Descriptions of Coordinate Systems in ODTK

The International Celestial Reference Frame axes are defined as the inertial (i.e., kinematically nonrotating) axes associated with a general relativity frame centered at the solar system barycenter (often called the BCRF). The IAU (International Astronomical Union) is the authority for the definition of the ICRF. The ICRF is the best realization of an inertial frame constructed to date and thus represents an improvement upon the theory behind the J2000 frame. While the ICRF and J2000 frames themselves are very close, they are not identical; moreover, the J2000 frame rotates (very slowly) over time with respect to the ICRF frame. Recent star catalogs and celestial body ephemerides are most often expressed natively with respect to the ICRF frame.

The ICRF frame is realized by its transformational algorithm between it and the Earth's Fixed frame. The current algorithm uses the P03 precession model, the IAU2006 nutation model, and the Earth rotation angle (expressed as a linear function of time in UT1) and became operational on 1 Jan 2009. There is no documentation available from IERS (the International Earth rotation and Reference systems Service) for the current operational model; AGI uses an implementation based upon code available from SOFA (Standards of Fundamental Astronomy), the same code used to produce values in the Astronomical Almanac. The IAU2006 nutation model and the Earth rotation angle are documented by IERS is its Technical Note No. 32, IERS Conventions 2003.

Within AGI products, the term "ICRF coordinate system" is not restricted to the system with its origin at the solar system barycenter. Rather, the term describes a coordinate system with an origin determined from context, i.e., for a central body, its center-of-mass location. The coordinate system's axes are aligned with the axes of the BCRF. In fact, the IAU uses the term GCRF to refer to the system with origin at the geocenter (i.e., Earth�s center-of-mass location), with axes parallel to the BCRF. In this case, "aligned" refers to directions in Euclidean space, not in a curved space governed by general relativity.

This is the Mean Equator and Mean Equinox of the J2000 epoch (JD 2451545.0 TDB, which is 1 Jan 2000 12:00:00.000 TDB). The J2000 axes were considered the best-realized inertial axes until the development of the ICRF. The J2000 frame is realized by the transformational algorithm (also known as the FK5 IAU76 theory) between it and the Earth's Fixed frame. The algorithm uses the 1976 IAU Theory of Precession, the 1980 Nutation model, and the Greenwich Mean apparent Sidereal Time (expressed as a function of time in UT1), updated by IERS Technical Note No. 21 to include an adjustment to the equation of the equinoxes.

Within AGI products, the term "J2000 coordinate system" is not restricted to the system having an origin at Earth’s center. Rather, the term describes a coordinate system having an origin determined from context (i.e., for a central body, its center-of-mass location), with axes parallel to the axes of the J2000 system defined at the Earth.

Inertial definition for all central bodies except the Moon

Each central body has its own Inertial frame, computed as a constant rotation from the ICRF frame. The Inertial frame for both Earth and Sun is ICRF itself (i.e., no rotation), so these bodies do not have an additional frame named Inertial. See the Inertial definition used by the Moon. For all other central bodies, Inertial is their TrueOfEpoch system at the J2000 epoch. Thus, the Inertial frames for different central bodies are not the same frame in general.

Many organizations use the term Inertial to refer to a frame with a different definition than that used by AGI. AGI does not recommend the use of the Inertial frame to share data with other organizations or software; instead, use a more definitive frame (e.g., ICRF or J2000).

Inertial definition for the Moon

Rather than using the Fixed Z axis to define the Inertial frame, the IAU2003 Z axis is used instead. The Inertial Z axis aligns with the IAU2003 Z axis, and the Inertial X axis aligns with the vector that is the cross product of the ICRF Z axis and the IAU2003 Z axis, evaluated the J2000 epoch. This frame is very close to the Moon�s TrueOfEpoch system evaluated at the J2000 epoch.

Fixed definition for all central bodies except Earth

The Fixed frame of a central body is the frame in which its topography is expressed.

For gaseous planets � Jupiter, Saturn, Uranus, and Neptune � the Fixed frame identifies the planet�s magnetic field instead. Earth realizes its Fixed frame from the transformation algorithm between it and the ICRF; Earth�s moon realizes its Fixed frame (by default) as its MeanEarth frame; all other central bodies realize their Fixed frames using the transformational algorithm and parameters contained in Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2015, B.A. Archinal et al., Celest. Mech Dyn Astr 130:22, 2018, https://doi.org/10.1007/s10569-017-9805-5. The algorithm uses three slowly varying Fourier series to represent:

• The right ascension
• the declination of the axis of rotation
• The rotation about the spin axis, where the right ascension and declination are measured with respect to the ICRF system at the central body

The parameter data for each central body using this model is contained in a Rotational Coefficients file (i.e., a file with the extension .rot) in the central body directory under \ODTKData\CentralBodies.

Fixed Definition for Earth

Earth realizes its Fixed frame from the transformation algorithm between it and the ICRF. The transformation includes precession, nutation, and rotation effects, as well as pole wander and frame corrections.

The International Terrestrial Reference Frame is only available for Earth. It is an alias for the Earth's Fixed frame.

This is a specific realization of the Earth's Fixed frame, as defined by the International Terrestrial Reference System (ITRS), specified by the four-digit year YYYY, e.g., ITRF2020.

This system is only available for Earth. See Earth Fixed Frames.

On export, ODTK will name an ITRF realization using the form ITRFYYYY, where YYYY represents for the four-digit year indicating the ITRF year version (e.g., the 2005 realization is called ITRF2005). On import, ODTK will accept both ITRFYYYY and ITRF-YYYY, where YYYY indicates the four-digit year. To be valid on import, the ITRFDefns.txt file must include a set of Helmert parameters describing the transformation between it and the current ITRF realization of Earth's Fixed (ITRF2020 as of February 2023). Additionally, ODTK accepts both ITRFYY and ITRF-YY as names during import, where YY is a two-digit designation of the four-digit year 19YY, with YY > 80 and YY <= 99 (e.g., ITRF-93).

TrueOfDate definition for all central bodies except Earth and Moon

The Z axis aligns with the Fixed Z axis, and the X axis aligns with the vector that is the cross product of the ICRF Z axis and the Fixed Z axis, evaluated at each given time.

If the cross product is zero, then the Y axis aligns with the cross product of the Fixed Z axis and the ICRF X axis.

TrueOfDate definition for Earth

This is the True Equator and True Equinox of date. The transformation between Earth’s MeanOfDate to Earth’s TrueOfDate axes uses the mean obliquity, the nutation in longitude, and the nutation in obliquity, computed according to the 1980 Nutation model. It then applies the update to the equation of the equinoxes. By default, the nutation values are obtained by interpolating values contained in the JPL DE file rather than evaluating the model directly. The TrueOfDate Z axis is the Earth’s spin axis if pole wander is ignored; the TrueOfDate X axis defines the true vernal equinox.

TrueOfDate definition for the Moon

The Z axis aligns with the Fixed Z axis, and the X axis aligns with the vector that is the cross product of the ICRF Z axis and the Fixed Z axis, evaluated at each given time. The TrueOfDate frame is very close to the Mean Lunar Equator and IAU Node of Date (Lunar Constants and Model Document, JPL D-32296, Sept 2005). If the Moon’s Fixed frame were set to use the IAU2003 frame, then the two frames would be identical.

This is the TrueOfDate system evaluated at some specified epoch, rather than at each given time. This frame does not rotate with respect to the ICRF frame.

MeanOfDate definition for all central bodies except Earth and Moon

This uses the same computation as TrueOfDate, except that when the Fixed frame Z axis is computed, any oscillatory terms in the formulas for the right ascension and declination are ignored.

MeanOfDate definition for Earth

This is the Mean Equator and Mean Equinox of date. The transformation between J2000 and MeanOfDate is computed using a sequence of Euler rotations. Rotation angles are computed using cubic polynomials of time past the J2000 epoch in JED according to the 1976 IAU Theory of Precession angles and rates, as found in the US Naval Observatory circular No. 163. The MeanOfDate Z axis is the Earth’s mean spin axis; the MeanOfDate X axis defines the mean vernal equinox.

MeanOfDate definition for the Moon

The Z axis aligns with the IAU2003 Z axis, and the X axis aligns with the vector that is the cross product of the ICRF Z axis and the IAU2003 Z axis, evaluated at each given time. However, when computing the IAU2003 Z axis, the oscillatory terms are ignored.

MeanOfEpoch definition for all central bodies except Earth

This is the MeanOfDate system evaluated at some specified epoch, rather than at each given time. This frame does not rotate with respect to the Inertial frame.

MeanOfEpoch definition for Earth

This is Earth’s MeanOfDate system evaluated at some specified epoch, rather than at each given time. This frame does not rotate with respect to the J2000 frame.

Earth System only

This is the True Equator and Mean Equinox of date. It is an intermediate frame associated with the transformation from Earth’s MeanOfDate to Earth’s TrueOfDate axes. The TEMEOfDate Z axis is aligned with the TrueOfDate Z axis; the TEMEOfDate X axis is close to (but not identical to) the MeanOfDate X axis.

This is the Fixed frame evaluated at some specified epoch, not at each given time. This frame does not rotate with respect to the Inertial frame.

Earth System only

True Equator and Mean Equinox of epoch. This is Earth�s TEMEOfDate frame evaluated at some specified epoch, rather than at each given time. The frame does not rotate with respect to the J2000 frame.

Earth System only

These axes were considered the best realized inertial axes until the development of J2000. These axes are associated with the FK4 star catalog and its theory modeling the mean equator and mean equinox. The epoch is the beginning of the Besselian year 1950, corresponding to 31 Dec 1949 22:09:46.866 or JD 2433282.4234591. The B1950 axes are realized by a constant rotation offset from the J2000 axes, using a formula available from the Explanatory Supplement to the Astronomical Almanac.

Moon system only

Principal Axes (PA) System: The principal axes frame is aligned with the principal inertia axes, with the Z axis along the maximum inertia and the X axis along the minimum inertia. This is sometimes referred to as the axis of figure frame. The PA frame is developed in conjunction with the development of the ephemerides for the Moon. Hence, the frame depends on the source JPL DE file being used. The PA 430 frame is defined through the use of the JPL DE430 file. The PA 430 frame is realized based on a transformation from the ICRF frame based on Euler angles provided as part of the DE430, if you select DE430 at the application level as the planetary ephemeris source. If the application is configured to use a different DE version, then the 430 PA frame is realized as a constant rotation from the MeanEarth frame.

The DE440 is the default Moon ephemeris.

Moon system only

Principal Axes (PA) System. The principal axes frame is aligned with the principal inertia axes with the Z axis along the maximum inertia and the X axis along the minimum inertia. This is sometimes referred to as the axis of figure frame. The PA frame is developed in conjunction with the development of the ephemerides for the Moon. Hence, the frame depends on the source JPL DE file being used. The PA 421 frame is defined through the use of the JPL DE421 file. The PA 421 frame is realized based on a transformation from the ICRF frame based on Euler angles provided as part of the DE421, if you select DE421 at the application level as the planetary ephemeris source. If the application is configured to use a different DE version, then the 421 PA frame is realized as a constant rotation from the MeanEarth frame.

Moon System only

Principal Axes (PA) System: The principal axes frame is aligned with the principal inertia axes, with the Z axis along the maximum inertia and the X axis along the minimum inertia. This is sometimes referred to as the axis of figure frame. The PA frame is developed in conjunction with the development of the ephemerides for the Moon. Hence, the frame depends on the source JPL DE file being used. The PA 403 frame is defined through the use of the JPL DE403 file. The PA 403 frame is realized based on a transformation from the ICRF frame based on Euler angles provided as part of the DE403, if you select DE403 at the application level as the planetary ephemeris source. If the application is configured to use a different DE version, then the 403 PA frame is realized as a constant rotation from the MeanEarth frame. The principal axis frame associated with the DE405 ephemeris is essentially identical to the 403 PA frame.

Moon system only

Principal Axes (PA) System: The principal axes frame is aligned with the principal inertia axes, with the Z axis along the maximum inertia and the X axis along the minimum inertia. This is sometimes referred to as the axis of figure frame. The PA frame is developed in conjunction with the development of the ephemerides for the Moon. Hence, the frame depends on the source JPL DE file being used. The PA 440 frame is defined through the use of the JPL DE440 file. The PA 440 frame is realized based on a transformation from the ICRF frame based on Euler angles provided as part of the DE440, if you select DE440 at the application level as the planetary ephemeris source. If the application is configured to use a different DE version, then the 440 PA frame is realized as a constant rotation from the MeanEarth frame.

This is the default Moon ephemeris.

Moon System only

Mean Earth / Polar Axis (ME) System. This is the preferred lunar frame for associating lunar topography and locating surface features. The Mean Earth frame is realized as a constant rotation from the PrincipalAxes frame associated with the version of the JPL/DE specified at the application level. The rotation, between the ME and PA frame is slightly different for different realizations of the PrincipalAxes frame. Values for the rotation for different DE file versions are contained in \ODTKData\CentralBodies\Moon\Moon.cb. By default, the DE 430 file is loaded so that the MeanEarth frame is defined as a rotation from the PrincipalAxes_430 frame.

Moon System only

These axes are realized using the transformational algorithm and parameters contained in �Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2015,� B.A. Archinal et al., Celest. Mech Dyn Astr 130:22, 2018, https://doi.org/10.1007/s10569-017-9805-5.

Moon System only

The X axis is aligned with the direction to the Earth from the Moon. The Z axis is aligned with the orbital momentum vector of the Moon, computed using the Moon’s relative position and velocity with respect to the Earth.

Sun System only

This is the mean ecliptic system evaluated at the J2000 epoch. The mean ecliptic plane is defined as the rotation of the J2000 X-Y plane about the J2000 X axis by the mean obliquity defined using FK5 IAU76 theory.

Sun System only

This is an ecliptic system defined as the rotation of the ICRF axes about the ICRF X axis by the FK5 IAU76 value of the mean obliquity at the J2000.0 epoch (i.e., 84381.448 arcsecs). The XY plane is the ecliptic plane.

Sun System only

This is the true ecliptic system, evaluated at each given time. The true ecliptic plane is defined as the rotation of the J2000 XY plane about the J2000 X axis by the true obliquity defined using FK5 IAU76 theory.