astro

Astronomical and nutation routines
Computes the basic astronomical mean longitudes and other fundamental orbital parameters
Computes the solar and lunar positions in Earth-Centered Earth-Fixed (ECEF) coordinates

Calling Sequence

import pyTMD.astro
S,H,P,N,PP = pyTMD.astro.mean_longitudes(MJD)

Source code

pyTMD.astro.mean_longitudes(MJD: ndarray, **kwargs)[source]

Computes the basic astronomical mean longitudes: \(S\), \(H\), \(P\), \(N\) and \(P_s\) [33, 51]

Note \(N\) is not \(N'\), i.e. \(N\) is decreasing with time.

Parameters:

MJD: np.ndarray

Modified Julian Day (MJD) of input date

method: str, default ‘Cartwright’

method for calculating mean longitudes

'Cartwright': use coefficients from David Cartwright

'Meeus': use coefficients from Meeus Astronomical Algorithms

'ASTRO5': use Meeus Astronomical coefficients from ASTRO5

'IERS': convert from IERS Delaunay arguments

Returns:

S: np.ndarray: mean longitude of moon (degrees)
H: np.ndarray: mean longitude of sun (degrees)
P: np.ndarray: mean longitude of lunar perigee (degrees)
N: np.ndarray: mean longitude of ascending lunar node (degrees)
Ps: np.ndarray: longitude of solar perigee (degrees)

pyTMD.astro.doodson_arguments(MJD: ndarray, equinox: bool = False, apply_correction: bool = True)[source]

Computes astronomical phase angles for the six Doodson Arguments: \(\tau\), \(S\), \(H\), \(P\), \(N'\), and \(P_s\) [16, 33]

Follows IERS conventions for the Doodson arguments [42]

Parameters:

MJD: np.ndarray: Modified Julian Day (MJD) of input date
equinox: bool, default False: use equinox method for calculating mean lunar time
apply_correction: bool, default True: Apply correction for mean lunar longitude

Returns:

TAU: np.ndarray: mean lunar time (radians)
S: np.ndarray: mean longitude of the moon (radians)
H: np.ndarray: mean longitude of the sun (radians)
P: np.ndarray: mean longitude of lunar perigee (radians)
Np: np.ndarray: negative mean longitude of the ascending node (radians)
Ps: np.ndarray: mean longitude of solar perigee (radians)

pyTMD.astro.delaunay_arguments(MJD: ndarray)[source]

Computes astronomical phase angles for the five primary Delaunay Arguments of Nutation: \(l\), \(l'\), \(F\), \(D\), and \(N\) [7, 33, 42]

Parameters:

MJD: np.ndarray: Modified Julian Day (MJD) of input date

Returns:

l: np.ndarray: mean anomaly of moon (radians)
lp: np.ndarray: mean anomaly of the sun (radians)
F: np.ndarray: mean argument of the moon (radians)
D: np.ndarray: mean elongation of the moon from the sun (radians)
N: np.ndarray: mean longitude of ascending lunar node (radians)

pyTMD.astro.schureman_arguments(P: ndarray, N: ndarray)[source]

Computes additional phase angles \(I\), \(\xi\), \(\nu\), \(R\), \(R_a\), \(\nu'\), and \(\nu''\) from Schureman [50]

See the explanation of symbols in appendix of Schureman [50]

Parameters:

P: np.ndarray: mean longitude of lunar perigee (radians)
N: np.ndarray: mean longitude of ascending lunar node (radians)

Returns:

I: np.ndarray: obliquity of lunar orbit with respect to Earth’s equator (radians)
xi: np.ndarray: longitude in the moon’s orbit of lunar intersection (radians)
nu: np.ndarray: right ascension of lunar intersection (radians)
Qa: np.ndarray: factor in amplitude for m1 constituent (radians)
Qu: np.ndarray: term in argument for m1 constituent (radians)
Ra: np.ndarray: factor in amplitude for l2 constituent (radians)
Ru: np.ndarray: term in argument for l2 constituent (radians)
nu_p: np.ndarray: term in argument for k1 constituent (radians)
nu_s: np.ndarray: term in argument for k2 constituent (radians)

pyTMD.astro.mean_obliquity(MJD: ndarray)[source]

Mean obliquity of the ecliptic [6, 7]

Parameters:

MJD: np.ndarray: Modified Julian Day (MJD) of input date

Returns:

epsilon: np.ndarray: mean obliquity of the ecliptic (radians)

pyTMD.astro.equation_of_time(MJD: ndarray)[source]

Approximate calcuation of the difference between apparent and mean solar times [33, 55]

Parameters:

MJD: np.ndarray: Modified Julian Day (MJD) of input date

Returns:

E: np.ndarray: equation of time (radians)

pyTMD.astro.solar_ecef(MJD: ndarray, **kwargs)[source]

Wrapper function for calculating the positional coordinates of the sun in an Earth-centric, Earth-Fixed (ECEF) frame [33, 34, 40]

Parameters:

MJD: np.ndarray

Modified Julian Day (MJD) of input date

ephemerides: str, default ‘approximate’

Method for calculating solar ephemerides

'approximate': low-resolution ephemerides

'JPL': computed ephemerides from JPL kernels

**kwargs: dict

Keyword options for ephemeris calculation

Returns:

X, Y, Z: np.ndarray: ECEF coordinates of the sun (meters)

pyTMD.astro.solar_approximate(MJD, **kwargs)[source]

Computes approximate positional coordinates of the sun in an Earth-centric, Earth-Fixed (ECEF) frame [33, 34]

Parameters:

MJD: np.ndarray: Modified Julian Day (MJD) of input date

Returns:

X, Y, Z: np.ndarray: ECEF coordinates of the sun (meters)

pyTMD.astro.solar_ephemerides(MJD: ndarray, **kwargs)[source]

Computes positional coordinates of the sun in an Earth-centric, Earth-Fixed (ECEF) frame using JPL ephemerides [33, 40]

Parameters:

MJD: np.ndarray: Modified Julian Day (MJD) of input date
kernel: str or pathlib.Path: Path to JPL ephemerides kernel file
include_aberration: bool, default False: Correct for aberration effects

Returns:

X, Y, Z: np.ndarray: ECEF coordinates of the sun (meters)

pyTMD.astro.lunar_ecef(MJD: ndarray, **kwargs)[source]

Wrapper function for calculating the positional coordinates of the moon in an Earth-centric, Earth-Fixed (ECEF) frame [33, 34, 40]

Parameters:

MJD: np.ndarray

Modified Julian Day (MJD) of input date

ephemerides: str, default ‘approximate’

Method for calculating lunar ephemerides

'approximate': low-resolution ephemerides

'JPL': computed ephemerides from JPL kernels

**kwargs: dict

Keyword options for ephemeris calculation

Returns:

X, Y, Z: np.ndarray: ECEF coordinates of the moon (meters)

pyTMD.astro.lunar_approximate(MJD, **kwargs)[source]

Computes approximate positional coordinates of the moon in an Earth-centric, Earth-Fixed (ECEF) frame [33, 34]

Parameters:

MJD: np.ndarray: Modified Julian Day (MJD) of input date

Returns:

X, Y, Z: np.ndarray: ECEF coordinates of the moon (meters)

pyTMD.astro.lunar_ephemerides(MJD: ndarray, **kwargs)[source]

Computes positional coordinates of the moon in an Earth-centric, Earth-Fixed (ECEF) frame using JPL ephemerides [33, 40]

Parameters:

MJD: np.ndarray: Modified Julian Day (MJD) of input date
kernel: str or pathlib.Path: Path to JPL ephemerides kernel file
include_aberration: bool, default False: Correct for aberration effects

Returns:

X, Y, Z: np.ndarray: ECEF coordinates of the moon (meters)

pyTMD.astro.gast(T: float | ndarray)[source]

Greenwich Apparent Sidereal Time (GAST) [6, 7, 42]

Parameters:

T: np.ndarray: Centuries since 2000-01-01T12:00:00

pyTMD.astro.itrs(T: float | ndarray, include_polar_motion: bool = True)[source]

International Terrestrial Reference System (ITRS) [6, 7, 42]

An Earth-centered Earth-fixed (ECEF) coordinate system combining the Earth’s true equator and equinox of date, the Earth’s rotation with respect to the stars, and the polar wobble of the crust with respect to the pole of rotation

Parameters:

T: np.ndarray: Centuries since 2000-01-01T12:00:00
include_polar_motion: bool, default True: Include polar motion in the rotation matrix

pyTMD.astro._eqeq_complement(T: float | ndarray)[source]

Compute complementary terms of the equation of the equinoxes [6, 7, 42]

These include the combined effects of precession and nutation [26, 42, 55]

Parameters:

T: np.ndarray: Centuries since 2000-01-01T12:00:00

pyTMD.astro._icrs_rotation_matrix(T: float | ndarray, include_polar_motion: bool = True)[source]

Rotation matrix for tranforming from the International Celestial Reference System (ICRS) to the International Terrestrial Reference System (ITRS) [6, 7, 42]

Parameters:

T: np.ndarray: Centuries since 2000-01-01T12:00:00
include_polar_motion: bool, default True: Include polar motion in the rotation matrix

pyTMD.astro._frame_bias_matrix()[source]: Frame bias rotation matrix for converting from a dynamical reference system to the International Celestial Reference System (ICRS) [42, 55]

pyTMD.astro._nutation_angles(T: float | ndarray)[source]

Calculate nutation rotation angles using tables from IERS Conventions [42]

Parameters:

T: np.ndarray: Centuries since 2000-01-01T12:00:00

Returns:

dpsi: np.ndarray: Nutation in longitude
deps: np.ndarray: Obliquity of the ecliptic

pyTMD.astro._nutation_matrix(mean_obliquity: float | ndarray, true_obliquity: float | ndarray, psi: float | ndarray)[source]

Nutation rotation matrix [27, 42]

Parameters:

mean_obliquity: np.ndarray: Mean obliquity of the ecliptic
true_obliquity: np.ndarray: True obliquity of the ecliptic
psi: np.ndarray: Nutation in longitude

pyTMD.astro._polar_motion_matrix(T: float | ndarray)[source]

Polar motion (Earth Orientation Parameters) rotation matrix [42, 55]

Parameters:

T: np.ndarray: Centuries since 2000-01-01T12:00:00

pyTMD.astro._precession_matrix(T: float | ndarray)[source]

Precession rotation matrix [6, 7, 28]

Parameters:

T: np.ndarray: Centuries since 2000-01-01T12:00:00

pyTMD.astro._correct_aberration(position, velocity)[source]

Correct a relative position for aberration effects [27]

Parameters:

position: np.ndarray: Position vector in astronomical units
velocity: np.ndarray: Velocity vector in astronomical units per day

pyTMD.astro._parse_table_5_2e()[source]: Parse table with expressions for Greenwich Sidereal Time provided in Chapter 5 of Petit and Luzum [42]

pyTMD.astro._parse_table_5_3a()[source]: Parse table with IAU 2000A lunisolar and planetary components of nutation in longitude provided in Chapter 5 of Petit and Luzum [42]

pyTMD.astro._parse_table_5_3b()[source]: Parse table with IAU 2000A lunisolar and planetary components of nutation in obliquity provided in Chapter 5 of Petit and Luzum [42]