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, ASTRO5=True)

Source code

pyTMD.astro.polynomial_sum(coefficients: list | numpy.ndarray, t: ndarray)[source]

Calculates the sum of a polynomial function of time

Parameters

coefficients: list or np.ndarray: leading coefficient of polynomials of increasing order
t: np.ndarray: delta time in units for a given astronomical longitudes calculation

pyTMD.astro.rotate(theta: float | numpy.ndarray, axis: str = 'x')[source]

Rotate a 3-dimensional matrix about a given axis

Parameters

theta: float or np.ndarray: Angle of rotation in radians
axis: str: Axis of rotation ('x', 'y', or 'z')

pyTMD.astro.mean_longitudes(MJD: ndarray, MEEUS: bool = False, ASTRO5: bool = False)[source]

Computes the basic astronomical mean longitudes: S, H, P, N and PP [1] [2]

Note N is not N’, i.e. N is decreasing with time.

Parameters

MJD: np.ndarray: Modified Julian Day (MJD) of input date
MEEUS: bool, default False: use additional coefficients from Meeus Astronomical Algorithms
ASTRO5: bool, default False: use Meeus Astronomical coefficients as implemented in ASTRO5

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)
PP: np.ndarray: longitude of solar perigee (degrees)

References

1

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

2

J. L. Simon, P. Bretagnon, J. Chapront, M. Chapront-Touzé, G. Francou, and J. Laskar “Numerical expressions for precession formulae and mean elements for the Moon and the planets”, Astronomy and Astrophysics, 282(2), 663–683, (1994). bibcode: 1994A%26A…282..663S

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, and N’, and Ps [1] [2]

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)

References

1

A. T. Doodson and H. Lamb, “The harmonic development of the tide-generating potential”, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 100(704), 305–329, (1921). doi: 10.1098/rspa.1921.0088

2

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

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 [1] [2] [3]

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)

References

1

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

2

G. Petit and B. Luzum (eds.), IERS Conventions (2010), International Earth Rotation and Reference Systems Service (IERS), IERS Technical Note No. 36

3

N. Capitaine, P. T. Wallace, and J. Chapront, “Expressions for IAU 2000 precession quantities”, Astronomy & Astrophysics, 412, 567–586, (2003). doi: 10.1051/0004-6361:20031539

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

Mean obliquity of the ecliptic

Parameters

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

References

1: N. Capitaine, P. T. Wallace, and J. Chapront, “Expressions for IAU 2000 precession quantities”, Astronomy & Astrophysics, 412, 567–586, (2003). doi: 10.1051/0004-6361:20031539
1: N. Capitaine, J. Chapront, S. Lambert, and P. T. Wallace, “Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession-nutation model”, Astronomy & Astrophysics, 400, 1145–1154, (2003). doi: 10.1051/0004-6361:20030077

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 [1] [2] [3]

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)

References

1

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

2

O. Montenbruck, Practical Ephemeris Calculations, 146 pp., (1989).

3

R. S. Park, W. M. Folkner, and J. G. Williams, and D. H. Boggs, “The JPL Planetary and Lunar Ephemerides DE440 and DE441”, The Astronomical Journal, 161(3), 105, (2021). doi: 10.3847/1538-3881/abd414

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

Computes approximate positional coordinates of the sun in an Earth-centric, Earth-Fixed (ECEF) frame [1] [2]

Parameters

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

Returns

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

References

1

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

2

O. Montenbruck, Practical Ephemeris Calculations, 146 pp., (1989).

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 [1] [2]

Parameters

MJD: np.ndarray: Modified Julian Day (MJD) of input date
kernel: str or pathlib.Path: Path to JPL ephemerides kernel file

Returns

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

References

1

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

2

R. S. Park, W. M. Folkner, and J. G. Williams, and D. H. Boggs, “The JPL Planetary and Lunar Ephemerides DE440 and DE441”, The Astronomical Journal, 161(3), 105, (2021). doi: 10.3847/1538-3881/abd414

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 [1] [2] [3]

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)

References

1

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

2

O. Montenbruck, Practical Ephemeris Calculations, 146 pp., (1989).

3

R. S. Park, W. M. Folkner, and J. G. Williams, and D. H. Boggs, “The JPL Planetary and Lunar Ephemerides DE440 and DE441”, The Astronomical Journal, 161(3), 105, (2021). doi: 10.3847/1538-3881/abd414

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

Computes approximate positional coordinates of the moon in an Earth-centric, Earth-Fixed (ECEF) frame [1] [2]

Parameters

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

Returns

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

References

1

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

2

O. Montenbruck, Practical Ephemeris Calculations, 146 pp., (1989).

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 [1] [2]

Parameters

MJD: np.ndarray: Modified Julian Day (MJD) of input date
kernel: str or pathlib.Path: Path to JPL ephemerides kernel file

Returns

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

References

1

Meeus, Astronomical Algorithms, 2nd edition, 477 pp., (1998).

2

R. S. Park, W. M. Folkner, and J. G. Williams, and D. H. Boggs, “The JPL Planetary and Lunar Ephemerides DE440 and DE441”, The Astronomical Journal, 161(3), 105, (2021). doi: 10.3847/1538-3881/abd414

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

Greenwich Apparent Sidereal Time (GAST) [1] [2] [3]

Parameters

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

References

1: N. Capitaine, P. T. Wallace, and J. Chapront, “Expressions for IAU 2000 precession quantities”, Astronomy & Astrophysics, 412, 567–586, (2003). doi: 10.1051/0004-6361:20031539
2: N. Capitaine, J. Chapront, S. Lambert, and P. T. Wallace, “Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession-nutation model”, Astronomy & Astrophysics, 400, 1145–1154, (2003). doi: 10.1051/0004-6361:20030077
3: G. Petit and B. Luzum (eds.), IERS Conventions (2010), International Earth Rotation and Reference Systems Service (IERS), IERS Technical Note No. 36

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

International Terrestrial Reference System (ITRS) [1] [2] [3]: 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

References

1: N. Capitaine, P. T. Wallace, and J. Chapront, “Expressions for IAU 2000 precession quantities”, Astronomy & Astrophysics, 412, 567–586, (2003). doi: 10.1051/0004-6361:20031539
2: N. Capitaine, J. Chapront, S. Lambert, and P. T. Wallace, “Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession-nutation model”, Astronomy & Astrophysics, 400, 1145–1154, (2003). doi: 10.1051/0004-6361:20030077
3: G. Petit and B. Luzum (eds.), IERS Conventions (2010), International Earth Rotation and Reference Systems Service (IERS), IERS Technical Note No. 36

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

Compute complementary terms of the equation of the equinoxes [1] [2] [3]

Parameters

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

References

1: N. Capitaine, P. T. Wallace, and J. Chapront, “Expressions for IAU 2000 precession quantities”, Astronomy & Astrophysics, 412, 567–586, (2003). doi: 10.1051/0004-6361:20031539
2: N. Capitaine, J. Chapront, S. Lambert, and P. T. Wallace, “Expressions for the Celestial Intermediate Pole and Celestial Ephemeris Origin consistent with the IAU 2000A precession-nutation model”, Astronomy & Astrophysics, 400, 1145–1154, (2003). doi: 10.1051/0004-6361:20030077
3: G. Petit and B. Luzum (eds.), IERS Conventions (2010), International Earth Rotation and Reference Systems Service (IERS), IERS Technical Note No. 36

pyTMD.astro._frame_bias_matrix()[source]: Frame bias rotation matrix

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

Calculate nutation rotation angles using tables from IERS Conventions [1]

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

References

1: G. Petit and B. Luzum (eds.), IERS Conventions (2010), International Earth Rotation and Reference Systems Service (IERS), IERS Technical Note No. 36

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

Nutation rotation matrix

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 | numpy.ndarray)[source]

Polar motion (Earth Orientation Parameters) rotation matrix

Parameters

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

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

Precession rotation matrix

Parameters

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

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

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 IERS Conventions

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 IERS Conventions