Technical Notes for Central Bodies

Central Body Configuration Settings

Each central body has its own configuration file that defaults many of its parameters. The central body configuration files are named <central body name>.cb and are located at <STK install folder>\STKData\CentralBodies\<central body name>. For example, Earth’s configuration is contained in \STKData\CentralBodies\Earth\Earth.cb while the Moon’s configuration is in \STKData\CentralBodies\Moon\Moon.cb. Editing these files to change parameter settings is recommended for advanced users only.

Settings Description
Gm Product of universal gravitational constant G and central body mass.
SystemGm Product of universal gravitational constant G and central body system mass. The system consists of a planet and its moons.
GravityModel

Default gravity model to be used with the central body. Used with the Twobody, J2Perturbation and J4Perturbation propagators; also used when gravity values are needed for element conversions (e.g., Cartesian to Keplerian elements). For consistency, the Gm value for the central body should match the Gm value for its default gravity model.

Gravity models are stored in the same directory as the central body *.cb file; they use file extension .grv.

Shape Values associated with the central body shape: parameters correspond to a sphere, oblate spheroid, or triaxial ellipsoid.
PathGenerator Propagators available for use with objects having this central body.
EphemerisData Sources of the ephemeris for this central body:
  • JplDE – uses JPL DE file (only for major planets, Sun, Moon)
  • JplSpice – uses SPICE files using bodyID given by JplSpiceID
  • File – uses external file (*.e or *.pe)
  • Analytic – uses slowly varying twobody algorithm
SpinData Source of the attitude definition. Usually, the attitude is specified using the algorithm and data from Report of the IAU/IAG Working Group on Cartographic Coordinates and Rotational Elements of the Planets and satellites: 2009, B.A. Archinal et al. (2011), Celest. Mech Dyn Astr 109 (2), 101-135 (DOI: 10.1007/s10569-010-9320-4). The data itself is contained in a rotational coefficients file (with file extension .rot) sitting in the same directory as the central body *.cb file. Earth uses the ICRF theory instead to define its attitude.

Central Body Shape

Central bodies are modeled as having an ellipsoid shape: either a tri-axial ellipsoid (the three semi-axes have different lengths); an oblate spheroid shape (where the z-axis is the axis of symmetry and also the minor axis; or a sphere (where all semi-axes have the same length). Prolate shapes are not considered (no major body of the solar system is prolate).

The size and shape for each central body is obtained from “Report of the IAU/IAG Working Group on cartographic coordinates and rotational elements: 2009,” B.A. Archinal et al., Celest. Mech Dyn Astr 109 (2), 101-135 (DOI: 10.1007/s10569-010-9320-4) . Values for each central body are specified in the central body configuration file, a file with extension .cb in the \STKData\CentralBodies\ directory (e.g., \STKData\CentralBodies\Earth\Earth.cb, \STKData\CentralBodies\Moon\Moon.cb, etc.).

Central Body Ephemerides

The ephemerides for most central bodies are obtained from a Developmental Ephemeris (DE) file or a JPL SPICE file. Such files contain tables of position and velocity over a large span of time.

The Developmental Ephemeris is generated from a numerical integration of the solar system, using barycentric positions for the outer planets and including effects from general relativity. The ephemerides is made available to the public from a JPL website. AGI has collected the ephemerides for the period 1960-2050 and packaged it into a JPL DE file, a particular binary format that allows software (publicly available from JPL) to extract and interpolate the data. By default, version 421 (i.e., DE421) dated Feb 2008 will be used. It contains the then-best ephemerides of the major planets, the Moon, and Sun and updates the attitude of the Moon over the older DE405/DE403 versions.

JPL also makes available ephemerides in another format called SPICE. There are SPICE files that contain the same developmental ephemerides as normally shipped in a DE file. In addition, JPL makes available ephemerides for a wide range of celestial bodies in the solar system. These ephemerides are usually developed during JPL mission planning and operations. Time spans for different bodies can vary widely; moreover, there may be several versions of ephemerides for the same celestial body developed at different times (and possibly incorporating different sets of data). For example, the ephemerides for the Jovian system was updated and improved during the Galileo mission.

SPICE files are read and interpolated using SPICE software available from JPL. The SPICE toolkit natively reports the position of planetary centers, rather than the barycenters of planetary systems from a DE file. AGI has incorporated those parts of SPICE dealing with ephemeris interpolation into its software, and collected the best ephemerides for many of the larger bodies of the solar system into a few SPICE files. Note that JPL does not publish a solar system SPICE file of its own.

Currently, all central bodies shipped with AGI software could be configured to use the ephemerides from SPICE files. However, the older DE file format provides better time precision when requesting interpolated ephemeris. Thus, the Earth, Moon, and Sun are configured to use the older DE file format as their ephemeris source; all other bodies (except for Ceres) use ephemerides contained in SPICE files. Ceres uses a simple analytic formula (essentially a two-body formula where the elements are modeled as drifting linearly in time).

All central body ephemerides span the period 1990 – 2048 (at least), except for the Jovian system which spans 2000-2048. A discussion of the numerical differences of using SPICE and DE files can be found here.

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 (whose axes are defined by a constant rotation from ICRF). Earth and Moon have the most number of 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; inertial systems do not, though they may rotate with respect to each other. 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 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 web site contains information referencing coordinate systems of central bodies.

Earth Frames

Many frames have been defined for Earth over the years. The newest frame is the ICRF frame, the best realization of an inertial frame constructed to date. (Although, an intermediate frame associated with the transformation from Fixed to ICRF has been identified, there is no known organization currently using such a frame, and it has not been made available to STK users.) Associated with the J2000 frame is a collection of intermediate frames (True Equator True Equinox [True], True Equator Mean Equinox [TEME], 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 basis of the J2000 frame.

Supported Coordinate Systems

Frame
Earth
Moon
Sun
All Others
ICRF
Y
Y
Y
Y
J2000
Y
Y
Y
Y
Inertial
Y*
Y
Y*
Y
Fixed
Y
Y
Y
Y
TrueOfDate
Y
Y
Y
Y
TrueOfEpoch
Y
Y
Y
Y
MeanOfDate
Y
Y
Y
Y
MeanOfEpoch
Y
Y
TEMEOfDate
Y
TEMEOfEpoch
Y
B1950
Y
AlignmentAtEpoch
Y
MeanEarth
Y
PrincipalAxes_403
Y
PrincipalAxes_421
Y
Fixed_IAU2003
Y
Fixed_NoLibration
Y
J2000_Ecliptic
Y
TrueEclipticOfDate
Y
Y* - The Inertial frame for the Earth and the Sun is ICRF—these bodies do not have a separate system named Inertial.

Coordinate System Descriptions

ICRF. International Celestial Reference Frame. The ICRF axes are defined as the inertial (i.e., kinematically non-rotating) 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 Fixed frame. The current algorithm uses the P03 precession model, the IAU2000A nutation model (as adjusted), and the Earth rotation angle (expressed as a linear function of time in UT1) and became operational on 1 Jan 2009. At present writing (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 IAU2000A 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 whose origin is at the solar system barycenter--- rather, the term describes a coordinate system whose origin is determined from context (i.e., for a central body, its center of mass location) whose 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. [Note that ‘aligned’ here refers to directions in Euclidean space – not in a curved space governed by general relativity.]

J2000. 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 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 whose origin is at Earth’s center--- rather, the term describes a coordinate system whose origin is determined from context (i.e., for a central body, its center of mass location) whose axes are parallel to the axes of the J2000 system defined at the Earth.

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

Note: 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, a more definitive frame should be used (e.g., ICRF, J2000).

Fixed. The Fixed frame of a central body is the frame in which its topography is expressed. For gaseous planets (Jupiter, Saturn, Uranus, 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: 2009, B.A. Archinal et al., Celest. Mech Dyn Astr 109 (2), 101-135 (DOI: 10.1007/s10569-010-9320-4) . The algorithm uses three slowly varying Fourier series to represent:

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 \STKData\CentralBodies.

TrueOfDate. 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.

TrueOfEpoch. 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. 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.

MeanOfEpoch. 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.

AlignmentAtEpoch. 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 Systems : Special Considerations

The generic definitions of the Fixed, True and Mean systems as defined above are not used for Earth. Instead, the following definitions apply.

Fixed. 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.

TrueOfDate. 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, and 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 would be the Earth’s spin axis if pole wander were ignored; the TrueOfDate X axis defines the true vernal equinox.

TrueOfEpoch. True Equator and True Equinox of epoch. Earth’s TrueOfDate system evaluated at some specified epoch, rather than at each given time. This frame does not rotate with respect to the J2000 frame.

MeanOfDate. 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.

MeanOfEpoch. 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.

Additional Earth Systems

TEMEOfDate. True Equator and Mean Equinox of date. This 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.

TEMEOfEpoch. True Equator and Mean Equinox of epoch. 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.

B1950. 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 Systems : Special Considerations

The generic definitions of the Inertial, TrueOfDate and MeanOfDate systems as defined above are not used for Moon. Instead, the following definitions apply.

Inertial. 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.

TrueOfDate. 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 to be set to use the IAU2003 frame, then the two frames would be identical.

MeanOfDate. 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.

Additional Moon Systems

PrincipalAxes_421. 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. By default, the DE 421 file is loaded and the attitude is obtained from it. If the DE403 or DE405 file is loaded instead, then the system is derived by first transforming to MeanEarth. Doing so, however, results in a slightly different attitude.

PrincipalAxes_403. 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 or DE 405 file (the data contained is virtually identical). By default, only the DE 421 file is loaded. Thus, this frame is realized by first transforming to the MeanEarth frame. Doing so, however, results in a slightly different attitude than simply using the DE403 or DE405 files directly. If the DE403 or DE405 file is loaded then the attitude is obtained directly from the DE file.

MeanEarth . Mean Earth / Polar Axis (ME) System. The preferred lunar frame for associating lunar topography. It is defined as a constant rotation from a PrincipalAxes frame. The rotation, however, depends on the PrincipalAxes frame. Values for the rotation for different DE file versions are contained in \STKData\CentralBodies\Moon\Moon.cb. By default, the DE 421 file is loaded so that the MeanEarth frame is defined as a rotation from the PrincipalAxes_421 frame.

Fixed_IAU2003. 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: 2009,” B.A. Archinal et al., Celest. Mech Dyn Astr 109 (2), 101-135 (DOI: 10.1007/s10569-010-9320-4) . The parameters are identical to the earlier report “Report of the IAU/IAG Working Group … 2000,” published in 2003.

Fixed_NoLibration 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.

Additional Sun Systems

J2000_Ecliptic. The mean ecliptic system evaluated at the J2000 epoch. The mean ecliptic plane is defined as the rotation of the J2000 XY plane about the J2000 X axis by the mean obliquity defined using FK5 IAU76 theory.

TrueEclipticOfDate. 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.

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 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.

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.

Note: The JPL DE does not provide the true obliquity directly. Instead, it provides a correction from the mean obliquity to the true obliquity.

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.

Note: 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 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 towards 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 towards 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.

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.

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.

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. The time between updates of the slowly varying data, which includes precession angles, nutation angles, and pole wander, may be specified by the user 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.