Coordinate Frame Transforms

Introduction

The satkit.frametransform module provides functions for transforming between various coordinate frames used in satellite tracking and orbit determination. These include multiple variations of “inertial” coordinate frames, and multiple versions of “Earth-fixed” coordinate frames.

Some notes:

• Most of the algorithms in this module are from the book `”Fundamentals of Astrodynamics and Applications”`` by David Vallado.

• The frame transforms are defined as arbitrary rotations in a 3-dimensional space. The rotations are a function of time, and are represented as Quaternions

• The rotation from the Geocentric Celestial Reference Frame (GCRF) to the Earth-Centered Inertial (ECI) frame is defined by the International Astronomical Union (IAU), available here. See IERS Technical Note 36 for the latest values.

API Reference

Transformations between coordinate frames, and associated utility functions

Coordinate frame transforms are mostly pulled from Vallado: https://www.google.com/books/edition/Fundamentals_of_satkitdynamics_and_Applic/PJLlWzMBKjkC?hl=en&gbpv=0

or the IERS: https://www.iers.org/

satkit.frametransform.disable_eop_time_warning()

Disable the warning printed to stderr when Earth Orientation Parameters (EOP) are not available for a given time.

Notes

• This function is used to disable the warning printed when EOP are not available for a given time.

• If not disabled, warning will be shown only once per library load,

satkit.frametransform.earth_orientation_params(time)

Get Earth Orientation Parameters at given instant

Args:

• time (satkit.time)

  Instant at which to query parameters

Returns:

• (float, float, float, float, float, float) | None

  Tuple with following elements:

• 0

  (UT1 - UTC) in seconds

• 1

  X polar motion in arcsecs

• 2

  Y polar motion in arcsecs

• 3

  LOD: instantaneous rate of change in (UT1-UTC), msec/day

• 4

  dX wrt IAU-2000A nutation, milli-arcsecs

• 5

  dY wrt IAU-2000A nutation, milli-arcsecs

Notes

• Returns None if the time is outside the range of available Earth Orientation Parameters (EOP)

• EOP are only available from 1962 to current, and predict to current + ~ 4 months

• If EOP are not available, all values will be set to zero, and a warning will be printed to stderr

• See: https://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html

satkit.frametransform.earth_rotation_angle(tm)

Earth Rotation Angle

Notes

• See: IERS Technical Note 36, Chapter 5, Equation 5.15

• Calculation Details:

  • Let t be UT1 Julian date

  • let f be fractional component of t (fraction of day)

  • ERA = 2𝜋 ((0.7790572732640 + f + 0.00273781191135448 * (t - 2451545.0)

Args:

• tm (npt.ArrayLike[datetime.datetime] | npt.ArrayLike[time])

  list, or numpy array of astro.time or datetime.datetime representing time at which to calculate output

Returns:

• npt.ArrayLike[np.float]

  Earth Rotation Angle at input time[s] in radians

satkit.frametransform.eqeq(tm)

Equation of Equinoxes

satkit.frametransform.gast(tm)

Greenwich Apparent Sidereal Time

Args:

• tm (npt.ArrayLike[datetime.datetime] | npt.ArrayLike[time])

  list, or numpy array of astro.time or datetime.datetime representing time at which to calculate output

Returns:

• npt.ArrayLike[np.float]

  Greenwich apparent sidereal time, radians, at input time(s)

satkit.frametransform.gmst(tm)

Greenwich Mean Sidereal Time

Notes

• GMST is the angle between the vernal equinox and the Greenwich meridian

• Vallado algorithm 15

• GMST = 67310.5481 + (876600h + 8640184.812866) * tᵤₜ₁ * (0.983104 + tᵤₜ₁ * −6.2e−6)

Args:

• tm (satkit.time | npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime])

  scalar, list, or numpy array of astro.time or datetime.datetime representing time at which to calculate output

Returns:

• float | npt.ArrayLike[np.float]

  Greenwich Mean Sidereal Time, radians, at input time(s)

satkit.frametransform.qcirs2gcrf(tm)

Rotation from Celestial Intermediate Reference System to Geocentric Celestial Reference Frame

Args:

• tm (npt.ArrayLike[satkit.time] | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from CIRS to GCRF at input time(s)

satkit.frametransform.qgcrf2itrf(tm)

Quaternion representing rotation from the Geocentric Celestial Reference Frame (GCRF) to the International Terrestrial Reference Frame (ITRF)

Notes

• Uses full IAU2010 Reduction

• See IERS Technical Note 36, Chapter 5

• Does not include solid tides, ocean tides

• Very computationally expensive

Args:

• tm (npt.ArrayLike[satkit.time] | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from GCRF to ITRF at input time(s)

satkit.frametransform.qgcrf2itrf_approx(tm)

Quaternion representing approximate rotation from the Geocentric Celestial Reference Frame (GCRF) to the International Terrestrial Reference Frame (ITRF)

Notes

• Accurate to approx. 1 arcsec

Args:

• tm (npt.ArrayLike[satkit.time] | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from GCRF to ITRF at input time(s)

satkit.frametransform.qitrf2gcrf(tm)

Quaternion representing rotation from the International Terrestrial Reference Frame (ITRF) to the Geocentric Celestial Reference Frame (GCRF)

Notes

• Uses full IAU2010 Reduction

• See IERS Technical Note 36, Chapter 5

• Does not include solid tides, ocean tides

• Very computationally expensive

Args:

• tm (npt.ArrayLike[satkit.time] | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from ITRF to GCRF at input time(s)

satkit.frametransform.qitrf2gcrf_approx(tm)

Quaternion representing approximate rotation from the International Terrestrial Reference Frame (ITRF) to the Geocentric Celestial Reference Frame (GCRF)

Notes

• Accurate to approx. 1 arcsec

Args:

• tm (npt.ArrayLike[satkit.time] | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from ITRF to GCRF at input time(s)

satkit.frametransform.qitrf2tirs(tm)

Rotation from Terrestrial Intermediate Reference System to Celestial Intermediate Reference Systems

Args:

• tm (npt.ArrayLike[satkit.time] | datetime.datetime | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from ITRF to TIRS at input time(s)

satkit.frametransform.qteme2gcrf(tm)

Rotation from True Equator Mean Equinox (TEME) to Geocentric Celestial Reference Frame (GCRF)

Args:

• tm (npt.ArrayLike[satkit.time] | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from TEME to GCRF at input time(s)

satkit.frametransform.qteme2itrf(tm)

Quaternion representing rotation from the True Equator Mean Equinox (TEME) frame to the International Terrestrial Reference Frame (ITRF)

Notes

• This is equation 3-90 in Vallado

• TEME is the output frame of the SGP4 propagator used to compute position from two-line element sets.

Args:

• tm (npt.ArrayLike[satkit.time] | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from TEME to ITRF at input time(s)

satkit.frametransform.qtirs2cirs(tm)

Rotation from Terrestrial Intermediate Reference System (TIRS) to the Celestial Intermediate Reference System (CIRS)

Args:

• tm (npt.ArrayLike[satkit.time] | npt.ArrayLike[datetime.datetime])

  Time[s] at which to calculate the quaternion

Returns:

• npt.ArrayLike[quaternion]

  Quaternion representing rotation from TIRS to CIRS at input time(s)