Reference frames and time systems

A state vector is meaningless without a frame and an epoch. Most large errors in practical astrodynamics are not errors of dynamics; they are a position expressed in one frame and interpreted in another, or a time tagged in one scale and used in another. Both mistakes are silent, and both produce results that look entirely plausible.

Why more than one frame

Newton’s laws hold in an inertial frame. The Earth is not one: it rotates, precesses and wobbles. But every ground station, every launch site and every map product is naturally expressed relative to the rotating Earth. Both frames are unavoidable, so the transformation between them is unavoidable too.

Inertial (ECI)

Where the equations of motion are integrated. Axes fixed relative to distant quasars.

Earth-fixed (ECEF)

Where ground stations, launch sites and geodetic coordinates live. Rotates with the planet.

Topocentric

Where an observation is made: azimuth, elevation and range from a specific site.

Orbital / local

Where relative motion and attitude are naturally described, moving with the spacecraft.

Inertial frames

The modern realisation is the International Celestial Reference Frame, defined by the measured positions of several hundred extragalactic radio sources. Because those sources have no detectable proper motion, the frame is kinematically non-rotating to microarcsecond precision. The GCRF is the geocentric version, with its origin at Earth’s centre of mass and axes aligned with the ICRF.

Older frames remain in wide use and are not identical to it:

The frame bias between J2000 and GCRF is negligible for mission design and significant for precise orbit determination. The TEME distinction is not negligible for anyone, and is dealt with below.

Earth-fixed frames

The International Terrestrial Reference Frame is realised by the coordinates and velocities of a global network of tracking stations, updated periodically as ITRF2014, ITRF2020 and so on. It accounts for plate tectonics, so a station’s coordinates carry an epoch and a velocity.

WGS 84 is the frame used by GPS. It agrees with the ITRF to within a few centimetres, which is below the noise for everything but geodesy.

Geodetic latitude, longitude and altitude are defined against the WGS 84 reference ellipsoid, with semi-major axis 6 378 137.0 m and flattening 1/298.257223563. The distinction between geodetic and geocentric latitude follows from that flattening and reaches 0.19° at mid-latitudes, which is about 21 km on the surface. Confusing them is a classic and expensive error.

The transformation chain

Going from GCRF to ITRF is not a single rotation. It is four, each accounting for a different physical motion of the Earth.

The current standard is the CIO-based IAU 2006/2000A model, which replaced the older equinox-based chain. Both are still encountered; the equinox route runs through mean-of-date and true-of-date frames and uses Greenwich Apparent Sidereal Time in place of the Earth Rotation Angle.

Polar motion and the UT1 offset cannot be predicted from theory. They are measured and published as Earth Orientation Parameters by the IERS, in Bulletins A and B. Software that does not ingest current EOP data is silently using stale values, which costs metres of geolocation accuracy.

Topocentric and orbital frames

Topocentric frames are centred on an observing site.

Orbital frames move with the spacecraft and are where relative motion is naturally expressed.

Conjunction assessment is always reported in RIC, for a good reason: orbit uncertainty is strongly anisotropic. Along-track error is typically an order of magnitude larger than radial or cross-track, because a small semi-major axis error integrates directly into a timing error. A covariance that looks alarming as a single number is often benign once resolved into RIC components.

Relative motion in the LVLH frame is the setting for the Clohessy-Wiltshire equations. See Rendezvous and docking.

Time scales

Five time scales are in routine use, and they measure genuinely different things.

The essential distinction is between atomic scales, which measure the passage of time uniformly, and UT1, which measures the orientation of the Earth. Earth’s rotation is irregular, so the two diverge and leap seconds are inserted to keep civil time aligned with the sky.

Use TT or TDB

For dynamics: propagating orbits, evaluating ephemerides

Use UT1

For Earth orientation: converting inertial to Earth-fixed

Use UTC

For logs, schedules and anything a human reads

Use GPS time

Inside GNSS processing, and convert at the boundary

Using UTC where UT1 is required introduces an error of up to 0.9 s of Earth rotation. At the equator that is about 420 m of ground position, which is catastrophic for geolocation and irrelevant for scheduling a pass. Knowing which case you are in is the whole skill.

Leap seconds

Leap seconds are inserted irregularly, announced only about six months ahead, and produce a UTC day of 86 401 seconds. Software that assumes a fixed 86 400-second day is wrong on those days, and the failure mode is a one-second discontinuity that is easy to miss and hard to trace.

No leap second has been inserted since 2016-12-31, because Earth’s rotation has been running slightly fast. In 2022 the CGPM resolved to stop inserting them by 2035, allowing UTC and UT1 to diverge freely and be reconciled by some future mechanism. Systems built now will outlive that change, so the offset should be read from a table, never hard-coded.

Julian dates

Continuous day counts avoid calendar arithmetic entirely.

The half-day offset in MJD exists so that the day boundary falls at midnight. Getting it backwards puts every result out by twelve hours.

Four mistakes worth naming

Treating TEME as J2000. SGP4 outputs TEME, the frame of the underlying theory. Precession since J2000 has accumulated to roughly 0.36° by the mid-2020s, so at a 7000 km orbital radius, propagating a TLE and then using the result as though it were J2000 puts the position out by more than 40 km. The conversion is short and there is no excuse for skipping it.

Mixing time scales across an interface. A state vector tagged in UTC and propagated as though it were TT is 69 seconds off. At 7.7 km/s that is 530 km of along-track error, which is large enough to be noticed immediately but is often misdiagnosed as a dynamics problem.

Geodetic against geocentric latitude. Up to 0.19° apart, about 21 km on the ground. Every ground station coordinate is geodetic; most textbook derivations are geocentric.

Ignoring Earth orientation parameters. Software that defaults to zero polar motion and zero UT1−UTC will be metres off in geolocation and will drift further as the values it is missing evolve.

References

• Petit, G. and Luzum, B. (eds.) IERS Conventions (2010), IERS Technical Note 36. The definitive specification of the frames and transformations.
• Vallado, D. A. Fundamentals of Astrodynamics and Applications, 5th ed., Microcosm Press, 2022, chapter 3.
• Vallado, D. A., Crawford, P., Hujsak, R. and Kelso, T. S. “Revisiting Spacetrack Report #3”, AIAA 2006-6753. The authoritative treatment of SGP4 and TEME.
• Kaplan, G. H. “The IAU Resolutions on Astronomical Reference Systems, Time Scales, and Earth Rotation Models”, USNO Circular 179, 2005.
• Seidelmann, P. K. (ed.) Explanatory Supplement to the Astronomical Almanac, 3rd ed., University Science Books, 2013.