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''