.472, 5.60736756575, 74.7815985673], [5.8, 2.25341847151, 1.4844727083], [4.309, 1.68126737666, 35.1640902212], [3.502, 2.39142672984, 114.3991069134], [2.649, 0.65061457644, 73.297125859], [1.518, 0.37600329684, 213.299095438], [1.223, 1.2311604303, 2.9689454166], [0.766, 5.45279753249, 453.424893819], [0.779, 2.07081431472, 529.6909650946], [0.496, 0.26552533921, 41.1019810544], [0.469, 5.87866293959, 77.7505439839], [0.482, 5.63056237954, 137.0330241624], [0.345, 1.80085651594, 71.8126531507], [0.274, 2.86650141006, 33.6796175129], [0.158, 4.63868656467, 206.1855484372], [0.166, 1.24877330835, 220.4126424388], [0.153, 2.87376446497, 111.4301614968], [0.116, 3.63838544843, 112.9146342051], [0.085, 0.43712705655, 4.4534181249], [0.104, 6.12597614674, 144.1465711632]], [[273.423, 1.01688979072, 38.1330356378], [2.274, 2.36805657126, 36.6485629295], [2.029, 5.33364321342, 76.2660712756], [2.393, 0.0, 0.0], [0.538, 3.21934211365, 39.6175083461], [0.242, 4.52650721578, 114.3991069134], [0.185, 1.04913770083, 74.7815985673], [0.155, 3.62376309338, 35.1640902212], [0.157, 3.9419536961, 1.4844727083]], [[5.728, 2.66872693322, 38.1330356378]], [[0.113, 4.70646877989, 38.1330356378]]]¶: This table contains Neptune’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in page 453.

pymeeus.Neptune.VSOP87_L = [[[531188633.047, 0.0, 0.0], [1798475.509, 2.9010127305, 38.1330356378], [1019727.662, 0.4858092366, 1.4844727083], [124531.845, 4.83008090682, 36.6485629295], [42064.45, 5.41054991607, 2.9689454166], [37714.589, 6.09221834946, 35.1640902212], [33784.734, 1.24488865578, 76.2660712756], [16482.741, 7.729261e-05, 491.5579294568], [9198.582, 4.93747059924, 39.6175083461], [8994.249, 0.27462142569, 175.1660598002], [4216.235, 1.98711914364, 73.297125859], [3364.818, 1.03590121818, 33.6796175129], [2284.8, 4.20606932559, 4.4534181249], [1433.512, 2.78340432711, 74.7815985673], [900.24, 2.07606702418, 109.9456887885], [744.996, 3.19032530145, 71.8126531507], [506.206, 5.74785370252, 114.3991069134], [399.552, 0.34972342569, 1021.2488945514], [345.195, 3.46186210169, 41.1019810544], [306.338, 0.49684039897, 0.5212648618], [287.322, 4.50523446022, 0.0481841098], [323.004, 2.24815188609, 32.1951448046], [340.323, 3.30369900416, 77.7505439839], [266.605, 4.88932609483, 0.9632078465], [227.079, 1.79713054538, 453.424893819], [244.722, 1.24693337933, 9.5612275556], [232.887, 2.50459795017, 137.0330241624], [282.17, 2.24565579693, 146.594251718], [251.941, 5.78166597292, 388.4651552382], [150.18, 2.99706110414, 5.9378908332], [170.404, 3.3239063065, 108.4612160802], [151.401, 2.1915309428, 33.9402499438], [148.295, 0.85948986145, 111.4301614968], [118.672, 3.67706204305, 2.4476805548], [101.821, 5.70539236951, 0.1118745846], [97.873, 2.80518260528, 8.0767548473], [103.054, 4.40441222, 70.3281804424], [103.305, 0.04078966679, 0.2606324309], [109.3, 2.41599378049, 183.2428146475], [73.938, 1.32805041516, 529.6909650946], [77.725, 4.16446516424, 4.192785694], [86.379, 4.22834506045, 490.0734567485], [81.536, 5.19908046216, 493.0424021651], [71.503, 5.29530386579, 350.3321196004], [64.418, 3.5454101605, 168.0525127994], [62.57, 0.15028731465, 182.279606801], [58.488, 3.50106873945, 145.1097790097], [48.276, 1.11259925628, 112.9146342051], [47.229, 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8.0767548473], [9.882, 6.08165628679, 7.1135470008], [7.632, 5.51307048241, 73.297125859], [6.992, 0.61688864282, 2.9207613068], [5.543, 2.24141557794, 46.2097904851], [4.845, 3.7105582375, 112.9146342051], [3.7, 5.25713252333, 111.4301614968], [3.233, 6.10303038418, 70.3281804424], [2.939, 4.86520586648, 98.8999885246], [2.403, 2.90637675099, 601.7642506762], [2.398, 1.04343654629, 6.592282139], [2.784, 4.95821114677, 108.4612160802], [2.894, 4.20148844767, 381.3516082374], [2.111, 5.93089610785, 25.6028626656], [2.075, 5.20632201951, 30.7106720963], [2.126, 0.54976393136, 41.0537969446], [2.235, 2.38045158073, 453.424893819], [1.859, 0.89409373259, 24.1183899573], [2.018, 3.42245274178, 31.019488637], [1.7, 3.91715254287, 11.0457002639], [1.776, 3.86571077241, 395.578702239], [1.644, 0.15855999051, 152.5321425512], [1.646, 3.34591387314, 44.7253177768], [1.876, 2.59784179105, 33.9402499438], [1.614, 0.42137145545, 175.1660598002], [1.468, 6.12983933526, 1550.939859646], [1.408, 6.13722948564, 490.0734567485], [1.207, 0.59525736062, 312.1990839626], [1.336, 3.28611928206, 493.0424021651], [1.176, 5.87266726996, 5.4166259714], [1.517, 3.12967210501, 491.5579294568], [1.053, 4.6037551683, 79.2350166922], [1.037, 4.89007314395, 1.2720243872], [1.034, 5.93741289103, 32.7164096664], [1.038, 1.13470380744, 1014.1353475506], [1.002, 1.85850922283, 5.1078094307], [0.983, 0.05345050384, 7.4223635415], [0.998, 1.73689827444, 1028.3624415522], [1.193, 4.63176675581, 60.7669528868], [0.94, 3.09103721222, 62.2514255951], [0.994, 4.11489180313, 4.665866446], [0.89, 0.87049255398, 31.2319369581], [0.852, 5.35508394316, 144.1465711632], [0.922, 5.12373360511, 145.1097790097], [0.789, 0.37496785039, 26.826702943], [0.828, 4.060351946, 115.8835796217], [0.711, 3.14189997439, 278.2588340188], [0.727, 1.39718382835, 213.299095438], [0.781, 0.10946327923, 173.6815870919], [0.793, 6.13086312116, 567.8240007324], [0.669, 4.50554989443, 27.0873353739], [0.825, 1.35568908148, 129.9194771616], [0.738, 3.5676601896, 176.6505325085], [0.714, 6.24797992301, 106.9767433719], [0.654, 1.13177751192, 68.8437077341], [0.624, 0.01567750666, 28.5718080822], [0.608, 4.60180625368, 189.3931538018], [0.595, 0.00857468445, 42.5864537627], [0.53, 5.61201247153, 12.5301729722], [0.521, 1.02371768017, 415.2918581812], [0.639, 0.68930265745, 529.6909650946], [0.526, 3.02138731705, 5.6290742925], [0.456, 4.44331571392, 43.2408450685], [0.524, 3.43316448349, 38.6543004996], [0.436, 2.41630174435, 82.8583534146], [0.424, 1.95736011325, 477.3308354552], [0.443, 3.39350946329, 357.4456666012], [0.383, 1.90232196422, 22.633917249], [0.479, 5.55141744216, 37.611770776], [0.462, 3.80436154644, 343.2185725996], [0.384, 5.60377408953, 594.6507036754], [0.369, 4.45577410338, 6.9010986797], [0.358, 3.69126616347, 3.9321532631], [0.352, 3.10952926034, 135.5485514541], [0.368, 3.53577440355, 40.5807161926], [0.424, 5.27159202779, 181.7583419392], [0.361, 0.29018303419, 72.0732855816], [0.39, 5.49512204296, 350.3321196004], [0.378, 2.74122401337, 488.3765357191], [0.372, 0.39980033572, 494.7393231945], [0.353, 1.10614174053, 20.6069278195], [0.296, 0.86351261285, 149.5631971346], [0.307, 5.39420288683, 160.9389657986], [0.395, 1.93577214824, 10137.0194749354], [0.288, 2.28755739359, 47.6942631934], [0.295, 2.4873753724, 19.1224551112], [0.29, 0.18636083306, 143.6253063014], [0.266, 3.09977370364, 69.3649725959], [0.266, 1.21002824826, 505.7850234584], [0.252, 3.12745026026, 460.5384408198], [0.328, 0.50849285663, 6206.8097787158], [0.257, 3.64119914774, 446.3113468182], [0.239, 5.54080102299, 911.042573332], [0.265, 0.62702473701, 253.5709950899], [0.287, 2.44403568436, 16.6747745564], [0.231, 2.47026250085, 454.9093665273], [0.23, 3.24571542922, 1066.49547719], [0.282, 1.48595620175, 983.1158589136], [0.212, 5.41931177641, 64.9597385808], [0.213, 1.64175339637, 1089.129394439], [0.238, 2.69801319489, 882.9438460018], [0.21, 4.53976756699, 1093.322180133], [0.22, 2.30038816175, 1052.2683831884], [0.256, 0.4207359846, 23.9059416362], [0.216, 5.4422591887, 39.0962434843], [0.201, 2.58746514605, 119.5069163441], [0.224, 4.43751392203, 639.897286314], [0.186, 2.50651218075, 487.3651437628], [0.189, 4.05785534221, 120.9913890524], [0.184, 2.24245977278, 815.0633461142], [0.202, 3.43517732411, 45.2465826386], [0.175, 4.49165234532, 171.2339065371], [0.171, 5.50633466316, 179.0982130633], [0.2, 6.12663205401, 14.2270940016], [0.173, 2.61090344107, 389.9496279465], [0.167, 3.94754384833, 77.2292791221], [0.166, 3.41009128748, 81.3738807063], [0.163, 3.88198848446, 556.5176680376], [0.164, 1.49614763046, 63.7358983034], [0.176, 3.86129425367, 148.3393568572], [0.161, 2.22215642318, 574.9375477332], [0.171, 0.66899426684, 179.3106613844], [0.161, 1.21480182441, 1024.4302882891], [0.155, 3.25842414799, 10251.4185818488], [0.183, 5.45168150656, 218.4069048687], [0.152, 3.35145509017, 285.3723810196], [0.152, 0.42398786475, 274.0660483248], [0.146, 5.70714579127, 419.4846438752], [0.156, 0.6432152487, 1029.8469142605], [0.147, 4.3095893074, 157.6399519819], [0.147, 1.8068917751, 377.4194549743], [0.14, 1.49826604627, 386.9806825299], [0.137, 2.14480243915, 563.6312150384], [0.127, 3.9872659971, 84.3428261229], [0.134, 4.16039455079, 169.5369855077], [0.121, 0.29300927469, 206.1855484372], [0.129, 2.6762505701, 180.7951340927], [0.134, 3.18868986487, 166.5680400911], [0.135, 5.0751756178, 426.598190876], [0.136, 1.8167245174, 151.0476698429], [0.129, 3.64795525602, 183.7640795093], [0.116, 6.06435563172, 220.4126424388], [0.123, 4.46641157829, 1022.7333672597], [0.112, 4.34485256988, 138.5174968707], [0.116, 5.58946529961, 35.685355083], [0.108, 1.03796693383, 488.5889840402], [0.108, 2.1037848588, 494.5268748734], [0.106, 0.87068583107, 1059.3819301892], [0.097, 0.74486741478, 485.9288551643], [0.095, 5.54259914856, 497.1870037493], [0.085, 3.16062141266, 522.5774180938], [0.097, 6.05634803604, 482.9599097477], [0.095, 0.2311185273, 500.1559491659], [0.084, 2.64687252518, 536.8045120954], [0.074, 3.90678924318, 1019.7644218431]], [[53892.649, 0.0, 0.0], [281.251, 1.19084538887, 38.1330356378], [295.693, 1.85520292248, 1.4844727083], [270.19, 5.72143228148, 76.2660712756], [23.023, 1.21035596452, 2.9689454166], [7.333, 0.5403330683, 2.4476805548], [9.057, 4.42544992035, 35.1640902212], [5.223, 0.67427930044, 168.0525127994], [5.201, 3.02338671812, 182.279606801], [4.288, 3.84351844003, 114.3991069134], [3.925, 3.53214557374, 484.444382456], [3.741, 5.90238217874, 498.6714764576], [2.966, 0.31002477611, 4.4534181249], [3.415, 0.55971639038, 74.7815985673], [3.255, 1.84921884906, 175.1660598002], [2.157, 1.89135758747, 388.4651552382], [2.211, 4.3799709224, 7.1135470008], [1.847, 3.48574435762, 9.5612275556], [2.451, 4.68586840176, 491.5579294568], [1.844, 5.12281562096, 33.6796175129], [2.204, 1.69321574906, 77.7505439839], [1.652, 2.55859494053, 36.6485629295], [1.309, 4.52400192922, 1021.2488945514], [1.124, 0.38710602242, 137.0330241624], [0.664, 0.88101734307, 4.192785694], [0.497, 2.24615784762, 395.578702239], [0.512, 6.22609200672, 381.3516082374], [0.582, 5.25716719826, 31.019488637], [0.446, 0.36647221351, 98.8999885246], [0.383, 5.48585528762, 5.9378908332], [0.375, 4.61250246774, 8.0767548473], [0.354, 1.30783918287, 601.7642506762], [0.259, 5.66033623678, 112.9146342051], [0.247, 2.89695614593, 189.3931538018], [0.245, 4.26572913391, 220.4126424388], [0.2, 0.52604535784, 64.9597385808], [0.191, 4.88786653062, 39.6175083461], [0.233, 3.16423779113, 41.1019810544], [0.248, 5.85877831382, 1059.3819301892], [0.194, 2.37949641473, 73.297125859], [0.227, 0.20028518978, 60.7669528868], [0.184, 3.01962045713, 1014.1353475506], [0.19, 5.57500985081, 343.2185725996], [0.172, 3.66036463613, 477.3308354552], [0.172, 0.59550457102, 46.2097904851], [0.182, 1.92429384025, 183.7640795093], [0.171, 1.61368476689, 357.4456666012], [0.173, 6.23717119485, 493.0424021651], [0.217, 1.46218158211, 71.8126531507], [0.178, 0.34928799031, 1028.3624415522], [0.169, 4.91086673212, 166.5680400911], [0.157, 5.89200571154, 169.5369855077], [0.182, 2.33457064554, 152.5321425512], [0.151, 3.81621340568, 146.594251718], [0.136, 2.75150881988, 144.1465711632], [0.104, 6.03262825314, 529.6909650946], [0.076, 0.20932812381, 453.424893819]], [[31.254, 0.0, 0.0], [12.461, 6.04431418812, 1.4844727083], [14.541, 1.35337075856, 76.2660712756], [11.547, 6.11257808366, 38.1330356378], [1.351, 4.93951495175, 2.9689454166], [0.741, 2.35936954597, 168.0525127994], [0.715, 1.27409542804, 182.279606801], [0.537, 5.23632185196, 484.444382456], [0.523, 4.16769839601, 498.6714764576], [0.664, 0.55871435877, 31.019488637], [0.301, 2.69253200796, 7.1135470008], [0.194, 2.05904114139, 137.0330241624], [0.206, 2.51012178002, 74.7815985673], [0.16, 5.63111039032, 114.3991069134], [0.149, 3.09327713923, 35.1640902212]], [[113.998, 3.14159265359, 0.0], [0.605, 3.18211885677, 76.2660712756]], [[0.874, 3.14159265359, 0.0]]]¶: This table contains Neptune’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 452-453.

pymeeus.Neptune.VSOP87_R = [[[3007013206.102, 0.0, 0.0], [27062259.49, 1.3299945893, 38.1330356378], [1691764.281, 3.25186138896, 36.6485629295], [807830.737, 5.18592836167, 1.4844727083], [537760.613, 4.52113902845, 35.1640902212], [495725.642, 1.57105654815, 491.5579294568], [274571.97, 1.84552256801, 175.1660598002], [135134.095, 3.37220607384, 39.6175083461], [121801.825, 5.79754444303, 76.2660712756], [100895.397, 0.37702748681, 73.297125859], [69791.722, 3.79617226928, 2.9689454166], [46687.838, 5.74937810094, 33.6796175129], [24593.778, 0.50801728204, 109.9456887885], [16939.242, 1.59422166991, 71.8126531507], [14229.686, 1.07786112902, 74.7815985673], [12011.825, 1.92062131635, 1021.2488945514], [8394.731, 0.67816895547, 146.594251718], [7571.8, 1.07149263431, 388.4651552382], [5720.852, 2.59059512267, 4.4534181249], [4839.672, 1.9068599107, 41.1019810544], [4483.492, 2.90573457534, 529.6909650946], [4270.202, 3.41343865825, 453.424893819], [4353.79, 0.6798566237, 32.1951448046], [4420.804, 1.74993796503, 108.4612160802], [2881.063, 1.98600105123, 137.0330241624], [2635.535, 3.09755943422, 213.299095438], [3380.93, 0.84810683275, 183.2428146475], [2878.942, 3.67415901855, 350.3321196004], [2306.293, 2.80962935724, 70.3281804424], [2530.149, 5.79839567009, 490.0734567485], [2523.132, 0.48630800015, 493.0424021651], [2087.303, 0.61858378281, 33.9402499438], [1976.522, 5.1170304456, 168.0525127994], [1905.254, 1.72186472126, 182.279606801], [1654.039, 1.92782545887, 145.1097790097], [1435.072, 1.70005157785, 484.444382456], [1403.029, 4.58914203187, 498.6714764576], [1499.193, 1.01623299513, 219.891377577], [1398.86, 0.7622031762, 176.6505325085], [1403.377, 6.07659416908, 173.6815870919], [1128.56, 5.96661179805, 9.5612275556], [1228.304, 1.59881465324, 77.7505439839], [835.414, 3.97066884218, 114.3991069134], [811.186, 3.0025888087, 46.2097904851], [731.925, 2.10447054189, 181.7583419392], [615.781, 2.97874625677, 106.9767433719], [704.778, 1.1873821088, 256.5399405065], [502.04, 1.38657803368, 5.9378908332], [530.357, 4.24059166485, 111.4301614968], [437.096, 2.27029212923, 1550.939859646], [400.25, 1.25609325435, 8.0767548473], [421.011, 1.89084929506, 30.7106720963], [382.457, 3.29965259685, 983.1158589136], [422.485, 5.53186169605, 525.4981794006], [355.389, 2.27847846648, 218.4069048687], [280.062, 1.54129714238, 98.8999885246], [314.499, 3.95932948594, 381.3516082374], [280.556, 4.54238271682, 44.7253177768], [267.738, 5.13323364247, 112.9146342051], [333.311, 5.75067616021, 39.0962434843], [291.625, 4.02398326341, 68.8437077341], [321.429, 1.50625025822, 454.9093665273], [309.196, 2.85452752153, 72.0732855816], [345.094, 1.35905860594, 293.188503436], [307.439, 0.31964571332, 601.7642506762], [251.356, 3.53992782846, 312.1990839626], [248.152, 3.41078346726, 37.611770776], [306.0, 2.72475094464, 6244.9428143536], [293.532, 4.89079857814, 528.2064923863], [234.479, 0.59231043427, 42.5864537627], [239.628, 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6.09373507569, 6246.4272870619], [3.659, 5.12211619716, 79.2350166922], [3.01, 0.24656411754, 91.7864415238], [2.861, 6.17465663902, 422.405405182], [2.989, 2.31620917965, 485.9288551643], [3.088, 2.29186342974, 110.2063212194], [3.03, 3.698661491, 532.6117264014], [3.02, 2.36422658177, 290.2195580194], [3.17, 1.23078934548, 10176.6369832815], [2.652, 3.35836234807, 148.0787244263], [2.673, 6.03366372927, 196.5067008026], [2.63, 0.46957619348, 1970.4245035212], [2.599, 4.86022081674, 439.1977998174], [2.878, 2.61946597178, 488.5889840402], [2.72, 1.71836225398, 364.559213602], [3.333, 3.25126857354, 30.0562807905], [3.053, 2.49346960035, 6243.4583416453], [3.062, 6.23776299963, 419.4846438752], [2.786, 0.83078219939, 497.1870037493], [2.834, 3.52926079424, 457.8783119439], [2.932, 1.80245810977, 500.1559491659], [3.03, 5.10152500393, 367.9701020033], [2.956, 5.76230870725, 986.0848043302], [3.116, 2.20042242739, 495.702531041], [2.554, 0.65945973992, 67.3592350258], [2.901, 3.91891656185, 10173.6680378649], [2.84, 1.34453183591, 482.9599097477], [2.458, 1.20012815574, 489.110248902], [2.556, 3.86921927085, 487.1045113319], [2.614, 1.51881085312, 463.5073862364], [2.386, 4.58400538443, 615.9913446778], [2.438, 5.19827220476, 501.1191570124], [2.537, 1.64802783144, 519.6084726772], [2.444, 3.87859489652, 185.2485522176], [2.795, 4.0426575258, 255.0554677982], [2.895, 3.26202698812, 1646.9190868638], [2.225, 5.75197574692, 605.9570363702], [2.324, 3.99503920129, 481.2629887183], [2.962, 1.74151265966, 2080.6308247406], [2.621, 1.74442251671, 35.212274331]], [[4247.412, 5.89910679117, 38.1330356378], [217.57, 0.3458182908, 1.4844727083], [163.025, 2.2387294713, 168.0525127994], [156.285, 4.59414467342, 182.279606801], [117.94, 5.10295026024, 484.444382456], [112.429, 1.19000583596, 498.6714764576], [127.141, 2.84786298079, 35.1640902212], [99.467, 3.41578558739, 175.1660598002], [64.814, 3.4621406484, 388.4651552382], [77.286, 0.01659281785, 491.5579294568], [49.509, 4.06995509133, 76.2660712756], [39.33, 6.09521855958, 1021.2488945514], [36.45, 5.17130059988, 137.0330241624], [37.08, 5.97288967681, 2.9689454166], [30.484, 3.58259801313, 33.6796175129], [21.099, 0.76843555176, 36.6485629295], [13.886, 3.59248623971, 395.578702239], [13.117, 5.09263515697, 98.8999885246], [11.379, 1.18060018898, 381.3516082374], [9.132, 2.34787658568, 601.7642506762], [8.527, 5.25134685897, 2.4476805548], [8.136, 4.96270726986, 4.4534181249], [7.417, 4.46775409796, 189.3931538018], [7.225, 1.92287508629, 9.5612275556], [7.289, 1.6551952578, 1028.3624415522], [8.076, 5.84268048311, 220.4126424388], [9.654, 0.0, 0.0], [6.554, 0.69397520733, 144.1465711632], [7.782, 1.14341656235, 1059.3819301892], [5.665, 6.25378258571, 74.7815985673], [5.628, 5.23383764266, 46.2097904851], [5.523, 4.59041448911, 1014.1353475506], [5.177, 5.23116646157, 477.3308354552], [5.503, 3.49522319102, 183.7640795093], [4.878, 3.52934357721, 39.6175083461], [4.787, 2.08260524745, 41.1019810544], [5.055, 0.19949888617, 166.5680400911], [4.751, 1.1805494827, 169.5369855077], [4.747, 1.50608965076, 73.297125859], [6.113, 6.18326155595, 71.8126531507], [4.606, 3.91970908886, 587.5371566746], [5.756, 2.23667359233, 176.6505325085], [4.536, 2.84337336954, 7.1135470008], [4.338, 0.51553847388, 446.3113468182], [3.891, 0.26338839265, 1550.939859646], [4.465, 3.01487041298, 129.9194771616], [3.727, 2.37977930658, 160.9389657986], [3.84, 3.7929038188, 111.4301614968], [4.142, 1.70293820961, 983.1158589136], [3.296, 1.07748822909, 505.7850234584], [4.008, 0.30663868827, 494.7393231945], [3.974, 5.9735178384, 488.3765357191], [3.925, 4.85736421123, 60.7669528868], [2.966, 2.01608546009, 822.176893115], [3.972, 1.07780371834, 374.2380612366], [3.843, 5.23002047199, 350.3321196004], [2.848, 6.17799253802, 704.8570248948], [3.527, 0.79317138165, 274.0660483248], [2.828, 1.32275775835, 386.9806825299], [2.773, 5.37132330836, 251.4321310758], [3.113, 5.1262228869, 426.598190876], [3.344, 5.61433537548, 1124.34166877], [2.597, 0.67759426519, 312.1990839626], [2.581, 3.55847612121, 567.8240007324], [2.578, 1.45603792456, 1035.475988553], [2.541, 5.19427579702, 1227.4344429886], [2.51, 4.12148891512, 171.2339065371], [2.511, 2.71606957319, 179.0982130633], [2.342, 0.96469916587, 1019.7644218431], [2.5, 0.7028227603, 707.7777862016], [2.48, 4.59623030219, 693.5506922], [2.253, 0.74334306011, 976.0023119128]], [[166.297, 4.55243893489, 38.1330356378], [22.38, 3.94830879358, 168.0525127994], [21.348, 2.86296778794, 182.279606801], [16.233, 0.54226725872, 484.444382456], [15.623, 5.75702251906, 498.6714764576], [11.867, 4.4028019271, 1.4844727083], [6.448, 5.19003066847, 31.019488637], [3.655, 5.91335292846, 1007.0218005498], [3.681, 1.62865545676, 388.4651552382], [3.198, 0.70197118575, 1558.0534066468], [3.243, 1.8803566598, 522.5774180938], [3.269, 2.94301808574, 76.2660712756], [2.688, 1.87062743473, 402.6922492398], [3.246, 0.79381356193, 536.8045120954], [2.65, 5.76858449026, 343.2185725996], [2.644, 4.64542905401, 500.1559491659], [2.541, 4.79217120822, 482.9599097477], [2.523, 1.7286988978, 395.578702239], [2.69, 2.21096415618, 446.3113468182], [2.355, 5.77381398401, 485.9288551643], [2.874, 6.1964334054, 815.0633461142], [2.278, 3.66579603119, 497.1870037493]], [[4.227, 2.40375758563, 477.3308354552], [4.333, 0.10459484545, 395.578702239], [3.545, 4.78431259422, 1028.3624415522], [3.154, 3.88192942366, 505.7850234584], [3.016, 1.03609346831, 189.3931538018], [2.294, 1.10879658603, 182.279606801], [2.295, 5.67776133184, 168.0525127994]]]¶: This table contains Neptune’s periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in page 454.

Pluto¶

Class to model Pluto planet.

pymeeus.Pluto.PLUTO_ARGUMENT = [(0.0, 0.0, 1.0), (0.0, 0.0, 2.0), (0.0, 0.0, 3.0), (0.0, 0.0, 4.0), (0.0, 0.0, 5.0), (0.0, 0.0, 6.0), (0.0, 1.0, -1.0), (0.0, 1.0, 0.0), (0.0, 1.0, 1.0), (0.0, 1.0, 2.0), (0.0, 1.0, 3.0), (0.0, 2.0, -2.0), (0.0, 2.0, -1.0), (0.0, 2.0, 0.0), (1.0, -1.0, 0.0), (1.0, -1.0, 1.0), (1.0, 0.0, -3.0), (1.0, 0.0, -2.0), (1.0, 0.0, -1.0), (1.0, 0.0, 0.0), (1.0, 0.0, 1.0), (1.0, 0.0, 2.0), (1.0, 0.0, 3.0), (1.0, 0.0, 4.0), (1.0, 1.0, -3.0), (1.0, 1.0, -2.0), (1.0, 1.0, -1.0), (1.0, 1.0, 0.0), (1.0, 1.0, 1.0), (1.0, 1.0, 3.0), (2.0, 0.0, -6.0), (2.0, 0.0, -5.0), (2.0, 0.0, -4.0), (2.0, 0.0, -3.0), (2.0, 0.0, -2.0), (2.0, 0.0, -1.0), (2.0, 0.0, 0.0), (2.0, 0.0, 1.0), (2.0, 0.0, 2.0), (2.0, 0.0, 3.0), (3.0, 0.0, -2.0), (3.0, 0.0, -1.0), (3.0, 0.0, 0.0)]¶: This table contains Pluto’s argument coefficients according to Table 37.A in Meeus’ book, page 265.

pymeeus.Pluto.PLUTO_LATITUDE = [(-5452852.0, -14974862), (3527812.0, 1672790.0), (-1050748.0, 327647.0), (178690.0, -292153.0), (18650.0, 100340.0), (-30697.0, -25823.0), (4878.0, 11248.0), (226.0, -64.0), (2030.0, -836.0), (69.0, -604.0), (-247.0, -567.0), (-57.0, 1.0), (-122.0, 175.0), (-49.0, -164.0), (-197.0, 199.0), (-25.0, 217.0), (589.0, -248.0), (-269.0, 711.0), (185.0, 193.0), (315.0, 807.0), (-130.0, -43.0), (5.0, 3.0), (2.0, 17.0), (2.0, 5.0), (2.0, 3.0), (3.0, 1.0), (2.0, -1.0), (1.0, -1.0), (0.0, -1.0), (0.0, 0.0), (0.0, -2.0), (2.0, 2.0), (-7.0, 0.0), (10.0, -8.0), (-3.0, 20.0), (6.0, 5.0), (14.0, 17.0), (-2.0, 0.0), (0.0, 0.0), (0.0, 0.0), (0.0, 1.0), (0.0, 0.0), (1.0, 0.0)]¶: This table contains the periodic terms to compute Pluto’s heliocentric latitude according to Table 37.A in Meeus’ book, page 265

pymeeus.Pluto.PLUTO_LONGITUDE = [(-19799805.0, 19850055.0), (897144.0, -4954829.0), (611149.0, 1211027.0), (-341243.0, -189585.0), (129287.0, -34992.0), (-38164.0, 30893.0), (20442.0, -9987.0), (-4063.0, -5071.0), (-6016.0, -3336.0), (-3956.0, 3039.0), (-667.0, 3572.0), (1276.0, 501.0), (1152.0, -917.0), (630.0, -1277.0), (2571.0, -459.0), (899.0, -1449.0), (-1016.0, 1043.0), (-2343.0, -1012.0), (7042.0, 788.0), (1199.0, -338.0), (418.0, -67.0), (120.0, -274.0), (-60.0, -159.0), (-82.0, -29.0), (-36.0, -29.0), (-40.0, 7.0), (-14.0, 22.0), (4.0, 13.0), (5.0, 2.0), (-1.0, 0.0), (2.0, 0.0), (-4.0, 5.0), (4.0, -7.0), (14.0, 24.0), (-49.0, -34.0), (163.0, -48.0), (9.0, -24.0), (-4.0, 1.0), (-3.0, 1.0), (1.0, 3.0), (-3.0, -1.0), (5.0, -3.0), (0.0, 0.0)]¶: This table contains the periodic terms to compute Pluto’s heliocentric longitude according to Table 37.A in Meeus’ book, page 265

pymeeus.Pluto.PLUTO_RADIUS_VECTOR = [(66865439.0, 68951812.0), (-11827535.0, -332538.0), (1593179.0, -1438890.0), (-18444.0, 483220.0), (-65977.0, -85431.0), (31174.0, -6032.0), (-5794.0, 22161.0), (4601.0, 4032.0), (-1729.0, 234.0), (-415.0, 702.0), (239.0, 723.0), (67.0, -67.0), (1034.0, -451.0), (-129.0, 504.0), (480.0, -231.0), (2.0, -441.0), (-3359.0, 265.0), (7856.0, -7832.0), (36.0, 45763.0), (8663.0, 8547.0), (-809.0, -769.0), (263.0, -144.0), (-126.0, 32.0), (-35.0, -16.0), (-19.0, -4.0), (-15.0, 8.0), (-4.0, 12.0), (5.0, 6.0), (3.0, 1.0), (6.0, -2.0), (2.0, 2.0), (-2.0, -2.0), (14.0, 13.0), (-63.0, 13.0), (136.0, -236.0), (273.0, 1065.0), (251.0, 149.0), (-25.0, -9.0), (9.0, -2.0), (-8.0, 7.0), (2.0, -10.0), (19.0, 35.0), (10.0, 3.0)]¶: This table contains the periodic terms to compute Pluto’s heliocentric radius vector according to Table 37.A in Meeus’ book, page 265

class pymeeus.Pluto.Pluto[source]¶

Class Pluto models that minor planet.

__weakref__¶: list of weak references to the object (if defined)

static geocentric_position(epoch)[source]¶

This method computes the geocentric position of Pluto (right ascension and declination) for the given epoch, for the standard equinox J2000.0.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension and the declination as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the 1885-2099 range.

>>> epoch = Epoch(1992, 10, 13.0)
>>> ra, dec = Pluto.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
15h 31' 43.7''
>>> print(dec.dms_str(n_dec=0))
-4d 27' 29.0''

static geometric_heliocentric_position(epoch)[source]¶

This method computes the geometric heliocentric position of planet Pluto for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute Pluto position, as an Epoch object
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the 1885-2099 range.

>>> epoch = Epoch(1992, 10, 13.0)
>>> l, b, r = Pluto.geometric_heliocentric_position(epoch)
>>> print(round(l, 5))
232.74071
>>> print(round(b, 5))
14.58782
>>> print(round(r, 6))
29.711111

Saturn¶

Class to model Saturn planet.

pymeeus.Saturn.ORBITAL_ELEM = [[50.077444, 1223.5110686, 0.00051908, -3e-08], [9.554909192, -2.139e-06, 4e-09, 0.0], [0.05554814, -0.000346641, -6.436e-07, 3.4e-09], [2.488879, -0.0037362, -1.519e-05, 8.7e-08], [113.665503, 0.877088, -0.00012176, -2.249e-06], [93.057237, 1.9637613, 0.00083753, 4.928e-06]]¶: This table contains the parameters to compute Saturn’s orbital elements for the mean equinox of date. Based in Table 31.A, page 213

pymeeus.Saturn.ORBITAL_ELEM_J2000 = [[50.077444, 1222.1138488, 0.00021004, -4.6e-08], [2.488879, 0.0025514, -4.906e-05, 1.7e-08], [113.665503, -0.2566722, -0.00018399, 4.8e-07], [93.057237, 0.5665415, 0.0005285, 4.912e-06]]¶: This table contains the parameters to compute Saturn’s orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 215

class pymeeus.Saturn.Saturn[source]¶

Class Saturn models that planet.

__weakref__¶: list of weak references to the object (if defined)

static apparent_heliocentric_position(epoch)[source]¶

This method computes the apparent heliocentric position of planet Saturn for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Saturn position, as an Epoch object
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

static conjunction(epoch)[source]¶

This method computes the time of the conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired conjunction
Returns:	The time when the conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2125, 6, 1.0)
>>> conj = Saturn.conjunction(epoch)
>>> y, m, d = conj.get_date()
>>> print(y)
2125
>>> print(m)
8
>>> print(round(d, 4))
26.4035

static geocentric_position(epoch)[source]¶

This method computes the geocentric position of Saturn (right ascension and declination) for the given epoch, as well as the elongation angle.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension, the declination and the elongation angle as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 20.0)
>>> ra, dec, elon = Saturn.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
21h 11' 41.8''
>>> print(dec.dms_str(n_dec=1))
-17d 15' 40.8''
>>> print(elon.dms_str(n_dec=1))
46d 51' 47.7''

static geometric_heliocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of planet Saturn for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Saturn position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2018, 10, 27.0)
>>> l, b, r = Saturn.geometric_heliocentric_position(epoch)
>>> print(round(l.to_positive(), 4))
279.5108
>>> print(round(b, 4))
0.6141
>>> print(round(r, 5))
10.06266

static magnitude(sun_dist, earth_dist, delta_U, B)[source]¶

This function computes the approximate magnitude of Saturn.

Parameters:	sun_dist (float) – Distance from Saturn to the Sun, in Astronomical Units earth_dist (float) – Distance from Saturn to Earth, in Astronomical Units delta_U (float, `Angle`) – Difference between the Saturnicentric longitudes of the Sun and the Earth, measured in the plane of the ring, in degrees B (float, `Angle`) – Saturnicentric latitude of the Earth refered to the plane of the ring, positive towards the north, in degrees
Returns:	Saturn’s magnitude
Return type:	float
Raises:	TypeError if input values are of wrong type.

Note

In order to obtain delta_U and B, please see method ring_parameters().

>>> sun_dist = 9.867882
>>> earth_dist = 10.464606
>>> delta_U = Angle(16.442)
>>> B = Angle(4.198)
>>> m = Saturn.magnitude(sun_dist, earth_dist, delta_U, B)
>>> print(m)
1.9

static opposition(epoch)[source]¶

This method computes the time of the opposition closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	The time when the opposition happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(-6, 9, 1.0)
>>> oppo = Saturn.opposition(epoch)
>>> y, m, d = oppo.get_date()
>>> print(y)
-6
>>> print(m)
9
>>> print(round(d, 4))
14.3709

static orbital_elements_j2000(epoch)[source]¶

This method computes the orbital elements of Saturn for the standard equinox J2000.0 for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Saturn.orbital_elements_j2000(epoch)
>>> print(round(l, 6))
130.28188
>>> print(round(a, 8))
9.55490779
>>> print(round(e, 7))
0.0553209
>>> print(round(i, 6))
2.490529
>>> print(round(ome, 5))
113.49736
>>> print(round(arg, 6))
-20.068943

static orbital_elements_mean_equinox(epoch)[source]¶

This method computes the orbital elements of Saturn for the mean equinox of the date for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Saturn.orbital_elements_mean_equinox(epoch)
>>> print(round(l, 6))
131.196871
>>> print(round(a, 8))
9.55490779
>>> print(round(e, 7))
0.0553209
>>> print(round(i, 6))
2.486426
>>> print(round(ome, 5))
114.23974
>>> print(round(arg, 6))
-19.896331

static passage_nodes(epoch, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of Saturn, nearest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch closest to the node passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2019, 1, 1)
>>> time, r = Saturn.passage_nodes(epoch)
>>> year, month, day = time.get_date()
>>> print(year)
2034
>>> print(month)
5
>>> print(round(day, 1))
30.2
>>> print(round(r, 4))
9.0546

static perihelion_aphelion(epoch, perihelion=True)[source]¶

This method computes the time of Perihelion (or Aphelion) closer to a given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired Perihelion (or Aphelion) peihelion – If True, the epoch of the closest Perihelion is computed, if False, the epoch of the closest Aphelion is found.
Returns:	The epoch of the desired Perihelion (or Aphelion)
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1944, 1, 1.0)
>>> e = Saturn.perihelion_aphelion(epoch)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
1944
>>> print(m)
9
>>> print(d)
8
>>> print(h)
1
>>> epoch = Epoch(2047, 1, 1.0)
>>> e = Saturn.perihelion_aphelion(epoch, perihelion=False)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
2047
>>> print(m)
7
>>> print(d)
15
>>> print(h)
0

static ring_inclination(epoch)[source]¶

This function computes the inclination of the plane of Saturn’s ring, referred to the ecliptic and mean equinox of the date.

Parameters:	epoch (`Epoch`) – Epoch to compute the ring inclination
Returns:	Inclination of the ring referred to the ecliptic and mean equinox of the date
Return type:	`Angle`
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 16.00068)
>>> i = Saturn.ring_inclination(epoch)
>>> print(round(i, 6))
28.076131

static ring_logitude_ascending_node(epoch)[source]¶

This function computes the longitude of the ascending node of the plane of Saturn’s ring, referred to the ecliptic and mean equinox of the date.

Parameters:	epoch (`Epoch`) – Epoch to compute the ring longitude of ascending node
Returns:	Longitude of the ascending node of the ring, referred to the ecliptic and mean equinox of the date
Return type:	`Angle`
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 16.00068)
>>> omega = Saturn.ring_logitude_ascending_node(epoch)
>>> print(round(omega, 6))
169.410243

static ring_parameters(epoch)[source]¶

This function computes the parameters related to Saturn’s ring, like geocentric position angle of the semiminor axis, the size of the major and minor axes of the outer edge, etc.

Parameters:	epoch (`Epoch`) – Epoch closest to the node passage
Returns:	Tuple containing: Saturnicentic latitude of the Earth referred to the plane of the ring (B), positive towards the north (`Angle`). Saturnicentric latitude of the Sun referred to the plane of the ring (B’), positive towards the north (`Angle`). Geocentric position angle of the northern semiminor axis of the apparent ellipse of the ring (P), measured from the north towards the east. This is also the position angle of the north pole of rotation of the planet (`Angle`). Difference between the Saturnicentric longitudes of the Sun and the Earth (delta_U), measured in the plane of the ring (`Angle`). The size of major axis of the outer edge of the outer ring (‘a’), in arcseconds (float). The size of minor axis of the outer edge of the outer ring (‘b’), in arcseconds (float).
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

Note

Factors by which the axes ‘a’ and ‘b’ of the outer edge of the outer ring are to be multiplied to obtain the axes of:

Inner edge of outer ring: 0.8801
Outer edge of inner ring: 0.8599
Inner edge of inner ring: 0.6650
Inner edge of dusky ring: 0.5486

>>> epoch = Epoch(1992, 12, 16.00068)
>>> B, Bprime, P, delta_U, a, b = Saturn.ring_parameters(epoch)
>>> print(round(B, 3))
16.442
>>> print(round(Bprime, 3))
14.679
>>> print(round(P, 3))
6.741
>>> print(round(delta_U, 3))
4.198
>>> print(round(a, 2))
35.87
>>> print(round(b, 2))
10.15

static station_longitude_1(epoch)[source]¶

This method computes the time of the 1st station in longitude (i.e. when the planet is stationary and begins to move westward - retrograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	Time when the 1st station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2018, 11, 1.0)
>>> sta1 = Saturn.station_longitude_1(epoch)
>>> y, m, d = sta1.get_date()
>>> print(y)
2018
>>> print(m)
4
>>> print(round(d, 4))
17.9433

static station_longitude_2(epoch)[source]¶

This method computes the time of the 2nd station in longitude (i.e. when the planet is stationary and begins to move eastward - prograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	Time when the 2nd station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2018, 11, 1.0)
>>> sta2 = Saturn.station_longitude_2(epoch)
>>> y, m, d = sta2.get_date()
>>> print(y)
2018
>>> print(m)
9
>>> print(round(d, 4))
6.4175

pymeeus.Saturn.VSOP87_B = [[[4330678.04, 3.60284428399, 213.299095438], [240348.303, 2.8523848939, 426.598190876], [84745.939, 0.0, 0.0], [30863.357, 3.48441504465, 220.4126424388], [34116.063, 0.57297307844, 206.1855484372], [14734.07, 2.1184659787, 639.897286314], [9916.668, 5.79003189405, 419.4846438752], [6993.564, 4.73604689179, 7.1135470008], [4807.587, 5.43305315602, 316.3918696566], [4788.392, 4.9651292742, 110.2063212194], [3432.125, 2.73255752123, 433.7117378768], [1506.129, 6.01304536144, 103.0927742186], [1060.298, 5.63099292414, 529.6909650946], [969.071, 5.20434966103, 632.7837393132], [942.05, 1.39646678088, 853.196381752], [707.645, 3.80302329547, 323.5054166574], [552.313, 5.13149109045, 202.2533951741], [399.675, 3.35891413961, 227.5261894396], [316.063, 1.99716764199, 647.0108333148], [319.38, 3.6257155098, 209.3669421749], [284.494, 4.88648481625, 224.3447957019], [314.225, 0.4651027241, 217.2312487011], [236.442, 2.13887472281, 11.0457002639], [215.354, 5.94982610103, 846.0828347512], [208.522, 2.12003893769, 415.5524906121], [178.958, 2.95361514672, 63.7358983034], [207.213, 0.73021462851, 199.0720014364], [139.14, 1.9982199094, 735.8765135318], [134.884, 5.24500819605, 742.9900605326], [140.585, 0.64417620299, 490.3340891794], [121.669, 3.11537140876, 522.5774180938], [139.24, 4.59535168021, 14.2270940016], [115.524, 3.10891547171, 216.4804891757], [114.218, 0.96261442133, 210.1177017003], [96.376, 4.48164339766, 117.3198682202], [80.593, 1.3169275015, 277.0349937414], [72.952, 3.0598848237, 536.8045120954], [69.261, 4.92378633635, 309.2783226558], [74.302, 2.8937653962, 149.5631971346], [68.04, 2.18002263974, 351.8165923087], [61.734, 0.67728106562, 1066.49547719], [56.598, 2.60963391288, 440.8252848776], [48.864, 5.78725874107, 95.9792272178], [48.243, 2.1821183743, 74.7815985673], [38.304, 5.29151303843, 1059.3819301892], [36.323, 1.63348365121, 628.8515860501], [35.055, 1.71279210041, 1052.2683831884], [34.27, 2.45740470599, 422.6660376129], [34.313, 5.97994514798, 412.3710968744], [33.787, 1.14073392951, 949.1756089698], [31.633, 4.14722153007, 437.6438911399], [36.833, 6.27769966148, 1162.4747044078], [26.98, 1.2715481681, 860.3099287528], [23.516, 2.74936525342, 838.9692877504], [23.46, 0.98962849901, 210.8514148832], [23.6, 4.11386961467, 3.9321532631], [23.631, 3.07427204313, 215.7467759928], [20.813, 3.51084686918, 330.6189636582], [19.509, 2.81857577372, 127.4717966068], [17.103, 3.89784279922, 214.2623032845], [17.635, 6.19715516746, 703.6331846174], [17.824, 2.28524493886, 388.4651552382], [20.935, 0.14356167048, 430.5303441391], [16.551, 1.66649120724, 38.1330356378], [19.1, 2.97699096081, 137.0330241624], [15.517, 4.54798410406, 956.2891559706], [17.065, 0.16611115812, 212.3358875915], [14.169, 0.48937283445, 213.3472795478], [19.027, 6.27326062836, 423.4167971383], [13.344, 2.37136126257, 429.7795846137], [12.565, 1.03178071173, 563.6312150384], [14.173, 3.57477564831, 213.2509113282], [11.374, 1.45300927024, 1368.660252845], [10.585, 6.1763342593, 200.7689224658], [10.6, 3.84358958373, 138.5174968707], [10.263, 2.17423692422, 76.2660712756], [10.072, 1.33197220789, 565.1156877467], [12.058, 0.441492427, 222.8603229936], [10.367, 1.85278552549, 350.3321196004], [8.706, 2.58144528603, 1155.361157407], [8.47, 1.97890349826, 625.6701923124], [8.518, 4.51649648578, 3.1813937377], [7.439, 4.92597321442, 212.7778305762], [7.409, 2.03506679104, 288.0806940053], [8.137, 3.98500592467, 85.8272988312], [7.985, 2.20794292064, 362.8622925726], [6.61, 6.14944028835, 417.0369633204], [7.753, 6.2366454907, 1478.8665740644], [6.318, 1.87388481013, 654.1243803156], [6.319, 1.17328438271, 1265.5674786264], [5.841, 2.35829915285, 750.1036075334], [5.808, 5.01602242794, 479.2883889155], [8.079, 0.42574715104, 554.0699874828], [6.014, 5.58952234348, 425.1137181677], [7.444, 5.41859596459, 213.8203602998], [7.567, 2.68446523795, 191.2076949102], [7.421, 4.19354269508, 9.5612275556], [5.466, 3.21737829505, 234.6397364404], [5.661, 1.4698271355, 265.9892934775], [5.851, 4.81776629912, 1.4844727083], [5.341, 3.45755372717, 203.7378678824], [4.96, 1.04628615559, 12.5301729722], [4.92, 3.85622235967, 173.9422195228], [4.883, 1.94823282939, 195.1398481733], [5.621, 0.81869581274, 52.6901980395], [5.2, 3.32827437636, 515.463871093], [4.927, 3.81806549732, 225.8292684102], [5.033, 0.10756875163, 252.6559713532], [4.416, 5.45506938037, 408.4389436113], [4.169, 1.21145214135, 1685.0521225016], [4.066, 6.24213578122, 1279.794572628], [3.972, 6.13850317719, 217.491881132], [5.398, 5.67212179194, 1375.7737998458], [3.916, 5.96105725915, 210.3783341312], [4.017, 0.99840226682, 842.1506814881], [3.899, 4.58983662507, 1272.6810256272], [3.764, 3.30663337976, 212.5483359126], [4.345, 3.1856224183, 414.0680179038], [3.565, 4.75262007127, 207.8824694666], [3.542, 2.30814954338, 1471.7530270636], [3.732, 1.61040235688, 635.9651330509], [3.709, 2.97082943086, 223.5940361765], [3.576, 3.83436862558, 483.2205421786], [4.053, 3.72105017218, 942.062061969], [3.756, 0.74987556308, 214.0498549634], [3.162, 3.64550741, 207.6700211455], [3.149, 2.27647454229, 728.762966531], [3.971, 4.37874143597, 216.2198567448], [3.541, 5.62281936827, 218.7157214094], [2.965, 3.40117024932, 650.9429865779], [3.949, 4.1972891245, 209.106309744], [2.853, 4.81077453523, 231.4583427027], [2.826, 0.86682282341, 217.964961884], [2.97, 5.75162134301, 160.6088973985], [2.724, 0.47941267764, 497.4476361802], [2.787, 4.03144791896, 62.2514255951], [2.605, 5.04152791794, 65.2203710117], [2.652, 0.30602610654, 424.1505103212], [2.543, 2.76499056123, 543.9180590962], [2.485, 5.78396049817, 99.1606209555], [3.209, 0.42853440759, 218.9281697305], [2.509, 2.947045891, 70.8494453042], [2.308, 0.63861650866, 251.4321310758], [2.869, 4.31959346745, 767.3690829208], [2.232, 0.56929937568, 1073.6090241908], [2.245, 1.69945964547, 488.8496164711], [2.283, 1.55589787463, 601.7642506762], [2.384, 4.4458349331, 21.3406410024], [2.096, 5.77425542767, 88.865680217], [2.363, 3.35786310868, 124.433415221], [2.162, 6.24029269257, 1795.258443721], [2.452, 3.19804814047, 208.633228992], [2.033, 4.87029603776, 327.4375699205], [1.95, 5.56004000293, 18.1592472647], [2.283, 4.22375355881, 22.0914005278], [2.217, 4.6143309463, 302.164775655], [1.888, 5.45600788064, 142.4496501338], [2.075, 3.55622165076, 1169.5882514086], [2.069, 2.75786819366, 491.8185618877], [1.81, 5.96568495526, 213.1872208534], [1.813, 1.39785500313, 211.8146227297], [1.843, 1.15001484281, 203.0041546995], [1.854, 1.41350087782, 1581.959348283], [1.697, 3.23613719814, 427.5613987225], [1.736, 5.45933992115, 916.9322800554], [1.714, 6.14729146384, 643.8294395771], [1.948, 5.70817363392, 425.6349830295], [1.77, 3.36194411768, 248.7238180901], [1.611, 0.97888081762, 2001.4439921582], [1.971, 2.59654430358, 429.0458714308], [1.628, 0.74011617198, 177.8743727859], [1.564, 2.04342011485, 1788.1448967202], [1.574, 6.01995224314, 426.6463749858], [1.939, 5.4813466938, 636.7158925763], [1.651, 4.61629429952, 621.7380390493], [1.552, 2.55542734908, 692.5874843535], [1.484, 6.17173980637, 56.6223513026], [1.782, 3.26122906302, 175.1660598002], [1.503, 2.59953333916, 228.276948965], [1.559, 0.36281050773, 776.9303104764], [1.799, 3.4639597697, 1258.4539316256], [1.521, 3.75588462293, 213.5115437591], [1.81, 4.37552745264, 213.4109700226], [1.521, 0.30214462385, 213.0866471169], [1.608, 4.66724132818, 269.9214467406], [1.525, 1.47939329423, 198.321241911], [1.408, 1.3849184675, 501.3797894433], [1.327, 0.23760037979, 148.0787244263], [1.305, 2.41201772023, 275.5505210331], [1.578, 2.82443242444, 426.5500067662], [1.203, 1.36928065935, 235.3904959658], [1.296, 5.75203277385, 1692.1656695024], [1.295, 3.04090959062, 831.8557407496], [1.283, 1.62400181159, 643.0786800517], [1.617, 1.31181209458, 214.7835681463], [1.346, 4.01262069353, 278.5194664497], [1.166, 0.09590019561, 340.7708920448], [1.115, 2.20460481017, 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[0.197, 5.39156324804, 316.3918696566], [0.236, 2.11084590211, 210.1177017003], [0.172, 6.09682874401, 860.3099287528], [0.159, 5.95049154821, 216.4804891757], [0.1, 1.98534903594, 625.6701923124], [0.112, 0.85526419268, 654.1243803156], [0.115, 5.03884718594, 117.3198682202], [0.115, 0.44589613974, 110.2063212194]], [[7.895, 2.81927558645, 206.1855484372], [1.014, 0.5118721027, 220.4126424388], [0.772, 2.99484124049, 199.0720014364], [0.967, 3.14159265359, 0.0], [0.583, 5.96456944075, 433.7117378768], [0.588, 0.78008666397, 227.5261894396], [0.445, 2.38630799074, 419.4846438752], [0.098, 5.10622131539, 647.0108333148], [0.091, 5.81659714144, 7.1135470008], [0.088, 6.17828532308, 440.8252848776], [0.089, 0.5839686453, 213.299095438]]]¶: This table contains Saturn’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 440-442.

pymeeus.Saturn.VSOP87_L = [[[87401354.029, 0.0, 0.0], [11107659.78, 3.96205090194, 213.299095438], [1414150.958, 4.58581515873, 7.1135470008], [398379.386, 0.52112025957, 206.1855484372], [350769.223, 3.30329903015, 426.598190876], [206816.296, 0.24658366938, 103.0927742186], [79271.288, 3.8400707853, 220.4126424388], [23990.338, 4.6697693486, 110.2063212194], [16573.583, 0.43719123541, 419.4846438752], [14906.995, 5.76903283845, 316.3918696566], [15820.3, 0.9380895376, 632.7837393132], [14609.562, 1.56518573691, 3.9321532631], [13160.308, 4.44891180176, 14.2270940016], [15053.509, 2.71670027883, 639.897286314], [13005.305, 5.98119067061, 11.0457002639], [10725.066, 3.12939596466, 202.2533951741], [5863.207, 0.23657028777, 529.6909650946], [5227.771, 4.2078316238, 3.1813937377], [6126.308, 1.76328499656, 277.0349937414], [5019.658, 3.17787919533, 433.7117378768], [4592.541, 0.61976424374, 199.0720014364], [4005.862, 2.24479893937, 63.7358983034], [2953.815, 0.98280385206, 95.9792272178], [3873.696, 3.22282692566, 138.5174968707], [2461.172, 2.03163631205, 735.8765135318], [3269.49, 0.77491895787, 949.1756089698], [1758.143, 3.26580514774, 522.5774180938], [1640.183, 5.50504966218, 846.0828347512], [1391.336, 4.02331978116, 323.5054166574], [1580.641, 4.3726631412, 309.2783226558], [1123.515, 2.83726793572, 415.5524906121], [1017.258, 3.71698151814, 227.5261894396], [848.643, 3.19149825839, 209.3669421749], [1087.237, 4.18343232481, 2.4476805548], [956.752, 0.50740889886, 1265.5674786264], [789.205, 5.00745123149, 0.9632078465], [686.965, 1.74714407827, 1052.2683831884], [654.47, 1.59889331515, 0.0481841098], [748.811, 2.14398149298, 853.196381752], [633.98, 2.29889903023, 412.3710968744], [743.584, 5.25276954625, 224.3447957019], [852.677, 3.42141350697, 175.1660598002], [579.857, 3.09259007048, 74.7815985673], [624.904, 0.97046831256, 210.1177017003], [529.861, 4.44938897119, 117.3198682202], [542.643, 1.51824320514, 9.5612275556], [474.279, 5.47527185987, 742.9900605326], [448.542, 1.28990416161, 127.4717966068], [546.358, 2.12678554211, 350.3321196004], [478.054, 2.96488054338, 137.0330241624], [354.944, 3.0128648303, 838.9692877504], [451.827, 1.04436664241, 490.3340891794], [347.413, 1.53928227764, 340.7708920448], [343.475, 0.24604039134, 0.5212648618], [309.001, 3.49486734909, 216.4804891757], [322.185, 0.96137456104, 203.7378678824], [372.308, 2.27819108625, 217.2312487011], [321.543, 2.57182354537, 647.0108333148], [330.196, 0.24715617844, 1581.959348283], [249.116, 1.47010534421, 1368.660252845], [286.688, 2.37043745859, 351.8165923087], [220.225, 4.20422424873, 200.7689224658], [277.775, 0.40020408926, 211.8146227297], [204.5, 6.010822066, 265.9892934775], [207.663, 0.48349820488, 1162.4747044078], [208.655, 1.34516255304, 625.6701923124], [182.454, 5.49122292426, 2.9207613068], [226.609, 4.91003163138, 12.5301729722], [207.659, 1.283022189, 39.3568759152], [173.914, 1.86305806814, 0.7507595254], [184.69, 3.50344404958, 149.5631971346], [183.511, 0.97254952728, 4.192785694], [146.068, 6.23102544071, 195.1398481733], [164.541, 0.4400551752, 5.4166259714], [147.526, 1.53529320509, 5.6290742925], [139.666, 4.29450260069, 21.3406410024], [131.283, 4.06828961903, 10.2949407385], [117.283, 2.67920400584, 1155.361157407], [149.299, 5.73594349789, 52.6901980395], [122.373, 1.97588777199, 4.665866446], [113.747, 5.59427544714, 1059.3819301892], [102.702, 1.19748124058, 1685.0521225016], [118.156, 5.340729339, 554.0699874828], [109.275, 3.43812715686, 536.8045120954], [110.399, 0.1660402409, 1.4844727083], [124.969, 6.27737805832, 1898.3512179396], [89.949, 5.80392934702, 114.1384744825], [103.956, 2.19210363069, 88.865680217], [112.437, 1.10502663534, 191.2076949102], [106.57, 4.01156608514, 956.2891559706], [91.43, 1.8752157751, 38.1330356378], [83.791, 5.48810655641, 0.1118745846], [83.461, 2.28972767279, 628.8515860501], [96.987, 4.53666595763, 302.164775655], [100.631, 4.96513666539, 269.9214467406], [75.491, 2.18045274099, 728.762966531], [96.33, 2.8331918921, 275.5505210331], [82.363, 3.05469876064, 440.8252848776], [73.888, 5.08914205084, 1375.7737998458], [71.633, 5.1094074343, 65.2203710117], [70.409, 4.86846451411, 0.2124483211], [69.76, 3.71029022489, 14.977853527], [88.772, 3.86334563977, 278.5194664497], [68.09, 0.7341546099, 1478.8665740644], [66.501, 0.02677580336, 70.8494453042], [65.682, 2.02165559602, 142.4496501338], [75.765, 1.61410487792, 284.1485407422], [63.153, 3.49493353034, 479.2883889155], [62.539, 2.58713611532, 422.6660376129], [69.313, 3.43979731402, 515.463871093], [79.021, 4.45154941586, 35.4247226521], [63.664, 3.31749528708, 62.2514255951], [52.939, 5.51392725227, 0.2606324309], [53.011, 3.18480701697, 8.0767548473], [54.492, 2.45674090515, 22.0914005278], [50.514, 4.26749346978, 99.1606209555], [55.17, 0.9679744615, 942.062061969], [49.288, 2.38641424063, 1471.7530270636], [47.199, 2.02515248245, 312.1990839626], [61.08, 1.50295092063, 210.8514148832], [45.126, 0.93109376473, 2001.4439921582], [60.556, 2.68715551585, 388.4651552382], [43.452, 2.52602011714, 288.0806940053], [42.544, 3.81793980322, 330.6189636582], [39.915, 5.713786529, 408.4389436113], [50.145, 6.03164759907, 2214.7430875962], [45.86, 0.54229721801, 212.3358875915], [54.165, 0.78154835399, 191.9584544356], [47.016, 4.59934671151, 437.6438911399], [42.362, 1.90070070955, 430.5303441391], [39.722, 1.63259419913, 1066.49547719], [36.345, 0.84756992711, 213.3472795478], [35.468, 4.18603772925, 215.7467759928], [36.344, 3.93295730315, 213.2509113282], [38.005, 0.31313803095, 423.4167971383], [44.746, 1.12488341174, 6.1503391543], [37.902, 1.19795851115, 2.7083129857], [43.402, 1.37363944007, 563.6312150384], [43.764, 3.93043802956, 525.4981794006], [34.825, 1.01566605408, 203.0041546995], [31.755, 1.69273634405, 0.1600586944], [30.88, 6.13525703832, 417.0369633204], [36.388, 6.00586032647, 18.1592472647], [29.032, 1.19660544505, 404.5067903482], [32.812, 0.53649479713, 107.0249274817], [30.433, 0.72335287989, 222.8603229936], [32.644, 0.81204701486, 1795.258443721], [37.769, 3.69666903716, 1272.6810256272], [27.679, 1.45663979401, 7.1617311106], [27.187, 1.89731951902, 1045.1548361876], [37.699, 4.51997049537, 24.3790223882], [34.885, 4.46095761791, 214.2623032845], [32.65, 0.66372395761, 692.5874843535], [30.324, 5.30369950147, 33.9402499438], [27.48, 6.22702216249, 1.2720243872], [26.657, 4.56713198392, 7.065362891], [31.745, 5.49798599565, 56.6223513026], [28.05, 5.64447420566, 128.9562693151], [24.277, 3.93966553574, 414.0680179038], [32.017, 5.22260660455, 92.0470739547], [26.976, 0.06705123981, 205.2223405907], [22.974, 3.6581775177, 207.6700211455], [31.775, 5.59198119173, 6069.7767545534], [23.153, 2.10054506119, 1788.1448967202], [31.025, 0.37190053329, 703.6331846174], [29.376, 0.14742155778, 131.4039498699], [22.562, 5.24009182383, 212.7778305762], [26.185, 5.41311252822, 140.001969579], [25.673, 4.36038885283, 32.2433289144], [20.392, 2.8241390926, 429.7795846137], [20.659, 0.67091805084, 2317.8358618148], [24.397, 3.08740396398, 145.6310438715], [23.735, 2.54365387567, 76.2660712756], [20.157, 5.06708675157, 617.8058857862], [23.307, 3.97357729211, 483.2205421786], [22.878, 6.10452832642, 177.8743727859], [22.978, 3.20140795404, 208.633228992], [20.638, 5.22128727027, 6.592282139], [21.446, 0.72034565528, 1258.4539316256], [18.034, 6.11382719947, 210.3783341312], [22.38, 5.92299908546, 173.9422195228], [19.128, 5.77772013766, 213.8203602998], [20.871, 5.79126331864, 2531.1349572528], [19.327, 1.64147367403, 565.1156877467], [16.806, 3.27953583323, 98.8999885246], [20.833, 2.01655935909, 860.3099287528], [17.939, 3.14329498012, 831.8557407496], [15.653, 3.10137669623, 106.2741679563], [18.235, 5.22595172482, 73.297125859], [19.302, 5.9394711405, 425.1137181677], [14.514, 2.75049388379, 1.2238402774], [14.562, 5.18795088579, 305.3461693927], [14.254, 3.88079504939, 54.1746707478], [14.594, 3.25016810034, 78.7137518304], [13.637, 2.55486219141, 405.2575498736], [13.914, 1.72356993808, 69.1525242748], [13.689, 2.37430586272, 125.9873238985], [13.496, 0.82683590985, 99.9113804809], [18.483, 0.73171264866, 9999.986450773], [13.542, 3.58584380924, 234.6397364404], [13.741, 6.18458356845, 245.5424243524], [16.944, 0.72200792996, 2111.6503133776], [17.441, 0.23803796878, 134.5853436076], [14.181, 4.51963935804, 59.8037450403], [13.598, 2.53776983965, 1.6969210294], [12.24, 2.11973445754, 28.3111756513], [11.988, 1.62114832786, 1361.5467058442], [11.974, 4.0737873512, 280.9671470045], [12.758, 5.31146919749, 344.7030453079], [16.051, 3.97093160336, 355.7487455718], [11.427, 5.51123470805, 192.6921676185], [13.133, 4.69168003518, 767.3690829208], [14.746, 3.28998910617, 1589.0728952838], [11.417, 1.81615681635, 2104.5367663768], [11.626, 2.79410384978, 362.8622925726], [13.234, 4.16642914717, 225.8292684102], [10.599, 5.50554288376, 199.2844497575], [10.558, 3.57501718639, 1.4362885985], [10.485, 2.84462532686, 85.8272988312], [10.296, 0.22225264071, 198.321241911], [10.552, 0.18716643576, 217.491881132], [11.853, 0.11584857323, 7.6348118626], [10.248, 0.2190415417, 144.1465711632], [10.403, 1.68776321208, 31.019488637], [10.313, 4.72132701805, 216.2198567448], [10.719, 2.60869377832, 339.2864193365], [9.636, 3.66746262954, 212.5483359126], [9.631, 3.34275630477, 223.5940361765], [9.684, 0.41556436593, 2634.2277314714], [9.885, 4.01798130416, 207.1487562837], [13.212, 6.00683506785, 214.7835681463], [11.346, 2.61898383052, 7.8643065262], [9.158, 5.39855118256, 342.2553647531], [11.882, 4.00188476744, 267.4737661858], [12.054, 3.59904816676, 124.433415221], [8.921, 4.22716773496, 6.3627874754], [10.142, 3.60807025662, 14.0146456805], [9.35, 0.72255756005, 347.8844390456], [10.529, 2.36779614951, 831.1049812242], [8.587, 4.48439552745, 1692.1656695024], [10.142, 3.93620624488, 207.8824694666], [9.147, 4.28032835242, 312.4597163935], [8.088, 0.81225752596, 264.5048207692], [9.241, 4.26402650779, 20.6069278195], [9.614, 0.64291347187, 218.9281697305], [8.537, 0.48756672382, 1574.8458012822], [7.986, 4.71088791079, 333.657345044], [8.951, 0.90641577433, 497.4476361802], [7.959, 2.73277594136, 4.1446015842], [9.133, 5.08250578843, 241.6102710893], [9.669, 1.60623316904, 0.8937718773], [8.883, 5.55491009279, 2847.5268269094], [8.926, 5.80857835271, 329.7251917809], [7.226, 0.60164771281, 206.233732547], [7.655, 5.53676341721, 116.4260963429], [7.118, 0.18747501525, 209.106309744], [7.507, 5.43555636173, 621.7380390493], [8.885, 5.36210591059, 343.2185725996], [7.056, 0.41911130648, 756.3233826569], [8.124, 4.05571025939, 237.6781178262], [8.964, 1.6502392713, 210.3301500214], [6.961, 3.17855200943, 543.0242872189], [8.916, 0.56503620503, 2428.0421830342], [6.926, 3.66869171435, 247.2393453818], [8.982, 4.25046722481, 46.470422916], [7.089, 5.14399672225, 231.4583427027], [7.381, 1.25092810119, 217.964961884], [7.134, 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429.0458714308], [0.17, 5.98716489326, 643.8294395771], [0.173, 5.19933564968, 1354.4331588434], [0.195, 4.50165508529, 2214.7430875962], [0.156, 4.16290662749, 572.2292347475], [0.153, 1.23776248578, 2413.8150890326], [0.15, 0.63076983213, 1478.8665740644], [0.169, 4.28090123029, 305.3461693927], [0.174, 6.23077892653, 3384.3313390048], [0.149, 3.13274908516, 9573.388259897], [0.162, 6.25601818345, 213.2509113282], [0.149, 4.81749019484, 945.9942152321], [0.162, 0.8861012919, 216.2198567448], [0.133, 2.31915371262, 156.6767441354], [0.165, 6.06456216591, 732.6951197941], [0.141, 6.14293754333, 1795.258443721], [0.133, 0.06530337135, 218.7157214094], [0.162, 3.17058130506, 213.3472795478], [0.125, 2.07143636845, 425.1137181677], [0.146, 1.88627500632, 211.8146227297], [0.113, 2.79541965778, 235.3904959658], [0.117, 0.76464798684, 479.2883889155], [0.108, 3.95650672786, 570.7447620392], [0.106, 0.12820734703, 188.0263011725], [0.134, 3.58244908862, 849.2642284889], [0.114, 0.25990388555, 398.1440028728], [0.112, 2.39181495831, 217.491881132], [0.091, 2.50716605179, 121.2520214833], [0.091, 1.75367948574, 213.8203602998], [0.088, 5.26121947108, 395.578702239], [0.096, 3.98832609364, 289.5651667136], [0.091, 0.35318362186, 312.4597163935], [0.112, 1.14387590923, 1802.3719907218], [0.082, 3.73605217214, 207.6700211455], [0.082, 6.06283262812, 210.3783341312], [0.084, 3.34470673492, 67.6680515665], [0.086, 2.7391730018, 5863.5912061162], [0.083, 2.81499116485, 776.9303104764], [0.091, 1.2616009317, 212.7778305762], [0.09, 2.08722491981, 2111.6503133776], [0.08, 2.13136842916, 421.93232443], [0.082, 4.16358350281, 9793.8009023358], [0.077, 2.96973341607, 431.264057322], [0.079, 3.42790361067, 417.0369633204], [0.079, 3.18693585419, 320.3240229197], [0.08, 0.78975763683, 204.7010757289], [0.077, 1.89354243952, 556.5176680376], [0.073, 4.85923277221, 2118.7638603784], [0.071, 3.64551577433, 198.321241911]], [[16038.734, 5.73945377424, 7.1135470008], [4249.793, 4.58539675603, 213.299095438], [1906.524, 4.76082050205, 220.4126424388], [1465.687, 5.91326678323, 206.1855484372], [1162.041, 5.61973132428, 14.2270940016], [1066.581, 3.60816533142, 426.598190876], [239.377, 3.86088273439, 433.7117378768], [236.975, 5.76826451465, 199.0720014364], [165.641, 5.11641150216, 3.1813937377], [131.409, 4.74327544615, 227.5261894396], [151.352, 2.73594641861, 639.897286314], [61.63, 4.74287052463, 103.0927742186], [63.365, 0.22850089497, 419.4846438752], [40.437, 5.47298059144, 21.3406410024], [40.205, 5.9642026672, 95.9792272178], [38.746, 5.83386199529, 110.2063212194], [28.025, 3.01235311514, 647.0108333148], [25.029, 0.9880817074, 3.9321532631], [18.101, 1.02506397063, 412.3710968744], [17.879, 3.31913418974, 309.2783226558], [16.208, 3.89825272754, 440.8252848776], [15.763, 5.61667809625, 117.3198682202], [19.014, 1.91614237463, 853.196381752], [18.262, 4.96738415934, 10.2949407385], [12.947, 1.18068953942, 88.865680217], [17.919, 4.20376505349, 216.4804891757], [11.453, 5.57520615096, 11.0457002639], [10.548, 5.92906266269, 191.9584544356], [10.389, 3.94838736947, 209.3669421749], [8.65, 3.39335369698, 302.164775655], [7.58, 4.87736913157, 323.5054166574], [6.697, 0.38198725552, 632.7837393132], [5.864, 1.05621157685, 210.1177017003], [5.449, 4.64268475485, 234.6397364404], [6.327, 2.25492722762, 522.5774180938], [3.602, 2.30677010956, 515.463871093], [3.229, 2.20309400066, 860.3099287528], [3.701, 3.14159265359, 0.0], [2.583, 4.93447677059, 224.3447957019], [2.543, 0.42393884183, 625.6701923124], [2.213, 3.19814958289, 202.2533951741], [2.421, 4.76621391814, 330.6189636582], [2.85, 0.5860439501, 529.6909650946], [1.965, 4.39525359412, 124.433415221], [2.154, 1.35488209144, 405.2575498736], [2.296, 3.34809165905, 429.7795846137], [2.018, 3.06693569701, 654.1243803156], [1.979, 1.02981005658, 728.762966531], [1.868, 3.09383546177, 422.6660376129], [1.846, 4.1522598545, 536.8045120954], [2.194, 1.18918501013, 1066.49547719], [2.09, 4.15631351317, 223.5940361765], [1.481, 0.37916705169, 316.3918696566], [1.72, 5.82865773356, 195.1398481733], [1.46, 1.57663426355, 81.7521332162], [1.623, 6.03706764648, 742.9900605326], [1.286, 1.66154726117, 63.7358983034], [1.304, 5.02409881054, 956.2891559706], [1.446, 2.10575519127, 838.9692877504], [1.245, 3.8810975277, 269.9214467406], [1.018, 3.72599601656, 295.0512286542], [1.323, 1.38492882986, 735.8765135318], [1.318, 2.33460998999, 217.2312487011], [0.943, 2.75813531246, 284.1485407422], [0.906, 0.71155526266, 846.0828347512], [0.886, 3.83754799777, 447.9388318784], [0.943, 3.31480217015, 18.1592472647], [0.8, 4.71386673963, 56.6223513026], [0.908, 2.02119147951, 831.8557407496], [0.787, 0.80410269937, 1045.1548361876], [0.709, 4.27064410504, 437.6438911399], [0.651, 6.17565900032, 942.062061969], [0.785, 2.40767785311, 203.0041546995], [0.702, 1.64585301418, 423.4167971383], [0.543, 2.86326941725, 184.8449074348], [0.532, 6.25762144463, 1059.3819301892], [0.521, 3.43013038466, 149.5631971346], [0.484, 4.8836606072, 1272.6810256272], [0.437, 5.40220619672, 408.4389436113], [0.388, 2.57589594168, 508.3503240922], [0.421, 4.05836524024, 543.9180590962], [0.375, 1.22747948298, 2324.9494088156], [0.347, 0.59237194522, 22.0914005278], [0.433, 1.69090148012, 1155.361157407], [0.389, 1.46170367972, 1073.6090241908], [0.307, 1.82185086955, 628.8515860501], [0.409, 1.21858750514, 1052.2683831884], [0.309, 0.33610530663, 6076.8903015542], [0.309, 1.42279282226, 6062.6632075526], [0.34, 1.8332577031, 1141.1340634054], [0.303, 2.4158474733, 127.4717966068], [0.305, 5.34154702988, 131.4039498699], [0.298, 2.28594631393, 635.9651330509], [0.372, 1.0372391139, 313.2104759189], [0.338, 0.69100012338, 1361.5467058442], [0.325, 1.78816356937, 1148.2476104062], [0.322, 1.1862880501, 721.6494195302], [0.271, 2.4566315646, 415.5524906121], [0.251, 3.12046701975, 1382.8873468466], [0.254, 3.00353256829, 618.5566453116], [0.295, 0.35280179538, 2730.2069586892], [0.242, 1.52154324392, 70.8494453042], [0.296, 0.89576757167, 2104.5367663768], [0.264, 3.00987438634, 661.2379273164], [0.267, 0.31623829657, 1677.9385755008], [0.27, 2.56774718753, 643.0786800517], [0.261, 1.55058302472, 1457.525933062], [0.246, 2.29214585472, 867.4234757536], [0.269, 3.18157515051, 750.1036075334], [0.272, 1.12208982319, 1788.1448967202], [0.256, 0.37673546414, 1279.794572628], [0.206, 1.81129778306, 497.4476361802], [0.251, 0.61933213502, 2413.8150890326], [0.237, 3.35941544147, 436.8931316145], [0.247, 0.10102936687, 99.9113804809], [0.247, 0.93125798111, 52.6901980395], [0.221, 2.07880035795, 824.7421937488], [0.197, 6.16682223437, 1258.4539316256], [0.229, 5.5791753484, 2943.5060541272], [0.227, 0.43324651601, 2737.32050569], [0.203, 4.12623986247, 337.732510659], [0.214, 3.57607524509, 934.9485149682], [0.212, 1.25688162158, 1773.9178027186], [0.215, 0.8886764788, 1038.0412891868], [0.244, 5.5157208457, 231.4583427027], [0.181, 2.13821830481, 416.3032501375], [0.21, 4.19139167658, 2221.856634597], [0.178, 2.91685344537, 74.7815985673], [0.201, 0.46214583002, 2854.6403739102], [0.236, 4.64388694899, 1905.4647649404], [0.199, 1.54991619669, 1471.7530270636], [0.199, 0.70725247497, 2420.9286360334], [0.162, 2.5148834502, 430.5303441391], [0.16, 1.23508694599, 1596.1864422846], [0.175, 4.14605894816, 2090.3096723752], [0.152, 0.05796022559, 32.2433289144], [0.176, 1.29002070623, 490.3340891794], [0.154, 3.60622857548, 650.9429865779], [0.185, 4.74969742128, 319.5732633943], [0.154, 1.54587199996, 1464.6394800628], [0.108, 4.251257865, 145.6310438715], [0.106, 1.04047809351, 1162.4747044078], [0.114, 2.640557371, 362.8622925726], [0.093, 3.36746275886, 483.2205421786], [0.091, 2.05796248692, 210.8514148832], [0.091, 4.53336314765, 241.7532834412], [0.072, 3.74361312157, 1485.9801210652], [0.076, 3.33892447677, 195.8906076987]], [[1661.894, 3.99826248978, 7.1135470008], [257.107, 2.98436499013, 220.4126424388], [236.344, 3.90241428075, 14.2270940016], [149.418, 2.74110824208, 213.299095438], [109.598, 1.51515739251, 206.1855484372], [113.953, 3.14159265359, 0.0], [68.39, 1.72120953337, 426.598190876], [37.699, 1.23795458356, 199.0720014364], [40.06, 2.04644897412, 433.7117378768], [31.219, 3.0109418409, 227.5261894396], [15.111, 0.82897064529, 639.897286314], [9.444, 3.71485300868, 21.3406410024], [5.69, 2.41995290633, 419.4846438752], [4.47, 1.45120818748, 95.9792272178], [5.608, 1.1560709574, 647.0108333148], [4.463, 2.11783225176, 440.8252848776], [3.229, 4.09278077834, 110.2063212194], [2.871, 2.77203153866, 412.3710968744], [2.796, 3.00730249564, 88.865680217], [2.638, 0.00255721254, 853.196381752], [2.574, 0.39246854091, 103.0927742186], [1.862, 5.07955457727, 309.2783226558], [2.225, 3.77689198137, 117.3198682202], [1.769, 5.19176876406, 302.164775655], [1.921, 2.82884328662, 234.6397364404], [1.805, 2.23816036743, 216.4804891757], [1.211, 1.54685246534, 191.9584544356], [0.765, 3.44501766503, 323.5054166574], [0.763, 4.83197222448, 210.1177017003], [0.613, 4.19052656353, 515.463871093], [0.648, 2.28591710303, 209.3669421749], [0.616, 4.03194472161, 522.5774180938], [0.63, 2.37952532019, 632.7837393132], [0.639, 0.29772678242, 860.3099287528], [0.559, 2.1711006053, 124.433415221], [0.442, 2.23500083592, 447.9388318784], [0.407, 5.4451597099, 1066.49547719], [0.469, 1.26889429317, 654.1243803156], [0.488, 3.20329778617, 405.2575498736], [0.415, 3.12435410343, 330.6189636582], [0.442, 3.38933498625, 81.7521332162], [0.332, 4.12464206608, 838.9692877504], [0.32, 3.18332026736, 529.6909650946], [0.312, 1.40962796637, 429.7795846137], [0.291, 3.18885372262, 1464.6394800628], [0.333, 2.94355912397, 728.762966531], [0.235, 3.67049647573, 1148.2476104062], [0.286, 2.57895004576, 1045.1548361876], [0.223, 3.57980034401, 1155.361157407], [0.261, 2.04564143519, 1677.9385755008], [0.218, 2.61967125327, 536.8045120954], [0.262, 2.48322150677, 625.6701923124], [0.191, 4.39064160974, 1574.8458012822], [0.176, 1.26161895188, 422.6660376129], [0.19, 2.326931712, 223.5940361765], [0.185, 1.08713469614, 742.9900605326], [0.168, 0.69946458053, 824.7421937488], [0.177, 5.02663339078, 203.0041546995], [0.218, 0.40426546037, 867.4234757536], [0.178, 3.67593243311, 831.8557407496], [0.175, 5.75326979098, 1073.6090241908], [0.156, 3.02120117572, 1781.0313497194], [0.148, 2.28313808274, 295.0512286542], [0.15, 3.48436135302, 956.2891559706], [0.152, 1.91404443241, 942.062061969], [0.146, 6.1651969664, 316.3918696566], [0.096, 2.93247663741, 224.3447957019], [0.088, 4.48383632427, 423.4167971383]], [[123.615, 2.25923345732, 7.1135470008], [34.19, 2.16250652689, 14.2270940016], [27.546, 1.19868150215, 220.4126424388], [5.818, 1.21584270184, 227.5261894396], [5.318, 0.23550400093, 433.7117378768], [3.677, 6.22669694355, 426.598190876], [3.057, 2.97372046322, 199.0720014364], [2.861, 4.28710932685, 206.1855484372], [1.617, 6.25265362286, 213.299095438], [1.279, 5.27612561266, 639.897286314], [0.932, 5.56741549127, 647.0108333148], [0.756, 6.17716234487, 191.9584544356], [0.76, 0.69475544472, 302.164775655], [1.038, 0.23516951637, 440.8252848776], [1.007, 3.14159265359, 0.0], [0.549, 4.87733288264, 88.865680217], [0.504, 4.77955496203, 419.4846438752], [0.346, 4.31847547394, 853.196381752], [0.392, 5.69922389094, 654.1243803156], [0.242, 2.05052677361, 323.5054166574], [0.266, 1.11384528244, 234.6397364404], [0.199, 0.88505901097, 309.2783226558], [0.258, 5.10074489186, 95.9792272178], [0.166, 2.40063312194, 515.463871093], [0.155, 4.70433216164, 860.3099287528], [0.089, 1.3637107038, 412.3710968744], [0.102, 0.49450039082, 117.3198682202]]]¶: This table contains Saturn’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 435-440.

pymeeus.Saturn.VSOP87_R = [[[955758135.801, 0.0, 0.0], [52921382.465, 2.39226219733, 213.299095438], [1873679.934, 5.23549605091, 206.1855484372], [1464663.959, 1.64763045468, 426.598190876], [821891.059, 5.93520025371, 316.3918696566], [547506.899, 5.01532628454, 103.0927742186], [371684.449, 2.27114833428, 220.4126424388], [361778.433, 3.13904303264, 7.1135470008], [140617.548, 5.70406652991, 632.7837393132], [108974.737, 3.29313595577, 110.2063212194], [69007.015, 5.94099622447, 419.4846438752], [61053.35, 0.94037761156, 639.897286314], [48913.044, 1.55733388472, 202.2533951741], [34143.794, 0.19518550682, 277.0349937414], [32401.718, 5.47084606947, 949.1756089698], [20936.573, 0.46349163993, 735.8765135318], [20839.118, 1.5210259064, 433.7117378768], [20746.678, 5.33255667599, 199.0720014364], [15298.457, 3.05943652881, 529.6909650946], [14296.479, 2.60433537909, 323.5054166574], [11993.314, 5.98051421881, 846.0828347512], [11380.261, 1.73105746566, 522.5774180938], [12884.128, 1.64892310393, 138.5174968707], [7752.769, 5.85191318903, 95.9792272178], [9796.061, 5.20475863996, 1265.5674786264], [6465.967, 0.17733160145, 1052.2683831884], [6770.621, 3.00433479284, 14.2270940016], [5850.443, 1.45519636076, 415.5524906121], [5307.481, 0.5973753405, 63.7358983034], [4695.746, 2.14919036956, 227.5261894396], [4043.988, 1.64010323863, 209.3669421749], [3688.132, 0.7801613317, 412.3710968744], [3376.457, 3.69528478828, 224.3447957019], [2885.348, 1.38764077631, 838.9692877504], [2976.033, 5.68467931117, 210.1177017003], [3419.551, 4.94549148887, 1581.959348283], [3460.943, 1.85088802878, 175.1660598002], [3400.616, 0.55386747515, 350.3321196004], [2507.63, 3.53851863255, 742.9900605326], [2448.325, 6.18412386316, 1368.660252845], [2406.138, 2.96559220267, 117.3198682202], [2881.181, 0.17960757891, 853.196381752], [2173.959, 0.01508587396, 340.7708920448], [2024.483, 5.05411271271, 11.0457002639], [1740.254, 2.34657043464, 309.2783226558], [1861.397, 5.93361638244, 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5.48677848094, 3995.7744373156], [0.872, 3.02363724703, 306.0969289181], [0.796, 0.4434366454, 486.4019359163], [0.878, 1.82164034386, 135.336103133], [0.791, 2.14989417962, 2620.0006374698], [0.881, 2.39697554334, 289.5651667136], [0.782, 4.50471317138, 980.6681783588], [0.783, 1.14229319753, 540.7366653585], [0.831, 0.69937251013, 421.93232443], [0.77, 2.40292326155, 576.1613880106], [0.95, 5.97621460162, 196.6243208816], [0.814, 4.19303098086, 831.1049812242], [0.969, 4.78071024754, 326.6868103951], [0.76, 0.4486053353, 425.6349830295], [0.907, 0.94781730418, 525.7588118315], [0.788, 0.14287187051, 916.9322800554], [0.801, 1.863831191, 3039.485281345], [0.801, 0.46947170994, 3466.083472221], [0.747, 6.05374861925, 211.8146227297], [0.968, 3.02618272726, 2634.2277314714], [0.739, 2.27110740297, 2303.6087678132], [0.75, 5.48554383902, 173.9422195228], [1.024, 1.9156492556, 229.9738699944], [0.816, 4.98990432666, 4209.0735327536], [0.728, 1.30997967935, 511.5317178299], [0.716, 3.74192651696, 3053.7123753466], [0.727, 0.39191881243, 1493.093668066], [0.717, 2.68899513085, 228.276948965], [0.739, 2.12749199443, 1176.7017984094], [0.805, 0.0718719391, 556.5176680376], [0.835, 3.482878557, 84.9335269539], [0.79, 0.48073040004, 4017.115078318], [0.725, 1.96643065215, 220.4608265486], [0.683, 2.68825142163, 151.0476698429], [0.739, 3.33688408107, 953.1077622329], [0.745, 6.22304530635, 1269.4996318895]], [[20315.005, 3.02186626038, 213.299095438], [8923.581, 3.19144205755, 220.4126424388], [6908.677, 4.35174889353, 206.1855484372], [4087.129, 4.22406927376, 7.1135470008], [3879.041, 2.01056445995, 426.598190876], [1070.788, 4.20360341236, 199.0720014364], [907.332, 2.28344368029, 433.7117378768], [606.121, 3.17458570534, 227.5261894396], [596.639, 4.13455753351, 14.2270940016], [483.181, 1.17345973258, 639.897286314], [393.174, 0.0, 0.0], [229.472, 4.69838526383, 419.4846438752], [188.25, 4.59003889007, 110.2063212194], [149.508, 3.201994444, 103.0927742186], [121.442, 3.76831374104, 323.5054166574], [101.215, 5.81884137755, 412.3710968744], [102.146, 4.70974422803, 95.9792272178], [93.078, 1.43531270909, 647.0108333148], [72.601, 4.15395598507, 117.3198682202], [84.347, 2.63462379693, 216.4804891757], [62.198, 2.31239345505, 440.8252848776], [45.145, 4.37317047297, 191.9584544356], [49.536, 2.38854232908, 209.3669421749], [54.829, 0.30526468471, 853.196381752], [40.498, 1.83836569765, 302.164775655], [38.089, 5.94455115525, 88.865680217], [32.243, 4.01146349387, 21.3406410024], [40.671, 0.6884518321, 522.5774180938], [28.209, 5.77193013961, 210.1177017003], [24.976, 3.06249709014, 234.6397364404], [20.824, 4.92570695678, 625.6701923124], [25.07, 0.73137425284, 515.463871093], [17.485, 5.73135068691, 728.762966531], [18.009, 1.45593152612, 309.2783226558], [16.927, 3.52771580455, 3.1813937377], [13.437, 3.36479898106, 330.6189636582], [11.09, 3.37212682914, 224.3447957019], [11.082, 3.41719974793, 956.2891559706], [9.978, 1.58791582772, 202.2533951741], [11.551, 5.99093726182, 735.8765135318], [10.5, 6.06911092266, 405.2575498736], [9.144, 2.93557421591, 124.433415221], [8.737, 4.65432480769, 632.7837393132], [10.023, 0.58247011625, 860.3099287528], [7.482, 4.50669216436, 942.062061969], [10.091, 0.28268774007, 838.9692877504], [9.243, 2.57034547708, 223.5940361765], [8.652, 1.75808100881, 429.7795846137], [7.564, 1.45635107202, 654.1243803156], [7.058, 5.47394786065, 1045.1548361876], [6.97, 1.51811695028, 422.6660376129], [8.067, 4.48457709292, 742.9900605326], [6.817, 4.83084424818, 316.3918696566], [7.693, 0.43769724671, 831.8557407496], [7.934, 4.20112367712, 195.1398481733], [6.119, 2.33960392135, 269.9214467406], [5.589, 1.14518720694, 284.1485407422], [5.564, 4.18123189068, 529.6909650946], [5.034, 2.12020038657, 295.0512286542], [6.556, 3.42459866876, 10.2949407385], [5.544, 2.46823271699, 536.8045120954], [6.189, 6.0143382752, 1066.49547719], [5.649, 0.82784598388, 217.2312487011], [4.264, 3.23245736673, 1272.6810256272], [4.45, 0.9247780859, 203.0041546995], [3.268, 4.32777516976, 1258.4539316256], [3.655, 0.05832123987, 81.7521332162], [3.951, 0.11124996745, 1155.361157407], [3.773, 6.01157059552, 1052.2683831884], [2.915, 5.64342950039, 3.9321532631], [3.019, 2.19411778004, 447.9388318784], [2.977, 1.8938734255, 149.5631971346], [3.146, 0.19215180096, 1148.2476104062], [2.763, 0.92363342001, 508.3503240922], [2.79, 4.97199778427, 1677.9385755008], [2.608, 2.99591016813, 1589.0728952838], [2.881, 5.40535671721, 1361.5467058442], [2.337, 1.30362271569, 184.8449074348], [2.536, 3.71412120849, 408.4389436113], [2.45, 3.22118361135, 319.5732633943], [2.585, 2.31346415454, 543.9180590962], [2.324, 5.87500715503, 721.6494195302], [1.99, 0.51565577383, 416.3032501375], [2.49, 4.24017800021, 1059.3819301892], [1.935, 2.41463084855, 337.732510659], [1.886, 0.53809070779, 635.9651330509], [1.893, 5.62352727352, 11.0457002639], [2.389, 5.73399981234, 313.2104759189], [1.9, 2.41000566465, 131.5469622218], [1.743, 4.57646237847, 1994.3304451574], [1.913, 5.17436386408, 2854.6403739102], [1.946, 6.23355845623, 1471.7530270636], [1.963, 6.17814558628, 1464.6394800628], [1.838, 5.59464577559, 1038.0412891868], [1.541, 0.60765337379, 210.8514148832], [1.617, 1.75479346067, 195.8906076987], [1.577, 0.55789908488, 2324.9494088156], [1.492, 0.26624235633, 497.4476361802], [1.659, 2.57526072926, 2090.3096723752], [1.809, 1.82317819973, 436.8931316145], [1.566, 6.15328100324, 490.3340891794], [1.771, 6.11741716855, 1073.6090241908], [1.456, 0.85374460914, 415.5524906121], [1.645, 2.95335775161, 437.6438911399], [1.391, 4.1202502856, 1574.8458012822], [1.585, 5.96841377266, 1781.0313497194], [1.507, 3.84895122542, 1251.3403846248], [1.442, 5.32547705924, 2538.2485042536], [1.805, 1.50973093681, 750.1036075334], [1.462, 3.28599831588, 1884.124123938], [1.482, 0.99340744053, 643.0786800517], [1.312, 3.79347668996, 1567.7322542814], [1.665, 0.02551523913, 423.4167971383], [1.469, 5.35285153471, 1354.4331588434], [1.352, 0.69945139243, 867.4234757536], [1.124, 1.79624810407, 618.5566453116], [1.126, 4.70052329245, 113.3877149571], [1.122, 3.9553722427, 1891.2376709388], [1.458, 1.50198846753, 430.5303441391], [1.145, 5.13093399117, 25.2727942655], [1.178, 2.97062300389, 241.7532834412], [1.274, 2.29089799814, 2420.9286360334], [1.071, 0.04888943982, 63.7358983034], [1.377, 5.58271514873, 1382.8873468466], [1.145, 3.10797488346, 2200.5159935946], [1.076, 0.79465514815, 127.4717966068], [1.046, 5.85060227045, 215.7467759928], [1.082, 3.7258944551, 131.4039498699], [1.218, 0.47504349592, 824.7421937488], [1.116, 3.78039049056, 1375.7737998458], [0.969, 5.90752273481, 265.9892934775], [1.23, 1.41325962069, 2634.2277314714], [1.07, 4.80334493874, 1987.2168981566], [0.946, 6.25968535931, 2015.6710861598], [1.03, 1.08973644893, 362.8622925726], [1.072, 5.41838042079, 1279.794572628], [0.88, 1.92224908504, 483.2205421786], [0.878, 2.96591300878, 934.9485149682], [0.879, 2.65659265685, 145.6310438715], [0.872, 6.26261969664, 2.4476805548], [1.082, 4.48298283322, 2214.7430875962], [0.959, 0.74479087918, 16.6747745564], [1.035, 4.05664979327, 231.4583427027], [0.851, 0.09360495322, 628.8515860501], [0.888, 5.98816755324, 2524.021410252], [0.866, 3.1625926563, 2207.6295405954], [0.843, 1.23731248821, 74.7815985673], [0.809, 2.89742868175, 2008.557539159], [0.779, 2.28434811609, 1478.8665740644], [0.99, 5.32604038017, 2428.0421830342], [0.795, 2.3817813581, 2228.9701815978], [0.765, 4.7003367494, 1670.8250285], [1.024, 4.23352869513, 1802.3719907218], [0.831, 5.87457134912, 1368.660252845], [0.717, 5.92144324994, 1685.0521225016], [0.772, 1.15596098579, 3053.7123753466], [0.691, 3.13193109668, 56.6223513026]], [[1202.05, 1.41499446465, 220.4126424388], [707.796, 1.16153570102, 213.299095438], [516.121, 6.2397356833, 206.1855484372], [426.664, 2.46924890293, 7.1135470008], [267.736, 0.18659206741, 426.598190876], [170.171, 5.95926972384, 199.0720014364], [145.113, 1.44211060143, 227.5261894396], [150.339, 0.4797016714, 433.7117378768], [121.033, 2.40527320817, 14.2270940016], [47.332, 5.56857488676, 639.897286314], [15.745, 2.90112466278, 110.2063212194], [16.668, 0.52920774279, 440.8252848776], [18.954, 5.85626429118, 647.0108333148], [14.074, 1.30343550656, 412.3710968744], [12.708, 2.09349305926, 323.5054166574], [14.724, 0.29905316786, 419.4846438752], [11.133, 2.4630482599, 117.3198682202], [11.32, 0.21785507019, 95.9792272178], [9.233, 2.28127318068, 21.3406410024], [9.246, 1.5649631283, 88.865680217], [8.97, 0.68301278041, 216.4804891757], [7.674, 3.59367715368, 302.164775655], [7.823, 4.48688804175, 853.196381752], [8.36, 1.27239488455, 234.6397364404], [9.552, 3.14159265359, 0.0], [4.834, 2.58836294602, 515.463871093], [6.059, 5.1677444874, 103.0927742186], [4.41, 0.02211643085, 191.9584544356], [4.364, 1.59622746023, 330.6189636582], [3.676, 3.29899839673, 210.1177017003], [4.364, 5.97349927933, 654.1243803156], [4.447, 4.97415112184, 860.3099287528], [3.22, 2.72684237392, 522.5774180938], [4.005, 1.59858435636, 405.2575498736], [3.099, 0.75235436533, 209.3669421749], [2.464, 1.19167306488, 124.433415221], [3.088, 1.32258934286, 728.762966531], [2.22, 3.28087994088, 203.0041546995], [2.127, 6.14648095022, 429.7795846137], [2.11, 0.75462855247, 295.0512286542], [2.02, 3.89394929749, 1066.49547719], [2.248, 0.49319150178, 447.9388318784], [2.18, 0.72761059998, 625.6701923124], [1.809, 0.09057839517, 942.062061969], [1.672, 1.39635398184, 224.3447957019], [1.641, 3.0246830755, 184.8449074348], [1.772, 0.81879250825, 223.5940361765], [1.902, 2.00472814984, 831.8557407496], [1.6, 5.41185167676, 824.7421937488], [1.505, 5.95520747253, 422.6660376129], [1.133, 1.11512973946, 838.9692877504], [1.19, 1.89600567803, 956.2891559706], [1.487, 2.11906469507, 529.6909650946], [1.409, 0.72254420236, 536.8045120954], [1.125, 0.89062692183, 721.6494195302], [1.301, 1.64867038984, 17.4084877393], [1.164, 5.9695798184, 195.1398481733], [0.95, 5.3608071329, 316.3918696566], [0.985, 3.05768671768, 1574.8458012822], [1.05, 1.59202481523, 735.8765135318], [0.817, 4.92838813598, 56.6223513026], [0.78, 2.72125404102, 508.3503240922], [0.969, 1.00708261792, 1045.1548361876], [0.716, 1.11042181341, 1169.5882514086]], [[128.612, 5.91282565136, 220.4126424388], [32.273, 0.69256228602, 7.1135470008], [26.698, 5.91428528629, 227.5261894396], [19.923, 0.67370653385, 14.2270940016], [20.223, 4.95136801768, 433.7117378768], [13.537, 1.45669521408, 199.0720014364], [14.097, 2.67074280191, 206.1855484372], [13.364, 4.5882699637, 426.598190876], [7.257, 4.62966127155, 213.299095438], [4.876, 3.61448275002, 639.897286314], [3.136, 4.65661021909, 191.9584544356], [2.917, 0.48665273315, 323.5054166574], [3.759, 4.89624165044, 440.8252848776], [3.303, 4.07190859545, 647.0108333148], [2.883, 3.18003019204, 419.4846438752], [2.338, 3.69553554327, 88.865680217], [1.95, 5.3272924778, 302.164775655], [2.052, 3.31663577368, 95.9792272178], [1.591, 2.67009215574, 853.196381752], [2.028, 0.56025552769, 117.3198682202], [1.56, 0.85608042681, 515.463871093], [1.678, 0.0, 0.0], [1.102, 5.98011943842, 3.1813937377], [1.285, 5.82563377753, 234.6397364404], [0.896, 5.22791858719, 216.4804891757], [1.141, 0.15741228205, 412.3710968744], [0.798, 0.37452846153, 28.4541880032], [0.837, 5.04769794123, 124.433415221]]]¶: This table contains Saturn’s periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 442-445.

Sun¶

Module holding functions to handle coordinates.

class pymeeus.Sun.Sun[source]¶

Class Sun handles the parameters related to the Sun.

__init__()[source]¶

Sun constructor.

Returns:	Sun object.
Return type:	`Sun`

__weakref__¶: list of weak references to the object (if defined)

static apparent_geocentric_position(epoch, nutation=True)[source]¶

This method computes the apparent geocentric position of the Sun for a given epoch, using the VSOP87 theory.

Parameters:	epoch (bool) – Epoch to compute Sun position, as an Epoch object nutation – Whether the nutation correction will be applied
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> lon, lat, r = Sun.apparent_geocentric_position(epoch)
>>> print(lon.to_positive().dms_str(n_dec=3))
199d 54' 21.548''
>>> print(lat.dms_str(n_dec=3))
0.721''
>>> print(round(r, 8))
0.99760852

static apparent_longitude_coarse(epoch)[source]¶

This method provides the Sun’s apparent longitude with a relatively low accuracy of about 0.01 degree.

Parameters:	epoch (`Epoch`) – Epoch to compute the position of the Sun
Returns:	A tuple containing the sun_apparent (ecliptical) longitude (as an Angle object) and the radius vector in astronomical units.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 10, 13)
>>> app_lon, r = Sun.apparent_longitude_coarse(epoch)
>>> print(app_lon.dms_str(n_dec=0))
199d 54' 32.0''
>>> print(round(r, 5))
0.99766

static apparent_rightascension_declination_coarse(epoch)[source]¶

This method provides the Sun’s apparent right ascension and declination with a relatively low accuracy of about 0.01 degree.

Parameters:	epoch (`Epoch`) – Epoch to compute the position of the Sun
Returns:	A tuple containing the right ascension and the declination (as Angle objects) and the radius vector in astronomical units.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epo = Epoch(1992, 10, 13)
>>> ra, delta, r = Sun.apparent_rightascension_declination_coarse(epo)
>>> print(ra.ra_str(n_dec=1))
13h 13' 31.4''
>>> print(delta.dms_str(n_dec=0))
-7d 47' 6.0''
>>> print(round(r, 5))
0.99766

static beginning_synodic_rotation(number)[source]¶

This method calculates the epoch when the Carrington’s synodic rotation No. ‘number’ starts.

Parameters:	number (int) – Number of Carrington’s synodic rotation
Returns:	Epoch when the provided rotation starts
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.

>>> epoch = Sun.beginning_synodic_rotation(1699)
>>> print(round(epoch(), 3))
2444480.723

static ephemeris_physical_observations(epoch)[source]¶

This method uses Carrington’s formulas to compute the following quantities:

P : position angle of the northern extremity of the axis of rotation
B0 : heliographic latitude of the center of the solar disk
L0 : heliographic longitude of the center of the solar disk

Parameters:	epoch (`Epoch`) – Epoch to compute the parameters
Returns:	Parameters P, B0 and L0, in a tuple
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 10, 13)
>>> p, b0, l0 = Sun.ephemeris_physical_observations(epoch)
>>> print(round(p, 2))
26.27
>>> print(round(b0, 2))
5.99
>>> print(round(l0, 2))
238.63

static equation_of_time(epoch)[source]¶

This method computes the equation of time for a given epoch, understood as the difference between apparent and mean time, or the difference between the hour angle of the true Sun and the mean Sun.

Parameters:	epoch (`Epoch`) – Epoch to compute the equation of time, as an Epoch object
Returns:	Difference between apparent and mean time, as a tuple, in minutes (int) and seconds (float) of time
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> m, s = Sun.equation_of_time(epoch)
>>> print(m)
13
>>> print(round(s, 1))
42.6

static geometric_geocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric geocentric position of the Sun for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Sun position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the geocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> l, b, r = Sun.geometric_geocentric_position(epoch, tofk5=False)
>>> print(round(l.to_positive(), 6))
199.907297
>>> print(b.dms_str(n_dec=3))
0.744''
>>> print(round(r, 8))
0.99760852

static get_equinox_solstice(year, target='spring')[source]¶

This method computes the times of the equinoxes or the solstices.

Parameters:	year (int) – Year we want to compute the equinox or solstice for target (str) – Corresponding equinox or solstice. It can be “spring”, “summer”, “autumn”, “winter”
Returns:	The instant of time when the equinox or solstice happens
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.
Raises:	ValueError if ‘target’ value is invalid.

>>> epoch = Sun.get_equinox_solstice(1962, target="summer")
>>> y, m, d, h, mi, s = epoch.get_full_date()
>>> print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 0)))
1962/6/21 21:24:42.0

static rectangular_coordinates_b1950(epoch)[source]¶

This method computes the rectangular geocentric equatorial coordinates (X, Y, Z) of the Sun, referred to the mean equinox of B1950.0. The X axis is directed towards the vernal equinox (longitude 0), the Y axis lies in the plane of the equator and is directed towards longitude 90, and the Z axis is directed towards the north celestial pole.

Parameters:	epoch (`Epoch`) – Epoch to compute Sun position, as an Epoch object
Returns:	A tuple with the X, Y, Z values in astronomical units
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> x, y, z = Sun.rectangular_coordinates_b1950(epoch)
>>> print(round(x, 8))
-0.94149557
>>> print(round(y, 8))
-0.30259922
>>> print(round(z, 8))
-0.11578695

static rectangular_coordinates_equinox(epoch, equinox_epoch)[source]¶

This method computes the rectangular geocentric equatorial coordinates (X, Y, Z) of the Sun, referred to an arbitrary mean equinox. The X axis is directed towards the vernal equinox (longitude 0), the Y axis lies in the plane of the equator and is directed towards longitude 90, and the Z axis is directed towards the north celestial pole.

Parameters:	epoch (`Epoch`) – Epoch to compute Sun position, as an Epoch object equinox_epoch (`Epoch`) – Epoch corresponding to the mean equinox
Returns:	A tuple with the X, Y, Z values in astronomical units
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> e_equinox = Epoch(2467616.0)
>>> x, y, z = Sun.rectangular_coordinates_equinox(epoch, e_equinox)
>>> print(round(x, 8))
-0.93368986
>>> print(round(y, 8))
-0.32235085
>>> print(round(z, 8))
-0.13977098

static rectangular_coordinates_j2000(epoch)[source]¶

This method computes the rectangular geocentric equatorial coordinates (X, Y, Z) of the Sun, referred to the standard equinox of J2000.0. The X axis is directed towards the vernal equinox (longitude 0), the Y axis lies in the plane of the equator and is directed towards longitude 90, and the Z axis is directed towards the north celestial pole.

Parameters:	epoch (`Epoch`) – Epoch to compute Sun position, as an Epoch object
Returns:	A tuple with the X, Y, Z values in astronomical units
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> x, y, z = Sun.rectangular_coordinates_j2000(epoch)
>>> print(round(x, 8))
-0.93740485
>>> print(round(y, 8))
-0.3131474
>>> print(round(z, 8))
-0.13577045

static rectangular_coordinates_mean_equinox(epoch)[source]¶

This method computes the rectangular geocentric equatorial coordinates (X, Y, Z) of the Sun, referred to the mean equinox of the date. The X axis is directed towards the vernal equinox (longitude 0), the Y axis lies in the plane of the equator and is directed towards longitude 90, and the Z axis is directed towards the north celestial pole.

Parameters:	epoch (`Epoch`) – Epoch to compute Sun position, as an Epoch object
Returns:	A tuple with the X, Y, Z values in astronomical units
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 10, 13.0)
>>> x, y, z = Sun.rectangular_coordinates_mean_equinox(epoch)
>>> print(round(x, 7))
-0.9379963
>>> print(round(y, 6))
-0.311654
>>> print(round(z, 7))
-0.1351207

static true_longitude_coarse(epoch)[source]¶

This method provides the Sun’s true longitude with a relatively low accuracy of about 0.01 degree.

Parameters:	epoch (`Epoch`) – Epoch to compute the position of the Sun
Returns:	A tuple containing the true (ecliptical) longitude (as an Angle object) and the radius vector in astronomical units.
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 10, 13)
>>> true_lon, r = Sun.true_longitude_coarse(epoch)
>>> print(true_lon.dms_str(n_dec=0))
199d 54' 36.0''
>>> print(round(r, 5))
0.99766

Uranus¶

Class to model Uranus planet.

pymeeus.Uranus.ORBITAL_ELEM = [[314.055005, 429.8640561, 0.0003039, 2.6e-08], [19.218446062, -3.72e-08, 9.8e-10, 0.0], [0.04638122, -2.7293e-05, 7.89e-08, 2.4e-10], [0.773197, 0.0007744, 3.749e-05, -9.2e-08], [74.005957, 0.5211278, 0.00133947, 1.8484e-05], [173.005291, 1.486379, 0.00021406, 4.34e-07]]¶: This table contains the parameters to compute Uranus’s orbital elements for the mean equinox of date. Based in Table 31.A, page 213

pymeeus.Uranus.ORBITAL_ELEM_J2000 = [[314.055005, 428.4669983, -4.86e-06, 6e-09], [0.773197, -0.0016869, 3.49e-06, 1.6e-08], [74.005957, 0.0741431, 0.00040539, 1.19e-07], [173.005291, 0.0893212, -9.47e-05, 4.14e-07]]¶: This table contains the parameters to compute Uranus’s orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 215

class pymeeus.Uranus.Uranus[source]¶

Class Uranus models that planet.

__weakref__¶: list of weak references to the object (if defined)

static apparent_heliocentric_position(epoch)[source]¶

This method computes the apparent heliocentric position of planet Uranus for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Uranus position, as an Epoch object
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

static conjunction(epoch)[source]¶

This method computes the time of the conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired conjunction
Returns:	The time when the conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> conj = Uranus.conjunction(epoch)
>>> y, m, d = conj.get_date()
>>> print(y)
1994
>>> print(m)
1
>>> print(round(d, 4))
12.7365

static geocentric_position(epoch)[source]¶

This method computes the geocentric position of Uranus (right ascension and declination) for the given epoch, as well as the elongation angle.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension, the declination and the elongation angle as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 20.0)
>>> ra, dec, elon = Uranus.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
19h 13' 48.7''
>>> print(dec.dms_str(n_dec=1))
-22d 46' 13.0''
>>> print(elon.dms_str(n_dec=1))
18d 44' 18.7''

static geometric_heliocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of planet Uranus for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Uranus position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2018, 10, 27.0)
>>> l, b, r = Uranus.geometric_heliocentric_position(epoch)
>>> print(round(l.to_positive(), 4))
30.5888
>>> print(round(b, 4))
-0.5315
>>> print(round(r, 5))
19.86964

static magnitude(sun_dist, earth_dist)[source]¶

This function computes the approximate magnitude of Uranus.

Parameters:	sun_dist (float) – Distance from Uranus to Sun, in Astronomical Units earth_dist (float) – Distance from Uranus to Earth, in Astronomical Units
Returns:	Uranus’s magnitude
Return type:	float
Raises:	TypeError if input values are of wrong type.

static opposition(epoch)[source]¶

This method computes the time of the opposition closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired opposition
Returns:	The time when the opposition happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1780, 12, 1.0)
>>> oppo = Uranus.opposition(epoch)
>>> y, m, d = oppo.get_date()
>>> print(y)
1780
>>> print(m)
12
>>> print(round(d, 4))
17.5998

static orbital_elements_j2000(epoch)[source]¶

This method computes the orbital elements of Uranus for the standard equinox J2000.0 for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Uranus.orbital_elements_j2000(epoch)
>>> print(round(l, 6))
234.602641
>>> print(round(a, 8))
19.21844604
>>> print(round(e, 7))
0.0463634
>>> print(round(i, 6))
0.772094
>>> print(round(ome, 5))
74.05468
>>> print(round(arg, 6))
99.009058

static orbital_elements_mean_equinox(epoch)[source]¶

This method computes the orbital elements of Uranus for the mean equinox of the date for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Uranus.orbital_elements_mean_equinox(epoch)
>>> print(round(l, 6))
235.517526
>>> print(round(a, 8))
19.21844604
>>> print(round(e, 7))
0.0463634
>>> print(round(i, 6))
0.77372
>>> print(round(ome, 5))
74.34776
>>> print(round(arg, 6))
99.630865

static passage_nodes(epoch, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of Uranus, nearest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch closest to the node passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(2019, 1, 1)
>>> time, r = Uranus.passage_nodes(epoch)
>>> year, month, day = time.get_date()
>>> print(year)
2028
>>> print(month)
8
>>> print(round(day, 1))
23.2
>>> print(round(r, 4))
19.3201

static perihelion_aphelion(epoch, perihelion=True)[source]¶

This method computes the time of Perihelion (or Aphelion) closer to a given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired Perihelion (or Aphelion) peihelion – If True, the epoch of the closest Perihelion is computed, if False, the epoch of the closest Aphelion is found.
Returns:	The epoch of the desired Perihelion (or Aphelion)
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.

Note

The solution provided by this method may have several days of error.

>>> epoch = Epoch(1880, 1, 1.0)
>>> e = Uranus.perihelion_aphelion(epoch)
>>> y, m, d = e.get_date()
>>> print(y)
1882
>>> print(m)
3
>>> print(int(d))
18
>>> epoch = Epoch(2090, 1, 1.0)
>>> e = Uranus.perihelion_aphelion(epoch, perihelion=False)
>>> y, m, d = e.get_date()
>>> print(y)
2092
>>> print(m)
11
>>> print(int(d))
22

pymeeus.Uranus.VSOP87_B = [[[1346277.639, 2.61877810545, 74.7815985673], [62341.405, 5.08111175856, 149.5631971346], [61601.203, 3.14159265359, 0.0], [9963.744, 1.61603876357, 76.2660712756], [9926.151, 0.57630387917, 73.297125859], [3259.455, 1.2611938596, 224.3447957019], [2972.318, 2.24367035538, 1.4844727083], [2010.257, 6.05550401088, 148.0787244263], [1522.172, 0.27960386377, 63.7358983034], [924.055, 4.03822927853, 151.0476698429], [760.624, 6.14000431923, 71.8126531507], [420.265, 5.21279984788, 11.0457002639], [430.668, 3.55445034854, 213.299095438], [436.843, 3.38082524317, 529.6909650946], [522.309, 3.3208519477, 138.5174968707], [434.625, 0.34065281858, 77.7505439839], [462.63, 0.74256727574, 85.8272988312], [232.649, 2.25716421383, 222.8603229936], [215.838, 1.5912170494, 38.1330356378], [244.698, 0.78795150326, 2.9689454166], [179.935, 3.72487952673, 299.1263942692], [174.895, 1.23550262213, 146.594251718], [173.667, 1.93654269131, 380.12776796], [160.368, 5.33635436463, 111.4301614968], [144.064, 5.96239326415, 35.1640902212], [102.049, 2.61876256513, 78.7137518304], [116.363, 5.73877190007, 70.8494453042], [106.441, 0.94103112994, 70.3281804424], [86.163, 0.70262506622, 39.6175083461], [72.617, 0.20564696113, 225.8292684102], [71.172, 0.83343269975, 109.9456887885], [57.502, 2.67039425415, 108.4612160802], [54.255, 3.35166579613, 184.7272873558], [44.47, 2.74408231138, 152.5321425512], [38.591, 5.17394663303, 202.2533951741], [39.157, 2.17108251341, 351.8165923087], [41.346, 3.22134319551, 160.6088973985], [35.14, 4.00111634363, 112.9146342051], [33.073, 3.61378095742, 221.3758502853], [31.315, 2.71969470781, 145.1097790097], [37.336, 4.02053241202, 52.6901980395], [32.028, 1.29160071142, 145.6310438715], [27.574, 3.7006426696, 36.6485629295], [24.277, 2.84989187496, 127.4717966068], [24.635, 1.11645461259, 3.9321532631], [24.315, 5.48987913644, 79.2350166922], [21.418, 0.63722900407, 277.0349937414], [19.826, 2.5933418223, 84.3428261229], [22.373, 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3.9321532631], [2.585, 0.41383633246, 71.6002048296], [1.394, 4.13126838571, 146.594251718], [1.183, 3.68471361409, 77.7505439839], [1.103, 5.54212014132, 222.8603229936], [1.205, 5.05109252937, 380.12776796], [1.146, 1.95280464754, 529.6909650946], [0.977, 1.52652616357, 77.962992305], [1.025, 4.33698643491, 2.9689454166], [0.858, 2.78728745263, 111.4301614968], [0.868, 5.55175791193, 38.1330356378], [0.633, 0.41074353315, 213.299095438], [0.596, 5.39265533517, 127.4717966068], [0.586, 4.00404667232, 109.9456887885], [0.543, 2.40369406419, 153.4953503977], [0.486, 2.05237757516, 299.1263942692], [0.557, 3.13408880388, 65.2203710117], [0.457, 3.96543219832, 454.9093665273], [0.481, 2.81511187371, 160.6088973985], [0.421, 2.16819778071, 56.6223513026], [0.326, 4.5292001243, 39.6175083461], [0.308, 6.26508780547, 70.3281804424], [0.338, 5.14594268587, 3.1813937377], [0.288, 0.47061435406, 22.0914005278], [0.336, 4.23512034174, 35.1640902212], [0.316, 3.93430525759, 52.6901980395], [0.306, 1.10359318443, 70.8494453042], [0.25, 3.58780257084, 202.2533951741], [0.239, 4.7767930608, 87.3117715395], [0.227, 2.74138067839, 12.5301729722], [0.263, 1.59203582407, 84.3428261229], [0.215, 3.88195737361, 131.4039498699], [0.216, 5.80700510713, 74.6697239827], [0.264, 1.82574036051, 77.2292791221], [0.222, 0.0411188355, 184.7272873558], [0.197, 1.56602555362, 9.5612275556], [0.193, 2.2641693816, 75.7448064138], [0.179, 2.69065316892, 145.6310438715], [0.17, 2.75844544119, 73.8183907208], [0.155, 3.5539324911, 18.1592472647], [0.174, 4.76111441901, 277.0349937414], [0.14, 0.2871476287, 221.3758502853], [0.134, 1.29065526326, 206.1855484372], [0.127, 6.17908901556, 62.2514255951], [0.116, 6.26646620658, 220.4126424388]], [[267.832, 1.25097888291, 74.7815985673], [11.048, 3.14159265359, 0.0], [6.154, 4.00663614486, 149.5631971346], [3.361, 5.77804694935, 73.297125859], [1.602, 1.05657834344, 63.7358983034], [1.265, 1.66795295537, 78.7137518304], [1.183, 2.58856450374, 138.5174968707], [1.087, 0.28687213135, 76.2660712756], [0.64, 1.87238784591, 71.6002048296], [0.59, 0.80206040001, 1.4844727083], [0.467, 4.42872012006, 148.0787244263], [0.272, 4.00684090176, 85.8272988312], [0.203, 0.60406901282, 71.8126531507], [0.18, 5.55657564049, 3.9321532631], [0.168, 4.67745630044, 70.8494453042], [0.17, 2.93672195979, 11.0457002639]], [[5.719, 2.85499529315, 74.7815985673], [0.3, 3.14159265359, 0.0]]]¶: This table contains Uranus’ periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 448-449.

pymeeus.Uranus.VSOP87_L = [[[548129294.299, 0.0, 0.0], [9260408.252, 0.8910642153, 74.7815985673], [1504247.826, 3.62719262195, 1.4844727083], [365981.718, 1.89962189068, 73.297125859], [272328.132, 3.35823710524, 149.5631971346], [70328.499, 5.39254431993, 63.7358983034], [68892.609, 6.09292489045, 76.2660712756], [61998.592, 2.26952040469, 2.9689454166], [61950.714, 2.85098907565, 11.0457002639], [26468.869, 3.14152087888, 71.8126531507], [25710.505, 6.11379842935, 454.9093665273], [21078.897, 4.36059465144, 148.0787244263], [17818.665, 1.74436982544, 36.6485629295], [14613.471, 4.73732047977, 3.9321532631], [11162.535, 5.82681993692, 224.3447957019], [10997.934, 0.48865493179, 138.5174968707], [9527.487, 2.95516893093, 35.1640902212], [7545.543, 5.23626440666, 109.9456887885], [4220.17, 3.23328535514, 70.8494453042], [4051.85, 2.27754158724, 151.0476698429], [3354.607, 1.06549008887, 4.4534181249], [2926.671, 4.62903695486, 9.5612275556], [3490.352, 5.48305567292, 146.594251718], [3144.093, 4.75199307603, 77.7505439839], [2922.41, 5.3523674338, 85.8272988312], [2272.79, 4.36600802756, 70.3281804424], [2051.209, 1.51773563459, 0.1118745846], [2148.599, 0.60745800902, 38.1330356378], [1991.726, 4.92437290826, 277.0349937414], [1376.208, 2.04281409054, 65.2203710117], [1666.91, 3.62744580852, 380.12776796], [1284.183, 3.11346336879, 202.2533951741], [1150.416, 0.93344454002, 3.1813937377], [1533.223, 2.58593414266, 52.6901980395], [1281.641, 0.54269869505, 222.8603229936], [1372.1, 4.19641615561, 111.4301614968], [1220.998, 0.19901396193, 108.4612160802], [946.195, 1.19249463066, 127.4717966068], [1150.993, 4.17898207045, 33.6796175129], [1244.342, 0.91612680579, 2.4476805548], [1072.008, 0.23564502877, 62.2514255951], [1090.461, 1.77501638912, 12.5301729722], [707.875, 5.18285226584, 213.299095438], [653.401, 0.96586909116, 78.7137518304], [627.562, 0.18210181975, 984.6003316219], [524.495, 2.01276706996, 299.1263942692], [559.37, 3.35776737704, 0.5212648618], [606.827, 5.43209728952, 529.6909650946], [404.891, 5.98689011389, 8.0767548473], [467.211, 0.41484068933, 145.1097790097], [471.288, 1.40664336447, 184.7272873558], [483.219, 2.10553990154, 0.9632078465], [395.614, 5.87039580949, 351.8165923087], [433.532, 5.52142978255, 183.2428146475], [309.885, 5.83301304674, 145.6310438715], [378.609, 2.34975805006, 56.6223513026], [398.996, 0.33810765436, 415.5524906121], [300.379, 5.64353974146, 22.0914005278], [249.229, 4.74617120584, 225.8292684102], [239.334, 2.35045874708, 137.0330241624], [294.172, 5.83916826225, 39.6175083461], [216.48, 4.77847481363, 340.7708920448], [251.792, 1.63696775578, 221.3758502853], [219.621, 1.92212987979, 67.6680515665], [201.963, 1.29693040865, 0.0481841098], [224.097, 0.51574863468, 84.3428261229], [216.549, 6.14211862702, 5.9378908332], [222.588, 2.84309380331, 0.2606324309], [207.828, 5.5802057004, 68.8437077341], [187.474, 1.31924326253, 0.1600586944], [158.028, 0.73811997211, 54.1746707478], [199.146, 0.9563415501, 152.5321425512], [168.648, 5.87874000882, 18.1592472647], [170.3, 3.67717520688, 5.4166259714], [193.652, 1.88800122606, 456.3938392356], [192.998, 0.91616058506, 453.424893819], [181.934, 3.53624029238, 79.2350166922], [173.145, 1.53860728054, 160.6088973985], [164.588, 1.42379714838, 106.9767433719], [171.968, 5.67952685533, 219.891377577], [162.792, 3.05029377666, 112.9146342051], [146.653, 1.26300172265, 59.8037450403], [139.453, 5.385977234, 32.1951448046], [138.585, 4.25994786673, 909.8187330546], [143.058, 1.29995487555, 35.4247226521], [123.84, 1.37359990336, 7.1135470008], [104.414, 5.02820888813, 0.7507595254], [103.277, 0.68095301267, 14.977853527], [94.741, 0.90674090409, 74.6697239827], [82.978, 2.92828718445, 265.9892934775], [110.163, 2.02685778976, 554.0699874828], [94.226, 3.9426632826, 74.8934731519], [79.858, 1.0144682918, 6.592282139], [109.376, 5.70581833286, 77.962992305], [85.876, 1.70649435603, 82.8583534146], [103.562, 1.45770270246, 24.3790223882], [74.667, 4.63177552576, 69.3649725959], [79.919, 3.00974084247, 297.6419215609], [84.502, 0.36887189574, 186.2117600641], [88.81, 0.52481330563, 181.7583419392], [70.303, 1.18986880009, 66.70484372], [69.965, 0.87476081875, 305.3461693927], [69.927, 3.76102749315, 131.4039498699], [84.604, 5.88725183325, 256.5399405065], [74.341, 6.24271323846, 447.7958195265], [62.31, 0.16901376623, 479.2883889155], [72.726, 2.84892775693, 462.0229135281], [69.06, 4.43934854374, 39.3568759152], [76.568, 4.5872111034, 6.2197751235], [73.387, 4.27603448634, 87.3117715395], [55.307, 1.49636544147, 71.6002048296], [57.291, 1.63015165542, 143.6253063014], [61.661, 3.18604743524, 77.2292791221], [57.634, 3.67180685401, 51.2057253312], [50.289, 1.12279384633, 20.6069278195], [53.744, 5.51890986247, 128.9562693151], [57.894, 2.66877593418, 381.6122406683], [58.112, 1.58629352171, 60.7669528868], [45.382, 0.48053933052, 14.0146456805], [37.581, 6.06822931932, 211.8146227297], [38.64, 3.43597050177, 153.4953503977], [46.087, 4.36201639577, 75.7448064138], [40.088, 4.57333927519, 46.2097904851], [34.229, 2.93967782207, 140.001969579], [38.669, 5.58941074168, 99.1606209555], [34.827, 1.02792863024, 203.7378678824], [40.024, 0.69889667397, 218.4069048687], [32.538, 4.21625657443, 200.7689224658], [31.865, 5.50961503408, 72.3339180125], [41.695, 3.82438031124, 81.0013736908], [34.795, 0.39363490236, 1.3725981237], [39.775, 6.05600836903, 293.188503436], [27.577, 2.18261286374, 125.9873238985], [36.279, 1.66586085405, 258.0244132148], [35.442, 1.96652806541, 835.0371344873], [35.361, 3.7225869003, 692.5874843535], [27.323, 2.10164372072, 209.3669421749], [26.53, 4.48265986115, 373.9079928365], [34.472, 1.07907945481, 191.2076949102], [29.915, 3.87358632506, 259.5088859231], [26.233, 3.63172504384, 490.3340891794], [25.848, 0.54461409359, 41.6444977756], [26.989, 6.27711247734, 28.5718080822], [26.391, 5.81110061049, 75.3028634291], [34.227, 6.05617272657, 275.5505210331], [24.279, 3.18776564878, 81.3738807063], [29.937, 1.88789751816, 269.9214467406], [26.235, 6.20105251336, 134.5853436076], [22.754, 0.92919725789, 288.0806940053], [25.18, 5.42547381962, 116.4260963429], [22.715, 0.53098783687, 1514.2912967165], [26.485, 4.77176167929, 284.1485407422], [27.008, 4.75281624832, 41.1019810544], [21.972, 4.58613057386, 404.5067903482], [22.012, 1.84389287183, 617.8058857862], [24.694, 4.7087519549, 378.6432952517], [28.949, 0.17127584792, 528.2064923863], [20.492, 0.10285646641, 195.1398481733], [20.696, 5.62143477633, 55.6591434561], [25.843, 0.74627159338, 278.5194664497], [22.99, 3.58378694661, 1.5963472929], [21.843, 0.05733533568, 173.9422195228], [19.05, 2.30351091243, 5.1078094307], [20.675, 2.64113858585, 105.4922706636], [21.856, 5.87352402691, 45.5766510387], [21.12, 1.98081790016, 114.3991069134], [19.279, 2.84304025179, 159.1244246902], [19.061, 0.50598371738, 67.3592350258], [20.434, 3.77601951414, 135.5485514541], [17.326, 4.47793157645, 120.358249606], [20.547, 0.88695598555, 255.0554677982], [19.32, 1.48569290504, 0.8937718773], [21.331, 2.7447002306, 28.3111756513], [17.582, 4.091396367, 296.1574488526], [15.918, 3.94525074972, 17.5261078183], [15.562, 0.92748407689, 300.6108669775], [16.439, 0.30868798605, 30.7106720963], [15.237, 4.93048601827, 7.4223635415], [19.284, 6.21950083268, 329.8370663655], [13.86, 0.56255266406, 144.1465711632], [16.206, 2.30292598693, 344.7030453079], [16.041, 0.19723295436, 103.0927742186], [14.414, 2.57606243208, 230.5645708254], [16.789, 4.93540052916, 565.1156877467], [17.052, 1.81844925116, 294.6729761443], [16.766, 0.2754218633, 73.8183907208], [15.428, 1.91577056305, 96.8729990951], [15.718, 3.87095025861, 98.8999885246], [11.923, 6.17545505441, 44.7253177768], [12.407, 6.22419970167, 80.1982245387], [13.04, 1.99652993223, 27.0873353739], [13.229, 3.43782440072, 227.3137411185], [11.669, 4.31526860843, 426.598190876], [14.378, 5.78353646474, 1059.3819301892], [15.879, 0.98454960055, 6208.2942514241], [11.158, 1.7441743069, 220.4126424388], [11.989, 5.8438865795, 13.3333221243], [11.386, 2.55925734515, 19.1224551112], [13.281, 5.39472153462, 391.1734682239], [12.295, 4.57340278496, 23.5758732361], [12.827, 1.7741026907, 180.2738692309], [11.651, 4.29138607818, 142.4496501338], [12.248, 2.44241346243, 100.3844612329], [12.421, 2.32591770919, 80.7194894005], [9.774, 0.39898140151, 7.8643065262], [13.172, 2.74099358938, 177.8743727859], [12.262, 5.42795591646, 831.1049812242], [10.272, 5.90194483926, 74.5209661364], [9.317, 3.75869700774, 74.8297826771], [10.701, 4.00709797731, 235.3904959658], [9.243, 5.38492199672, 20.4468691251], [9.461, 2.60126707172, 92.3077063856], [12.066, 5.5216310022, 74.2603337055], [10.836, 1.88393779293, 241.6102710893], [9.317, 1.1648365831, 74.7334144575], [10.718, 5.50310449842, 187.6962327724], [12.057, 6.0212005039, 154.0166152595], [9.124, 1.15458738606, 0.6331394464], [11.526, 6.26425302826, 155.7829722581], 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0.97965330777, 113.8778420516], [0.319, 5.09582764612, 285.6330134505], [0.28, 0.67871105907, 312.4597163935], [0.3, 5.93285242876, 124.2904028691], [0.32, 4.86151247369, 448.6895914038], [0.31, 3.75000484412, 253.5709950899], [0.311, 5.5968659072, 271.4059194489], [0.316, 1.89686876211, 228.276948965], [0.269, 0.14585942744, 142.6620984549], [0.27, 2.12904548682, 778.4147831847], [0.267, 0.96560769114, 90.8232336773], [0.308, 1.38454900684, 375.3924655448], [0.298, 3.99595366039, 451.9404211107], [0.278, 3.38339026214, 346.3999663373], [0.287, 1.01918432834, 905.8865797915], [0.263, 0.16921968622, 124.5028511902], [0.283, 5.95865378023, 362.8622925726], [0.266, 3.07331582044, 193.655375465], [0.264, 5.47114459575, 133.1008708993], [0.288, 2.78232740152, 1812.5239191084], [0.361, 4.30140629884, 198.321241911], [0.257, 1.60206491208, 369.0820676961], [0.274, 2.88347680082, 233.9060232575], [0.267, 4.90554019072, 681.5417840896], [0.305, 1.55983861329, 49.7212526229], [0.253, 0.50457429429, 316.3918696566], [0.258, 5.81453094409, 630.3360587584], [0.28, 1.15452517706, 986.0848043302], [0.265, 4.93584097286, 831.1049812242], [0.246, 1.2518623362, 134.0640787458], [0.298, 5.75927878031, 902.7051860538], [0.24, 2.84888261768, 44.0921783304], [0.261, 2.20643594285, 73.0846775379], [0.321, 3.4686482782, 372.4235201282], [0.234, 6.06783988023, 147.1155165798], [0.262, 2.69623862046, 167.7224443993], [0.267, 4.05985113852, 75.3028634291], [0.24, 0.48471871511, 172.1971143836], [0.261, 4.64354183979, 535.3200393871], [0.244, 5.85987959874, 507.5995645668], [0.237, 4.79666486485, 377.1588225434], [0.224, 1.94589447357, 593.426863398], [0.226, 3.71637531808, 449.2802922348], [0.233, 5.98739382153, 219.891377577], [0.24, 2.71609791875, 227.5261894396], [0.221, 2.23218400256, 460.5384408198], [0.226, 2.74516124394, 446.3113468182], [0.221, 3.2033980767, 463.5073862364], [0.203, 5.04975483055, 457.8783119439], [0.191, 4.24841510229, 4.665866446], [0.169, 0.59358171769, 983.1158589136], [0.179, 4.12060524413, 310.1720945331], [0.187, 6.22165475247, 294.6729761443], [0.162, 1.30776665222, 248.7238180901], [0.219, 4.17413407057, 303.8616966844], [0.193, 1.64715944768, 91.4563731237], [0.17, 2.18067759964, 66.1835788582], [0.156, 4.92094728667, 68.1893164283], [0.145, 5.51404722738, 280.9671470045], [0.144, 5.81835834612, 75.5323580927], [0.153, 0.48549989656, 144.8973306886], [0.168, 5.81402201452, 149.45132255], [0.15, 4.66632209585, 306.830642101], [0.131, 1.01359934164, 175.1660598002], [0.174, 3.03279013213, 298.2326223919], [0.163, 1.97665571311, 221.1634019642], [0.144, 2.5905808501, 217.2312487011], [0.167, 2.7460416758, 69.1525242748], [0.129, 2.87574897902, 156.1554792736]], [[53033.277, 0.0, 0.0], [2357.636, 2.26014661705, 74.7815985673], [769.129, 4.52561041823, 11.0457002639], [551.533, 3.25814281023, 63.7358983034], [541.532, 2.27573907424, 3.9321532631], [529.473, 4.92348433826, 1.4844727083], [257.521, 3.69059216858, 3.1813937377], [238.835, 5.85806638405, 149.5631971346], [181.904, 6.21763603405, 70.8494453042], [49.401, 6.03101301723, 56.6223513026], [53.504, 1.44225240953, 76.2660712756], [38.222, 1.78467827781, 52.6901980395], [44.753, 3.90904910523, 2.4476805548], [44.53, 0.81152639478, 85.8272988312], [37.403, 4.46228598032, 2.9689454166], [33.029, 0.86388149962, 9.5612275556], [24.292, 2.10702559049, 18.1592472647], [29.423, 5.09818697708, 73.297125859], [22.135, 4.81730808582, 78.7137518304], [22.491, 5.99320728691, 138.5174968707], [17.226, 2.53537183199, 145.6310438715], [21.392, 2.39880709309, 77.962992305], [20.578, 2.16918786539, 224.3447957019], [16.777, 3.46631344086, 12.5301729722], [12.012, 0.01941361902, 22.0914005278], [10.466, 4.45556032593, 62.2514255951], [11.01, 0.0849627437, 127.4717966068], [8.668, 4.25550086984, 7.1135470008], [10.476, 5.16453084068, 71.6002048296], [7.16, 1.24903906391, 5.4166259714], [8.387, 5.50115930045, 67.6680515665], [6.087, 5.44611674384, 65.2203710117], [6.013, 4.51836836347, 151.0476698429], [5.718, 1.8293391534, 202.2533951741], [6.109, 3.36320161279, 447.7958195265], [6.003, 5.72500086735, 462.0229135281], [5.111, 3.52374555791, 59.8037450403], [5.155, 1.05810305746, 131.4039498699], [5.969, 5.61147374852, 148.0787244263], [5.065, 3.36477113418, 4.4534181249], [4.845, 1.20298837109, 71.8126531507], [3.979, 0.67629577193, 77.7505439839], [3.673, 1.76315074166, 351.8165923087], [3.149, 3.83590892865, 45.5766510387], [3.036, 3.32062892682, 160.6088973985], [3.033, 6.14532331482, 77.2292791221], [3.596, 4.57256025582, 454.9093665273], [2.664, 5.36121614612, 269.9214467406], [2.498, 1.04819496324, 69.3649725959], [2.307, 2.69282373897, 84.3428261229], [2.249, 5.07693376112, 14.977853527], [2.228, 1.38937510191, 284.1485407422], [2.064, 4.34647674542, 984.6003316219], [2.105, 2.32047802326, 120.358249606], [1.864, 5.70354779393, 54.1746707478], [2.005, 3.87177765185, 195.1398481733], [1.622, 5.07964536529, 209.3669421749], [1.597, 0.48807990368, 137.0330241624], [1.583, 2.90536212187, 51.2057253312], [1.725, 6.25703202673, 41.6444977756], [2.073, 1.24032244487, 35.1640902212], [1.543, 2.15414338268, 70.3281804424], [1.671, 6.28283232471, 277.0349937414], [1.494, 6.04572758571, 87.3117715395], [1.418, 1.15843502159, 213.299095438], [1.239, 4.63223076077, 92.940845832], [1.238, 2.65969680342, 134.5853436076], [1.273, 5.87964059822, 60.5545045657], [1.16, 1.03320781667, 153.4953503977], [1.43, 4.68022239016, 299.1263942692], [1.117, 2.62506108047, 72.3339180125], [1.142, 4.64615099782, 152.7445908723], [0.974, 2.85233132493, 222.8603229936], [1.046, 4.81299135934, 116.4260963429], [0.872, 3.49659835508, 340.7708920448], [0.952, 2.1083748084, 20.6069278195], [0.964, 2.46471453524, 380.12776796], [0.843, 6.12869288891, 49.5088043018], [0.821, 0.27134156683, 191.2076949102], [0.813, 4.08930465981, 14.2270940016], [0.796, 6.170668463, 344.7030453079], [0.924, 2.11096444289, 14.0146456805], [0.791, 2.38927423348, 58.1068240109], [0.781, 0.7422311595, 408.4389436113], [0.759, 3.77564054479, 80.1982245387], [0.884, 1.99930014838, 265.9892934775], [0.722, 3.10001033669, 422.6660376129], [0.75, 2.33167721991, 358.9301393095], [0.687, 2.0286634204, 33.6796175129], [0.603, 1.10391172652, 55.1378785943], [0.655, 3.85415269764, 16.6747745564], [0.606, 0.15052747979, 28.3111756513], [0.639, 5.16714934188, 23.5758732361], [0.658, 0.75636229109, 76.4785195967], [0.59, 1.73778850095, 8.0767548473], [0.565, 4.92645232089, 35.4247226521], [0.656, 2.34273264083, 38.1330356378], [0.542, 5.97968975563, 146.594251718], [0.518, 3.19086220901, 152.5321425512], [0.536, 4.52808465499, 220.4126424388], [0.489, 4.80633294199, 159.1244246902], [0.491, 0.85765309118, 565.1156877467], [0.483, 3.52583593251, 144.1465711632], [0.521, 5.21561656321, 206.1855484372], [0.477, 4.25420753202, 365.9006739584], [0.466, 5.13219663072, 297.6419215609], [0.557, 0.98387565952, 225.8292684102], [0.531, 4.2253465745, 29.2049475286], [0.5, 3.49663062387, 128.9562693151], [0.445, 2.60797570173, 96.8729990951], [0.466, 6.05585106742, 70.1157321213], [0.425, 1.04692398351, 19.643719973], [0.491, 2.2612339868, 152.0108776894], [0.455, 5.45520675, 333.657345044], [0.458, 0.91654899383, 373.0142209592], [0.52, 5.72828536642, 111.4301614968], [0.432, 1.04604024916, 125.9873238985], [0.387, 2.82547341355, 200.7689224658], [0.383, 1.91679738697, 5.6290742925], [0.504, 1.95816731769, 415.5524906121], [0.37, 3.21958844151, 387.2413149608], [0.379, 2.75940848661, 81.8951455681], [0.345, 2.98021540638, 429.7795846137], [0.368, 6.20331898497, 456.3938392356], [0.335, 5.29062985955, 13.3333221243], [0.32, 0.74685222907, 347.8844390456], [0.307, 1.65925943351, 99.1606209555], [0.284, 2.0943747648, 129.9194771616], [0.275, 0.62680026669, 31.492569389], [0.339, 1.65968150805, 142.4496501338], [0.27, 2.7937834555, 977.4867846211], [0.284, 2.42261530322, 546.956440482], [0.271, 0.59225635449, 1894.4190646765], [0.263, 3.49309481771, 440.6822725257], [0.295, 0.43376026627, 373.9079928365], [0.271, 4.82853730065, 561.1835344836], [0.248, 5.72379940676, 79.2350166922], [0.252, 5.76728095309, 235.3904959658], [0.261, 0.09830739366, 991.7138786227], [0.238, 1.14634663242, 288.0806940053], [0.22, 5.12386263728, 479.2883889155], [0.263, 1.52366362053, 146.3818033969], [0.23, 0.35983101967, 109.9456887885], [0.214, 2.9438876558, 184.7272873558], [0.198, 5.72107161521, 453.424893819], [0.185, 4.7160017678, 108.4612160802], [0.139, 2.99832472431, 211.8146227297], [0.167, 1.1964352228, 39.6175083461], [0.18, 0.80198096578, 183.2428146475], [0.131, 2.73236351123, 522.5774180938], [0.142, 5.03489222377, 536.8045120954]], [[120.936, 0.02418789918, 74.7815985673], [68.064, 4.12084267733, 3.9321532631], [52.828, 2.3896406126, 11.0457002639], [43.754, 2.95965039734, 1.4844727083], [45.3, 2.0442379841, 3.1813937377], [45.806, 0.0, 0.0], [24.969, 4.88741307918, 63.7358983034], [21.061, 4.54511486862, 70.8494453042], [19.897, 2.31320314136, 149.5631971346], [8.901, 1.57548871761, 56.6223513026], [4.271, 0.22777319552, 18.1592472647], [3.613, 5.39244611308, 76.2660712756], [3.488, 4.97622811775, 85.8272988312], [3.479, 4.12969359977, 52.6901980395], [3.572, 0.95052448578, 77.962992305], [2.328, 0.85770961794, 145.6310438715], [2.696, 0.37287796344, 78.7137518304], [1.946, 2.67997393431, 7.1135470008], [2.156, 5.65647821519, 9.5612275556], [1.363, 4.86983744746, 224.3447957019], [1.333, 1.25032115614, 12.5301729722], [1.613, 0.48764377311, 71.6002048296], [1.475, 5.19957293069, 73.297125859], [1.225, 3.93406822032, 22.0914005278], [0.911, 2.18921999026, 127.4717966068], [0.811, 3.98323855938, 462.0229135281], [0.808, 5.06374463008, 447.7958195265], [0.718, 0.34600103024, 5.6290742925], [0.722, 1.05856935832, 138.5174968707], [0.687, 2.93752748595, 131.4039498699], [0.463, 1.58927254512, 151.0476698429], [0.414, 4.3290428762, 120.358249606], [0.372, 0.73596518002, 269.9214467406], [0.345, 3.05968942771, 561.1835344836], [0.338, 5.94221536204, 284.1485407422], [0.343, 4.01891371998, 546.956440482], [0.382, 5.93515231196, 45.5766510387], [0.276, 3.44212110991, 202.2533951741], [0.3, 1.13119175675, 160.6088973985], [0.268, 3.24615387968, 536.8045120954], [0.358, 1.10916640253, 333.657345044], [0.236, 4.65292396535, 387.2413149608], [0.204, 5.81663798296, 373.0142209592], [0.145, 2.75632381347, 92.940845832], [0.149, 0.13764106563, 71.8126531507], [0.156, 2.90936922804, 153.4953503977]], [[113.855, 3.14159265359, 0.0], [5.599, 4.57882424417, 74.7815985673], [3.203, 0.34623003207, 11.0457002639], [1.217, 3.42199121826, 56.6223513026], [0.634, 4.65759668097, 18.1592472647], [0.171, 3.80393539303, 149.5631971346], [0.133, 4.35519131657, 63.7358983034]], [[0.873, 3.14159265359, 0.0]]]¶: This table contains Uranus’ periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 445-448.

pymeeus.Uranus.VSOP87_R = [[[1921264847.881, 0.0, 0.0], [88784984.055, 5.60377526994, 74.7815985673], [3440835.545, 0.32836098991, 73.297125859], [2055653.495, 1.78295170028, 149.5631971346], [649321.851, 4.52247298119, 76.2660712756], [602248.144, 3.86003820462, 63.7358983034], [496404.171, 1.40139934716, 454.9093665273], [338525.522, 1.58002682946, 138.5174968707], [243508.222, 1.57086595074, 71.8126531507], [190521.915, 1.99809364502, 1.4844727083], [161858.251, 2.79137863469, 148.0787244263], [143705.902, 1.38368574483, 11.0457002639], [93192.359, 0.17437193645, 36.6485629295], [71424.265, 4.24509327405, 224.3447957019], [89805.842, 3.66105366329, 109.9456887885], [39009.624, 1.66971128869, 70.8494453042], [46677.322, 1.39976563936, 35.1640902212], [39025.681, 3.36234710692, 277.0349937414], [36755.16, 3.88648934736, 146.594251718], [30348.875, 0.70100446346, 151.0476698429], [29156.264, 3.18056174556, 77.7505439839], [20471.584, 1.555889615, 202.2533951741], [25620.36, 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[3.336, 0.89628904042, 469.1364605289], [3.274, 3.86236880159, 42.5382696529], [3.201, 2.76459652868, 248.7238180901], [3.184, 0.07709843451, 1514.2912967165], [3.783, 5.29835962126, 369.0820676961], [3.266, 2.24754480216, 73.8183907208], [3.055, 2.60120354415, 433.7117378768], [3.051, 4.54953369151, 980.6681783588], [3.062, 1.27089879603, 200.7689224658], [3.055, 1.70878161343, 639.897286314], [3.11, 3.63187411723, 16.6747745564], [3.472, 4.93521260607, 411.620337349], [3.531, 4.49372794858, 881.5075574033], [3.284, 5.59170577331, 472.1748419147], [3.015, 6.02967446446, 291.262087743], [3.467, 2.17484439267, 554.0699874828], [3.138, 0.52367930477, 1094.8066528413], [3.257, 2.49339546514, 451.7279727896], [2.881, 0.50481204892, 305.3461693927], [3.082, 4.20145474081, 146.594251718], [2.883, 2.44983947531, 135.336103133], [2.965, 0.3929499553, 25.2727942655], [2.831, 2.52728803131, 867.2804634017], [2.728, 5.29491477549, 125.9873238985], [2.857, 4.71106805785, 218.9281697305], [2.763, 4.27510031656, 350.3321196004], [2.73, 1.98552777251, 82.8583534146], [2.857, 3.08706426922, 216.4804891757], [3.365, 3.67691210011, 661.0949149645], [2.925, 1.43646759644, 381.6122406683], [2.753, 0.39468041761, 33.6796175129], [2.756, 4.6267249884, 1357.6145525811], [3.45, 2.12911756067, 685.4739373527], [2.571, 5.92862393284, 89.7594520943], [2.677, 0.76342313946, 486.4019359163], [2.689, 4.16436463826, 235.3904959658], [2.646, 3.81808560938, 550.8885937451], [3.369, 3.17071565345, 108.4612160802], [2.613, 5.68333838067, 24.3790223882], [2.736, 1.87107584495, 529.6909650946], [2.606, 4.36605237304, 1080.5795588397], [2.407, 3.07343136742, 391.1734682239], [2.446, 5.7384638154, 535.9107402181], [2.334, 5.18878243102, 1059.3819301892], [2.568, 1.09886876369, 913.0001267923], [2.236, 6.10115874045, 140.001969579], [3.053, 5.35047433775, 681.5417840896]], [[1164.382, 4.73453291602, 74.7815985673], [212.367, 3.34255734999, 63.7358983034], [196.408, 2.98004616318, 70.8494453042], [104.527, 0.95807937648, 11.0457002639], [71.681, 0.02528455665, 56.6223513026], [72.54, 0.99701907912, 149.5631971346], [54.875, 2.59436811267, 3.9321532631], [34.029, 3.81553325635, 76.2660712756], [32.081, 3.5982517784, 131.4039498699], [29.641, 3.44111535957, 85.8272988312], [36.377, 5.65035573017, 77.962992305], [27.663, 0.4283600147, 3.1813937377], [27.464, 2.55126467481, 52.6901980395], [24.569, 5.14034173566, 78.7137518304], [19.39, 5.13477648625, 18.1592472647], [15.767, 0.37116951743, 447.7958195265], [15.441, 5.57271837433, 462.0229135281], [15.232, 3.85998573509, 73.297125859], [15.475, 2.97496547327, 145.6310438715], [17.951, 0.0, 0.0], [15.958, 5.19915553904, 71.6002048296], [11.056, 6.03152659562, 138.5174968707], [10.529, 3.58261852497, 224.3447957019], [7.606, 1.44542030704, 1.4844727083], [8.121, 2.61579604319, 22.0914005278], [7.107, 5.43946774526, 269.9214467406], [6.459, 4.37142319461, 284.1485407422], [6.818, 0.01396812984, 151.0476698429], [8.101, 0.29563819537, 127.4717966068], [5.768, 4.22672716593, 373.0142209592], [5.022, 1.84154937974, 202.2533951741], [4.692, 2.7840457544, 120.358249606], [5.087, 0.77745294804, 62.2514255951], [4.16, 1.83820502779, 72.3339180125], [3.922, 1.88900691473, 209.3669421749], [5.201, 4.15791319343, 195.1398481733], [3.636, 1.99709010456, 65.2203710117], [3.582, 3.92592140377, 124.2904028691], [3.808, 1.04818660873, 92.940845832], [4.241, 3.95755998904, 9.5612275556], [3.497, 1.54139696251, 148.0787244263], [3.195, 2.98608971329, 387.2413149608], [3.95, 1.85721204932, 152.7445908723], [3.277, 1.40881404192, 351.8165923087], [3.605, 1.17366167402, 153.4953503977], [2.94, 6.03594958459, 12.5301729722], [2.744, 5.64674283515, 134.5853436076], [2.8, 0.79480255927, 572.2292347475], [3.054, 5.84310939105, 160.6088973985], [2.662, 1.98593312104, 450.9772132642], [2.7, 2.77036653988, 213.299095438], [2.323, 1.67918985468, 358.9301393095], [2.254, 5.77129530133, 84.3428261229], [2.291, 4.814246016, 536.8045120954], [2.213, 2.20360299816, 465.9550667912]], [[52.996, 3.00838033088, 74.7815985673], [9.887, 1.91399083603, 56.6223513026], [7.008, 5.08677527404, 11.0457002639], [6.728, 5.42924958121, 149.5631971346], [3.703, 5.22728581851, 131.4039498699], [3.361, 1.29695290266, 85.8272988312], [2.664, 0.44064577837, 63.7358983034], [2.309, 0.92380720934, 145.6310438715], [2.383, 6.21390585593, 358.9301393095], [2.288, 2.23425399117, 440.6822725257], [3.093, 3.14159265359, 0.0]]]¶: This table contains Uranus’ periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 449-451.

Venus¶

Class to model Venus planet.

pymeeus.Venus.ORBITAL_ELEM = [[181.979801, 58519.2130302, 0.00031014, 1.5e-08], [0.72332982, 0.0, 0.0, 0.0], [0.00677192, -4.7765e-05, 9.81e-08, 4.6e-10], [3.394662, 0.0010037, -8.8e-07, -7e-09], [76.67992, 0.9011206, 0.00040618, -9.3e-08], [131.563703, 1.4022288, -0.00107618, -5.678e-06]]¶: This table contains the parameters to compute Venus’ orbital elements for the mean equinox of date. Based in Table 31.A, page 212

pymeeus.Venus.ORBITAL_ELEM_J2000 = [[181.979801, 58517.815676, 1.65e-06, -2e-09], [3.394662, -0.0008568, -3.244e-05, 9e-09], [76.67992, -0.2780134, -0.00014257, -1.64e-07], [131.563703, 0.0048746, -0.00138467, -5.695e-06]]¶: This table contains the parameters to compute Venus’ orbital elements for the standard equinox J2000.0. Based on Table 31.B, page 214

pymeeus.Venus.VSOP87_B = [[[5923638.472, 0.26702775813, 10213.285546211], [40107.978, 1.14737178106, 20426.571092422], [32814.918, 3.14159265359, 0.0], [1011.392, 1.08946123021, 30639.856638633], [149.458, 6.25390296069, 18073.7049386502], [137.788, 0.86020146523, 1577.3435424478], [129.973, 3.67152483651, 9437.762934887], [119.507, 3.70468812804, 2352.8661537718], [107.971, 4.53903677647, 22003.9146348698], [92.029, 1.53954562706, 9153.9036160218], [52.982, 2.28138172277, 5507.5532386674], [45.617, 0.72319641722, 10239.5838660108], [38.855, 2.93437865147, 10186.9872264112], [43.491, 6.14015776699, 11790.6290886588], [41.7, 5.99126845246, 19896.8801273274], [39.644, 3.86842095901, 8635.9420037632], [39.175, 3.94960351174, 529.6909650946], [33.32, 4.83194909595, 14143.4952424306], [23.711, 2.90646621218, 10988.808157535], [23.5, 2.00770618322, 13367.9726311066], [21.809, 2.69701424951, 19651.048481098], [20.653, 0.98666685459, 775.522611324], [16.976, 4.13711782135, 10021.8372800994], [17.835, 5.96268643102, 25934.1243310894], [14.949, 5.61075168206, 10404.7338123226], [18.579, 1.80529277514, 40853.142184844], [15.407, 3.29563855296, 11015.1064773348], [12.936, 5.42651448496, 29580.4747084438], [11.962, 3.57604253827, 10742.9765113056], [11.827, 1.190709196, 8624.2126509272], [11.466, 5.12780364967, 6283.0758499914], [9.484, 2.75167834335, 191.4482661116], [13.129, 5.70735942511, 9683.5945811164], [8.583, 0.43182249199, 9786.687355335], [9.763, 0.14614896296, 20618.0193585336], [8.148, 1.30548515603, 15720.8387848784], [6.05, 6.26541665966, 11322.6640983044], [5.955, 4.92235372433, 1059.3819301892], [6.983, 3.44920932146, 17298.1823273262], [6.228, 1.13312070908, 29864.334027309], [6.186, 4.92498052443, 19367.1891622328], [6.155, 2.424139469, 4705.7323075436], [5.204, 3.42528906628, 9103.9069941176], [6.0, 3.57639095526, 3154.6870848956], [4.796, 3.86676184909, 7860.4193924392], [5.289, 4.99182712443, 7084.8967811152], [4.07, 5.5879814474, 12566.1516999828], [3.942, 5.68758787835, 10206.1719992102], [3.797, 3.89520601076, 10192.5101507186], [3.798, 6.06410995916, 10234.0609417034], [3.579, 0.73789669235, 4551.9534970588], [3.641, 2.61501257205, 15874.6175953632], [3.266, 0.97517223854, 23581.2581773176], [2.813, 0.29951755546, 9411.4646150872], [3.048, 2.5108514699, 33794.5437235286], [2.559, 4.58043833032, 801.8209311238], [2.462, 5.05790874754, 29050.7837433492], [2.593, 5.73113176751, 20213.271996984], [2.625, 4.24272906574, 213.299095438], [2.246, 0.82112963936, 28286.9904848612], [2.229, 2.22457598233, 10426.584641649], [1.742, 1.48394229233, 7058.5984613154], [1.66, 5.42775825275, 32217.2001810808], [1.491, 4.64883377941, 1109.3785520934], [2.01, 0.75702888128, 9999.986450773], [1.562, 3.93962080463, 37724.7534197482], [1.538, 2.1730957708, 21535.9496445154], [1.546, 4.70759186462, 14945.3161735544], [1.2, 1.48282382657, 9830.3890139878], [1.224, 5.55090394449, 5661.3320491522], [1.111, 1.20276209213, 9573.388259897], [1.064, 1.98891375536, 26.2983197998], [1.041, 5.38535116069, 7.1135470008], [1.036, 1.16719443387, 8662.240323563], [1.143, 3.20596958337, 3532.0606928114], [1.201, 0.81913312536, 8094.5216858326], [1.005, 2.38429892132, 27511.4678735372], [1.047, 4.56525030769, 20419.45754542119], [0.968, 6.18496721871, 25158.6017197654], [1.044, 1.98055689074, 10596.1820784342], [0.962, 0.48573513747, 23958.6317852334], [0.846, 0.01566400887, 3128.3887650958], [0.792, 5.39686899735, 24356.7807886416], [0.858, 5.34692750735, 41654.9631159678], [0.757, 6.25904553773, 20452.8694122218], [0.801, 4.62406152514, 9929.4262273458], [0.802, 5.3723489252, 10497.1448650762], [0.75, 3.85219782842, 21228.3920235458], [0.7, 1.98097957188, 3930.2096962196], [0.719, 6.11596800207, 10218.8084705184], [0.672, 6.23429601219, 14765.2390432698], [0.639, 5.37566437358, 1589.0728952838], [0.605, 2.4233039112, 10251.3132188468], [0.726, 6.16683781802, 18875.525869774], [0.613, 5.9973118069, 4732.0306273434], [0.72, 3.84286345199, 10207.7626219036], [0.637, 6.17053891156, 10220.3990932118], [0.515, 1.03001478293, 22779.4372461938], [0.574, 0.43813688572, 17085.9586657222], [0.51, 1.41065159851, 9161.0171630226], [0.569, 3.34601425125, 3340.6124266998], [0.608, 1.25236241968, 10175.2578735752], [0.524, 2.3979424867, 26087.9031415742], [0.542, 1.34665646732, 29088.811415985], [0.527, 4.01994270827, 18849.2275499742], [0.569, 1.65498800378, 39264.0692895602], [0.518, 4.96996115446, 30213.258447757], [0.514, 5.78413007838, 12592.4500197826], [0.538, 4.56198493922, 10063.7223490764], [0.484, 4.18538027381, 14919.0178537546], [0.493, 4.79939382739, 9146.790069021], [0.427, 3.76876868949, 11272.6674764002], [0.495, 0.49175293655, 45585.1728121874], [0.494, 3.74345863918, 31441.6775697568], [0.524, 0.97991794166, 30110.1656735384], [0.483, 1.87898057316, 51066.427731055], [0.505, 3.70047474212, 20400.2727726222], [0.351, 4.3402657449, 10137.0194749354], [0.355, 5.56672554631, 18837.49819713819], [0.328, 3.7842737891, 6681.2248533996], [0.349, 4.20550749672, 20956.2620575166], [0.333, 4.44969281739, 28521.0927782546], [0.296, 2.83205515646, 17277.4069318338], [0.311, 2.57334132897, 20809.4676246452], [0.294, 0.75089224483, 3149.1641605882], [0.377, 3.98143308775, 21202.093703746], [0.272, 5.56183082489, 16496.3613962024], [0.314, 0.02584607093, 13745.3462390224], [0.263, 0.55328410985, 36147.4098773004], [0.286, 5.16408902215, 426.598190876], [0.279, 4.29871615943, 19999.97290154599], [0.28, 1.92925047377, 49515.382508407], [0.265, 4.81168402147, 20235.1228263104], [0.273, 5.12740051559, 35371.8872659764], [0.306, 5.28903794869, 382.8965322232], [0.223, 2.50591724074, 26709.6469424134], [0.235, 5.96522395118, 10198.033075026], [0.234, 3.52866583267, 10228.538017396], [0.224, 6.24561979789, 7064.1213856228], [0.251, 2.84739274245, 33019.0211122046], [0.196, 1.5061039379, 31749.2351907264], [0.192, 1.69321442572, 13341.6743113068], [0.18, 6.19353087076, 39793.7602546548], [0.199, 1.1643332188, 22805.7355659936], [0.18, 3.72646417141, 1551.045222648], [0.173, 3.35235705827, 53445.5922046266], [0.195, 1.51901264131, 43232.3066584156], [0.174, 2.84049662693, 9967.4538999816], [0.163, 4.29160537719, 36949.2308084242], [0.169, 0.37000676558, 10459.1171924404], [0.137, 5.61149803116, 10529.6774158676], [0.139, 0.87847805052, 16522.6597160022], [0.139, 4.12576475427, 36301.18868778519], [0.127, 5.14447758616, 5481.2549188676], [0.131, 3.11317801589, 9896.8936765544], [0.131, 0.89697384735, 3442.5749449654], [0.121, 1.32802112907, 38734.3783244656], [0.122, 1.59017183044, 10110.1927719924], [0.123, 2.33714216061, 10316.3783204296], [0.133, 2.90682399304, 9793.8009023358], [0.111, 2.5207763476, 13936.794505134], [0.12, 0.36076947165, 536.8045120954], [0.115, 2.53355582059, 26735.9452622132], [0.108, 2.65839634325, 10232.95530711079], [0.108, 0.55230439694, 10193.61578531121], [0.138, 1.0691923924, 65236.2212932854], [0.101, 3.17012502017, 19317.1925403286], [0.127, 5.63110477712, 10288.0671447783], [0.127, 3.86278127025, 10138.5039476437], [0.137, 2.9335065946, 47162.5163546352], [0.095, 5.03917884334, 52175.8062831484], [0.094, 0.71308489207, 38500.2760310722], [0.092, 5.46204624886, 11764.330768859], [0.096, 1.52914774412, 9690.7081281172], [0.101, 0.83318284426, 6489.776587288], [0.115, 3.76443612245, 522.5774180938], [0.089, 2.53312656681, 10735.8629643048], [0.082, 0.85628515615, 2379.1644735716], [0.103, 5.2268323762, 103.0927742186], [0.09, 2.12423586627, 28313.288804661], [0.09, 0.39668501735, 9580.5018068978], [0.074, 6.0268009555, 3723.508958923], [0.081, 5.25045057985, 10419.4710946482], [0.08, 4.23724598221, 10007.0999977738], [0.091, 2.48874147947, 10846.0692855242], [0.085, 3.82784790321, 51868.2486621788], [0.081, 2.26235214191, 3903.9113764198], [0.097, 0.772950916, 18307.8072320436], [0.094, 0.17063414792, 6872.6731195112], [0.08, 5.62254102739, 29999.959352319], [0.068, 2.7176293667, 16983.9961474566], [0.066, 0.76731351736, 20.7753954924], [0.075, 0.36155638007, 39302.096962196], [0.075, 2.27327165974, 8521.1198767086], [0.058, 2.14482855875, 8631.326197928], [0.064, 5.83569051301, 2118.7638603784], [0.058, 2.98524209824, 19889.76658032659], [0.054, 1.78260668333, 40077.61957352], [0.055, 4.70485939861, 639.897286314], [0.06, 5.8966189292, 41962.5207369374], [0.066, 2.24746237999, 74.7815985673], [0.061, 3.40726181591, 27490.6924780448], [0.051, 3.07811180039, 24150.080051345], [0.057, 2.30081371235, 20529.66386664059], [0.052, 2.37192464233, 29573.361161443], [0.052, 4.76610409132, 57375.8019008462], [0.047, 1.61630288856, 30831.3049047446], [0.054, 5.89684197257, 19903.99367432819], [0.04, 5.32101847424, 42430.4857272918], [0.051, 5.29186795569, 29587.5882554446]], [[513347.602, 1.80364310797, 10213.285546211], [4380.1, 3.38615711591, 20426.571092422], [196.586, 2.53001197486, 30639.856638633], [199.162, 0.0, 0.0], [14.031, 2.27087044687, 9437.762934887], [12.958, 1.50735622957, 18073.7049386502], [11.941, 5.60462450426, 1577.3435424478], [10.324, 5.24224313355, 2352.8661537718], [9.294, 6.07545631303, 22003.9146348698], [7.441, 1.50257909439, 11790.6290886588], [8.031, 0.29371105198, 9153.9036160218], [7.514, 5.0808188599, 10186.9872264112], [4.669, 3.87801635015, 10239.5838660108], [4.399, 3.58872736593, 40853.142184844], [3.975, 1.28397121206, 10404.7338123226], [4.657, 0.75073886819, 5507.5532386674], [3.783, 4.33004753984, 19651.048481098], [3.39, 4.88976070903, 10988.808157535], [3.555, 1.25927550356, 19896.8801273274], [3.479, 5.5079700216, 529.6909650946], [2.884, 0.08549582037, 14143.4952424306], [1.786, 0.37134513186, 13367.9726311066], [1.6, 1.68378002982, 20618.0193585336], [1.539, 1.21683853657, 25934.1243310894], [1.341, 2.90077139758, 15720.8387848784], [0.993, 1.74681248965, 11322.6640983044], [1.165, 6.13437155401, 7860.4193924392], [1.115, 0.6674369038, 29580.4747084438], [0.923, 2.25384969096, 10021.8372800994], [0.965, 1.36425494833, 9683.5945811164], [0.973, 0.39071758442, 6283.0758499914], [0.805, 0.53331923557, 8624.2126509272], [0.913, 0.76046003719, 8635.9420037632], [0.991, 0.5531987933, 19367.1891622328], [0.609, 2.62364470139, 23581.2581773176], [0.532, 5.10925676528, 9786.687355335], [0.476, 6.17672999981, 11015.1064773348], [0.472, 1.696726292, 17298.1823273262], [0.503, 2.65840772485, 29864.334027309], [0.456, 5.01205315518, 10742.9765113056], [0.478, 3.94100005156, 775.522611324], [0.477, 3.71554345922, 10596.1820784342], [0.347, 2.3455106268, 9411.4646150872], [0.458, 2.31894399069, 9999.986450773], [0.374, 3.76878356974, 21228.3920235458], [0.44, 4.33400244581, 15874.6175953632], [0.349, 1.31468836511, 10234.0609417034], [0.31, 5.45422332781, 10192.5101507186], [0.346, 0.94242286364, 1059.3819301892], [0.308, 4.90145899142, 3930.2096962196], [0.331, 4.89498986674, 10206.1719992102], [0.269, 2.39650266204, 801.8209311238], [0.269, 0.00589873499, 9830.3890139878], [0.261, 3.48196147279, 7058.5984613154], [0.29, 0.10953964861, 29050.7837433492], [0.283, 6.12133736787, 20419.45754542119], [0.232, 3.0784585003, 12566.1516999828], [0.265, 4.02431894973, 33794.5437235286], [0.22, 2.37315851889, 4551.9534970588], [0.247, 3.07626728158, 28286.9904848612], [0.202, 3.56872121409, 21535.9496445154], [0.225, 5.7688889632, 213.299095438], [0.217, 0.88382111135, 20213.271996984], [0.172, 6.12653050186, 9161.0171630226], [0.195, 5.472408554, 37724.7534197482], [0.153, 4.07656151671, 27511.4678735372], [0.174, 1.33676849359, 32217.2001810808], [0.157, 5.98474214437, 26.2983197998], [0.163, 5.4551913476, 10426.584641649], [0.129, 2.08748660996, 3128.3887650958], [0.131, 1.51959002513, 10218.8084705184], [0.139, 4.42330401713, 10220.3990932118], [0.126, 2.62296638037, 22779.4372461938], [0.146, 4.69869606856, 25158.6017197654], [0.172, 6.13435208788, 18837.49819713819], [0.157, 5.44507403858, 4705.7323075436], [0.117, 6.18296175153, 20400.2727726222], [0.164, 3.30849473132, 51066.427731055], [0.113, 3.64412860654, 7.1135470008], [0.109, 5.21220660788, 8662.240323563], [0.133, 1.78047296245, 191.4482661116], [0.117, 0.14681677884, 9146.790069021], [0.116, 0.61940521198, 41654.9631159678], [0.096, 1.49631428731, 7084.8967811152], [0.096, 1.21744230443, 10198.033075026], [0.082, 1.45863866349, 10207.7626219036], [0.085, 6.04057728058, 21202.093703746], [0.083, 0.19985600927, 14919.0178537546], [0.077, 5.5013231061, 5661.3320491522], [0.077, 2.00173927326, 10228.538017396], [0.093, 1.85466268819, 45585.1728121874], [0.066, 3.25826124156, 1109.3785520934], [0.089, 0.64100435648, 3154.6870848956], [0.061, 3.80043027736, 11272.6674764002], [0.077, 1.8551635895, 3532.0606928114], [0.062, 0.81341290651, 382.8965322232], [0.072, 2.35312965005, 9103.9069941176], [0.053, 3.21969389511, 20452.8694122218], [0.067, 1.42090542131, 24356.7807886416], [0.056, 2.97733070198, 30110.1656735384], [0.051, 4.22406663447, 20809.4676246452], [0.058, 6.20761936031, 29088.811415985], [0.061, 3.27309494322, 49515.382508407], [0.046, 5.49443476235, 31441.6775697568], [0.05, 4.16651052942, 13341.6743113068], [0.047, 1.25473247769, 33019.0211122046], [0.047, 2.03402044389, 23958.6317852334], [0.036, 5.24409311105, 3149.1641605882], [0.038, 4.15337829669, 18849.2275499742], [0.042, 0.43005959574, 1589.0728952838], [0.041, 1.21289342964, 12592.4500197826], [0.038, 5.91928287144, 28521.0927782546], [0.033, 3.98241699279, 4732.0306273434], [0.035, 2.24417218267, 16496.3613962024], [0.04, 6.13293942728, 26087.9031415742], [0.044, 1.7812329486, 426.598190876], [0.041, 3.16744909855, 39264.0692895602], [0.033, 4.96183427323, 536.8045120954], [0.034, 0.12963030501, 30213.258447757], [0.036, 5.41167321573, 522.5774180938], [0.027, 4.44250239485, 17277.4069318338], [0.034, 5.94541303751, 9929.4262273458], [0.033, 0.40689057274, 10497.1448650762], [0.023, 2.59067946967, 10175.2578735752], [0.022, 0.69625017371, 19999.97290154599], [0.023, 3.76162101633, 10251.3132188468], [0.023, 0.62711494266, 35371.8872659764], [0.022, 4.64142978776, 19889.76658032659], [0.02, 4.01315480107, 26709.6469424134], [0.02, 4.0334440068, 29573.361161443], [0.023, 0.90416640595, 8094.5216858326], [0.022, 1.92092469688, 17085.9586657222], [0.019, 5.04938942644, 6681.2248533996]], [[22377.665, 3.38509143877, 10213.285546211], [281.739, 0.0, 0.0], [173.164, 5.25563766915, 20426.571092422], [26.945, 3.87040891568, 30639.856638633], [1.174, 0.09768632072, 10186.9872264112], [0.685, 3.19139067811, 11790.6290886588], [0.788, 4.36515965295, 10239.5838660108], [0.592, 5.22270440328, 40853.142184844], [0.515, 6.12821215207, 10988.808157535], [0.538, 0.57550272342, 2352.8661537718], [0.54, 3.11657836329, 18073.7049386502], [0.454, 2.79306867629, 10404.7338123226], [0.374, 6.10468482446, 9437.762934887], [0.431, 4.00778431184, 1577.3435424478], [0.36, 6.0174784232, 19651.048481098], [0.375, 1.31319959789, 22003.9146348698], [0.354, 5.12509281266, 9153.9036160218], [0.15, 4.58623687118, 15720.8387848784], [0.164, 5.41790158607, 5507.5532386674], [0.159, 2.78191550878, 19896.8801273274], [0.157, 0.65774905071, 529.6909650946], [0.155, 2.54824315372, 9683.5945811164], [0.109, 2.01866665583, 14143.4952424306], [0.106, 2.28289033017, 6283.0758499914], [0.115, 3.23636374193, 20618.0193585336], [0.128, 5.32400510939, 13367.9726311066], [0.087, 3.28265082435, 11322.6640983044], [0.09, 5.23585072275, 10596.1820784342], [0.055, 4.82369879741, 7058.5984613154], [0.044, 0.58444963462, 10206.1719992102], [0.044, 2.34401612969, 19367.1891622328], [0.038, 4.55053233088, 9999.986450773], [0.039, 5.84340580032, 10220.3990932118], [0.036, 4.41006216127, 51066.427731055], [0.039, 3.14348236386, 9411.4646150872], [0.033, 4.5574866034, 10742.9765113056], [0.037, 2.79630938717, 25934.1243310894], [0.034, 0.55287110072, 11015.1064773348], [0.034, 2.25809144959, 29580.4747084438], [0.038, 1.88638747393, 801.8209311238], [0.034, 1.22706917271, 10021.8372800994], [0.027, 4.83867137637, 9830.3890139878], [0.027, 4.3114017935, 23581.2581773176], [0.027, 2.17187621336, 8635.9420037632], [0.02, 5.66581696952, 21228.3920235458], [0.024, 2.1720810785, 18849.2275499742], [0.02, 5.29318634138, 775.522611324], [0.019, 2.73486845601, 3128.3887650958], [0.013, 3.40362915274, 1059.3819301892], [0.014, 0.05074160195, 7860.4193924392], [0.014, 5.43035907265, 26.2983197998], [0.012, 3.24834347355, 9103.9069941176], [0.013, 5.04826725887, 7.1135470008], [0.015, 1.42027402522, 29050.7837433492], [0.01, 4.9813806749, 10426.584641649], [0.011, 0.85773045784, 17298.1823273262], [0.011, 4.23048200054, 29864.334027309], [0.01, 0.26447399758, 3930.2096962196], [0.011, 1.46728576671, 20419.45754542119]], [[646.671, 4.99166565277, 10213.285546211], [19.952, 3.14159265359, 0.0], [5.54, 0.77376923951, 20426.571092422], [2.526, 5.4449376302, 30639.856638633], [0.079, 1.51447613604, 10186.9872264112], [0.056, 0.63647808442, 40853.142184844], [0.058, 5.7073117655, 10239.5838660108], [0.031, 4.72523061067, 11790.6290886588], [0.026, 1.02068113372, 10988.808157535], [0.025, 5.60599130442, 9437.762934887], [0.017, 2.05293621864, 2352.8661537718], [0.011, 4.33056892256, 10404.7338123226], [0.009, 1.36283915068, 19651.048481098], [0.007, 4.69592781899, 18073.7049386502], [0.006, 2.97926526705, 22003.9146348698]], [[14.102, 0.31537190181, 10213.285546211], [0.19, 2.35466404492, 20426.571092422], [0.164, 0.74476215141, 30639.856638633], [0.214, 3.14159265359, 0.0], [0.004, 2.34190883009, 40853.142184844]], [[0.239, 2.05201727566, 10213.285546211], [0.039, 0.0, 0.0], [0.011, 3.82500275251, 20426.571092422], [0.009, 2.32953116868, 30639.856638633]]]¶: This table contains Venus’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric latitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in page 420.

pymeeus.Venus.VSOP87_L = [[[317614666.774, 0.0, 0.0], [1353968.419, 5.59313319619, 10213.285546211], [89891.645, 5.30650048468, 20426.571092422], [5477.201, 4.41630652531, 7860.4193924392], [3455.732, 2.69964470778, 11790.6290886588], [2372.061, 2.99377539568, 3930.2096962196], [1317.108, 5.18668219093, 26.2983197998], [1664.069, 4.2501893503, 1577.3435424478], [1438.322, 4.15745043958, 9683.5945811164], [1200.521, 6.15357115319, 30639.856638633], [761.38, 1.9501470212, 529.6909650946], [707.676, 1.06466707214, 775.522611324], [584.836, 3.99839884762, 191.4482661116], [769.314, 0.81629615911, 9437.762934887], [499.915, 4.12340210074, 15720.8387848784], [326.221, 4.59056473097, 10404.7338123226], [429.498, 3.58642859752, 19367.1891622328], [326.967, 5.67736583705, 5507.5532386674], [231.937, 3.16251057072, 9153.9036160218], [179.695, 4.65337915578, 1109.3785520934], [128.263, 4.22604493736, 20.7753954924], [155.464, 5.57043888948, 19651.048481098], [127.907, 0.96209822685, 5661.3320491522], [105.547, 1.53721191253, 801.8209311238], [85.722, 0.35589249966, 3154.6870848956], [99.121, 0.83288185132, 213.299095438], [98.804, 5.39389655503, 13367.9726311066], [82.094, 3.21596990826, 18837.49819713819], [88.031, 3.88868860307, 9999.986450773], [71.577, 0.11145739345, 11015.1064773348], [56.122, 4.24039855475, 7.1135470008], [70.239, 0.67458813282, 23581.2581773176], [50.796, 0.24531603049, 11322.6640983044], [46.111, 5.31576465717, 18073.7049386502], [44.574, 6.06282201966, 40853.142184844], [42.594, 5.3287333721, 2352.8661537718], [42.635, 1.7995542168, 7084.8967811152], [41.177, 0.36240972161, 382.8965322232], [35.749, 2.70448479296, 10206.1719992102], [33.893, 2.02347322198, 6283.0758499914], [29.138, 3.59230925768, 22003.9146348698], [28.479, 2.22375414002, 1059.3819301892], [29.85, 4.02176977477, 10239.5838660108], [33.252, 2.10025596509, 27511.4678735372], [30.172, 4.9419191989, 13745.3462390224], [29.252, 3.51392387787, 283.8593188652], [24.424, 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1.76206343944, 19374.3027092336], [0.575, 2.38792087791, 6770.7106012456], [0.538, 1.52023264138, 25934.1243310894], [0.69, 4.01873754171, 19651.048481098], [0.532, 4.4157613089, 574.3447983348], [0.54, 2.15936134728, 16496.3613962024], [0.576, 5.41170044566, 206.1855484372], [0.482, 0.40815793538, 3340.6124266998], [0.501, 3.08578363577, 245.8316462294], [0.488, 5.22311611589, 25158.6017197654], [0.45, 0.212798446, 11.729352836], [0.432, 1.32004964493, 103.0927742186], [0.434, 5.91094755233, 19786.67380610799], [0.564, 0.38776462529, 19367.1891622328], [0.421, 2.71057839701, 13936.794505134], [0.549, 6.08792865644, 3930.2096962196], [0.478, 4.70234715828, 14945.3161735544], [0.408, 4.80890663927, 32217.2001810808], [0.404, 2.85003595942, 29864.334027309], [0.407, 2.94002049006, 10220.3990932118], [0.359, 0.72354778897, 419.4846438752], [0.449, 1.44520508753, 8662.240323563], [0.353, 2.22195492336, 51066.427731055], [0.324, 1.40308439067, 29580.4747084438], [0.443, 1.93864353398, 9146.790069021], [0.314, 0.96837035284, 20618.0193585336], [0.324, 5.10759068171, 24356.7807886416], [0.324, 1.80146948625, 18830.38465013739], [0.37, 6.16895004656, 2218.7571041868], [0.278, 2.20429108375, 18844.61174413899], [0.286, 3.08459438435, 17277.4069318338], [0.383, 0.13890934755, 4732.0306273434], [0.292, 0.43528982259, 29088.811415985], [0.273, 5.84415407168, 9573.388259897], [0.324, 2.14144542781, 9999.986450773], [0.264, 5.20407029554, 220.4126424388], [0.254, 0.34411959301, 28521.0927782546], [0.3, 3.76014360906, 8094.5216858326], [0.301, 3.64457981649, 20400.2727726222], [0.287, 1.84003536598, 1589.0728952838], [0.206, 0.97167234723, 10234.0609417034], [0.212, 0.241736776, 36.0278666774], [0.216, 5.8861892303, 18875.525869774], [0.198, 1.89506914939, 20452.8694122218], [0.258, 6.27611355094, 1551.045222648], [0.197, 2.09222675324, 9683.5945811164], [0.217, 5.79472589364, 9103.9069941176], [0.188, 0.39123199129, 19573.37471066999], [0.195, 6.23142464829, 30110.1656735384], [0.187, 5.49670351645, 170.6728706192], [0.178, 4.90042854659, 10787.6303445458], [0.188, 1.62614804098, 9161.0171630226], [0.211, 2.71884568392, 15720.8387848784], [0.177, 1.88170417337, 33019.0211122046], [0.209, 2.66033422116, 3442.5749449654], [0.164, 4.92240093026, 10426.584641649], [0.186, 5.13678812068, 7255.5696517344], [0.177, 5.70206821967, 9992.8729037722], [0.214, 2.70027196648, 3128.3887650958], [0.208, 3.38876526854, 17778.11626694899], [0.147, 4.25008782855, 16983.9961474566], [0.148, 3.4640441813, 21202.093703746], [0.189, 1.43553862242, 2379.1644735716], [0.139, 2.99154379541, 110.2063212194], [0.159, 5.23851679605, 10007.0999977738], [0.136, 0.88942869764, 22805.7355659936], [0.155, 5.90500835975, 12592.4500197826], [0.151, 0.03422618975, 27991.40181316], [0.153, 4.01743770323, 33794.5437235286], [0.121, 0.51392111799, 21535.9496445154], [0.109, 2.25388616761, 26735.9452622132], [0.109, 0.78612823474, 6681.2248533996], [0.122, 4.84805105466, 19992.85935454519], [0.112, 3.31796669604, 36949.2308084242], [0.106, 3.34507236765, 10103.0792249916], [0.114, 4.36384000196, 20007.0864485468], [0.098, 5.07711736751, 135.62532501], [0.12, 5.41870615047, 37724.7534197482], [0.103, 2.62610244425, 20213.271996984], [0.085, 5.04808202087, 9830.3890139878], [0.103, 2.01549383816, 45585.1728121874], [0.088, 2.62613816931, 21.8508293264], [0.084, 3.50355880173, 639.897286314], [0.099, 0.61079620895, 41654.9631159678], [0.088, 3.63836700262, 49515.382508407], [0.098, 2.42401801881, 23581.2581773176], [0.081, 0.46468679835, 77.673770428], [0.092, 4.82530051729, 29043.67019634839], [0.102, 4.27051236894, 15874.6175953632], [0.09, 4.34075776744, 29057.89729034999], [0.081, 0.01896422336, 24150.080051345], [0.093, 1.79250830018, 12432.0426503978], [0.087, 5.25157021446, 14128.2427712456], [0.089, 5.65756996753, 377.3736079158], [0.097, 5.67942873241, 227.476132789], [0.076, 2.93363913259, 38204.687359371], [0.091, 2.60544242067, 1052.2683831884], [0.087, 3.82284200928, 27511.4678735372], [0.073, 4.75280755154, 40879.4405046438], [0.067, 3.54815262526, 30666.1549584328], [0.067, 5.81350818057, 20809.4676246452], [0.064, 4.24772678145, 153.7788104848], [0.064, 2.99454749109, 27197.2816936676], [0.07, 4.03868009742, 56600.2792895222], [0.071, 4.3362880685, 39264.0692895602], [0.069, 1.73648747605, 37410.5672398786], [0.065, 1.08206062736, 68050.42387851159], [0.062, 4.7769845465, 3914.9572250346], [0.061, 4.96121014691, 34596.3646546524], [0.063, 5.04865067599, 53445.5922046266], [0.058, 3.74010494151, 1066.49547719], [0.057, 5.39355890141, 20419.45754542119], [0.057, 3.59399518494, 735.8765135318], [0.065, 2.10322000074, 74.7815985673], [0.073, 1.31083648835, 11272.6674764002], [0.055, 1.33161298098, 18300.69368504279], [0.065, 4.21150522641, 49.7570254718], [0.061, 5.66161679402, 17468.8551979454], [0.053, 4.30231233835, 18849.2275499742], [0.055, 2.63906959481, 52670.0695933026], [0.05, 5.69803054279, 39793.7602546548], [0.049, 0.77345264124, 35371.8872659764], [0.048, 6.00565977593, 283.8593188652], [0.047, 2.63299859494, 51868.2486621788], [0.046, 0.05105081843, 38526.574350872], [0.05, 4.37549274002, 28513.97923125379], [0.046, 2.93422086586, 27682.1407441564], [0.051, 5.45979584751, 60530.4889857418], [0.045, 5.59492908223, 467.9649903544], [0.045, 2.34680401001, 9411.4646150872], [0.045, 0.02999265111, 44809.6502008634], [0.043, 5.62725673544, 14.2270940016], [0.047, 3.73567275749, 64460.6986819614], [0.046, 0.12586526756, 57375.8019008462], [0.044, 2.03114426076, 18314.9207790444], [0.039, 0.99375127466, 94138.32702008578], [0.053, 0.41974404621, 30831.3049047446], [0.055, 1.38351566741, 38500.2760310722], [0.041, 4.47012768909, 40077.61957352], [0.041, 0.36665992484, 19999.97290154599], [0.04, 3.06358586355, 813.5502839598], [0.04, 2.16802870803, 59728.668054618], [0.037, 1.08739100421, 17085.9586657222], [0.039, 1.31040309875, 48739.859897083], [0.036, 1.43280677914, 42456.7840470916], [0.037, 0.14190533464, 29050.7837433492], [0.037, 3.66792179278, 20956.2620575166], [0.025, 3.38876180652, 7058.5984613154], [0.031, 6.16829805337, 10192.5101507186]], [[54127.076, 0.0, 0.0], [3891.46, 0.34514360047, 10213.285546211], [1337.88, 2.02011286082, 20426.571092422], [23.836, 2.04592119012, 26.2983197998], [19.331, 3.53527371458, 30639.856638633], [9.984, 3.97130221102, 775.522611324], [7.046, 1.51962593409, 1577.3435424478], [6.014, 0.99926757893, 191.4482661116], [3.163, 4.36095475762, 9437.762934887], [2.125, 2.65810625752, 40853.142184844], [1.934, 3.39287946981, 382.8965322232], [1.46, 6.04899046273, 529.6909650946], [1.346, 2.94633106219, 5507.5532386674], [1.025, 1.40598904981, 10404.7338123226], [1.221, 3.73339139385, 3154.6870848956], [1.033, 3.52858472904, 11015.1064773348], [0.955, 5.11133878923, 801.8209311238], [0.742, 1.49198584483, 1109.3785520934], [0.525, 3.32087042103, 213.299095438], [0.578, 0.92614279843, 10239.5838660108], [0.602, 5.19220099775, 7084.8967811152], [0.431, 2.67159914364, 13367.9726311066], [0.389, 4.14116341739, 8635.9420037632], [0.355, 1.12061570874, 9153.9036160218], [0.301, 3.90047984197, 10596.1820784342], [0.212, 5.32697688872, 18837.49819713819], [0.26, 0.22761369281, 2352.8661537718], [0.243, 4.70747902991, 6283.0758499914], [0.196, 4.10467294392, 11790.6290886588], [0.194, 6.0119775947, 7860.4193924392], [0.14, 4.97015671653, 14143.4952424306], [0.134, 4.10529011674, 17298.1823273262], [0.119, 3.39375528828, 11322.6640983044], [0.126, 0.0985451614, 18073.7049386502], [0.122, 5.92478855457, 574.3447983348], [0.107, 0.35660030184, 1059.3819301892], [0.108, 2.25352052666, 12566.1516999828], [0.093, 5.48716819776, 10021.8372800994], [0.084, 4.89744332968, 18307.8072320436], [0.074, 2.35354025573, 426.598190876], [0.093, 4.99316908815, 14945.3161735544], [0.069, 3.8640906586, 51066.427731055], [0.082, 5.36280178643, 10186.9872264112], [0.077, 3.75728548372, 3723.508958923], [0.063, 5.39882267787, 21228.3920235458], [0.056, 4.11564786973, 7064.1213856228], [0.056, 6.26920407723, 32217.2001810808], [0.06, 5.02186497542, 19367.1891622328], [0.058, 5.1326370967, 20400.2727726222], [0.051, 4.52870390511, 22003.9146348698], [0.041, 3.83822107919, 16496.3613962024], [0.041, 3.36020411807, 4705.7323075436], [0.043, 5.98371820588, 15720.8387848784], [0.047, 0.18498155367, 18875.525869774], [0.038, 0.52232581277, 1551.045222648], [0.039, 5.05391675878, 10742.9765113056], [0.036, 3.16242472203, 20452.8694122218], [0.035, 5.17462577483, 29088.811415985], [0.035, 3.47325394141, 24356.7807886416], [0.031, 4.74511706231, 28521.0927782546], [0.029, 0.19383091192, 19896.8801273274], [0.033, 1.80059867635, 20618.0193585336], [0.024, 0.14022912457, 21202.093703746], [0.022, 4.73565067573, 10988.808157535], [0.018, 0.74039763161, 25158.6017197654], [0.019, 1.53770232218, 28286.9904848612], [0.014, 1.49084059765, 30110.1656735384], [0.013, 4.72171283479, 29864.334027309], [0.013, 5.79700427846, 29580.4747084438], [0.014, 3.6920522501, 27511.4678735372]], [[135.742, 4.80389020993, 10213.285546211], [77.846, 3.66876371591, 20426.571092422], [26.023, 0.0, 0.0], [1.214, 5.31970006917, 30639.856638633], [0.254, 4.15021671822, 40853.142184844], [0.008, 5.55523563261, 51066.427731055], [0.008, 1.40501229148, 1577.3435424478], [0.006, 1.27791479726, 10404.7338123226], [0.006, 5.76447068962, 10239.5838660108]], [[114.016, 3.14159265359, 0.0], [3.209, 5.20514170164, 20426.571092422], [1.714, 2.51099591706, 10213.285546211], [0.05, 0.71356059861, 30639.856638633], [0.023, 5.68127607034, 40853.142184844]], [[0.874, 3.14159265359, 0.0], [0.117, 0.54643013, 20426.571092422], [0.118, 1.90548541922, 10213.285546211], [0.002, 1.07734277826, 40853.142184844], [0.002, 1.89460223529, 30639.856638633]]]¶: This table contains Venus’s periodic terms (all of them) from the planetary theory VSOP87 for the heliocentric longitude at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 418-420.

pymeeus.Venus.VSOP87_R = [[[72334820.905, 0.0, 0.0], [489824.185, 4.02151832268, 10213.285546211], [1658.058, 4.90206728012, 20426.571092422], [1632.093, 2.84548851892, 7860.4193924392], [1378.048, 1.128465906, 11790.6290886588], [498.399, 2.58682187717, 9683.5945811164], [373.958, 1.42314837063, 3930.2096962196], [263.616, 5.5293818592, 9437.762934887], [237.455, 2.55135903978, 15720.8387848784], [221.983, 2.01346776772, 19367.1891622328], [119.467, 3.01975365264, 10404.7338123226], [125.896, 2.72769833559, 1577.3435424478], [76.178, 1.59577224486, 9153.9036160218], [85.336, 3.98607953754, 19651.048481098], [74.347, 4.11957854039, 5507.5532386674], [41.904, 1.64273363458, 18837.49819713819], [42.493, 3.81864530735, 13367.9726311066], [39.43, 5.39019422358, 23581.2581773176], [29.042, 5.67739528728, 5661.3320491522], [27.555, 5.72392407794, 775.522611324], [27.283, 4.82151812709, 11015.1064773348], [31.274, 2.31806719544, 9999.986450773], [19.7, 4.96157560245, 11322.6640983044], [19.809, 0.53189326492, 27511.4678735372], [13.567, 3.75530870628, 18073.7049386502], [12.921, 1.13381083556, 10206.1719992102], [16.215, 0.5645383429, 529.6909650946], [11.821, 5.09025877427, 3154.6870848956], [11.728, 0.23432298744, 7084.8967811152], [13.079, 5.24353197586, 17298.1823273262], [13.18, 3.37207825651, 13745.3462390224], [9.097, 3.07004895769, 1109.3785520934], [10.818, 2.45024712908, 10239.5838660108], [11.438, 4.56838894696, 29050.7837433492], [8.377, 5.78327612352, 30639.856638633], [8.193, 1.9502311186, 22003.9146348698], [9.308, 1.61615909286, 2352.8661537718], [10.652, 1.9552839614, 31441.6775697568], [10.357, 1.20234990061, 15874.6175953632], [9.585, 1.46639856228, 19999.97290154599], [6.506, 2.17390732263, 14143.4952424306], [7.562, 1.13789564977, 8624.2126509272], [6.434, 0.84419623033, 6283.0758499914], [5.898, 0.0109373111, 8635.9420037632], [5.632, 3.94956548631, 12566.1516999828], [5.523, 1.27394296557, 18307.8072320436], [4.488, 2.47835729057, 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15874.6175953632], [0.04, 3.93466530964, 20419.45754542119], [0.051, 0.79154899901, 23581.2581773176], [0.038, 1.77428239418, 10103.0792249916], [0.049, 1.12423644455, 3442.5749449654], [0.04, 5.22874487975, 21535.9496445154], [0.038, 1.12473430132, 7.1135470008], [0.038, 0.11510547453, 11272.6674764002], [0.036, 2.02476324983, 7058.5984613154], [0.047, 0.0558943239, 12432.0426503978], [0.034, 3.45481114998, 9830.3890139878], [0.045, 4.59817214088, 10192.5101507186], [0.037, 4.93959675364, 3340.6124266998], [0.044, 0.70533027806, 20213.271996984], [0.034, 2.16487642765, 64460.6986819614], [0.031, 1.57612397319, 36147.4098773004], [0.028, 2.56454760402, 94138.32702008578], [0.033, 1.08907268562, 29864.334027309], [0.029, 0.59718407064, 59728.668054618], [0.031, 3.04423979263, 40879.4405046438], [0.035, 0.32247158762, 1589.0728952838], [0.031, 3.27727318906, 19992.85935454519], [0.027, 5.83705748551, 17085.9586657222], [0.032, 2.6426078826, 41962.5207369374], [0.028, 4.90613317287, 29050.7837433492], [0.025, 4.55050389739, 14919.0178537546], [0.028, 3.58851614957, 40853.142184844], [0.029, 2.79705093386, 20007.0864485468], [0.033, 0.93862065616, 15720.8387848784], [0.024, 2.74970637101, 18947.7045183576], [0.024, 4.38966861409, 46386.9937433112], [0.024, 0.73361964525, 9411.4646150872], [0.028, 4.19559784013, 37674.9963942764], [0.023, 1.00023735538, 22779.4372461938], [0.026, 0.46990555736, 13745.3462390224], [0.028, 4.65181292126, 1551.045222648], [0.025, 4.18690270765, 44007.8292697396], [0.022, 0.98102807789, 426.598190876], [0.03, 1.24986033487, 27461.7108480654], [0.027, 3.94986823486, 17468.8551979454], [0.021, 6.09897508157, 18300.69368504279], [0.025, 4.75875623888, 27991.40181316], [0.022, 2.95281481673, 40077.61957352], [0.028, 6.12038264955, 38500.2760310722], [0.022, 4.11184201321, 19779.56025910719], [0.027, 3.7244644608, 19793.7873531088], [0.02, 4.27086627368, 31441.6775697568], [0.022, 4.99040169444, 31022.7531708562], [0.023, 1.33505132122, 65236.2212932854], [0.021, 4.46897353468, 53285.1848352418], [0.02, 4.15140915983, 2218.7571041868], [0.025, 2.18447182965, 27511.4678735372], [0.019, 1.43653410349, 27197.2816936676], [0.027, 1.22555218015, 42430.4857272918], [0.019, 3.65054338893, 49515.382508407], [0.022, 5.88380811711, 10218.8084705184], [0.018, 2.29853355765, 19264.0963880142], [0.017, 5.44429906531, 6681.2248533996], [0.02, 3.68116637773, 14128.2427712456], [0.021, 4.30316190532, 44809.6502008634], [0.02, 2.48583613985, 33794.5437235286], [0.017, 3.02735393984, 28528.2063252554], [0.019, 5.92656850674, 22805.7355659936], [0.022, 5.30827572791, 10207.7626219036], [0.02, 0.75829381378, 18314.9207790444], [0.017, 5.63315744126, 16522.6597160022], [0.016, 1.71021408448, 536.8045120954], [0.015, 5.27016880041, 53445.5922046266], [0.017, 5.61443395877, 47938.0389659592], [0.015, 5.81110284451, 43071.8992890308], [0.015, 4.96237667003, 19999.97290154599], [0.018, 0.55618686515, 14765.2390432698], [0.014, 3.48144272414, 29786.660256881], [0.015, 5.84132627836, 10228.538017396], [0.016, 1.05720065324, 26735.9452622132], [0.014, 6.08462030302, 35371.8872659764], [0.014, 2.8453287189, 574.3447983348], [0.015, 5.3451771514, 10198.033075026], [0.013, 0.45004137509, 20452.8694122218]], [[1406.587, 5.0636639519, 10213.285546211], [15.529, 5.47321687981, 20426.571092422], [13.059, 0.0, 0.0], [1.099, 2.78883988292, 9437.762934887], [0.488, 6.27806914496, 1577.3435424478], [0.361, 6.11914188253, 10404.7338123226], [0.31, 1.38984998403, 5507.5532386674], [0.389, 1.95017779915, 11015.1064773348], [0.372, 2.33222828423, 775.522611324], [0.207, 5.63406721595, 10239.5838660108], [0.168, 1.10765197296, 13367.9726311066], [0.175, 6.1667465295, 30639.856638633], [0.168, 3.64495311632, 7084.8967811152], [0.12, 5.85815843789, 9153.9036160218], [0.16, 2.21564938463, 3154.6870848956], [0.118, 2.62358866565, 8635.9420037632], [0.112, 2.36235956804, 10596.1820784342], [0.092, 0.72664449269, 12566.1516999828], [0.067, 3.76089669118, 18837.49819713819], [0.065, 2.4798370999, 11790.6290886588], [0.048, 4.26620187144, 2352.8661537718], [0.048, 5.5089818955, 191.4482661116], [0.048, 2.54730918293, 17298.1823273262], [0.046, 3.40293459332, 14143.4952424306], [0.041, 1.83997113019, 11322.6640983044], [0.037, 6.17871126027, 1109.3785520934], [0.039, 4.77190210316, 18073.7049386502], [0.035, 3.10133256432, 4705.7323075436], [0.046, 3.30090415967, 6283.0758499914], [0.034, 3.91721765773, 10021.8372800994], [0.034, 3.24663787383, 22003.9146348698], [0.042, 3.39360926939, 14945.3161735544], [0.044, 4.42979374073, 7860.4193924392], [0.034, 2.16381407025, 16496.3613962024], [0.031, 0.45714618479, 26.2983197998], [0.035, 3.62868651241, 801.8209311238], [0.032, 1.84138997078, 382.8965322232], [0.025, 3.32908650295, 18307.8072320436], [0.026, 3.64313769818, 29088.811415985], [0.029, 3.8296717881, 10186.9872264112], [0.022, 3.17741520378, 28521.0927782546], [0.021, 2.52643834111, 529.6909650946], [0.025, 5.71401244457, 21202.093703746], [0.021, 3.77813434325, 21228.3920235458], [0.019, 5.24505118517, 19896.8801273274], [0.018, 4.62463651925, 19651.048481098], [0.016, 3.35893297896, 28286.9904848612], [0.015, 5.05571633205, 33019.0211122046], [0.014, 2.83786355803, 19786.67380610799], [0.014, 1.79922718553, 9830.3890139878], [0.014, 3.14801263138, 19367.1891622328], [0.014, 3.57896195191, 10988.808157535], [0.013, 3.06303088617, 10742.9765113056], [0.013, 5.43981998532, 25158.6017197654], [0.015, 4.83166312889, 18875.525869774], [0.012, 2.54141086214, 7064.1213856228], [0.012, 4.45255110769, 15720.8387848784], [0.01, 1.87933121728, 24356.7807886416], [0.011, 2.58708635685, 9103.9069941176], [0.01, 2.179013099, 3723.508958923], [0.008, 3.63520673832, 1059.3819301892], [0.008, 4.67523115598, 25934.1243310894], [0.009, 5.97856553283, 9683.5945811164]], [[49.582, 3.2226355452, 10213.285546211], [0.831, 3.21219077104, 20426.571092422], [0.112, 3.14159265359, 0.0], [0.013, 3.77448689585, 30639.856638633], [0.009, 4.19802043629, 10239.5838660108], [0.006, 0.20714935358, 10186.9872264112], [0.005, 0.68781956122, 8635.9420037632]], [[0.573, 0.9222969782, 10213.285546211], [0.04, 0.95468912157, 20426.571092422], [0.006, 3.14159265359, 0.0]], [[0.045, 0.30032866722, 10213.285546211], [0.002, 5.29627718483, 20426.571092422]]]¶: This table contains Venus’s periodic terms (all of them) from the planetary theory VSOP87 for the radius vector at the equinox of date (taken from the ‘D’ solution). In Meeus’ book a shortened version can be found in pages 420-421.

class pymeeus.Venus.Venus[source]¶

Class Venus models that planet.

__weakref__¶: list of weak references to the object (if defined)

static apparent_heliocentric_position(epoch)[source]¶

This method computes the apparent heliocentric position of planet Venus for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Venus position, as an Epoch object
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

static eastern_elongation(epoch)[source]¶

This method computes the time of the eastern elongation closest to the given epoch, as well as the corresponding maximum elongation angle.

Parameters:	epoch (`Epoch`) – Epoch close to the desired eastern elongation
Returns:	A tuple with the time when the eastern elongation happens, as an Epoch, and the maximum elongation angle, as an Angle
Return type:	tuple
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2019, 10, 1.0)
>>> time, elongation = Venus.eastern_elongation(epoch)
>>> y, m, d = time.get_date()
>>> print(y)
2020
>>> print(m)
3
>>> print(round(d, 4))
24.9179
>>> print(round(elongation, 4))
46.078

static geocentric_position(epoch)[source]¶

This method computes the geocentric position of Venus (right ascension and declination) for the given epoch, as well as the elongation angle.

Parameters:	epoch (`Epoch`) – Epoch to compute geocentric position, as an Epoch object
Returns:	A tuple containing the right ascension, the declination and the elongation angle as Angle objects
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 20.0)
>>> ra, dec, elon = Venus.geocentric_position(epoch)
>>> print(ra.ra_str(n_dec=1))
21h 4' 41.5''
>>> print(dec.dms_str(n_dec=1))
-18d 53' 16.8''
>>> print(elon.dms_str(n_dec=1))
44d 46' 8.9''

static geometric_heliocentric_position(epoch, tofk5=True)[source]¶

This method computes the geometric heliocentric position of planet Venus for a given epoch, using the VSOP87 theory.

Parameters:	epoch (`Epoch`) – Epoch to compute Venus position, as an Epoch object tofk5 (bool) – Whether or not the small correction to convert to the FK5 system will be applied or not
Returns:	A tuple with the heliocentric longitude and latitude (as `Angle` objects), and the radius vector (as a float, in astronomical units), in that order
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(1992, 12, 20.0)
>>> l, b, r = Venus.geometric_heliocentric_position(epoch, tofk5=False)
>>> print(round(l, 5))
26.11412
>>> print(round(b, 4))
-2.6206
>>> print(round(r, 6))
0.724602

static illuminated_fraction(epoch)[source]¶

This function computes an approximation of the illuminated fraction of Venus disk, as seen from Earth.

Parameters:	epoch (`Epoch`) – Epoch to compute the illuminated fraction
Returns:	Illuminated fraction of Venus disk
Return type:	float
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1992, 12, 20)
>>> k = Venus.illuminated_fraction(epoch)
>>> print(round(k, 2))
0.64

static inferior_conjunction(epoch)[source]¶

This method computes the time of the inferior conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired inferior conjunction
Returns:	The time when the inferior conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1882, 12, 1.0)
>>> conjunction = Venus.inferior_conjunction(epoch)
>>> y, m, d = conjunction.get_date()
>>> print(y)
1882
>>> print(m)
12
>>> print(round(d, 1))
6.7

static magnitude(sun_dist, earth_dist, phase_angle)[source]¶

This function computes the approximate magnitude of Venus.

Parameters:	sun_dist (float) – Distance from Venus to the Sun, in Astronomical Units earth_dist (float) – Distance from Venus to Earth, in Astronomical Units phase_angle (float, `Angle`) – Venus phase angle
Returns:	Venus’ magnitude
Return type:	float
Raises:	TypeError if input values are of wrong type.

>>> sun_dist = 0.724604
>>> earth_dist = 0.910947
>>> phase_angle = Angle(72.96)
>>> m = Venus.magnitude(sun_dist, earth_dist, phase_angle)
>>> print(m)
-3.8

static orbital_elements_j2000(epoch)[source]¶

This method computes the orbital elements of Venus for the standard equinox J2000.0 for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Venus.orbital_elements_j2000(epoch)
>>> print(round(l, 6))
337.731227
>>> print(round(a, 8))
0.72332982
>>> print(round(e, 7))
0.0067407
>>> print(round(i, 6))
3.394087
>>> print(round(ome, 5))
76.49782
>>> print(round(arg, 6))
55.068476

static orbital_elements_mean_equinox(epoch)[source]¶

This method computes the orbital elements of Venus for the mean equinox of the date for a given epoch.

Parameters:	epoch (`Epoch`) – Epoch to compute orbital elements, as an Epoch object
Returns:	A tuple containing the following six orbital elements: - Mean longitude of the planet (Angle) - Semimajor axis of the orbit (float, astronomical units) - eccentricity of the orbit (float) - inclination on the plane of the ecliptic (Angle) - longitude of the ascending node (Angle) - argument of the perihelion (Angle)
Return type:	tuple
Raises:	TypeError if input value is of wrong type.

>>> epoch = Epoch(2065, 6, 24.0)
>>> l, a, e, i, ome, arg = Venus.orbital_elements_mean_equinox(epoch)
>>> print(round(l, 6))
338.646306
>>> print(round(a, 8))
0.72332982
>>> print(round(e, 7))
0.0067407
>>> print(round(i, 6))
3.395319
>>> print(round(ome, 5))
77.27012
>>> print(round(arg, 6))
55.211257

static passage_nodes(epoch, ascending=True)[source]¶

This function computes the time of passage by the nodes (ascending or descending) of Venus, nearest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch closest to the node passage ascending (bool) – Whether the time of passage by the ascending (True) or descending (False) node will be computed
Returns:	Tuple containing: - Time of passage through the node (`Epoch`) - Radius vector when passing through the node (in AU, float)
Return type:	tuple
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1979, 1, 1)
>>> time, r = Venus.passage_nodes(epoch)
>>> year, month, day = time.get_date()
>>> print(year)
1978
>>> print(month)
11
>>> print(round(day, 1))
27.4
>>> print(round(r, 4))
0.7205

static perihelion_aphelion(epoch, perihelion=True)[source]¶

This method computes the time of Perihelion (or Aphelion) closer to a given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired Perihelion (or Aphelion) peihelion – If True, the epoch of the closest Perihelion is computed, if False, the epoch of the closest Aphelion is found.
Returns:	The epoch of the desired Perihelion (or Aphelion)
Return type:	`Epoch`
Raises:	TypeError if input values are of wrong type.

>>> epoch = Epoch(1978, 10, 15.0)
>>> e = Venus.perihelion_aphelion(epoch)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
1978
>>> print(m)
12
>>> print(d)
31
>>> print(h)
4
>>> epoch = Epoch(1979, 2, 1.0)
>>> e = Venus.perihelion_aphelion(epoch, perihelion=False)
>>> y, m, d, h, mi, s = e.get_full_date()
>>> print(y)
1979
>>> print(m)
4
>>> print(d)
22
>>> print(h)
12

static station_longitude_1(epoch)[source]¶

This method computes the time of the 1st station in longitude (i.e. when the planet is stationary and begins to move westward - retrograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired inferior conjunction
Returns:	Time when the 1st station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2018, 12, 1.0)
>>> sta1 = Venus.station_longitude_1(epoch)
>>> y, m, d = sta1.get_date()
>>> print(y)
2018
>>> print(m)
10
>>> print(round(d, 4))
5.7908

static station_longitude_2(epoch)[source]¶

This method computes the time of the 1st station in longitude (i.e. when the planet is stationary and begins to move eastward - prograde - among the starts) closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired inferior conjunction
Returns:	Time when the 2nd station in longitude happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2018, 12, 1.0)
>>> sta2 = Venus.station_longitude_2(epoch)
>>> y, m, d = sta2.get_date()
>>> print(y)
2018
>>> print(m)
11
>>> print(round(d, 4))
16.439

static superior_conjunction(epoch)[source]¶

This method computes the time of the superior conjunction closest to the given epoch.

Parameters:	epoch (`Epoch`) – Epoch close to the desired superior conjunction
Returns:	The time when the superior conjunction happens, as an Epoch
Return type:	`Epoch`
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(1993, 10, 1.0)
>>> conjunction = Venus.superior_conjunction(epoch)
>>> y, m, d = conjunction.get_date()
>>> print(y)
1994
>>> print(m)
1
>>> print(round(d, 2))
17.05

static western_elongation(epoch)[source]¶

This method computes the time of the western elongation closest to the given epoch, as well as the corresponding maximum elongation angle.

Parameters:	epoch (`Epoch`) – Epoch close to the desired western elongation
Returns:	A tuple with the time when the western elongation happens, as an Epoch, and the maximum elongation angle, as an Angle
Return type:	tuple
Raises:	TypeError if input value is of wrong type.
Raises:	ValueError if input epoch outside the -2000/4000 range.

>>> epoch = Epoch(2019, 1, 1.0)
>>> time, elongation = Venus.western_elongation(epoch)
>>> y, m, d = time.get_date()
>>> print(y)
2019
>>> print(m)
1
>>> print(round(d, 4))
6.1895
>>> print(round(elongation, 4))
46.9571

Examples¶

Usage Examples¶

Properly Using PyMeeus¶

It is very common to try to run PyMeeus like this:

import pymeeus

mydate = pymeeus.Epoch(1992, 10, 13.0)

But if you do that, you’ll get an error like this:

Traceback (most recent call last):
  File "/home/user/test/test.py", line 3, in <module>
    epoch = pymeeus.Epoch(1992, 10, 13.0)
AttributeError: module 'pymeeus' has no attribute 'Epoch'

This issue points to a misunderstanding that is very common in the Python world. The keyword import is used to import MODULES… but PyMeeus is NOT a module: It is a LIBRARY composed of MULTIPLE modules (Angle, Epoch, Coordinates, etc). As of today, the library Pymeeus has 19 different modules (if you look into the directory where pip stores the library, you’ll find one “.py” file per module).

Therefore if you want to use, for example, the module Angle you should use:

import pymeeus.Angle

I.e., your module is pymeeus.Angle, and not just Angle.

But there is more! When you use import to fetch a module, you must then use the dot notation to access the components of the module (classes, functions, etc). For instance:

import pymeeus.Angle

i = pymeeus.Angle.Angle(11.94524)

In this case, you are telling the Python interpreter that you want to use the class Angle (with parameter ‘11.94524’) from the module Angle belonging to the library pymeeus.

There is, however, a more practical (and common) way to handle modules using the statement from <MODULE> import <COMPONENT>. For instance:

from pymeeus.Angle import Angle
from pymeeus.Epoch import Epoch, JDE2000
from math import sin, cos, tan, acos, atan2, sqrt, radians, log10

This way is preferred because, among other reasons, only the required components are loaded into memory instead of the whole module. Also, now the component is directly added to your execution environment, which means that you no longer need to use the dot notation.

Therefore, the script at the beginning would become:

from pymeeus.Epoch import Epoch

mydate = Epoch(1992, 10, 13.0)

Angle examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

Create an Angle object, providing degrees, minutes and seconds:

a = Angle(-23.0, 26.0, 48.999983999)

First we print using the __call__() method (note the extra parentheses):

print_me("The angle 'a()' is", a())

# The angle 'a()' is: -23.44694444

Second we print using the __str__() method (no extra parentheses needed):

print_me("The angle 'a' is", a)

# The angle 'a' is: -23.44694444

Use the copy constructor:

b = Angle(a)

print_me("Angle 'b', which is a copy of 'a', is", b)

# Angle 'b', which is a copy of 'a', is: -23.44694444

Use the static deg2dms() method to carry out conversions:

d, m, s, sign = Angle.deg2dms(23.44694444)

val = "{}d {}' {}''".format(int(sign*d), m, s)

print_me("{Deg}d {Min}' {Sec}''", val)

# {Deg}d {Min}' {Sec}'': 23d 26' 48.999984''

We can print Angle a directly in sexagesimal format.

In fancy format:

print_me("{Deg}d {Min}' {Sec}''", a.dms_str())

# {Deg}d {Min}' {Sec}'': -23d 26' 48.999983999''

In plain format:

print_me("{Deg}:{Min}:{Sec}", a.dms_str(False))

# {Deg}:{Min}:{Sec}: -23:26:48.999983999

Print directly as a tuple:

a = Angle(23.44694444)

print_me("a.dms_tuple()", a.dms_tuple())

# a.dms_tuple(): (23, 26, 48.999983999997596, 1.0)

print_me("a.ra_tuple()", a.ra_tuple())

# a.ra_tuple(): (1, 33, 47.26666559999941, 1.0)

Redefine Angle a several times:

a.set(-0.44694444)

print_me("   a.dms_str()", a.dms_str())

#    a.dms_str(): -26' 48.999984''

a.set(0, 0, -46.31)

print_me("   a.dms_str(False)", a.dms_str(False))

#    a.dms_str(False): 0:0:-46.31

We can use decimals in degrees/minutes. They are converted automatically:

a.set(0, -46.25, 0.0)

print_me("   a.dms_str()", a.dms_str())

#    a.dms_str(): -46' 15.0''

a.set(0, 0, 0.0)

print_me("   a.dms_str()", a.dms_str())

#    a.dms_str(): 0d 0' 0.0''

We can define the angle as in radians. It will be converted to degrees:

b = Angle(pi, radians=True)

print_me("b = Angle(pi, radians=True); print(b)", b)

# b = Angle(pi, radians=True); print(b): 180.0

And we can easily carry out the degrees to radians conversion:

print_me("print(b.rad())", b.rad())

# print(b.rad()): 3.14159265359

We can also specify the angle as a Right Ascension. Angle can be given as a Right Ascension: Hours, Minutes, Seconds:

a.set_ra(9, 14, 55.8)

print_me("   print(a)", a)

#    print(a): 138.7325

b = Angle(9, 14, 55.8, ra=True)

print_me("   print(b)", b)

#    print(b): 138.7325

We can print the Angle as Right Ascension, as a float and as string:

a = Angle(138.75)

print_me("   print(a.get_ra())", a.get_ra())

#    print(a.get_ra()): 9.25

print_me("   print(a.ra_str())", a.ra_str())

#    print(a.ra_str()): 9h 15' 0.0''

print_me("   print(a.ra_str(False))", a.ra_str(False))

#    print(a.ra_str(False)): 9:15:0.0

Use the to_positive() method to get the positive version of an angle:

a = Angle(-87.32)

print_me("   print(a.to_positive())", a.to_positive())

#    print(a.to_positive()): 272.68

Call the __repr__() method to get a string defining the current object. This string can then be fed to the eval() function to generate the object:

print_me("print(b.__repr__())", b.__repr__())

# print(b.__repr__()): Angle(138.7325)

c = eval(repr(b))

print_me("c = eval(repr(b)); print(c)", c)

# c = eval(repr(b)); print(c): 138.7325

Let’s now work with some useful operators and functions:

print_me("c", c)

# c: 138.7325

Negate an angle:

d = Angle(13, 30)

print_me("d", d)

# d: 13.5

e = -d

print_me("   e = -d", e)

#   e = -d: -13.5

Get the absolute value of an angle:

e = abs(e)

print_me("   e = abs(e)", e)

#    e = abs(e): 13.5

Module operation on an angle:

d = Angle(17.0)

print_me("d", d)

# d: 17.0

e = c % d

print_me("   e = c % d", e)

#    e = c % d: 2.7325

Convert the angle to an integer:

d = Angle(13.95)

print_me("d", d)

# d: 13.95

print_me("   int(d)", int(d))

#    int(d): 13

d = Angle(-4.95)

print_me("d", d)

# d: -4.95

print_me("   int(d)", int(d))

#   int(d): -4

Convert the angle to a float:

print_me("   float(d)", float(d))

#    float(d): -4.95

Round the angle to a float:

e = Angle(-4.951648)

print_me("e", e)

# e: -4.951648

print_me("   round(e)", round(e))

#    round(e): -5.0

print_me("   round(e, 2)", round(e, 2))

#    round(e, 2): -4.95

print_me("   round(e, 3)", round(e, 3))

#    round(e, 3): -4.952

print_me("   round(e, 4)", round(e, 4))

#    round(e, 4): -4.9516

Comparison operators:

print_me("   d == e", d == e)

#    d == e: False

print_me("   d != e", d != e)

#    d != e: True

print_me("   d > e", d > e)

#    d > e: True

print_me("   c >= 180.0", c >= 180.0)

#    c >= 180.0: False

print_me("   c < 180.0", c < 180.0)

#    c < 180.0: True

print_me("   c <= 180.0", c <= 180.0)

#    c <= 180.0: True

It is very easy to add Angles to obtain a new Angle:

e = c + d

print_me("   c + d", e)

#    c + d: 133.7825

We can also directly add a decimal angle:

e = c + 11.5

print_me("   c + 11.5", e)

#    c + 11.5: 150.2325

Types allowed are int, float and Angle:

print('e = c + "32.5"')

# e = c + "32.5"

try:

    e = c + "32.5"

except TypeError:

    print("TypeError!: Valid types are int, float, and Angle, not string!")

# TypeError!: Valid types are int, float, and Angle, not string!

Subtraction:

e = c - d

print_me("   c - d", e)

#    c - d: 143.6825

Multiplication:

c.set(150.0)

d.set(5.0)

print_me("c", c)

# c: 150.0

print_me("d", d)

# d: 5.0

e = c * d

print_me("   c * d", e)

#    c * d: 30.0

Division:

c.set(150.0)

d.set(6.0)

print_me("d", d)

# d: 6.0

e = c / d

print_me("   c / d", e)

#    c / d: 25.0

Division by zero is not allowed:

d.set(0.0)

print_me("d", d)

# d: 0.0

print('e = c / d')

# e = c / d

try:

    e = c / d

except ZeroDivisionError:

    print("ZeroDivisionError!: Division by zero is not allowed!")

# ZeroDivisionError!: Division by zero is not allowed!

Power:

d.set(2.2)

print_me("d", d)

# d: 2.2

e = c ** d

print_me("   c ** d", e)

#    c ** d: 91.5733670999

Accumulative module operation:

d.set(17.0)

print_me("d", d)

# d: 17.0

e %= d

print_me("   e %= d", e)

#    e %= d: 6.57336709993

Accumulative addition:

c += d

print_me("   c += d", c)

#    c += d: 167.0

Accumulative subtraction:

print_me("b", b)

# b: 138.7325

c -= b

print_me("   c -= b", c)

#    c -= b: 28.2675

Accumulative multiplication:

print_me("b", b)

# b: 138.7325

c *= b

print_me("   c *= b", c)

#    c *= b: 321.62094375

Accumulative division:

print_me("b", b)

# 138.7325

d.set(6.0)

print_me("d", d)

# d: 6.0

b /= d

print_me("   b /= d", b)

#    b /= d: 23.1220833333

Accumulative power:

d.set(2.2)

print_me("d", d)

# d: 2.2

c = abs(c)

print_me("   c = abs(c)", c)

#    c = abs(c): 321.62094375

c **= d

print_me("   c **= d", c)

#    c **= d: 254.307104203

The same operations, but by the right side:

e = 3.5 + b

print_me("   e = 3.5 + b", e)

#    e = 3.5 + b: 26.6220833333

e = 3.5 - b

print_me("   e = 3.5 - b", e)

#    e = 3.5 - b: -19.6220833333

e = 3.5 * b

print_me("   e = 3.5 * b", e)

#    e = 3.5 * b: 80.9272916667

e = 3.5 / b

print_me("   e = 3.5 / b", e)

#    e = 3.5 / b: 0.151370443119

e = 3.5 ** b

print_me("   e = 3.5 ** b", e)

#    e = 3.5 ** b: 260.783691406

Base examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

Let’s print the tolerance:

print_me("The default value for the tolerance is", TOL)

The default value for the tolerance is: 1e-10

Find the accuracy of this computer:

j, d = machine_accuracy()

print_me("Number of significant BITS in the mantissa\t", j)

Number of significant BITS in the mantissa  : 52.0

print_me("Number of significant DIGITS in a decimal number", d)

Number of significant DIGITS in a decimal number: 15

Print the suffixes for some ordinal numbers:

print_me("The suffix for ordinal 2 is", get_ordinal_suffix(2))

The suffix for ordinal 2 is: nd

print_me("The suffix for ordinal 16 is", get_ordinal_suffix(16))

The suffix for ordinal 16 is: th

Coordinates examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

It follows a series of important parameters related to the angle between Earth’s rotation axis and the ecliptic.

The mean angle between Earth rotation axis and ecliptic axis is the mean obliquity:

e0 = Earth.mean_obliquity(1987, 4, 10)

print_me("Mean obliquity for 1987/4/10", e0.dms_str())

# Mean obliquity for 1987/4/10: 23d 26' 27.4066466278''

If we take into account the nutation effect on the obliquity, we get the true obliquity:

epsilon = Earth.true_obliquity(1987, 4, 10)

print_me("True obliquity for 1987/4/10", epsilon.dms_str())

# True obliquity for 1987/4/10: 23d 26' 36.8491882378''

epsilon = Earth.true_obliquity(2018, 7, 29)

print_me("True obliquity for 2018/7/29", epsilon.dms_str())

# True obliquity for 2018/7/29: 23d 26' 7.21570241139''

The nutation effect is separated in two components: One parallel to the ecliptic (nutation in longitude) and other perpendicular to the ecliptic (nutation in obliquity):

dpsi = Earth.nutation_longitude(1987, 4, 10)

print_me("Nutation in longitude for 1987/4/10", dpsi.dms_str(n_dec=3))

# Nutation in longitude for 1987/4/10: -3.788''

depsilon = Earth.nutation_obliquity(1987, 4, 10)

print_me("Nutation in obliquity for 1987/4/10", depsilon.dms_str(n_dec=3))

# Nutation in obliquity for 1987/4/10: 9.443''

We can compute the effects of precession on the equatorial coordinates of a given star, taking also into account its proper motion:

start_epoch = JDE2000

final_epoch = Epoch(2028, 11, 13.19)

alpha0 = Angle(2, 44, 11.986, ra=True)

delta0 = Angle(49, 13, 42.48)

print_me("Initial right ascension", alpha0.ra_str(n_dec=3))

# Initial right ascension: 2h 44' 11.986''

print_me("Initial declination", delta0.dms_str(n_dec=2))

# Initial declination: 49d 13' 42.48''

pm_ra = Angle(0, 0, 0.03425, ra=True)

pm_dec = Angle(0, 0, -0.0895)

alpha, delta = Earth.precession_equatorial(start_epoch, final_epoch,

                                           alpha0, delta0, pm_ra, pm_dec)

print_me("Final right ascension", alpha.ra_str(n_dec=3))

# Final right ascension: 2h 46' 11.331''

print_me("Final declination", delta.dms_str(n_dec=2))

# Final declination: 49d 20' 54.54''

Something similar can also be done with the ecliptical coordinates:

start_epoch = JDE2000

final_epoch = Epoch(-214, 6, 30.0)

lon0 = Angle(149.48194)

lat0 = Angle(1.76549)

print_me("Initial ecliptical longitude", round(lon0(), 5))

# Initial ecliptical longitude: 149.48194

print_me("Initial ecliptical latitude", round(lat0(), 5))

# Initial ecliptical latitude: 1.76549

lon, lat = Earth.precession_ecliptical(start_epoch, final_epoch, lon0, lat0)

print_me("Final ecliptical longitude", round(lon(), 3))

# Final ecliptical longitude: 118.704

print_me("Final ecliptical latitude", round(lat(), 3))

# Final ecliptical latitude: 1.615

Additionally, module Coordinates provides a function to compute the true movement of a star through the sky relative to the Sun:

ra = Angle(6, 45, 8.871, ra=True)

dec = Angle(-16.716108)

pm_ra = Angle(0, 0, -0.03847, ra=True)

pm_dec = Angle(0, 0, -1.2053)

dist = 2.64

vel = -7.6

alpha, delta = motion_in_space(ra, dec, dist, vel, pm_ra, pm_dec, -4000.0)

print(alpha.ra_str(False, 2))

# 6:47:39.91

print(delta.dms_str(False, 1))

# -15:23:30.6

This module Coordinates also provides a series of functions to convert between equatorial, ecliptical, horizontal and galactic coordinates.

Equatorial to ecliptical coordinates:

ra = Angle(7, 45, 18.946, ra=True)

dec = Angle(28, 1, 34.26)

epsilon = Angle(23.4392911)

lon, lat = equatorial2ecliptical(ra, dec, epsilon)

print_me("Equatorial to ecliptical. Longitude", round(lon(), 5))

# Equatorial to ecliptical. Longitude: 113.21563

print_me("Equatorial to ecliptical. Latitude", round(lat(), 5))

# Equatorial to ecliptical. Latitude: 6.68417

Ecliptical to equatorial coordinates:

lon = Angle(113.21563)

lat = Angle(6.68417)

epsilon = Angle(23.4392911)

ra, dec = ecliptical2equatorial(lon, lat, epsilon)

print_me("Ecliptical to equatorial. Right ascension", ra.ra_str(n_dec=3))

# Ecliptical to equatorial. Right ascension: 7h 45' 18.946''

print_me("Ecliptical to equatorial. Declination", dec.dms_str(n_dec=2))

# Ecliptical to equatorial. Declination: 28d 1' 34.26''

Equatorial to horizontal coordinates:

lon = Angle(77, 3, 56)

lat = Angle(38, 55, 17)

ra = Angle(23, 9, 16.641, ra=True)

dec = Angle(-6, 43, 11.61)

theta0 = Angle(8, 34, 57.0896, ra=True)

eps = Angle(23, 26, 36.87)

# Compute correction to convert from mean to apparent sidereal time

delta = Angle(0, 0, ((-3.868*cos(eps.rad()))/15.0), ra=True)

theta0 += delta

h = theta0 - lon - ra

azi, ele = equatorial2horizontal(h, dec, lat)

print_me("Equatorial to horizontal: Azimuth", round(azi, 3))

# Equatorial to horizontal: Azimuth: 68.034

print_me("Equatorial to horizontal: Elevation", round(ele, 3))

# Equatorial to horizontal: Elevation: 15.125

Horizontal to equatorial coordinates:

azi = Angle(68.0337)

ele = Angle(15.1249)

lat = Angle(38, 55, 17)

h, dec = horizontal2equatorial(azi, ele, lat)

print_me("Horizontal to equatorial. Hour angle", round(h, 4))

# Horizontal to equatorial. Hour angle: 64.3521

print_me("Horizontal to equatorial. Declination", dec.dms_str(n_dec=0))

# Horizontal to equatorial. Declination: -6d 43' 12.0''

Equatorial to galactic coordinates:

ra = Angle(17, 48, 59.74, ra=True)

dec = Angle(-14, 43, 8.2)

lon, lat = equatorial2galactic(ra, dec)

print_me("Equatorial to galactic. Longitude", round(lon, 4))

# Equatorial to galactic. Longitude: 12.9593

print_me("Equatorial to galactic. Latitude", round(lat, 4))

# Equatorial to galactic. Latitude: 6.0463

Galactic to equatorial coordinates:

lon = Angle(12.9593)

lat = Angle(6.0463)

ra, dec = galactic2equatorial(lon, lat)

print_me("Galactic to equatorial. Right ascension", ra.ra_str(n_dec=1))

# Galactic to equatorial. Right ascension: 17h 48' 59.7''

print_me("Galactic to equatorial. Declination", dec.dms_str(n_dec=0))

# Galactic to equatorial. Declination: -14d 43' 8.0''

In addition, there is a function to compute the ecliptic longitudes of the two points of the ecliptic which are on the horizon, as well as the angle between the ecliptic and the horizon:

sidereal_time = Angle(5.0, ra=True)

lat = Angle(51.0)

epsilon = Angle(23.44)

lon1, lon2, i = ecliptic_horizon(sidereal_time, lat, epsilon)

print_me("Longitude of ecliptic point #1 on the horizon", lon1.dms_str(n_dec=1))

# Longitude of ecliptic point #1 on the horizon: 169d 21' 29.9''

print_me("Longitude of ecliptic point #2 on the horizon", lon2.dms_str(n_dec=1))

# Longitude of ecliptic point #2 on the horizon: 349d 21' 29.9''

print_me("Angle between the ecliptic and the horizon", round(i, 0))

# Angle between the ecliptic and the horizon: 62.0

Also, it is possible to compute the angle of the diurnal path of a celestial body relative to the horizon at the time of rising and setting:

dec = Angle(23.44)

lat = Angle(40.0)

j = diurnal_path_horizon(dec, lat)

print_me("Diurnal path vs. horizon angle at time of rising and setting", j.dms_str(n_dec=1))

# Diurnal path vs. horizon angle at time of rising and setting: 45d 31' 28.4''

The times (in hours of the day) of rising, transit and setting of a given celestial body can be computed with the appropriate function:

longitude = Angle(71, 5, 0.0)

latitude = Angle(42, 20, 0.0)

alpha1 = Angle(2, 42, 43.25, ra=True)

delta1 = Angle(18, 2, 51.4)

alpha2 = Angle(2, 46, 55.51, ra=True)

delta2 = Angle(18, 26, 27.3)

alpha3 = Angle(2, 51, 7.69, ra=True)

delta3 = Angle(18, 49, 38.7)

h0 = Angle(-0.5667)

delta_t = 56.0

theta0 = Angle(11, 50, 58.1, ra=True)

rising, transit, setting = times_rise_transit_set(longitude, latitude,alpha1, delta1, \

                                                  alpha2, delta2, alpha3, delta3, h0, \

                                                  delta_t, theta0)

print_me("Time of rising (hours of day)", round(rising, 4))

# Time of rising (hours of day): 12.4238

print_me("Time of transit (hours of day)", round(transit, 3))

# Time of transit (hours of day): 19.675

print_me("Time of setting (hours of day, next day)", round(setting, 3))

# Time of setting (hours of day, next day): 2.911

The air in the atmosphere introduces an error in the elevation due to the refraction. We can compute the true (airless) elevation from the apparent elevation, and viceversa.

Apparent elevation to true (airless) elevation:

apparent_elevation = Angle(0, 30, 0.0)

true_elevation = refraction_apparent2true(apparent_elevation)

print_me("True elevation for an apparent elevation of 30'",

         true_elevation.dms_str(n_dec=1))

# True elevation for an apparent elevation of 30': 1' 14.7''

True elevation to apparent elevation:

true_elevation = Angle(0, 33, 14.76)

apparent_elevation = refraction_true2apparent(true_elevation)

print_me("Apparent elevation for a true elevation of 33' 14.76''",

         apparent_elevation.dms_str(n_dec=2))

# Apparent elevation for a true elevation of 33' 14.76'': 57' 51.96''

This module provides a function to compute the angular separation between two celestial bodies:

alpha1 = Angle(14, 15, 39.7, ra=True)

delta1 = Angle(19, 10, 57.0)

alpha2 = Angle(13, 25, 11.6, ra=True)

delta2 = Angle(-11, 9, 41.0)

sep_ang = angular_separation(alpha1, delta1, alpha2, delta2)

print_me("Angular separation between two given celestial bodies, in degrees",
         round(sep_ang, 3))

# Angular separation between two given celestial bodies, in degrees: 32.793

We can compute the minimum angular separation achieved between two celestial objects. For that, we must provide the positions at three equidistant epochs:

# EPOCH: Sep 13th, 1978, 0h TT:

alpha1_1 = Angle(10, 29, 44.27, ra=True)

delta1_1 = Angle(11, 2, 5.9)

alpha2_1 = Angle(10, 33, 29.64, ra=True)

delta2_1 = Angle(10, 40, 13.2)

# EPOCH: Sep 14th, 1978, 0h TT:

alpha1_2 = Angle(10, 36, 19.63, ra=True)

delta1_2 = Angle(10, 29, 51.7)

alpha2_2 = Angle(10, 33, 57.97, ra=True)

delta2_2 = Angle(10, 37, 33.4)

# EPOCH: Sep 15th, 1978, 0h TT:

alpha1_3 = Angle(10, 43, 1.75, ra=True)

delta1_3 = Angle(9, 55, 16.7)

alpha2_3 = Angle(10, 34, 26.22, ra=True)

delta2_3 = Angle(10, 34, 53.9)

a = minimum_angular_separation(alpha1_1, delta1_1, alpha1_2, delta1_2,

                               alpha1_3, delta1_3, alpha2_1, delta2_1,

                               alpha2_2, delta2_2, alpha2_3, delta2_3)

print_me("Minimum angular separation, epoch fraction", round(a[0], 6))

# Minimum angular separation, epoch fraction: -0.370726

# NOTE: Given that 'n' is negative, and Sep 14th is the middle epoch (n=0),

# then the minimum angular separation is achieved on Sep 13th, specifically

# at: 1.0 - 0.370726 = 0.629274 => Sep 13.629274 = Sep 13th, 15h 6' 9''

print_me("Minimum angular separation", a[1].dms_str(n_dec=0))

# Minimum angular separation: 3' 44.0''

There is a function to compute the relative position angle P of a body with respect to another body. In this example, given that the two bodies have the same right ascension, then the relative position angle between them must be 0 (or 180):

alpha1 = Angle(14, 15, 39.7, ra=True)

delta1 = Angle(19, 10, 57.0)

alpha2 = Angle(14, 15, 39.7, ra=True)                   # Same as alpha1

delta2 = Angle(-11, 9, 41.0)

pos_ang = relative_position_angle(alpha1, delta1, alpha2, delta2)

print_me("Relative position angle", round(pos_ang, 1))

# Relative position angle: 0.0

Planetary conjunctions may be computed with the appropriate function:

alpha1_1 = Angle(10, 24, 30.125, ra=True)

delta1_1 = Angle(6, 26, 32.05)

alpha1_2 = Angle(10, 25,  0.342, ra=True)

delta1_2 = Angle(6, 10, 57.72)

alpha1_3 = Angle(10, 25, 12.515, ra=True)

delta1_3 = Angle(5, 57, 33.08)

alpha1_4 = Angle(10, 25,  6.235, ra=True)

delta1_4 = Angle(5, 46, 27.07)

alpha1_5 = Angle(10, 24, 41.185, ra=True)

delta1_5 = Angle(5, 37, 48.45)

alpha2_1 = Angle(10, 27, 27.175, ra=True)

delta2_1 = Angle(4,  4, 41.83)

alpha2_2 = Angle(10, 26, 32.410, ra=True)

delta2_2 = Angle(3, 55, 54.66)

alpha2_3 = Angle(10, 25, 29.042, ra=True)

delta2_3 = Angle(3, 48,  3.51)

alpha2_4 = Angle(10, 24, 17.191, ra=True)

delta2_4 = Angle(3, 41, 10.25)

alpha2_5 = Angle(10, 22, 57.024, ra=True)

delta2_5 = Angle(3, 35, 16.61)

alpha1_list = [alpha1_1, alpha1_2, alpha1_3, alpha1_4, alpha1_5]

delta1_list = [delta1_1, delta1_2, delta1_3, delta1_4, delta1_5]

alpha2_list = [alpha2_1, alpha2_2, alpha2_3, alpha2_4, alpha2_5]

delta2_list = [delta2_1, delta2_2, delta2_3, delta2_4, delta2_5]

pc = planetary_conjunction(alpha1_list, delta1_list, alpha2_list, delta2_list)

print_me("Epoch fraction 'n' for planetary conjunction", round(pc[0], 5))

# Epoch fraction 'n' for planetary conjunction: 0.23797

print_me("Difference in declination at conjunction", pc[1].dms_str(n_dec=1))

# Difference in declination at conjunction: 2d 8' 21.8''

If the planetary conjunction is with a star, it is a little bit simpler:

alpha_1 = Angle(15,  3, 51.937, ra=True)

delta_1 = Angle(-8, 57, 34.51)

alpha_2 = Angle(15,  9, 57.327, ra=True)

delta_2 = Angle(-9,  9,  3.88)

alpha_3 = Angle(15, 15, 37.898, ra=True)

delta_3 = Angle(-9, 17, 37.94)

alpha_4 = Angle(15, 20, 50.632, ra=True)

delta_4 = Angle(-9, 23, 16.25)

alpha_5 = Angle(15, 25, 32.695, ra=True)

delta_5 = Angle(-9, 26,  1.01)

alpha_star = Angle(15, 17, 0.446, ra=True)

delta_star = Angle(-9, 22, 58.47)

alpha_list = [alpha_1, alpha_2, alpha_3, alpha_4, alpha_5]

delta_list = [delta_1, delta_2, delta_3, delta_4, delta_5]

pc = planet_star_conjunction(alpha_list, delta_list, alpha_star, delta_star)

print_me("Epoch fraction 'n' for planetary conjunction with star", round(pc[0], 4))

# Epoch fraction 'n' for planetary conjunction with star: 0.2551

print_me("Difference in declination with star at conjunction", pc[1].dms_str(n_dec=0))

# Difference in declination with star at conjunction: 3' 38.0''

It is possible to compute when a planet and two other stars will be in a straight line:

alpha_1 = Angle(7, 55, 55.36, ra=True)

delta_1 = Angle(21, 41,  3.0)

alpha_2 = Angle(7, 58, 22.55, ra=True)

delta_2 = Angle(21, 35, 23.4)

alpha_3 = Angle(8,  0, 48.99, ra=True)

delta_3 = Angle(21, 29, 38.2)

alpha_4 = Angle(8,  3, 14.66, ra=True)

delta_4 = Angle(21, 23, 47.5)

alpha_5 = Angle(8,  5, 39.54, ra=True)

delta_5 = Angle(21, 17, 51.4)

alpha_star1 = Angle(7, 34, 16.40, ra=True)

delta_star1 = Angle(31, 53, 51.2)

alpha_star2 = Angle(7, 45,  0.10, ra=True)

delta_star2 = Angle(28,  2, 12.5)

alpha_list = [alpha_1, alpha_2, alpha_3, alpha_4, alpha_5]

delta_list = [delta_1, delta_2, delta_3, delta_4, delta_5]

n = planet_stars_in_line(alpha_list, delta_list, alpha_star1, delta_star1,

                         alpha_star2, delta_star2)

print_me("Epoch fraction 'n' when bodies are in a straight line", round(n, 4))

# Epoch fraction 'n' when bodies are in a straight line: 0.2233

The function ‘straight_line()’ computes if three celestial bodies are in line providing the angle with which the bodies differ from a great circle:

alpha1 = Angle(5, 32,  0.40, ra=True)

delta1 = Angle(0, -17, 56.9)

alpha2 = Angle(5, 36, 12.81, ra=True)

delta2 = Angle(-1, 12,  7.0)

alpha3 = Angle(5, 40, 45.52, ra=True)

delta3 = Angle(-1, 56, 33.3)

psi, omega = straight_line(alpha1, delta1, alpha2, delta2, alpha3, delta3)

print_me("Angle deviation from a straight line", psi.dms_str(n_dec=0))

# Angle deviation from a straight line: 7d 31' 1.0''

print_me("Angular distance of central point to the straight line", omega.dms_str(n_dec=0))

# Angular distance of central point to the straight line: -5' 24.0''

Now let’s compute the size of the smallest circle that contains three given celestial bodies:

alpha1 = Angle(12, 41,  8.63, ra=True)

delta1 = Angle(-5, 37, 54.2)

alpha2 = Angle(12, 52,  5.21, ra=True)

delta2 = Angle(-4, 22, 26.2)

alpha3 = Angle(12, 39, 28.11, ra=True)

delta3 = Angle(-1, 50,  3.7)

d = circle_diameter(alpha1, delta1, alpha2, delta2, alpha3, delta3)

print_me("Diameter of smallest circle containing three celestial bodies", d.dms_str(n_dec=0))

# Diameter of smallest circle containing three celestial bodies: 4d 15' 49.0''

Let’s find the apparent position of a star (Theta Persei) for a given epoch:

epoch = Epoch(2028, 11, 13.19)

alpha = Angle(2, 46, 11.331, ra=True)

delta = Angle(49, 20, 54.54)

sun_lon = Angle(231.328)

app_alpha, app_delta = apparent_position(epoch, alpha, delta, sun_lon)

print_me("Apparent right ascension", app_alpha.ra_str(n_dec=2))

# Apparent right ascension: 2h 46' 14.39''

print_me("Apparent declination", app_delta.dms_str(n_dec=2))

# Apparent declination: 49d 21' 7.45''

Convert orbital elements of a celestial object from one equinox to another:

epoch0 = Epoch(2358042.5305)

epoch = Epoch(2433282.4235)

i0 = Angle(47.122)

arg0 = Angle(151.4486)

lon0 = Angle(45.7481)

i1, arg1, lon1 = orbital_equinox2equinox(epoch0, epoch, i0, arg0, lon0)

print_me("New inclination", round(i1(), 3))

# New inclination: 47.138

print_me("New argument of perihelion", round(arg1(), 4))

# New argument of perihelion: 151.4782

print_me("New longitude of ascending node", round(lon1(), 4))

# New longitude of ascending node: 48.6037

Compute the eccentric and true anomalies using Kepler’s equation:

eccentricity = 0.1

mean_anomaly = Angle(5.0)

e, v = kepler_equation(eccentricity, mean_anomaly)

print_me("Eccentric anomaly, Case #1", round(e(), 6))

# Eccentric anomaly, Case #1: 5.554589

print_me("True anomaly, Case #1", round(v(), 6))

# True anomaly, Case #1: 6.139762

e, v = kepler_equation(0.99, Angle(0.2, radians=True))

print_me("Eccentric anomaly, Case #2", round(e(), 8))

# Eccentric anomaly, Case #2: 61.13444578

print_me("True anomaly, Case #2", round(v(), 6))

# True anomaly, Case #2: 166.311977

Compute the velocity of a body in a given point of its (unperturbated elliptic) orbit, in this case the comet Halley in 1986:

r = 1.0

a = 17.9400782

v = velocity(r, a)

print_me("Velocity ar 1 AU", round(v, 2))

# Velocity at 1 AU: 41.53

Compute the velocity at perihelion:

a = 17.9400782

e = 0.96727426

vp = velocity_perihelion(e, a)

print_me("Velocity at perihelion", round(vp, 2))

# Velocity at perihelion: 54.52

And now compute the velocity at aphelion:

a = 17.9400782

e = 0.96727426

va = velocity_aphelion(e, a)

print_me("Velocity at aphelion", round(va, 2))

# Velocity at aphelion: 0.91

Calculate the length of the orbit for the same comet Halley:

a = 17.9400782

e = 0.96727426

length = length_orbit(e, a)

print_me("Length of the orbit (AU)", round(length, 2))

# Length of the orbit (AU): 77.06

Compute passage through the nodes of an elliptic orbit:

omega = Angle(111.84644)

e = 0.96727426

a = 17.9400782

t = Epoch(1986, 2, 9.45891)

time, r = passage_nodes_elliptic(omega, e, a, t)

y, m, d = time.get_date()

d = round(d, 2)

print("Time of passage through ascending node: {}/{}/{}".format(y, m, d))

# Time of passage through ascending node: 1985/11/9.16

print("Radius vector at ascending node: {}".format(round(r, 4)))

# Radius vector at ascending node: 1.8045

Passage through the nodes of a parabolic orbit:

omega = Angle(154.9103)

q = 1.324502

t = Epoch(1989, 8, 20.291)

time, r = passage_nodes_parabolic(omega, q, t, ascending=False)

y, m, d = time.get_date()

d = round(d, 2)

print("Time of passage through descending node: {}/{}/{}".format(y, m, d))

# Time of passage through descending node: 1989/9/17.64

print("Radius vector at descending node: {}".format(round(r, 4)))

# Radius vector at descending node: 1.3901

Compute the phase angle:

sun_dist = 0.724604

earth_dist = 0.910947

sun_earth_dist = 0.983824

angle = phase_angle(sun_dist, earth_dist, sun_earth_dist)

print_me("Phase angle", round(angle, 2))

# Phase angle: 72.96

Now, let’s compute the illuminated fraction of the disk:

k = illuminated_fraction(sun_dist, earth_dist, sun_earth_dist)

print_me("Illuminated fraction of planet disk", round(k, 3))

# Illuminated fraction of planet disk: 0.647

CurveFitting examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

Now let’s work with the CurveFitting class. First, create a CurveFitting object:

cf1 = CurveFitting([73.0, 38.0, 35.0, 42.0, 78.0, 68.0, 74.0, 42.0, 52.0,

                    54.0, 39.0, 61.0, 42.0, 49.0, 50.0, 62.0, 44.0, 39.0,

                    43.0, 54.0, 44.0, 37.0],

                   [90.4, 125.3, 161.8, 143.4, 52.5, 50.8, 71.5, 152.8,

                    131.3, 98.5, 144.8, 78.1, 89.5, 63.9, 112.1, 82.0,

                    119.8, 161.2, 208.4, 111.6, 167.1, 162.1])

Let’s use linear_fitting():

a, b = cf1.linear_fitting()

print("   a = {}\tb = {}".format(round(a, 2), round(b, 2)))

#    a = -2.49      b = 244.18

Use the copy constructor:

cf2 = CurveFitting(cf1)

a, b = cf2.linear_fitting()

print("   a = {}\tb = {}".format(round(a, 2), round(b, 2)))

#    a = -2.49      b = 244.18

Get the number of value pairs internally stored:

print_me("Number of value pairs inside 'cf2'", len(cf2))

# Number of value pairs inside 'cf2': 22

Compute the correlation coefficient:

r = cf1.correlation_coeff()

print_me("   r", round(r, 3))

#    r: -0.767

Define a new CurveFitting object:

cf2 = CurveFitting([-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0],

                   [-9.372, -3.821, 0.291, 3.730, 5.822, 8.324, 9.083,

                    6.957, 7.006, 0.365, -1.722])

Now use quadratic_fitting():

a, b, c = cf2.quadratic_fitting()

# Original curve: y = -2.0*x*x + 3.5*x + 7.0 + noise

print("   a = {}\tb = {}\tc = {}".format(round(a, 2), round(b, 2), round(c, 2)))

#    a = -2.22      b = 3.76        c = 6.64

Yet another CurveFitting object:

cf4 = CurveFitting([3, 20, 34, 50, 75, 88, 111, 129, 143, 160, 183, 200,

                    218, 230, 248, 269, 290, 303, 320, 344],

                   [0.0433, 0.2532, 0.3386, 0.3560, 0.4983, 0.7577, 1.4585,

                    1.8628, 1.8264, 1.2431, -0.2043, -1.2431, -1.8422,

                    -1.8726, -1.4889, -0.8372, -0.4377, -0.3640, -0.3508,

                    -0.2126])

Let’s define the three functions to be used for fitting:

def sin1(x): return sin(radians(x))

def sin2(x): return sin(radians(2.0*x))

def sin3(x): return sin(radians(3.0*x))

Use general_fitting() here:

a, b, c = cf4.general_fitting(sin1, sin2, sin3)

# General fitting with f0 = sin(x), f1 = sin(2*x), f2 = sin(3*x)

print("   a = {}\tb = {}\tc = {}".format(round(a, 2), round(b, 2), round(c, 2)))

#    a = 1.2        b = -0.77       c = 0.39

A final example:

cf5 = CurveFitting([0, 1.2, 1.4, 1.7, 2.1, 2.2])

a, b, c = cf5.general_fitting(sqrt)

# General fitting with f0 = sqrt(x), f1 = 0.0 and f2 = 0.0

print("   a = {}\tb = {}\t\tc = {}".format(round(a, 3), round(b, 3), round(c, 3)))

#   a = 1.016       b = 0.0         c = 0.0

Earth examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

An important concept are the reference ellipsoids, comprising information about the Earth global model we are going to use.

A very important reference ellipsoid is WGS84, predefined here:

print_me("WGS84", WGS84)

# WGS84: 6378137.0:0.00335281066475:7.292115e-05

# First field is equatorial radius, second field is the flattening, and the

# third field is the angular rotation velocity, in radians per second

Let’s print the semi-minor axis (polar radius):

print_me("Polar radius, b", WGS84.b())

# Polar radius, b: 6356752.31425

And now, let’s print the eccentricity of Earth’s meridian:

print_me("Eccentricity, e", WGS84.e())

# Eccentricity, e: 0.0818191908426

We create an Earth object with a given reference ellipsoid. By default, it is WGS84, but we can use another:

e = Earth(IAU76)

Print the parameters of reference ellipsoid being used:

print_me("'e' Earth object parameters", e)

# 'e' Earth object parameters: 6378140.0:0.0033528131779:7.292114992e-05

Compute the distance to the center of the Earth from a given point at sea level, and at a certain latitude. It is given as a fraction of equatorial radius:

lat = Angle(65, 45, 30.0)               # We can use an Angle for this

print_me("Distance to Earth's center, from latitude 65d 45' 30''", e.rho(lat))

# Distance to Earth's center, from latitude 65d 45' 30'': 0.997216343095

Parameters rho*sin(lat) and rho*cos(lat) are useful for different astronomical applications:

height = 650.0

print_me("rho*sin(lat)", e.rho_sinphi(lat, height))

# rho*sin(lat): 0.908341718779

print_me("rho*cos(lat)", e.rho_cosphi(lat, height))

# rho*cos(lat): 0.411775501279

Compute the radius of the parallel circle at a given latitude:

print_me("Radius of parallel circle at latitude 65d 45' 30'' (meters)", e.rp(lat))

# Radius of parallel circle at latitude 65d 45' 30'' (meters): 2626094.91467

Compute the radius of curvature of the Earth’s meridian at given latitude:

print_me("Radius of Earth's meridian at latitude 65d 45' 30'' (meters)", e.rm(lat))

# Radius of Earth's meridian at latitude 65d 45' 30'' (meters): 6388705.74543

It is easy to compute the linear velocity at different latitudes:

print_me("Linear velocity at the Equator (meters/second)", e.linear_velocity(0.0))

# Linear velocity at the Equator (meters/second): 465.101303151

print_me("Linear velocity at latitude 65d 45' 30'' (meters/second)", e.linear_velocity(lat))

# Linear velocity at latitude 65d 45' 30'' (meters/second): 191.497860977

And now, let’s compute the distance between two points on the Earth:

Bangkok: 13d 14’ 09’’ North, 100d 29’ 39’’ East
Buenos Aires: 34d 36’ 12’’ South, 58d 22’ 54’’ West

Note

We will consider that positions ‘East’ and ‘South’ are negative

Here we will take advantage of facilities provided by Angle class:

lon_ban = Angle(-100, 29, 39.0)

lat_ban = Angle(13, 14, 9.0)

lon_bai = Angle(58, 22, 54.0)

lat_bai = Angle(-34, 36, 12.0)

dist, error = e.distance(lon_ban, lat_ban, lon_bai, lat_bai)

print_me("The distance between Bangkok and Buenos Aires is (km)", round(dist/1000.0, 2))

# The distance between Bangkok and Buenos Aires is (km): 16832.89

print_me("The approximate error of the estimation is (meters)", round(error, 0))

# The approximate error of the estimation is (meters): 189.0

Let’s now compute the geometric heliocentric position for a given epoch:

epoch = Epoch(1992, 10, 13.0)

lon, lat, r = Earth.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", lon.to_positive())

# Geometric Heliocentric Longitude: 19.9072721503

print_me("Geometric Heliocentric Latitude", lat.dms_str(n_dec=3))

# Geometric Heliocentric Latitude: -0.721''

print_me("Radius vector", r)

# Radius vector: 0.997608520236

And now, compute the apparent heliocentric position for the same epoch:

epoch = Epoch(1992, 10, 13.0)

lon, lat, r = Earth.apparent_heliocentric_position(epoch)

print_me("Apparent Heliocentric Longitude", lon.to_positive())

# Apparent Heliocentric Longitude: 19.9059856939

print_me("Apparent Heliocentric Latitude", lat.dms_str(n_dec=3))

# Apparent Heliocentric Latitude: -0.721''

print_me("Radius vector", r)

# Radius vector: 0.997608520236

Print mean orbital elements for Earth at 2065.6.24:

epoch = Epoch(2065, 6, 24.0)

l, a, e, i, ome, arg = Earth.orbital_elements_mean_equinox(epoch)

print_me("Mean longitude of the planet", round(l, 6))

# Mean longitude of the planet: 272.716028

print_me("Semimajor axis of the orbit (UA)", round(a, 8))

# Semimajor axis of the orbit (UA): 1.00000102

print_me("Eccentricity of the orbit", round(e, 7))

# Eccentricity of the orbit: 0.0166811

print_me("Inclination on plane of the ecliptic", round(i, 6))

# Inclination on plane of the ecliptic: 0.0

print_me("Longitude of the ascending node", round(ome, 5))

# Longitude of the ascending node: 174.71534

print_me("Argument of the perihelion", round(arg, 6))

# Argument of the perihelion: -70.651889

Find the epoch of the Perihelion closer to 2008/02/01:

epoch = Epoch(2008, 2, 1.0)

e = Earth.perihelion_aphelion(epoch)

y, m, d, h, mi, s = e.get_full_date()

peri = str(y) + '/' + str(m) + '/' + str(d) + ' ' + str(h) + ':' + str(mi)

print_me("The Perihelion closest to 2008/2/1 happened on", peri)

# The Perihelion closest to 2008/2/1 happened on: 2008/1/2 23:53

Compute the time of passage through an ascending node:

epoch = Epoch(2019, 1, 1)

time, r = Earth.passage_nodes(epoch)

y, m, d = time.get_date()

d = round(d, 1)

print("Time of passage through ascending node: {}/{}/{}".format(y, m, d))

# Time of passage through ascending node: 2019/3/15.0

print("Radius vector at ascending node: {}".format(round(r, 4)))

# Radius vector at ascending node: 0.9945

Compute the parallax correction:

right_ascension = Angle(22, 38, 7.25, ra=True)

declination = Angle(-15, 46, 15.9)

latitude = Angle(33, 21, 22)

distance = 0.37276

hour_angle = Angle(288.7958)

top_ra, top_dec = Earth.parallax_correction(right_ascension, declination,

                                            latitude, distance, hour_angle)

print_me("Corrected topocentric right ascension: ", top_ra.ra_str(n_dec=2))

# Corrected topocentric right ascension: : 22h 38' 8.54''

print_me("Corrected topocentric declination", top_dec.dms_str(n_dec=1))

# Corrected topocentric declination: -15d 46' 30.0''

Compute the parallax correction in ecliptical coordinates:

longitude = Angle(181, 46, 22.5)

latitude = Angle(2, 17, 26.2)

semidiameter = Angle(0, 16, 15.5)

obs_lat = Angle(50, 5, 7.8)

obliquity = Angle(23, 28, 0.8)

sidereal_time = Angle(209, 46, 7.9)

distance = 0.0024650163

topo_lon, topo_lat, topo_diam = \

    Earth.parallax_ecliptical(longitude, latitude, semidiameter, obs_lat,

                              obliquity, sidereal_time, distance)

print_me("Corrected topocentric longitude", topo_lon.dms_str(n_dec=1))

# Corrected topocentric longitude: 181d 48' 5.0''

print_me("Corrected topocentric latitude", topo_lat.dms_str(n_dec=1))

# Corrected topocentric latitude: 1d 29' 7.1''

print_me("Corrected topocentric semidiameter", topo_diam.dms_str(n_dec=1))

# Corrected topocentric semidiameter: 16' 25.5''

Epoch examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

Let’s start creating and Epoch object, and printing it:

e = Epoch(1987, 6, 19.5)

print_me("JDE for 1987/6/19.5", e)

# JDE for 1987/6/19.5: 2446966.00064

Redefine the Epoch object:

e.set(333, 'Jan', 27, 12)

print_me("JDE for 333/1/27.5", e)

# JDE for 333/1/27.5: 1842713.0

We can create an Epoch from a date or datetime object:

d = datetime.datetime(837, 4, 10, 7, 12, 0, 0)

f = Epoch(d)

print_me("JDE for 837/4/10.3", f)

# JDE for 837/4/10.3: 2026871.8

Let’s check if a given date belong to the Julian or the Gregorian calendar:

print_me("Is 1590/4/21.4 a Julian date?", Epoch.is_julian(1590, 4, 21.4))

# Is 1590/4/21.4 a Julian date?: False

We can also check if a given year is leap or not:

print_me("Is -1000 a leap year?", Epoch.is_leap(-1000))

# Is -1000 a leap year?: True

print_me("Is 1800 a leap year?", Epoch.is_leap(1800))

# Is 1800 a leap year?: False

print_me("Is 2012 a leap year?", Epoch.is_leap(2012))

# Is 2012 a leap year?: True

Get the Day Of Year (DOY) corresponding to a given date:

print_me("Day Of Year (DOY) of 1978/11/14", Epoch.get_doy(1978, 11, 14))

# Day Of Year (DOY) of 1978/11/14: 318.0

print_me("Day Of Year (DOY) of -400/2/29.9", Epoch.get_doy(-400, 2, 29.9))

# Day Of Year (DOY) of -400/2/29.9: 60.9

Now the opposite: Get a date from a DOY:

t = Epoch.doy2date(2017, 365.7)

s = str(t[0]) + "/" + str(t[1]) + "/" + str(round(t[2], 2))

print_me("Date from DOY 2017:365.7", s)

# Date from DOY 2017:365.7: 2017/12/31.7

t = Epoch.doy2date(-4, 60)

s = str(t[0]) + "/" + str(t[1]) + "/" + str(round(t[2], 2))

print_me("Date from DOY -4:60", s)

# Date from DOY -4:60: -4/2/29.0

There is an internal table which we can use to get the leap seconds:

print_me("Number of leap seconds applied up to July 1983", Epoch.leap_seconds(1983, 7))

# Number of leap seconds applied up to July 1983: 12

We can convert the internal JDE value back to a date:

e = Epoch(2436116.31)

y, m, d = e.get_date()

s = str(y) + "/" + str(m) + "/" + str(round(d, 2))

print_me("Date from JDE 2436116.31", s)

# Date from JDE 2436116.31: 1957/10/4.81

It is possible to get the day of the week corresponding to a given date:

e = Epoch(2018, 'Feb', 15)

print_me("The day of week of 2018/2/15 is", e.dow(as_string=True))

# The day of week of 2018/2/15 is: Thursday

In some cases it is useful to get the Modified Julian Day (MJD):

e = Epoch(1923, 'August', 23)

print_me("Modified Julian Day for 1923/8/23", round(e.mjd(), 2))

# Modified Julian Day for 1923/8/23: 23654.0

If your system is appropriately configured, you can get the difference in seconds between your local time and UTC:

print_me("From local system time to UTC you must add/subtract" +

         " this amount of seconds", Epoch.utc2local())

# From local system time to UTC you must add/subtract this amount of seconds: 7200.0

Compute DeltaT = TT - UT differences for various dates:

print_me("DeltaT (TT - UT) for Feb/333", round(Epoch.tt2ut(333, 2), 1))

# DeltaT (TT - UT) for Feb/333: 7358.5

print_me("DeltaT (TT - UT) for Jan/1642", round(Epoch.tt2ut(1642, 1), 1))

# DeltaT (TT - UT) for Jan/1642: 62.1

print_me("DeltaT (TT - UT) for Feb/1928", round(Epoch.tt2ut(1928, 1), 1))

# DeltaT (TT - UT) for Feb/1928: 24.2

print_me("DeltaT (TT - UT) for Feb/1977", round(Epoch.tt2ut(1977, 2), 1))

# DeltaT (TT - UT) for Feb/1977: 47.7

print_me("DeltaT (TT - UT) for Jan/1998", round(Epoch.tt2ut(1998, 1), 1))

# DeltaT (TT - UT) for Jan/1998: 63.0

The difference between civil day and sidereal day is almost 4 minutes:

e = Epoch(1987, 4, 10)

st1 = round(e.mean_sidereal_time(), 9)

e = Epoch(1987, 4, 11)

st2 = round(e.mean_sidereal_time(), 9)

ds = (st2 - st1)*DAY2MIN

msg = "{}m {}s".format(INT(ds), (ds % 1)*60.0)

print_me("Difference between sidereal time 1987/4/11 and 1987/4/10", msg)

# Difference between sidereal time 1987/4/11 and 1987/4/10: 3m 56.555424s

When correcting for nutation-related effects, we get the apparent sidereal time:

e = Epoch(1987, 4, 10)

print_me("e.apparent_sidereal_time(23.44357, (-3.788)/3600.0)",

         e.apparent_sidereal_time(23.44357, (-3.788)/3600.0))

# e.apparent_sidereal_time(23.44357, (-3.788)/3600.0): 0.549145082637

Epoch class can also provide the date of Easter for a given year. Let’s spice up the output a little bit, calling dow() and get_month():

month, day = Epoch.easter(2019)

e = Epoch(2019, month, day)

s = e.dow(as_string=True) + ", " + str(day) + get_ordinal_suffix(day) + \

    " of " + Epoch.get_month(month, as_string=True)

print_me("Easter day for 2019", s)

# Easter day for 2019: Sunday, 21st of April

Compute the date of the Jewish Easter (Pesach) for a given year:

month, day = Epoch.jewish_pesach(1990)

s = str(day) + get_ordinal_suffix(day) + " of " + Epoch.get_month(month, as_string=True)

print_me("Jewish Pesach day for 1990", s)

# Jewish Pesach day for 1990: 10th of April

Now, let’s convert a date in the Moslem calendar to the Gregorian calendar:

y, m, d = Epoch.moslem2gregorian(1421, 1, 1)

print_me("The date 1421/1/1 in the Moslem calendar is, in Gregorian " +

         "calendar", "{}/{}/{}".format(y, m, d))

# The date 1421/1/1 in the Moslem calendar is, in Gregorian calendar: 2000/4/6

y, m, d = Epoch.moslem2gregorian(1439, 9, 1)

print_me("The start of Ramadan month (9/1) for Gregorian year 2018 is",

         "{}/{}/{}".format(y, m, d))

# The start of Ramadan month (9/1) for Gregorian year 2018 is: 2018/5/16

We can go from the Gregorian calendar back to the Moslem calendar too:

print_me("Date 1991/8/13 in Gregorian calendar is, in Moslem calendar",

         "{}/{}/{}".format(*Epoch.gregorian2moslem(1991, 8, 13)))

# Date 1991/8/13 in Gregorian calendar is, in Moslem calendar: 1412/2/2

Note

The * before Epoch will unpack the tuple into components

It is possible to carry out some algebraic operations with Epochs.

Add 10000 days to a given date:

a = Epoch(1991, 7, 11)

b = a + 10000

y, m, d = b.get_date()

s = str(y) + "/" + str(m) + "/" + str(round(d, 2))

print_me("1991/7/11 plus 10000 days is", s)

# 1991/7/11 plus 10000 days is: 2018/11/26.0

Subtract two Epochs to find the number of days between them:

a = Epoch(1986, 2, 9.0)

b = Epoch(1910, 4, 20.0)

print_me("The number of days between 1986/2/9 and 1910/4/20 is", round(a - b, 2))

# The number of days between 1986/2/9 and 1910/4/20 is: 27689.0

We can also subtract a given amount of days from an Epoch:

a = Epoch(2003, 12, 31.0)

b = a - 365.5

y, m, d = b.get_date()

s = str(y) + "/" + str(m) + "/" + str(round(d, 2))

print_me("2003/12/31 minus 365.5 days is", s)

# 2003/12/31 minus 365.5 days is: 2002/12/30.5

Accumulative addition and subtraction of days is also allowed:

a = Epoch(2003, 12, 31.0)

a += 32.5

y, m, d = a.get_date()

s = str(y) + "/" + str(m) + "/" + str(round(d, 2))

print_me("2003/12/31 plus 32.5 days is", s)

# 2003/12/31 plus 32.5 days is: 2004/2/1.5

a = Epoch(2001, 12, 31.0)

a -= 2*365

y, m, d = a.get_date()

s = str(y) + "/" + str(m) + "/" + str(round(d, 2))

print_me("2001/12/31 minus 2*365 days is", s)

# 2001/12/31 minus 2*365 days is: 2000/1/1.0

It is also possible to add days from the right:

a = Epoch(2004, 2, 27.8)

b = 2.2 + a

y, m, d = b.get_date()

s = str(y) + "/" + str(m) + "/" + str(round(d, 2))

print_me("2.2 days plus 2004/2/27.8 is", s)

# 2.2 days plus 2004/2/27.8 is: 2004/3/1.0

Comparison operadors between epochs are also defined:

a = Epoch(2007, 5, 20.0)

b = Epoch(2007, 5, 20.000001)

print_me("2007/5/20.0 == 2007/5/20.000001", a == b)

# 2007/5/20.0 == 2007/5/20.000001: False

print_me("2007/5/20.0 != 2007/5/20.000001", a != b)

# 2007/5/20.0 != 2007/5/20.000001: True

print_me("2007/5/20.0 > 2007/5/20.000001", a > b)

# 2007/5/20.0 > 2007/5/20.000001: False

print_me("2007/5/20.0 <= 2007/5/20.000001", a <= b)

# 2007/5/20.0 <= 2007/5/20.000001: True

Compute the time of rise and setting of the Sun in a given day:

e = Epoch(2018, 5, 2)

print("On May 2nd, 2018, Sun rising/setting times in Munich were (UTC):")

latitude = Angle(48, 8, 0)

longitude = Angle(11, 34, 0)

altitude = 520.0

rising, setting = e.rise_set(latitude, longitude, altitude)

y, m, d, h, mi, s = rising.get_full_date()

print("Rising time: {}:{}".format(h, mi))

# Rising time: 3:50

y, m, d, h, mi, s = setting.get_full_date()

print("Setting time: {}:{}".format(h, mi))

# Setting time: 18:33

Compute the hash of a given Epoch:

h = e.__hash__()

print("Hash of Epoch({}): {}".format(e, h))

# Hash of Epoch(2458240.5): 1152921504609305216

Interpolation examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

Declare an Interpolation object:

i = Interpolation([5, 3, 6, 1, 2, 4, 9], [10, 6, 12, 2, 4, 8])

print(i)

# X: [1, 2, 3, 4, 5, 6]

# Y: [2, 4, 6, 8, 10, 12]

Note

They are ordered in ‘x’
The extra value in ‘x’ was dropped

Use the copy constructor. We can easily make a copy of an Interpolation object:

j = Interpolation(i)

print(j)

# X: [1, 2, 3, 4, 5, 6]

# Y: [2, 4, 6, 8, 10, 12]

j = Interpolation([0.0, 1.0, 3.0], [-1.0, -2.0, 2.0])

print(j)

# X: [0.0, 1.0, 3.0]

# Y: [-1.0, -2.0, 2.0]

print_me("j(2)", j(2))

# j(2): -1.0

print_me("j(0.5)", j(0.5))

# j(0.5): -1.75

Test with a value already in the data table:
```
print_me("j(1)", j(1))

# j(1): -2.0
```

Get the number of interpolation points internally stored:

print_me("Number or interpolation points in 'j'", len(j))

# Number or interpolation points in 'j': 3

We can interpolate Angles too:

k = Interpolation([27.0, 27.5, 28.0, 28.5, 29.0],

                  [Angle(0, 54, 36.125), Angle(0, 54, 24.606),

                   Angle(0, 54, 15.486), Angle(0, 54, 8.694),

                   Angle(0, 54, 4.133)])

print_me("k(28.278)", Angle(k(28.278)).dms_str())

# k(28.278): 54' 11.4279073579''

Let’s work with a new Interpolation object:

m = Interpolation([-1.0, 0.0, 1.0], [-2.0, 3.0, 2.0])

print(m)

# X: [-1.0, 0.0, 1.0]

# Y: [-2.0, 3.0, 2.0]

Get some interpolated values:

print_me("m(-0.5)", m(-0.5))

# m(-0.5): 1.25

print_me("m(0.5)", m(0.5))

# m(0.5): 3.25

Get derivatives:

print_me("m'(-1.0)", m.derivative(-1.0))

# m'(-1.0): 8.0

print_me("m'(-0.5)", m.derivative(-0.5))

# m'(-0.5): 5.0

print_me("m'(0.0)", m.derivative(0.0))

# m'(0.0): 2.0

print_me("m'(0.5)", m.derivative(0.5))

# m'(0.5): -1.0

print_me("m'(1.0)", m.derivative(1.0))

# m'(1.0): -4.0

Get the root within the interval:

print_me("m.root()", m.root())

# m.root(): -0.720759220056

Get the extremum within the interval:

print_me("m.minmax()", m.minmax())

# m.minmax(): 0.333333333333

Let’s work now with the interpolation of sine function:

m = Interpolation([29.43, 30.97, 27.69, 28.11, 31.58, 33.05],

                  [0.4913598528, 0.5145891926, 0.4646875083,

                   0.4711658342, 0.5236885653, 0.5453707057])

print_me("sin(29.5)\t", m(29.5))

# sin(29.5) : 0.492423560118

print_me("sin(30.0)\t", m(30.0))

# sin(30.0) : 0.500000000018

print_me("sin(30.5)\t", m(30.5))

# sin(30.5) : 0.507538362978

Derivatives must be adjusted because degrees were used instead of radians:

print_me("sin'(29.5)\t", degrees(m.derivative(29.5)))

# sin'(29.5)        : 0.870355696916

print_me("sin'(30.0)\t", degrees(m.derivative(30.0)))

# sin'(30.0)        : 0.866025403791

print_me("sqrt(3.0)/2.0\t", sqrt(3.0)/2.0)

# sqrt(3.0)/2.0     : 0.866025403784

print_me("sin'(30.5)\t", degrees(m.derivative(30.5)))

# sin'(30.5)        : 0.861629160353

Jupiter examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

We can compute the geometric heliocentric position for a given epoch:

epoch = Epoch(2018, 10, 27.0)

lon, lat, r = Jupiter.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", lon.to_positive())

# Geometric Heliocentric Longitude: 241.5873

print_me("Geometric Heliocentric Latitude", lat)

# Geometric Heliocentric Latitude: 0.8216

print_me("Radius vector", r)

# Radius vector: 5.36848

Compute the geocentric position for 1992/12/20:

epoch = Epoch(1992, 12, 20.0)

ra, dec, elon = Jupiter.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 12h 47' 9.6''

print_me("Declination", dec.dms_str(n_dec=1))

# Declination: -3d 41' 55.3''

print_me("Elongation", elon.dms_str(n_dec=1))

# Elongation: 76d 2' 26.0''

Print mean orbital elements for Jupiter at 2065.6.24:

epoch = Epoch(2065, 6, 24.0)

l, a, e, i, ome, arg = Jupiter.orbital_elements_mean_equinox(epoch)

print_me("Mean longitude of the planet", round(l, 6))

# Mean longitude of the planet: 222.433723

print_me("Semimajor axis of the orbit (UA)", round(a, 8))

# Semimajor axis of the orbit (UA): 5.20260333

print_me("Eccentricity of the orbit", round(e, 7))

# Eccentricity of the orbit: 0.0486046

print_me("Inclination on plane of the ecliptic", round(i, 6))

# Inclination on plane of the ecliptic: 1.29967

print_me("Longitude of the ascending node", round(ome, 5))

# Longitude of the ascending node: 101.13309

print_me("Argument of the perihelion", round(arg, 6))

# Argument of the perihelion: -85.745532

Compute the time of the conjunction close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

conj = Jupiter.conjunction(epoch)

y, m, d = conj.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Conjunction date", date)

# Conjunction date: 1993/10/18.3341

Compute the time of the opposition close to -6/9/1:

epoch = Epoch(-6, 9, 1.0)

oppo = Jupiter.opposition(epoch)

y, m, d = oppo.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Opposition date", date)

# Opposition date: -6/9/15.2865

Compute the time of the station in longitude #1 close to 2018/11/1:

epoch = Epoch(2018, 11, 1.0)

sta1 = Jupiter.station_longitude_1(epoch)

y, m, d = sta1.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #1", date)

# Date of station in longitude #1: 2018/3/9.1288

Compute the time of the station in longitude #2 close to 2018/11/1:

epoch = Epoch(2018, 11, 1.0)

sta2 = Jupiter.station_longitude_2(epoch)

y, m, d = sta2.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #2", date)

# Date of station in longitude #2: 2018/7/10.6679

Find the epoch of the Aphelion closer to 1981/6/1:

epoch = Epoch(1981, 6, 1.0)

e = Jupiter.perihelion_aphelion(epoch, perihelion=False)

y, m, d, h, mi, s = e.get_full_date()

peri = str(y) + '/' + str(m) + '/' + str(d) + ' at ' + str(h) + ' hours'

print_me("The Aphelion closest to 1981/6/1 will happen on", peri)

# The Aphelion closest to 1981/6/1 will happen on: 1981/7/28 at 6 hours

Compute the time of passage through an ascending node:

epoch = Epoch(2019, 1, 1)

time, r = Jupiter.passage_nodes(epoch)

y, m, d = time.get_date()

d = round(d, 1)

print("Time of passage through ascending node: {}/{}/{}".format(y, m, d))

# Time of passage through ascending node: 2025/9/15.6

print("Radius vector at ascending node: {}".format(round(r, 4)))

# Radius vector at ascending node: 5.1729

JupiterMoons examples¶

Let’s define a small helper function:

def print_me(msg, val):
    print("{}: {}".format(msg, val))

Lets compute the ascending node of Jupiter as well as the longitude of the node of the equator of Jupiter on the ecliptic (psi):

utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
psi_corrected, OMEGA_ascending_node_jupiter = JupiterMoons.jupiter_system_angles(utc_1992_12_16_00_00_00)
print("Ascending node of Jupiter: ", OMEGA_ascending_node_jupiter)
#100.39249942976576
print("Longitude of the node of the eauator of Jupiter on the ecliptic (psi):", psi_corrected)
#317.1058009213959

Lets compute the corrected rectangular geocentric position of Jupiter’s satellites for a given epoch, using the E5-theory:

utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)

io, europa, ganymede, callisto = JupiterMoons.rectangular_positions_jovian_equatorial(utc_1992_12_16_00_00_00)

print("Corrected rectangular geocentric position of Io [X, Y , Z]: ", io)
#(-3.450168811390241, 0.21370246960509387, -4.818966623735296)

print("Corrected rectangular geocentric position of Europa [X, Y , Z]: ", europa)
#(7.441869121153001, 0.27524463479625677, -5.747104399729193)

print("Corrected rectangular geocentric position of Ganymede [X, Y , Z]: ", ganymede)
#(1.201111684800708, 0.5899903274317162, -14.940581367576527)

print("Corrected rectangular geocentric position of Callisto [X, Y , Z]: ", callisto)
#(7.071943240286434, 1.0289562923230684, -25.224137724734955)

Lets compute the uncorrected rectangular geocentric position of Jupiter’s satellites for a given epoch, using the E5-theory:

#So the effects of different light-time and perspective described in Pymeeus page 313 - 314 are neglected

utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
io_uncorrected, europa_uncorrected, ganymede_uncorrected, callisto_uncorrected = \
    JupiterMoons.rectangular_positions_jovian_equatorial(utc_1992_12_16_00_00_00, do_correction=False)

print("Uncorrected rectangular geocentric position of Io [X, Y , Z]: ", io_uncorrected)
# (-3.4489935969836503, 0.21361563816963675, -4.818966623735296)

print("Uncorrected rectangular geocentric position of Europa [X, Y , Z]: ", europa_uncorrected)
# (7.438101803124541, 0.2751112576349763, -5.747104399729193)

print("Uncorrected rectangular geocentric position of Ganymede [X, Y , Z]: ", ganymede_uncorrected)
# (1.1990581804888616, 0.589247092847632, -14.940581367576527)

print("Uncorrected rectangular geocentric position of Callisto [X, Y , Z]: ", callisto_uncorrected)
# (7.056237832405445, 1.0267678919629089, -25.224137724734955)

Lets calculate the distance between Earth and Jupiter (DELTA) for a given epoch:

utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
delta, tau, l, b, r = JupiterMoons.calculate_delta(utc_1992_12_16_00_00_00)

print("Distance between Earth and Jupiter in AU: ", delta)
#5.6611211815432645

print("Light-time from Earth to Jupiter in d (day): ", tau)
#0.03269590898252075

Lets calculate the perspective distance in Jupiter radii of all satellites for an eclipse of Io:

io_ecc_start_2021_02_12_14_19_14 = Epoch(2021, 2, 12.5966898148148)

result_matrix = JupiterMoons.check_phenomena(io_ecc_start_2021_02_12_14_19_14)

#structure of result_matrix
# Row 0: Io          Column 0: perspective distance as seen from the Earth
# Row 1: Europa      Column 1: perspective distance as seen from the Sun
# Row 2: Ganymede    Column 2: No use
# Row 3: Callisto

# print Row 0
print("(perspective distance of Io (Earth View), perspective distance of Io (Sun view), No use): ")
print(result_matrix[0])
#[1.1926058680144362, 0.856027716233023, 0.0]

# print Row 1
print("(perspective distance of Europa (Earth View), perspective distance of Europa (Sun view), No use): ")
print(result_matrix[1])
#[-8.739720236890856, -8.893094092124032, 0.0]

# print Row 2
print("(perspective distance of Ganymede (Earth View), perspective distance of Ganymede (Sun view), No use): ")
print(result_matrix[2])
#[14.069121992481382, 13.8323491767871, 0.0]

# print Row 3
print("(perspective distance of Callisto (Earth View), perspective distance of Callisto (Sun view), No use): ")
print(result_matrix[3])
#[-2.934134686233644, -3.9904786452498144, 0.0]

Lets check if an eclipse orand occultation for any of the four Galilean satellites is detected for a given epoch:

io_ecc_start_2021_02_12_14_19_14 = Epoch(2021, 2, 12.5966898148148)

#Structure of result matrix
# Row 0: Io          Column 0: Occultation True\False
# Row 1: Europa      Column 1: Eclipse True\False
# Row 2: Ganymede    Column 2: No use
# Row 3: Callisto

result_matrix = JupiterMoons.is_phenomena(io_ecc_start_2021_02_12_14_19_14)

#print Row 0
print("(Occultation of Io, Eclipse of Io, No use): ")
print(result_matrix[0])
#[False, True, False]

# print Row 1
print(" (Occultation of Europa, Eclipse of Europa, No use): ")
print(result_matrix[1])
#[False, False, False]

# print Row 2
print(" (Occultation of Ganymede, Eclipse of Gaymede, No use): ")
print(result_matrix[2])
#[False,False,False]

# print Row 3
print("(Occultation of Callisto, Eclipse of Callisto, No use): ")
print(result_matrix[3])
#[False,False,False]

Calculation of the perspective distance ot the planet Io to the center of Jupiter for December 16 at 0h UTC as seen from the Sun:

utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)

result_matrix = JupiterMoons.rectangular_positions_jovian_equatorial(utc_1992_12_16_00_00_00, solar=True)

#Structure of result matrix
# Row 0: Io          Column 0: X coordinate of satellite in Jupiter radii
# Row 1: Europa      Column 1: Y coordinate of satellite in Jupiter radii
# Row 2: Ganymede    Column 2: Z coordinate of satellite in Jupiter radii
# Row 3: Callisto

io_radius_to_center_of_jupiter_sun = JupiterMoons.check_coordinates(result_matrix[0][0], result_matrix[0][1])

print("Perspective distance of Io as seen from the Sun in Jupiter radii: ", io_radius_to_center_of_jupiter_sun)
#3.457757270630766

Calculation of the perspective distance ot the planet Io to the center of Jupiter for December 16 at 0h UTC as seen from the Earth:

utc_1992_12_16_00_00_00 = Epoch(1992, 12, 16, utc=True)
result_matrix = JupiterMoons.rectangular_positions_jovian_equatorial(utc_1992_12_16_00_00_00, solar=False)

#Structure of result matrix
# Row 0: Io          Column 0: X coordinate of satellite in Jupiter radii
# Row 1: Europa      Column 1: Y coordinate of satellite in Jupiter radii
# Row 2: Ganymede    Column 2: Z coordinate of satellite in Jupiter radii
# Row 3: Callisto

io_radius_to_center_of_jupiter_earth = JupiterMoons.check_coordinates(result_matrix[0][0], result_matrix[0][1])

print("Perspective distance of Io as seen from the Earth in Jupiter radii: ", io_radius_to_center_of_jupiter_earth)
# 2.553301264153796

Mars examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

We can compute the geometric heliocentric position for a given epoch:

epoch = Epoch(2018, 10, 27.0)

lon, lat, r = Mars.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", lon.to_positive())

# Geometric Heliocentric Longitude: 2.0015

print_me("Geometric Heliocentric Latitude", lat)

# Geometric Heliocentric Latitude: -1.3683

print_me("Radius vector", r)

# Radius vector: 1.39306

Compute the geocentric position for 1992/12/20:

epoch = Epoch(1992, 12, 20.0)

ra, dec, elon = Mars.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 7h 48' 35.4''

print_me("Declination", dec.dms_str(n_dec=1))

# Declination: 24d 35' 33.9''

print_me("Elongation", elon.dms_str(n_dec=1))

# Elongation: 153d 35' 1.6''

Print mean orbital elements for Mars at 2065.6.24:

epoch = Epoch(2065, 6, 24.0)

l, a, e, i, ome, arg = Mars.orbital_elements_mean_equinox(epoch)

print_me("Mean longitude of the planet", round(l, 6))

# Mean longitude of the planet: 288.855211

print_me("Semimajor axis of the orbit (UA)", round(a, 8))

# Semimajor axis of the orbit (UA): 1.52367934

print_me("Eccentricity of the orbit", round(e, 7))

# Eccentricity of the orbit: 0.0934599

print_me("Inclination on plane of the ecliptic", round(i, 6))

# Inclination on plane of the ecliptic: 1.849338

print_me("Longitude of the ascending node", round(ome, 5))

# Longitude of the ascending node: 50.06365

print_me("Argument of the perihelion", round(arg, 6))

# Argument of the perihelion: 287.202108

Compute the time of the conjunction close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

conj = Mars.conjunction(epoch)

y, m, d = conj.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Conjunction date", date)

# Conjunction date: 1993/12/27.0898

Compute the time of the opposition close to 2729/10/1:

epoch = Epoch(2729, 10, 1.0)

oppo = Mars.opposition(epoch)

y, m, d = oppo.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Opposition date", date)

# Opposition date: 2729/9/9.1412

Compute the time of the station in longitude #1 close to 1997/3/1:

epoch = Epoch(1997, 3, 1.0)

sta1 = Mars.station_longitude_1(epoch)

y, m, d = sta1.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #1", date)

# Date of station in longitude #1: 1997/2/6.033

Compute the time of the station in longitude #2 close to 1997/3/1:

epoch = Epoch(1997, 3, 1.0)

sta2 = Mars.station_longitude_2(epoch)

y, m, d = sta2.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #2", date)

# Date of station in longitude #2: 1997/4/27.7553

Find the epoch of the Aphelion closer to 2032/1/1:

epoch = Epoch(2032, 1, 1.0)

e = Mars.perihelion_aphelion(epoch, perihelion=False)

y, m, d, h, mi, s = e.get_full_date()

peri = str(y) + '/' + str(m) + '/' + str(d) + ' at ' + str(h) + ' hours'

print_me("The Aphelion closest to 2032/1/1 will happen on", peri)

# The Aphelion closest to 2032/1/1 will happen on: 2032/10/24 at 22 hours

Compute the time of passage through an ascending node:

epoch = Epoch(2019, 1, 1)

time, r = Mars.passage_nodes(epoch)

y, m, d = time.get_date()

d = round(d, 1)

print("Time of passage through ascending node: {}/{}/{}".format(y, m, d))

# Time of passage through ascending node: 2019/1/15.2

print("Radius vector at ascending node: {}".format(round(r, 4)))

# Radius vector at ascending node: 1.4709

Mercury examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

We can compute the geometric heliocentric position for a given epoch:

epoch = Epoch(2018, 10, 27.0)

lon, lat, r = Mercury.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", lon.to_positive())

# Geometric Heliocentric Longitude: 287.4887

print_me("Geometric Heliocentric Latitude", lat)

# Geometric Heliocentric Latitude: -6.0086

print_me("Radius vector", r)

# Radius vector: 0.45113

Compute the geocentric position for 1992/12/20:

epoch = Epoch(1992, 12, 20.0)

ra, dec, elon = Mercury.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 16h 33' 59.3''

print_me("Declination", dec.dms_str(n_dec=1))

# Declination: -20d 53' 31.6''

print_me("Elongation", elon.dms_str(n_dec=1))

# Elongation: 18d 24' 29.8''

Print mean orbital elements for Mercury at 2065.6.24:

epoch = Epoch(2065, 6, 24.0)

l, a, e, i, ome, arg = Mercury.orbital_elements_mean_equinox(epoch)

print_me("Mean longitude of the planet", round(l, 6))

# Mean longitude of the planet: 203.494701

print_me("Semimajor axis of the orbit (UA)", round(a, 8))

# Semimajor axis of the orbit (UA): 0.38709831

print_me("Eccentricity of the orbit", round(e, 7))

# Eccentricity of the orbit: 0.2056451

print_me("Inclination on plane of the ecliptic", round(i, 6))

# Inclination on plane of the ecliptic: 7.006171

print_me("Longitude of the ascending node", round(ome, 5))

# Longitude of the ascending node: 49.10765

print_me("Argument of the perihelion", round(arg, 6))

# Argument of the perihelion: 29.367732

Compute the time of the inferior conjunction close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

conjunction = Mercury.inferior_conjunction(epoch)

y, m, d = conjunction.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Inferior conjunction date", date)

# Inferior conjunction date: 1993/11/6.1449

Compute the time of the superior conjunction close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

conjunction = Mercury.superior_conjunction(epoch)

y, m, d = conjunction.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Superior conjunction date", date)

# Superior conjunction date: 1993/8/29.3301

Compute the time and angle of the western elongation close to 1993/11/1:

epoch = Epoch(1993, 11, 1.0)

time, elongation = Mercury.western_elongation(epoch)

y, m, d = time.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Western elongation date", date)

# Western elongation date: 1993/11/22.6386

elong = round(elongation, 4)

print_me("Maximum western elongation angle", elong)

# Maximum western elongation angle: 19.7506

Compute the time and angle of the eastern elongation close to 1990/8/1:

epoch = Epoch(1990, 8, 1.0)

time, elongation = Mercury.eastern_elongation(epoch)

y, m, d = time.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Eastern elongation date", date)

# Eastern elongation date: 1990/8/11.8514

elong = round(elongation, 4)

print_me("Maximum eastern elongation angle", elong)

# Maximum eastern elongation angle: 27.4201

Compute the time of the station in longitude #1 close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

sta1 = Mercury.station_longitude_1(epoch)

y, m, d = sta1.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #1", date)

# Date of station in longitude #1: 1993/10/25.9358

Compute the time of the station in longitude #2 close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

sta2 = Mercury.station_longitude_2(epoch)

y, m, d = sta2.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #2", date)

# Date of station in longitude #2: 1993/11/15.0724

Find the epoch of the Perihelion closer to 2000/01/01:

epoch = Epoch(2000, 1, 1.0)

e = Mercury.perihelion_aphelion(epoch)

y, m, d, h, mi, s = e.get_full_date()

peri = str(y) + '/' + str(m) + '/' + str(d) + ' at ' + str(h) + ' hours'

print_me("The Perihelion closest to 2000/1/1 happened on", peri)

# The Perihelion closest to 2000/1/1 happened on: 2000/2/15 at 18 hours

Compute the time of passage through an ascending node:

epoch = Epoch(2019, 1, 1)

time, r = Mercury.passage_nodes(epoch)

y, m, d = time.get_date()

d = round(d, 1)

print("Time of passage through ascending node: {}/{}/{}".format(y, m, d))

# Time of passage through ascending node: 2018/11/24.7

print("Radius vector at ascending node: {}".format(round(r, 4)))

# Radius vector at ascending node: 0.3143

Minor examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

Let’s compute the equatorial coordinates of comet Encke:

a = 2.2091404

e = 0.8502196

q = a * (1.0 - e)

i = Angle(11.94524)

omega = Angle(334.75006)

w = Angle(186.23352)

t = Epoch(1990, 10, 28.54502)

epoch = Epoch(1990, 10, 6.0)

minor = Minor(q, e, i, omega, w, t)

ra, dec, elong = minor.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 10h 34' 13.7''

print_me("Declination", dec.dms_str(n_dec=0))

# Declination: 19d 9' 32.0''

print_me("Elongation", round(elong, 2))

# Elongation: 40.51

Now compute the heliocentric ecliptical coordinates:

a = 2.2091404

e = 0.8502196

q = a * (1.0 - e)

i = Angle(11.94524)

omega = Angle(334.75006)

w = Angle(186.23352)

t = Epoch(1990, 10, 28.54502)

epoch = Epoch(1990, 10, 6.0)

minor = Minor(q, e, i, omega, w, t)

lon, lat = minor.heliocentric_ecliptical_position(epoch)

print_me("Heliocentric ecliptical longitude", lon.dms_str(n_dec=1))

# Heliocentric ecliptical longitude: 66d 51' 57.8''

print_me("Heliocentric ecliptical latitude", lat.dms_str(n_dec=1))

# Heliocentric ecliptical latitude: 11d 56' 14.4''

Moon examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

Let’s now compute the Moon geocentric ecliptical position for a given epoch:

epoch = Epoch(1992, 4, 12.0)

Lambda, Beta, Delta, ppi = Moon.geocentric_ecliptical_pos(epoch)

print_me("Longitude (Lambda)", round(Lambda, 6))

# Longitude (Lambda): 133.162655

print_me("Latitude (Beta)", round(Beta, 6))

# Latitude (Beta): -3.229126

print_me("Distance (Delta)", round(Delta, 1))

# Distance (Delta): 368409.7

print_me("Equatorial horizontal parallax (Pi)", round(ppi, 6))

# Equatorial horizontal parallax (Pi): 0.99199

Now let’s compute the apparent ecliptical position:

epoch = Epoch(1992, 4, 12.0)

Lambda, Beta, Delta, ppi = Moon.apparent_ecliptical_pos(epoch)

print_me("Longitude (Lambda)", round(Lambda, 6))

# Longitude (Lambda): 133.167264

print_me("Latitude (Beta)", round(Beta, 6))

# Latitude (Beta): -3.229126

print_me("Distance (Delta)", round(Delta, 1))

# Distance (Delta): 368409.7

print_me("Equatorial horizontal parallax (Pi)", round(ppi, 6))

# Equatorial horizontal parallax (Pi): 0.99199

Get the apparent equatorial position:

epoch = Epoch(1992, 4, 12.0)

ra, dec, Delta, ppi = Moon.apparent_equatorial_pos(epoch)

print_me("Right Ascension (ra)", round(ra, 6))

# Right Ascension (ra): 134.688469

print_me("Declination (dec)", round(dec, 6))

# Declination (dec): 13.768367

print_me("Distance (Delta)", round(Delta, 1))

# Distance (Delta): 368409.7

print_me("Equatorial horizontal parallax (Pi)", round(ppi, 6))

# Equatorial horizontal parallax (Pi): 0.99199

Compute the longitude of the Moon’s mean ascending node:

epoch = Epoch(1913, 5, 27.0)

Omega = Moon.longitude_mean_ascending_node(epoch)

print_me("Longitude of the mean ascending node", round(Omega, 1))

# Longitude of the mean ascending node: 0.0

epoch = Epoch(1959, 12, 7.0)

Omega = Moon.longitude_mean_ascending_node(epoch)

print_me("Longitude of the mean ascending node", round(Omega, 1))

# Longitude of the mean ascending node: 180.0

Get the longitude of the Moonś true ascending node:

epoch = Epoch(1913, 5, 27.0)

Omega = Moon.longitude_true_ascending_node(epoch)

print_me("Longitude of the true ascending node", round(Omega, 4))

# Longitude of the true ascending node: 0.8763

Compute the longitude of the Moon’s mean perigee:

epoch = Epoch(2021, 3, 5.0)

Pi = Moon.longitude_mean_perigee(epoch)

print_me("Longitude of the mean perigee", round(Pi, 5))

# Longitude of the mean perigee: 224.89194

Compute the approximate illuminated fraction of the Moon’s disk:

epoch = Epoch(1992, 4, 12.0)

k = Moon.illuminated_fraction_disk(epoch)

print_me("Approximate illuminated fraction of Moon's disk", round(k, 2))

# Approximate illuminated fraction of Moon's disk: 0.68

Compute the position angle of the bright limb of the Moon:

epoch = Epoch(1992, 4, 12.0)

xi = Moon.position_bright_limb(epoch)

print_me("Position angle of the bright limb of the Moon", round(xi, 1))

# Position angle of the bright limb of the Moon: 285.0

Calculate the instant of a New Moon:

epoch = Epoch(1977, 2, 15.0)

new_moon = Moon.moon_phase(epoch, target="new")

y, m, d, h, mi, s = new_moon.get_full_date()

print("New Moon: {}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s)))

# New Moon: 1977/2/18 3:37:42

Calculate the time of a Last Quarter:

epoch = Epoch(2044, 1, 15.0)

new_moon = Moon.moon_phase(epoch, target="last")

y, m, d, h, mi, s = new_moon.get_full_date()

print("Last Quarter: {}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s)))

# Last Quarter: 2044/1/21 23:48:17

Compute the time and parallax of apogee:

epoch = Epoch(1988, 10, 1.0)

apogee, parallax = Moon.moon_perigee_apogee(epoch, target="apogee")

y, m, d, h, mi, s = apogee.get_full_date()

print("Apogee epoch: {}/{}/{} {}:{}".format(y, m, d, h, mi))

# Apogee epoch: 1988/10/7 20:30

print_me("Equatorial horizontal parallax", parallax.dms_str(n_dec=3))

# Equatorial horizontal parallax: 54' 0.679''

Compute the time of passage by the ascending node:

epoch = Epoch(1987, 5, 15.0)

passage = Moon.moon_passage_nodes(epoch, target="ascending")

y, m, d, h, mi, s = passage.get_full_date()

mi += s/60.0

print("Passage by the ascending node: {}/{}/{} {}:{}".format(y, m, d, h, round(mi)))

# Passage by the ascending node: 1987/5/23 6:26

Compute the epoch and amplitude of maximum southern declination:

epoch = Epoch(2049, 4, 15.0)

epo, dec = Moon.moon_maximum_declination(epoch, target='southern')

y, m, d, h, mi, s = epo.get_full_date()

print("Epoch of maximum declination: {}/{}/{} {}:{}".format(y, m, d, h, mi))

# Epoch of maximum declination: 2049/4/21 14:0

print_me("Amplitude of maximum declination", dec.dms_str(n_dec=0))

# Amplitude of maximum declination: -22d 8' 18.0''

Compute the librations of the Moon:

epoch = Epoch(1992, 4, 12.0)

lopt, bopt, lphys, bphys, ltot, btot = Moon.moon_librations(epoch)

print_me("Optical libration in longitude", round(lopt, 3))

# Optical libration in longitude: -1.206

print_me("Optical libration in latitude", round(bopt, 3))

# Optical libration in latitude: 4.194

print_me("Physical libration in longitude", round(lphys, 3))

# Physical libration in longitude: -0.025

print_me("Physical libration in latitude", round(bphys, 3))

# Physical libration in latitude: 0.006

print_me("Total libration in longitude", round(lphys, 2))

# Total libration in longitude: -1.23

print_me("Total libration in latitude", round(bphys, 3))

# Total libration in latitude: 4.2

Let’s calculate the position angle of the Moon’s axis of rotation:

epoch = Epoch(1992, 4, 12.0)

p = Moon.moon_position_angle_axis(epoch)

print_me("Position angle of Moon's axis of rotation", round(p, 2))

# Position angle of Moon's axis of rotation: 15.08

Neptune examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

We can compute the geometric heliocentric position for a given epoch:

epoch = Epoch(2018, 10, 27.0)

lon, lat, r = Neptune.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", lon.to_positive())

# Geometric Heliocentric Longitude: 345.3776

print_me("Geometric Heliocentric Latitude", lat)

# Geometric Heliocentric Latitude: -0.9735

print_me("Radius vector", r)

# Radius vector: 29.93966

Compute the geocentric position for 1992/12/20:

epoch = Epoch(1992, 12, 20.0)

ra, dec, elon = Neptune.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 19h 17' 14.5''

print_me("Declination", dec.dms_str(n_dec=1))

# Declination: -21d 34' 15.1''

print_me("Elongation", elon.dms_str(n_dec=1))

# Elongation: 19d 44' 59.6''

Print mean orbital elements for Neptune at 2065.6.24:

epoch = Epoch(2065, 6, 24.0)

l, a, e, i, ome, arg = Neptune.orbital_elements_mean_equinox(epoch)

print_me("Mean longitude of the planet", round(l, 6))

# Mean longitude of the planet: 88.321947

print_me("Semimajor axis of the orbit (UA)", round(a, 8))

# Semimajor axis of the orbit (UA): 30.11038676

print_me("Eccentricity of the orbit", round(e, 7))

# Eccentricity of the orbit: 0.0094597

print_me("Inclination on plane of the ecliptic", round(i, 6))

# Inclination on plane of the ecliptic: 1.763855

print_me("Longitude of the ascending node", round(ome, 5))

# Longitude of the ascending node: 132.46986

print_me("Argument of the perihelion", round(arg, 6))

# Argument of the perihelion: -83.415521

Compute the time of the conjunction close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

conj = Neptune.conjunction(epoch)

y, m, d = conj.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Conjunction date", date)

# Conjunction date: 1994/1/11.3057

Compute the time of the opposition close to 1846/8/1:

epoch = Epoch(1846, 8, 1)

oppo = Neptune.opposition(epoch)

y, m, d = oppo.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Opposition date", date)

# Opposition date: 1846/8/20.1623

Pluto examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

We can compute the geometric heliocentric position for a given epoch:

epoch = Epoch(2018, 10, 27.0)

lon, lat, r = Pluto.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", lon.to_positive())

# Geometric Heliocentric Longitude: 232.740711423

print_me("Geometric Heliocentric Latitude", lat)

# Geometric Heliocentric Latitude: 14.5878173017

print_me("Radius vector", r)

# Radius vector: 29.711110981

Compute the geocentric position for 1992/12/20:

epoch = Epoch(1992, 12, 20.0)

ra, dec, elon = Pluto.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 15h 31' 43.7''

print_me("Declination", dec.dms_str(n_dec=1))

# Declination: -4d 27' 28.8''

Saturn examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

We can compute the geometric heliocentric position for a given epoch:

epoch = Epoch(2018, 10, 27.0)

lon, lat, r = Saturn.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", lon.to_positive())

# Geometric Heliocentric Longitude: 279.5108

print_me("Geometric Heliocentric Latitude", lat)

# Geometric Heliocentric Latitude: 0.6141

print_me("Radius vector", r)

# Radius vector: 10.06266

Compute the geocentric position for 1992/12/20:

epoch = Epoch(1992, 12, 20.0)

ra, dec, elon = Saturn.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 21h 11' 41.8''

print_me("Declination", dec.dms_str(n_dec=1))

# Declination: -17d 15' 40.8''

print_me("Elongation", elon.dms_str(n_dec=1))

# Elongation: 46d 51' 47.7''

Print mean orbital elements for Saturn at 2065.6.24:

epoch = Epoch(2065, 6, 24.0)

l, a, e, i, ome, arg = Saturn.orbital_elements_mean_equinox(epoch)

print_me("Mean longitude of the planet", round(l, 6))

# Mean longitude of the planet: 131.196871

print_me("Semimajor axis of the orbit (UA)", round(a, 8))

# Semimajor axis of the orbit (UA): 9.55490779

print_me("Eccentricity of the orbit", round(e, 7))

# Eccentricity of the orbit: 0.0553209

print_me("Inclination on plane of the ecliptic", round(i, 6))

# Inclination on plane of the ecliptic: 2.486426

print_me("Longitude of the ascending node", round(ome, 5))

# Longitude of the ascending node: 114.23974

print_me("Argument of the perihelion", round(arg, 6))

# Argument of the perihelion: -19.896331

Compute the time of the conjunction close to 2125/6/1:

epoch = Epoch(2125, 6, 1.0)

conj = Saturn.conjunction(epoch)

y, m, d = conj.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Conjunction date", date)

# Conjunction date: 2125/8/26.4035

Compute the time of the opposition close to -6/9/1:

epoch = Epoch(-6, 9, 1.0)

oppo = Saturn.opposition(epoch)

y, m, d = oppo.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Opposition date", date)

# Opposition date: -6/9/14.3709

Compute the time of the station in longitude #1 close to 2018/11/1:

epoch = Epoch(2018, 11, 1.0)

sta1 = Saturn.station_longitude_1(epoch)

y, m, d = sta1.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #1", date)

# Date of station in longitude #1: 2018/4/17.9433

Compute the time of the station in longitude #2 close to 2018/11/1:

epoch = Epoch(2018, 11, 1.0)

sta2 = Saturn.station_longitude_2(epoch)

y, m, d = sta2.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #2", date)

# Date of station in longitude #2: 2018/9/6.4175

Find the epoch of the Perihelion closer to 2000/1/1:

epoch = Epoch(2000, 1, 1.0)

e = Saturn.perihelion_aphelion(epoch)

y, m, d, h, mi, s = e.get_full_date()

peri = str(y) + '/' + str(m) + '/' + str(d) + ' at ' + str(h) + ' hours'

print_me("The Perihelion closest to 2000/1/1 happened on", peri)

# The Perihelion closest to 2000/1/1 happened on: 2003/7/26 at 15 hours

Compute the time of passage through an ascending node:

epoch = Epoch(2019, 1, 1)

time, r = Saturn.passage_nodes(epoch)

y, m, d = time.get_date()

d = round(d, 1)

print("Time of passage through ascending node: {}/{}/{}".format(y, m, d))

# Time of passage through ascending node: 2034/5/30.2

print_me("Radius vector at ascending node", round(r, 4))

# Radius vector at ascending node: 9.0546

Compute the approximate magnitude of Saturn:

sun_dist = 9.867882

earth_dist = 10.464606

delta_u = Angle(16.442)

b = Angle(4.198)

m = Saturn.magnitude(sun_dist, earth_dist, delta_u, b)

printi_me("Approximate magnitude of Saturn", m)

# Approximate magnitude of Saturn: 1.9

Compute the ring inclination:

epoch = Epoch(1992, 12, 16.00068)

i = Saturn.ring_inclination(epoch)

print_me("Saturn's ring inclination", round(i, 6))

# Saturn's ring inclination: 28.076131

Compute the longitude of the ascending node of the ring:

epoch = Epoch(1992, 12, 16.00068)

omega = Saturn.ring_logitude_ascending_node(epoch)

print_me("Saturn's ring longitude of the ascending node", round(omega, 6))

# Saturn's ring longitude of the ascending node: 169.410243

Compute the parameters related to the ring:

epoch = Epoch(1992, 12, 16.00068)

B, Bprime, P, delta_U, a, b = Saturn.ring_parameters(epoch)

print_me("Saturnicentric latitude of the Earth", round(B, 3))

# Saturnicentric latitude of the Earth: 16.442

print_me("Saturnicentric latitude of the Sun", round(Bprime, 3))

# Saturnicentric latitude of the Sun: 14.679

print_me("Geocentric position angle of nothern semiminor axis", round(P, 3))

# Geocentric position angle of nothern semiminor axis: 6.741

print_me("Difference in Saturnicentric longitudes of Sun and Earth", round(delta_U, 3))

# Difference in Saturnicentric longitudes of Sun and Earth: 4.198

print_me("Size of major axis of outer ring", round(a, 2))

# Size of major axis of outer ring: 35.87

print_me("Size of minor axis of outer ring", round(b, 2))

# Size of minor axis of outer ring: 10.15

Sun examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

It is possible to compute an approximation of the Sun’s true ecliptical longitude:

epoch = Epoch(1992, 10, 13)

true_lon, r = Sun.true_longitude_coarse(epoch)

print_me("Sun's approximate true longitude", true_lon.dms_str(n_dec=0))

# Sun's approximate true longitude: 199d 54' 36.0''

print_me("Sun's radius vector", round(r, 5))

# Sun's radius vector: 0.99766

Now let’s compute the Sun’s approximate apparent ecliptical longitude:

epoch = Epoch(1992, 10, 13)

app_lon, r = Sun.apparent_longitude_coarse(epoch)

print_me("Sun's approximate apparent longitude", app_lon.dms_str(n_dec=0))

# Sun's approximate apparent longitude: 199d 54' 32.0''

And now is the turn for the apparent right ascension and declination:

epoch = Epoch(1992, 10, 13)

ra, delta, r = Sun.apparent_rightascension_declination_coarse(epoch)

print_me("Sun's apparent right ascension", ra.ra_str(n_dec=1))

# Sun's apparent right ascension: 13h 13' 31.4''

print_me("Sun's apparent declination", delta.dms_str(n_dec=0))

# Sun's apparent declination: -7d 47' 6.0''

Now, let’s compute Sun’s true (geometric) position again, but more accurately:

epoch = Epoch(1992, 10, 13.0)

l, b, r = Sun.geometric_geocentric_position(epoch, toFK5=False)

print_me("Geometric Geocentric Longitude", round(l.to_positive(), 6))

# Geometric Geocentric Longitude: 199.907297

print_me("Geometric Geocentric Latitude", b.dms_str(n_dec=3))

# Geometric Geocentric Latitude: 0.744''

print_me("Radius vector", round(r, 8))

# Radius vector: 0.99760852

Compute Sun’s apparent postion accurately:

epoch = Epoch(1992, 10, 13.0)

l, b, r = Sun.apparent_geocentric_position(epoch)

print_me("Apparent Geocentric Longitude", l.to_positive().dms_str(n_dec=3))

# Apparent Geocentric Longitude: 199d 54' 21.548''

print_me("Apparent Geocentric Latitude", b.dms_str(n_dec=3))

# Apparent Geocentric Latitude; 0.721''

print_me("Radius vector", round(r, 8))

# Radius vector: 0.99760852

We can compute rectangular coordinates referred to mean equinox of date:

epoch = Epoch(1992, 10, 13.0)

x, y, z = Sun.rectangular_coordinates_mean_equinox(epoch)

print_me("X", round(x, 7))

# X: -0.9379963

print_me("Y", round(y, 6))

# Y: -0.311654

print_me("Z", round(z, 7))

# Z: -0.1351207

Now, compute rectangular coordinates w.r.t. standard equinox J2000.0:

epoch = Epoch(1992, 10, 13.0)

x, y, z = Sun.rectangular_coordinates_j2000(epoch)

print_me("X", round(x, 8))

# X: -0.93740485

print_me("Y", round(y, 8))

# Y: -0.3131474

print_me("Z", round(z, 8))

# Z: -0.12456646

Compute rectangular coordinates w.r.t. mean equinox of B1950.0:

epoch = Epoch(1992, 10, 13.0)

x, y, z = Sun.rectangular_coordinates_b1950(epoch)

print_me("X", round(x, 8))

# X: -0.94149557

print_me("Y", round(y, 8))

# Y: -0.30259922

print_me("Z", round(z, 8))

# Z: -0.11578695

And compute rectangular coordinates w.r.t. an arbitrary mean equinox:

epoch = Epoch(1992, 10, 13.0)

e_equinox = Epoch(2467616.0)

x, y, z = Sun.rectangular_coordinates_equinox(epoch, e_equinox)

print_me("X", round(x, 8))

# X: -0.93373777

print_me("Y", round(y, 8))

# Y: -0.32235109

print_me("Z", round(z, 8))

# Z: -0.12856709

It is possible to compute the date of equinoxes and solstices:

epoch = Sun.get_equinox_solstice(1962, target="summer")

y, m, d, h, mi, s = epoch.get_full_date()

print("The summer solstice of 1962:")

print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 0)))

# 1962/6/21 21:24:42.0

The equation of time, i.e., the difference between apparent and mean time, can be easily computed:

epoch = Epoch(1992, 10, 13.0)

m, s = Sun.equation_of_time(epoch)

print("Equation of time difference: {} min {} secs".format(m, round(s, 1)))

# Equation of time difference: 13 min 42.6 secs

Compute the ephemeris of physical observations of the Sun using Carrington’s formulas:

epoch = Epoch(1992, 10, 13)

p, b0, l0 = Sun.ephemeris_physical_observations(epoch)

print("Ephemeris of physical observations of the Sun:")

print_me("P ", round(p, 2))

# P : 26.27

print_me("B0", round(b0, 2))

# B0: 5.99

print_me("L0", round(l0, 2))

# L0: 238.63

Get the epoch when the Carrington’s synodic rotation No. ‘number’ starts:

epoch = Sun.beginning_synodic_rotation(1699)

print_me("Epoch for Carrington's synodic rotation No. 1699", round(epoch(), 3))

# Epoch for Carrington's synodic rotation No. 1699: 2444480.723

Uranus examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

We can compute the geometric heliocentric position for a given epoch:

epoch = Epoch(2018, 10, 27.0)

lon, lat, r = Uranus.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", lon.to_positive())

# Geometric Heliocentric Longitude: 30.5888

print_me("Geometric Heliocentric Latitude", lat)

# Geometric Heliocentric Latitude: -0.5315

print_me("Radius vector", r)

# Radius vector: 19.86964

Compute the geocentric position for 1992/12/20:

epoch = Epoch(1992, 12, 20.0)

ra, dec, elon = Uranus.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 19h 13' 48.7''

print_me("Declination", dec.dms_str(n_dec=1))

# Declination: -22d 46' 13.0''

print_me("Elongation", elon.dms_str(n_dec=1))

# Elongation: 18d 44' 18.7''

Print mean orbital elements for Uranus at 2065.6.24:

epoch = Epoch(2065, 6, 24.0)

l, a, e, i, ome, arg = Uranus.orbital_elements_mean_equinox(epoch)

print_me("Mean longitude of the planet", round(l, 6))

# Mean longitude of the planet: 235.517526

print_me("Semimajor axis of the orbit (UA)", round(a, 8))

# Semimajor axis of the orbit (UA): 19.21844604

print_me("Eccentricity of the orbit", round(e, 7))

# Eccentricity of the orbit: 0.0463634

print_me("Inclination on plane of the ecliptic", round(i, 6))

# Inclination on plane of the ecliptic: 0.77372

print_me("Longitude of the ascending node", round(ome, 5))

# Longitude of the ascending node: 74.34776

print_me("Argument of the perihelion", round(arg, 6))

# Argument of the perihelion: 99.630865

Compute the time of the conjunction close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

conj = Uranus.conjunction(epoch)

y, m, d = conj.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Conjunction date", date)

# Conjunction date: 1994/1/12.7365

Compute the time of the opposition close to 1780/12/1:

epoch = Epoch(1780, 12, 1.0)

oppo = Uranus.opposition(epoch)

y, m, d = oppo.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Opposition date", date)

# Opposition date: 1780/12/17.5998

Find the epoch of the Perihelion closer to 1780/1/1:

epoch = Epoch(1780, 1, 1.0)

e = Uranus.perihelion_aphelion(epoch)

y, m, d = e.get_date()

peri = str(y) + '/' + str(m) + '/' + str(int(d))

print_me("The Perihelion closest to 1780/1/1 happened on", peri)

# The Perihelion closest to 1780/1/1 happened on: 1798/2/26

Compute the time of passage through an ascending node:

epoch = Epoch(2019, 1, 1)

time, r = Uranus.passage_nodes(epoch)

y, m, d = time.get_date()

d = round(d, 1)

print("Time of passage through ascending node: {}/{}/{}".format(y, m, d))

# Time of passage through ascending node: 2028/8/23.2

print("Radius vector at ascending node: {}".format(round(r, 4)))

# Radius vector at ascending node: 19.3201

Venus examples¶

Let’s define a small helper function:

def print_me(msg, val):

    print("{}: {}".format(msg, val))

We can compute the geometric heliocentric position for a given epoch:

epoch = Epoch(1992, 12, 20.0)

lon, lat, r = Venus.geometric_heliocentric_position(epoch)

print_me("Geometric Heliocentric Longitude", round(lon.to_positive(), 5))

# Geometric Heliocentric Longitude: 26.11428

print_me("Geometric Heliocentric Latitude", round(lat, 4))

# Geometric Heliocentric Latitude: -2.6207

print_me("Radius vector", round(r, 6))

# Radius vector: 0.724603

Compute the geocentric position for 1992/12/20:

epoch = Epoch(1992, 12, 20.0)

ra, dec, elon = Venus.geocentric_position(epoch)

print_me("Right ascension", ra.ra_str(n_dec=1))

# Right ascension: 21h 4' 41.5''

print_me("Declination", dec.dms_str(n_dec=1))

# Declination: -18d 53' 16.8''

print_me("Elongation", elon.dms_str(n_dec=1))

# Elongation: 44d 46' 8.9''

Print mean orbital elements for Venus at 2065.6.24:

epoch = Epoch(2065, 6, 24.0)

l, a, e, i, ome, arg = Venus.orbital_elements_mean_equinox(epoch)

print_me("Mean longitude of the planet", round(l, 6))

# Mean longitude of the planet: 338.646306

print_me("Semimajor axis of the orbit (UA)", round(a, 8))

# Semimajor axis of the orbit (UA): 0.72332982

print_me("Eccentricity of the orbit", round(e, 7))

# Eccentricity of the orbit: 0.0067407

print_me("Inclination on plane of the ecliptic", round(i, 6))

# Inclination on plane of the ecliptic: 3.395319

print_me("Longitude of the ascending node", round(ome, 5))

# Longitude of the ascending node: 77.27012

print_me("Argument of the perihelion", round(arg, 6))

# Argument of the perihelion: 55.211257

Compute the time of the inferior conjunction close to 1882/12/1.0:

epoch = Epoch(1882, 12, 1.0)

conjunction = Venus.inferior_conjunction(epoch)

y, m, d = conjunction.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Inferior conjunction date", date)

# Inferior conjunction date: 1882/12/6.6912

Compute the time of the superior conjunction close to 1993/10/1:

epoch = Epoch(1993, 10, 1.0)

conjunction = Venus.superior_conjunction(epoch)

y, m, d = conjunction.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Superior conjunction date", date)

# Superior conjunction date: 1994/1/17.0465

Compute the time and angle of the western elongation close to 2019/1/1:

epoch = Epoch(2019, 1, 1.0)

time, elongation = Venus.western_elongation(epoch)

y, m, d = time.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Western elongation date", date)

# Western elongation date: 2019/1/6.1895

elong = round(elongation, 4)

print_me("Maximum western elongation angle", elong)

# Maximum western elongation angle: 46.9571

Compute the time and angle of the eastern elongation close to 2019/10/1:

epoch = Epoch(2019, 10, 1.0)

time, elongation = Venus.eastern_elongation(epoch)

y, m, d = time.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Eastern elongation date", date)

# Eastern elongation date: 2020/3/24.9179

elong = round(elongation, 4)

print_me("Maximum eastern elongation angle", elong)

# Maximum eastern elongation angle: 46.078

Compute the time of the station in longitude #1 close to 2018/12/1:

epoch = Epoch(2018, 12, 1.0)

sta1 = Venus.station_longitude_1(epoch)

y, m, d = sta1.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #1", date)

# Date of station in longitude #1: 2018/10/5.7908

Compute the time of the station in longitude #2 close to 2018/12/1:

epoch = Epoch(2018, 12, 1.0)

sta2 = Venus.station_longitude_2(epoch)

y, m, d = sta2.get_date()

d = round(d, 4)

date = "{}/{}/{}".format(y, m, d)

print_me("Date of station in longitude #2", date)

# Date of station in longitude #2: 2018/11/16.439

Find the epoch of the Perihelion closer to 1978/10/15:

epoch = Epoch(1978, 10, 15.0)

e = Venus.perihelion_aphelion(epoch)

y, m, d, h, mi, s = e.get_full_date()

peri = str(y) + '/' + str(m) + '/' + str(d) + ' at ' + str(h) + ' hours'

print_me("The Perihelion closest to 1978/10/15 happened on", peri)

# The Perihelion closest to 1978/10/15 happened on: 1978/12/31 at 4 hours

Compute the time of passage through an ascending node:

epoch = Epoch(1979, 1, 1)

time, r = Venus.passage_nodes(epoch)

y, m, d = time.get_date()

d = round(d, 1)

print("Time of passage through ascending node: {}/{}/{}".format(y, m, d))

# Time of passage through ascending node: 1978/11/27.4

print("Radius vector at ascending node: {}".format(round(r, 4)))

# Radius vector at ascending node: 0.7205

Compute the (approximate) illuminated fraction of Venus disk for an Epoch:

epoch = Epoch(1992, 12, 20)

k = Venus.illuminated_fraction(epoch)

print_me("Approximate illuminated fraction of Venus", round(k, 2))

# Approximate illuminated fraction of Venus: 0.64

Compute the magnitude of Venus:

sun_dist = 0.724604

earth_dist = 0.910947

phase_angle = Angle(72.96)

m = Venus.magnitude(sun_dist, earth_dist, phase_angle)

print_me("Venus' magnitude", round(m, 1))

# Venus' magnitude: -3.8

PyMeeus¶

Installation¶

Properly Using PyMeeus¶

Meta¶

Contributing¶

Contributors¶

What’s new¶

API¶

Core¶

Base¶

Angle¶

Coordinates¶

Curveffiting¶

Epoch¶

Interpolation¶

Bodies¶

Earth¶

JupiterMoons¶

Jupiter¶

Mars¶

Mercury¶

Minor¶

Moon¶

Neptune¶

Pluto¶

Saturn¶

Sun¶

Uranus¶

Venus¶

Examples¶

Usage Examples¶

Properly Using PyMeeus¶

Angle examples¶

Base examples¶

Coordinates examples¶

CurveFitting examples¶

Earth examples¶

Epoch examples¶

Interpolation examples¶

Jupiter examples¶

JupiterMoons examples¶

Mars examples¶

Mercury examples¶

Minor examples¶

Moon examples¶

Neptune examples¶

Pluto examples¶

Saturn examples¶

Sun examples¶

Uranus examples¶

Venus examples¶

Indices and tables¶