# Source code for pymeeus.Sun

```# -*- coding: utf-8 -*-

# PyMeeus: Python module implementing astronomical algorithms.
# Copyright (C) 2018  Dagoberto Salazar
#
# This program is free software: you can redistribute it and/or modify
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program.  If not, see <https://www.gnu.org/licenses/>.

from math import sin, cos, tan, atan, atan2, asin

from pymeeus.Angle import Angle
from pymeeus.Epoch import Epoch, JDE2000
from pymeeus.Coordinates import (
mean_obliquity,
true_obliquity,
nutation_longitude,
ecliptical2equatorial,
)
from pymeeus.Earth import Earth

"""
.. module:: Sun
:synopsis: Module including functions regarding Sun position

.. moduleauthor:: Dagoberto Salazar
"""

[docs]class Sun(object):
"""
Class Sun handles the parameters related to the Sun.
"""

[docs]    def __init__(self):
"""Sun constructor.

:returns: Sun object.
:rtype: :py:class:`Sun`
"""

[docs]    @staticmethod
def true_longitude_coarse(epoch):
"""This method provides the Sun's true longitude with a relatively low

:param epoch: Epoch to compute the position of the Sun
:type epoch: :py:class:`Epoch`

:returns: A tuple containing the true (ecliptical) longitude (as an
Angle object) and the radius vector in astronomical units.
:rtype: 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
"""

# First check that input values are of correct types
if not (isinstance(epoch, Epoch)):
raise TypeError("Invalid input type")
# Compute the time in Julian centuries
t = (epoch - JDE2000) / 36525.0
# Compute the geometric mean longitude of the Sun
l0 = 280.46646 + t * (36000.76983 + t * 0.0003032)
l0 = Angle(l0)
l0.to_positive()
# Now, compute the mean anomaly of the Sun
m = 357.52911 + t * (35999.05029 - t * 0.0001537)
m = Angle(m)
# The eccentricity of the Earth's orbit
e = 0.016708634 - t * (0.000042037 + t * 0.0000001267)
# Equation of the center
c = (
(1.914602 - t * (0.004817 + t * 0.000014)) * sin(mrad)
+ (0.019993 - t * 0.000101) * sin(2.0 * mrad)
+ 0.000289 * sin(3.0 * mrad)
)
c = Angle(c)
true_lon = l0 + c
true_anom = m + c
r = (1.000001018 * (1.0 - e * e)) / (1.0 + e * cos(true_anom.rad()))
return (true_lon, r)

[docs]    @staticmethod
def apparent_longitude_coarse(epoch):
"""This method provides the Sun's apparent longitude with a relatively
low accuracy of about 0.01 degree.

:param epoch: Epoch to compute the position of the Sun
:type epoch: :py:class:`Epoch`

:returns: A tuple containing the sun_apparent (ecliptical) longitude
(as an Angle object) and the radius vector in astronomical units.
:rtype: 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
"""

# First check that input values are of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# First find the true longitude
sun = Sun()
true_lon, r = sun.true_longitude_coarse(epoch)
# Compute the time in Julian centuries
t = (epoch - JDE2000) / 36525.0
# Then correct for nutation and aberration
omega = 125.04 - 1934.136 * t
omega = Angle(omega)
lambd = true_lon - 0.00569 - 0.00478 * sin(omega.rad())
return (lambd, r)

[docs]    @staticmethod
def apparent_rightascension_declination_coarse(epoch):
"""This method provides the Sun's apparent right ascension and
declination with a relatively low accuracy of about 0.01 degree.

:param epoch: Epoch to compute the position of the Sun
:type epoch: :py:class:`Epoch`

:returns: A tuple containing the right ascension and the declination
(as Angle objects) and the radius vector in astronomical units.
:rtype: 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
"""

# First check that input values are of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Second, find the apparent longitude
sun = Sun()
app_lon, r = sun.apparent_longitude_coarse(epoch)
# Compute the obliquity of the ecliptic
e0 = mean_obliquity(epoch)
# Compute the time in Julian centuries
t = (epoch - JDE2000) / 36525.0
# Then correct for nutation and aberration
omega = 125.04 - 1934.136 * t
omega = Angle(omega)
# Correct the obliquity
e = e0 + 0.00256 * cos(omega.rad())
alpha.to_positive()
return (alpha, delta, r)

[docs]    @staticmethod
def geometric_geocentric_position(epoch, tofk5=True):
"""This method computes the geometric geocentric position of the Sun
for a given epoch, using the VSOP87 theory.

:param epoch: Epoch to compute Sun position, as an Epoch object
:type epoch: :py:class:`Epoch`
:param tofk5: Whether or not the small correction to convert to the FK5
system will be applied or not
:type tofk5: bool

:returns: A tuple with the geocentric longitude and latitude (as
:py:class:`Angle` objects), and the radius vector (as a float,
in astronomical units), in that order
:rtype: 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
"""

# NOTE: In page 169, Meeus gives a different value for the LONGITUDE
# (199.907372 degrees) as the one presented above (199.907297 degrees).
# After many checks and tests, I came to the conclusion that the result
# above is the right one, and Meeus' result is wrong.
# On the other hand, the difference in LATITUDE may be due to the fact
# that this software uses the complete set of VSOP87C terms, instead of
# the abridged version in Meeus' book.

# First check that input values are of correct types
if not isinstance(epoch, Epoch) and not isinstance(tofk5, bool):
raise TypeError("Invalid input types")
# Use Earth heliocentric position to compute Sun's geocentric position
lon, lat, r = Earth.geometric_heliocentric_position(epoch, tofk5)
lon = lon.to_positive() + 180.0
lat = -lat
return lon, lat, r

[docs]    @staticmethod
def apparent_geocentric_position(epoch, nutation=True):
"""This method computes the apparent geocentric position of the Sun
for a given epoch, using the VSOP87 theory.

:param epoch: Epoch to compute Sun position, as an Epoch object
:type epoch: :py:class:`Epoch`
:param nutation: Whether the nutation correction will be applied
:type epoch: bool

:returns: A tuple with the heliocentric longitude and latitude (as
:py:class:`Angle` objects), and the radius vector (as a float,
in astronomical units), in that order
:rtype: 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
"""

# NOTE: In page 169, Meeus gives a different value for the LONGITUDE
# (199d 54' 21.818'') as the one presented above (199d 54' 21.548'').
# After many checks and tests, I came to the conclusion that the result
# above is the right one, and Meeus' result is wrong.
# On the other hand, the difference in LATITUDE may be due to the fact
# that this software uses the complete set of VSOP87C terms, instead of
# the abridged version in Meeus' book.

# First check that input values are of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Use Earth heliocentric position to compute Sun's geocentric position
lon, lat, r = Earth.apparent_heliocentric_position(epoch, nutation)
lon = lon.to_positive() + 180.0
lat = -lat
return lon, lat, r

[docs]    @staticmethod
def rectangular_coordinates_mean_equinox(epoch):
"""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.

:param epoch: Epoch to compute Sun position, as an Epoch object
:type epoch: :py:class:`Epoch`

:returns: A tuple with the X, Y, Z values in astronomical units
:rtype: 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
"""

# NOTE: In page 172, Meeus gives slightly different values for x, y, z
# as the ones internally computed in the example above. After many
# checks and tests, I came to the conclusion that the results above are
# the right ones, and Meeus' results are wrong.

# First check that input values are of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Get Sun's geocentric position with reduction to FK5 system
lon, lat, r = Sun.geometric_geocentric_position(epoch)
# Get the mean obliquity
epsilon0 = mean_obliquity(epoch)
# Compute the results
x = r * cos(ll)
y = r * (sin(ll) * cos(e) - sin(b) * sin(e))
z = r * (sin(ll) * sin(e) + sin(b) * cos(e))
return x, y, z

[docs]    @staticmethod
def rectangular_coordinates_j2000(epoch):
"""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.

:param epoch: Epoch to compute Sun position, as an Epoch object
:type epoch: :py:class:`Epoch`

:returns: A tuple with the X, Y, Z values in astronomical units
:rtype: 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
"""

# First check that input values are of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Second, compute Earth heliocentric position referred to J2000.0
lon, lat, r = Earth.geometric_heliocentric_position_j2000(epoch)
# Third, convert from Earth's heliocentric to Sun's geocentric
lon = lon.to_positive() + 180.0
lat = -lat
x0 = x + 0.00000044036 * y - 0.000000190919 * z
y0 = -0.000000479966 * x + 0.917482137087 * y - 0.397776982902 * z
z0 = 0.397776982902 * y + 0.917482137087 * z
return x0, y0, z0

[docs]    @staticmethod
def rectangular_coordinates_b1950(epoch):
"""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.

:param epoch: Epoch to compute Sun position, as an Epoch object
:type epoch: :py:class:`Epoch`

:returns: A tuple with the X, Y, Z values in astronomical units
:rtype: 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
"""

# First check that input values are of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Second, compute Earth heliocentric position referred to J2000.0
lon, lat, r = Earth.geometric_heliocentric_position_j2000(epoch)
# Third, convert from Earth's heliocentric to Sun's geocentric
lon = lon.to_positive() + 180.0
lat = -lat
x = 0.999925702634 * x + 0.012189716217 * y + 0.000011134016 * z
y = -0.011179418036 * x + 0.917413998946 * y - 0.397777041885 * z
z = -0.004859003787 * x + 0.397747363646 * y + 0.917482111428 * z
return x, y, z

[docs]    @staticmethod
def rectangular_coordinates_equinox(epoch, equinox_epoch):
"""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.

:param epoch: Epoch to compute Sun position, as an Epoch object
:type epoch: :py:class:`Epoch`
:param equinox_epoch: Epoch corresponding to the mean equinox
:type equinox_epoch: :py:class:`Epoch`

:returns: A tuple with the X, Y, Z values in astronomical units
:rtype: 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
"""

# First check that input values are of correct types
if (not isinstance(epoch, Epoch)
and not isinstance(equinox_epoch, Epoch)):
raise TypeError("Invalid input types")
# Second, compute Sun's rectangular coordinates w.r.t. J2000.0
x0, y0, z0 = Sun.rectangular_coordinates_j2000(epoch)
# Third, computed auxiliary angles
t = (equinox_epoch - JDE2000) / 36525.0
tt = (epoch - equinox_epoch) / 36525.0
# Compute the conversion parameters
zeta = t * (
(2306.2181 + tt * (1.39656 - 0.000139 * tt))
+ t * ((0.30188 - 0.000344 * tt) + 0.017998 * t)
)
z = t * (
(2306.2181 + tt * (1.39656 - 0.000139 * tt))
+ t * ((1.09468 + 0.000066 * tt) + 0.018203 * t)
)
theta = t * (
2004.3109
+ tt * (-0.85330 - 0.000217 * tt)
+ t * (-(0.42665 + 0.000217 * tt) - 0.041833 * t)
)
# Redefine the former values as Angles, and compute them in radians
zeta = Angle(0, 0, zeta)
z = Angle(0, 0, z)
theta = Angle(0, 0, theta)
xx = cos(zetar) * cos(zr) * cos(thetar) - sin(zetar) * sin(zr)
xy = sin(zetar) * cos(zr) + cos(zetar) * sin(zr) * cos(thetar)
xz = cos(zetar) * sin(thetar)
yx = -cos(zetar) * sin(zr) - sin(zetar) * cos(zr) * cos(thetar)
yy = cos(zetar) * cos(zr) - sin(zetar) * sin(zr) * cos(thetar)
yz = -sin(zetar) * sin(thetar)
zx = -cos(zr) * sin(thetar)
zy = -sin(zr) * sin(thetar)
zz = cos(thetar)
xp = xx * x0 + yx * y0 + zx * z0
yp = xy * x0 + yy * y0 + zy * z0
zp = xz * x0 + yz * y0 + zz * z0
return xp, yp, zp

[docs]    @staticmethod
def get_equinox_solstice(year, target="spring"):
"""This method computes the times of the equinoxes or the solstices.

:param year: Year we want to compute the equinox or solstice for
:type year: int
:param target: Corresponding equinox or solstice. It can be "spring",
"summer", "autumn", "winter"
:type target: str

:returns: The instant of time when the equinox or solstice happens
:rtype: :py:class:`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
"""

# NOTE: The results from the previous example are computed using the
# complete VSOP87 theory. The results provided by Meeus in exercises
# 27.a and 27.b are computed with lower accuracy

# First check that input values are of correct types
if not (isinstance(year, int) and isinstance(target, str)):
raise TypeError("Invalid input types")
# Second, check that the target is correct
if (
(target != "spring")
and (target != "summer")
and (target != "autumn")
and (target != "winter")
):
raise ValueError("'target' value is invalid")
# Now we can start computing an approximate value (Tables 27.A, 27.B)
if (year >= -1000) and (year < 1000):
y = year / 1000.0
if target == "spring":
jde0 = 1721139.29189 + y * (
365242.1374 + y * (0.06134 + y * (0.00111 - y * 0.00071))
)
elif target == "summer":
jde0 = 1721233.25401 + y * (
365241.72562 + y * (-0.05323 + y * (0.00907 + y * 0.00025))
)
elif target == "autumn":
jde0 = (1721325.70455
+ y * (365242.49558
+ y * (-0.11677 + y * (-0.00297
+ y * 0.00074))))
elif target == "winter":
jde0 = (1721414.39987
+ y * (365242.88257 + y * (-0.00769
+ y * (-0.00933
- y * 0.00006))))
elif (year >= 1000) and (year <= 3000):
y = (year - 2000.0) / 1000.0
if target == "spring":
jde0 = 2451623.80984 + y * (
365242.37404 + y * (0.05169 + y * (-0.00411 - y * 0.00057))
)
elif target == "summer":
jde0 = 2451716.56767 + y * (
365241.62603 + y * (0.00325 + y * (0.00888 - y * 0.0003))
)
elif target == "autumn":
jde0 = 2451810.21715 + y * (
365242.01767 + y * (-0.11575 + y * (0.00337 + y * 0.00078))
)
elif target == "winter":
jde0 = (2451900.05952
+ y * (365242.74049
+ y * (-0.06223 + y * (-0.00823
+ y * 0.00032))))
else:
raise ValueError("'year' value out of range")
k = ["spring", "summer", "autumn", "winter"].index(target)
epoch = Epoch(jde0)
corr = 1.0
while abs(corr) > 0.0000025:
lon, lat, r = Sun.apparent_geocentric_position(epoch)
arg = k * 90.0 - lon.to_positive()
arg = Angle(arg)
epoch += corr
epoch -= corr
return epoch

[docs]    @staticmethod
def equation_of_time(epoch):
"""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.

:param epoch: Epoch to compute the equation of time, as an Epoch object
:type epoch: :py:class:`Epoch`

:returns: Difference between apparent and mean time, as a tuple, in
minutes (int) and seconds (float) of time
:rtype: 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
"""

# First check that input values are of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Compute time in Julian millenia from J2000.0
t = (epoch - JDE2000) / 365250
l0 = (280.4664567
+ t * (360007.6982779
+ t * (0.03032028
+ t * (1.0 / 49931.0
+ t * (-1.0 / 15300.0
- t * 1.0 / 2000000.0)))))
l0 = Angle(l0)
l0 = l0.to_positive()
# Compute the apparent position of the Sun
lon, lat, r = Sun.apparent_geocentric_position(epoch)
# Now, get the true obliquity
epsilon = true_obliquity(epoch)
# Transform from eclliptical to equatorial coordinates
alpha, dec = ecliptical2equatorial(lon, lat, epsilon)
alpha = alpha.to_positive()
# Now we need the nutation in longitude
deltapsi = nutation_longitude(epoch)
e = l0() - 0.0057183 - alpha + deltapsi * cos(epsilon.rad())
# The following line is a fix devised by janbredenbeek to a problem
# that arises in cases where alpha was just past the spring equinox
# but l0 was still < 360. In those cases e was incorrectly calculated
# The solution is to keep e as a float and reduce to range -180..+180
e = e - 360.0 * round(e / 360.0)
e *= 4.0
# Extract seconds
s = (abs(e()) % 1) * 60.0
m = int(e())
return m, s

[docs]    @staticmethod
def ephemeris_physical_observations(epoch):
"""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

:param epoch: Epoch to compute the parameters
:type epoch: :py:class:`Epoch`

:returns: Parameters P, B0 and L0, in a tuple
:rtype: 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
"""

# First check that input values are of correct types
if not isinstance(epoch, Epoch):
raise TypeError("Invalid input type")
# Compute the auxiliary parameters
epoch += 0.00068
theta = (epoch() - 2398220.0) * 360.0 / 25.38
theta = Angle(theta)
i = Angle(7.25)
k = 73.6667 + 1.3958333 * (epoch() - 2396758.0) / 36525.0
k = Angle(k)
lon, lat, r = Sun.apparent_geocentric_position(epoch, nutation=False)
eps = true_obliquity(epoch)
dpsi = nutation_longitude(epoch)
lonp = lon + dpsi
delta = lon - k
p = x + y
l0 = eta - theta
return p, b0, l0.to_positive()

[docs]    @staticmethod
def beginning_synodic_rotation(number):
"""This method calculates the epoch when the Carrington's synodic
rotation No. 'number' starts.

:param number: Number of Carrington's synodic rotation
:type number: int

:returns: Epoch when the provided rotation starts
:rtype: :py:class:`Epoch`
:raises: TypeError if input value is of wrong type.

>>> epoch = Sun.beginning_synodic_rotation(1699)
>>> print(round(epoch(), 3))
2444480.723
"""

# First check that input values are of correct types
if not isinstance(number, int):
raise TypeError("Invalid input type")
# Apply formula (29.1)
jde = 2398140.227 + 27.2752316*number
# Now, find the correction using formula (29.2)
m = 281.96 + 26.882476*number
m = Angle(m)
delta = 0.1454 * sin(m) - 0.0085 * sin(2.0 * m) - 0.0141 * cos(2.0 * m)
# Apply the correction
jde += delta
return Epoch(jde)

def main():

# Let's define a small helper function
def print_me(msg, val):
print("{}: {}".format(msg, val))

# Let's show some uses of Sun functions
print("\n" + 35 * "*")
print("*** Use of Sun class")
print(35 * "*" + "\n")

# Compute an approximation of the Sun's true 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))
# 199d 54' 36.0''
print_me("Sun's radius vector", round(r, 5))  # 0.99766

print("")

# Now let's compute the Sun's approximate apparent longitude
app_lon, r = Sun.apparent_longitude_coarse(epoch)
print_me("Sun's approximate apparent longitude", app_lon.dms_str(n_dec=0))
# 199d 54' 32.0''

print("")

# And now is the turn for the apparent right ascension and declination
ra, delta, r = Sun.apparent_rightascension_declination_coarse(epoch)
print_me("Sun's apparent right ascension", ra.ra_str(n_dec=1))
# 13h 13' 31.4''
print_me("Sun's apparent declination", delta.dms_str(n_dec=0))
# -7d 47' 6.0''

print("")

# Let's compute Sun's postion, 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))
# 199.906016
print_me("Geometric Geocentric Latitude", b.dms_str(n_dec=3))
# 0.644''
# 0.99760775

print("")

# Compute Sun's apparent postion
l, b, r = Sun.apparent_geocentric_position(epoch)
print_me("Apparent Geocentric Longitude", l.to_positive().dms_str(n_dec=3))
# 199d 54' 16.937''
print_me("Apparent Geocentric Latitude", b.dms_str(n_dec=3))
# 0.621''
# 0.99760775

print("")

# We can compute rectangular coordinates referred to mean equinox of date
x, y, z = Sun.rectangular_coordinates_mean_equinox(epoch)
print("Rectangular coordinates referred to mean equinox of date:")
print_me("X", round(x, 7))  # -0.9379963
print_me("Y", round(y, 6))  # -0.311654
print_me("Z", round(z, 7))  # -0.1351207

print("")

# Now, compute rectangular coordinates w.r.t. standard equinox J2000.0
x, y, z = Sun.rectangular_coordinates_j2000(epoch)
print("Rectangular coordinates w.r.t. standard equinox J2000.0:")
print_me("X", round(x, 8))  # -0.93740485
print_me("Y", round(y, 8))  # -0.3131474
print_me("Z", round(z, 8))  # -0.12456646

print("")

# Compute rectangular coordinates w.r.t. mean equinox of B1950.0
x, y, z = Sun.rectangular_coordinates_b1950(epoch)
print("Rectangular coordinates w.r.t. mean equinox of B1950.0:")
print_me("X", round(x, 8))  # -0.94149557
print_me("Y", round(y, 8))  # -0.30259922
print_me("Z", round(z, 8))  # -0.11578695

print("")

# Compute rectangular coordinates w.r.t. an arbitrary mean equinox
e_equinox = Epoch(2467616.0)
x, y, z = Sun.rectangular_coordinates_equinox(epoch, e_equinox)
print("Rectangular coordinates w.r.t. an arbitrary mean equinox:")
print_me("X", round(x, 8))  # -0.93373777
print_me("Y", round(y, 8))  # -0.32235109
print_me("Z", round(z, 8))  # -0.12856709

print("")

# We can 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

epoch = Sun.get_equinox_solstice(2018, target="autumn")
y, m, d, h, mi, s = epoch.get_full_date()
print("The autumn equinox of 2018:")
print("{}/{}/{} {}:{}:{}".format(y, m, d, h, mi, round(s, 0)))
# 2018/9/23 1:55:14.0

print("")

# 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)))
# 13m 42.6s

print("")

# 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))     # 26.27
print_me("B0", round(b0, 2))    # 5.99
print_me("L0", round(l0, 2))    # 238.63

print("")

# 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))     # 2444480.723

if __name__ == "__main__":

main()
```