Technical Overview of Central Body Reference Frames (Coordinate Systems)
This topic provides an overview of the reference frames (coordinate systems) and the FK5 IAU76 Theory for central bodies in STK.
For central bodies defined using the Vector Geometry tool, see Vector Geometry Tool Reference Frames.
Central body coordinate systems
Each central body has an associated list of supported coordinate systems. The origin of each coordinate system is the location of the center of mass of the central body; thus, the distinguishing feature of each system is the reference axes that are used.
All central bodies support a Fixed coordinate system (i.e., a coordinate system in which its topography has no motion), an ICRF coordinate system, a J2000 coordinate system, and an Inertial coordinate system (with axes defined by a constant rotation from ICRF). Earth and Moon have the most supported systems; the Moon has many different versions of the Fixed system. The Sun includes ecliptic-based frames; other central bodies have a generic set of systems.
There are two classes of coordinate systems: fixed and inertial.
Fixed systems nominally rotate with the central body’s topography:
Earth rotation with Earth Fixed frame axes
Inertial systems do not rotate in conjunction with the Earth, though they may rotate with respect to each other:
Earth rotation with Earth Inertial frame axes
Thus, from an observer on Earth's surface, fixed frames will look stationary while inertial frames will appear to rotate. Typically, frames for a central body arise from modeling the fixed-to-inertial transformation, where intermediate frames may be constructed as part of the modeling.
The fixed-frame attitude is computed as decomposition into the motion of the spin axis (usually modeled as right ascension and declination) and the motion about the spin axis (rotation). Inertial frames depend on the spin axis motion only. Typically, the motion of the spin axis's orientation consists of two parts:
Mean frames only account for precession, while True frames account for both precession and nutation; neither accounts for rotation.
The U.S. Geological Survey International Astronomical Union website contains information referencing coordinate systems of central bodies.
Earth inertial frames
Several inertial frames have been defined for Earth over the years. The latest frame is the ICRF frame, the best realization of an inertial frame constructed to date. Prior to ICRF, the best inertial frame was the J2000 frame (i.e., Mean Equator and Mean Equinox of the J2000 epoch). The difference between ICRF and J2000 frames is a very small rotation having a magnitude within the uncertainty of the J2000 frame itself. Associated with the J2000 frame is a collection of intermediate frames (True Equator True Equinox [True], True Equator Mean Equinox [TEME], and Mean Equator Mean Equinox [Mean]) that have wide usage throughout the community. These frames continue to be provided and are defined using the FK5 IAU76 theory that is the basis of the J2000 frame. Although an intermediate frame associated with the ICRF theory has been identified, there is no known organization currently using such a frame, and it has not been made available for use to STK users.
Earth-fixed frames
The Fixed frame for Earth is intended to be a frame attached to the surface of the Earth and rotating with it. This frame is realized by modeling the inertial frame, where star data is collected from stations around the globe. Since the motions of stars in an inertial frame centered at the solar system barycenter can be very accurately modeled, using either the ICRF or FK5 IAU76 theory, the location of ground stations can be accurately determined. Processing data from these stations leads to the determination of Earth Orientation Parameters (EOP), which are small deviations from the inertial model for the location of the rotation axis and the rotation rate about that axis. These represent unmodeled physical effects in the inertial theory, mostly the effects on the Earth's attitude arising from its internal motions that are difficult to model well. Combining the data from different stations provides a best fit for the determination of EOP that realizes the orientation of the Earth's Fixed frame.
Data collection has improved greatly over the years, so much so that the phrase "attached to the surface of the Earth" is no longer a precise specification. While ground stations are attached to the ground, the ground itself moves because of plate tectonic motion and even because of large earthquakes. This motion can be detected when taking data from the same station over long periods of time. To deal with this, the International Terrestrial Reference System (ITRS) has been defined, with its realization as an International Terrestrial Reference Frame (ITRF) to represent the Earth's Fixed frame as of some date. The earliest ITRF frames were defined in the late 1980s, while the latest one (as of January 2023) is ITRF2020.
The International Earth Rotation and Reference Systems Service (IERS) publishes EOP data daily, providing values for historical and future dates. The STK installation includes a current version of an EOP file. You can update this file to the latest version at any time using the Data Update Utility. On February 14, 2023, the published EOP data from IERS was updated to be consistent with ITRF2020. Thus, AGI has adjusted the data in the EOP file to be consistent with ITRF2020. As of version 12.7, STK is set to associate its Earth-fixed frame with ITRF2020 and assumes that the EOP file contains data consistent with ITRF2020. You can change this association to use a different ITRF frame by editing the EOP_ITRF_FRAME setting in the following directory: <STK install>/STKData/CentralBodies/Earth/Earth.cb.
For historical EOP data for dates earlier than February 14, 2023, currently published values differ from EOP files published before February 14, 2023, because earlier files were consistent with ITRF2014. AGI's published EOP files prior to March 2023 are marked as being consistent with ITRF2008, but that marking is incorrect.
Different ITRF frames are related to each other by a transformation using Helmert parameters, which is a set of 14 values describing a translation offset and its rate, a rotational offset and its rate, and a scaling offset and its rate. You can obtain a file (ITRFDefns.txt) containing these parameters for the latest ITRF frames in the DynamicEarthData folder in C:\ProgramData\AGI\STK 12. The differences in position computed using different ITRF frames are very often less than 10 cm. Older frames (e.g., ITRF1993) may differ by up to 1 m from newer ones (i.e., ITRF2020), while newer ones may differ on the order of millimeters from each other.
In most applications, the difference between locating a position in one ITRF frame versus locating it using another ITRF frame is umimportant because the difference is on the order of millimeters: the difference doesn't really represent information pertinent to a problem. For that reason, STK only supports the different ITRF realizations when importing or exporting external data that requires a coordinate frame. Examples of this are loading and exporting an external ephemeris file and also importing and exporting an initial state. In STK, Earth's fixed frame is simply named "Fixed" and not the ITRF realization equivalent. STK also uses the generic name "ITRF" as simply being an alias for Earth's Fixed frame.
On export, STK 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, STK 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, STK 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).
FK5 IAU76 theory: The theory behind the J2000 frame
The name "J2000 frame" is shorthand for the frame defined by 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 (International Celestial Reference Frame). 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.
Obliquity
Mean obliquity of the ecliptic is computed using the cubic polynomial of time past the J2000 epoch represented in JED (Julian Ephemeris Date). The coefficients of the polynomial are converted from the 1996 IERS Conventions. True obliquity is computed as the sum of the mean obliquity and the nutation in obliquity.
The JPL DE does not provide the true obliquity directly. Instead, it provides a correction from the mean obliquity to the true obliquity.
Nutation
Nutation in celestial longitude and in obliquity are computed using the 1980 IAU Theory of Nutation JPL DE (developmental ephemeris), if available. Otherwise, they are computed using series expansions of 106 terms.
Equation of equinox
The equation of equinox is updated based on IERS Technical Note 21, which includes periodic terms depending on the longitude of the ascending node of the Moon.
Mean J2000 to Mean of Date transformation
The Mean J2000 to Mean of Date (MOD) matrix is computed via the sequence of Euler rotations. Rotation angles are computed using cubic polynomials of time past the J2000 epoch in JED according to the 1976 IAU Precession angles and rates. The coefficients of the polynomials are converted from US Naval Observatory circular No. 163.
The matrix reduces to identity at the J2000 epoch and, therefore, all of the angles become zero at that epoch. In other words, 0 power coefficients of all cubic polynomials are zero.
Mean J2000 to Mean of Date Transformation
Mean of Date to True of Date transformation
The MOD to True of Date (TOD) matrix is also computed using the sequence of Euler rotations. The first rotation is about the MOD X axis, which points toward the mean equinox of date, by the mean obliquity. This rotation moves the XY plane of the moving axes from the mean equator of date to the mean ecliptic of date. The second rotation is about the new Z axis (perpendicular to the mean ecliptic plane) by the negative of the nutation in longitude. This rotation moves the X axis into its final position pointing toward the true equinox of date. The last rotation is about this new X axis by the negative of the true obliquity, which moves the XY plane from the mean ecliptic of date to the true equator of date.
Mean of Date to True of Date Transformation
True of Date to Earth Centered Pseudo-Fixed transformation (Pseudo ECF)
The transformation from TOD to Earth Centered Pseudo-Fixed (Pseudo ECF) involves a single rotation about the Z axis by the apparent Greenwich hour angle (i.e., Greenwich Mean apparent Sidereal Time). The angle is computed as the sum of the mean Greenwich hour angle and the equation of equinox. The former is also computed as the sum of the mean Greenwich hour angle at zero hour UT1 and the offset angle. The mean Greenwich hour angle is computed using a cubic polynomial in Julian Date (JD) UT1 time past the J2000 epoch. The coefficients of the polynomial are converted from US Naval Observatory circular No. 163, the Document CG-SCF-225C Code Ident 23892 and from the Explanatory supplement to the Astronomical Almanac. The offset angle is based on the Earth rotation rate, which is updated linearly as a function of zero-hour JD past the J2000 epoch. The computation of the zero-hour UT1 also requires tabulated values of UT1-UTC from the Earth Orientation Parameter (EOP) table.
True of Date to Earth Centered Pseudo-Fixed Transformation
Pole Wander transformation
Transformation from the Pseudo ECF reference frame to the Earth Centered Fixed reference frame is based on two small angles taking into account continental drift. The angles are obtained from the Earth Orientation Parameters (EOP) table, which is constructed based on data from the US Naval Observatory. This transformation is the motion of the rotational pole.
Earth Centered Pseudo-Fixed to Earth Centered Fixed Transformation
Mean J2000 to Earth Centered Fixed transformation
This transformation is a combination of the Mean J2000 to MOD, MOD to TOD, TOD to pseudo ECF, and Pole Wander (user-selectable option). If the application of pole wander is turned off, the Pseudo ECF and ECF frames are equivalent. Slowly varying parts of the Mean J2000 to ECF transformation may be cached and not necessarily computed for the exact time of the transformation. You may specify the time between updates of the slowly varying data, which includes precession angles, nutation angles, and pole wander, in terms of the Nutation Update Interval in the Earth.cb file. A nutation update interval of 0 (the default) will cause all quantities to be updated at the exact time of the transformation.