IERS Conventions (2010): Coordinate Frames

Source

Petit, G. and Luzum, B. (eds.), 2010, “IERS Conventions (2010),” IERS Technical Note, No. 36, Bureau International des Poids et Mesures (BIPM). ISBN: 3-89888-989-6.

How IERS 2010 Relates to SGP4’s TEME Frame

SGP4 outputs satellite positions and velocities in a frame called TEME (True Equator, Mean Equinox). TEME is not a standard IAU reference frame. It is a computational artifact of the SGP4 algorithm, defined by:

• The true equator of date (the CIP equator after applying nutation)
• A “mean equinox” that is not the standard IAU mean equinox, but an equinox computed using only a truncated 4-term nutation series for the equation of the equinoxes

To convert SGP4 output to standard frames (ITRF for ground positions, GCRS/J2000 for inertial work), the relationship between TEME’s simplified rotation model and the full IAU 2006/2000A model must be understood.

TEME to ITRF (Earth-Fixed)

$$[\text{PEF}] = R_3(\theta_{GAST_{TEME}}) \cdot [\text{TEME}]$$ $$[\text{ITRF}] = W(t) \cdot [\text{PEF}]$$

where:

• $\theta_{GAST_{TEME}}$ is Greenwich Apparent Sidereal Time computed using the same truncated nutation that SGP4 uses internally
• $W(t)$ is the polar motion matrix using observed $x_p$, $y_p$ values

TEME to GCRS/J2000

$$[\text{GCRS}] = Q(t) \cdot R_3(\theta_{GAST_{FULL}}) \cdot R_3(-\theta_{GAST_{TEME}}) \cdot [\text{TEME}]$$

Or equivalently, using the two-step approach employed by Skyfield:

1. Convert TEME to ITRF using the simplified rotation
2. Convert ITRF to GCRS using the full IERS 2010 model

Nutation model consistency is critical

You must use the same nutation model as SGP4 when computing GAST for the TEME-to-PEF step. Using the full IAU 2006/2000A nutation here would introduce systematic errors of several hundred meters. The 4-term truncated nutation defines TEME — it is not an approximation to be improved upon, but part of the frame’s definition.

The Nutation Model Hierarchy

Model	Terms	Accuracy	Used by
1980 IAU (4-term subset)	4	~1 arcsec	SGP4/SDP4 (TEME frame)
1980 IAU (full)	106	~10 mas	IERS Conventions 1996
IAU 2000B	<80	~1 mas	Quick applications (1995—2050)
IAU 2000A (MHB2000)	1365	~0.1 mas	IERS Conventions 2010 (current)

SGP4’s 4-Term Nutation

SGP4’s internal nutation computation uses only the four largest terms from the 1980 IAU series. From Vallado et al. (2006):

1. 18.6-year ($\Omega$). The Moon’s ascending node regression. Amplitude ~17.2” in longitude, ~9.2” in obliquity. By far the dominant nutation term.
2. Semi-annual. ~1.3” in longitude.
3. Fortnightly. ~0.2” in longitude.
4. Monthly. ~0.1” in longitude.

These four terms capture approximately 98% of the total nutation amplitude but leave systematic errors of ~1 arcsecond (about 30 meters at LEO altitudes when converted to position error). This residual is consistent with TLE accuracy limits of ~1 km.

The IAU 2000A Full Model

The IERS 2010 standard adopts IAU 2000A_R06:

• 678 lunisolar nutation terms (in-phase and out-of-phase)
• 687 planetary nutation terms
• Total of 1365 terms
• Includes time-varying coefficients (Poisson terms)
• Small adjustments for IAU 2006 precession compatibility ($J_2$ rate effect, obliquity correction)
• Accuracy limited to ~0.3 mas by the unpredictable Free Core Nutation

Five Largest Nutation Terms

Period	Amplitude ($\Delta\psi$)	Source
18.613 yr	$-17206.4$ mas	Lunar node ($\Omega$)
182.6 days	$-1317.1$ mas	Solar semi-annual ($2F - 2D + 2\Omega$)
13.66 days	$-227.6$ mas	Lunar fortnightly ($2F + 2\Omega$)
9.13 days	$207.5$ mas	Lunar ($2\Omega$)
365.26 days	$147.6$ mas	Solar annual ($l'$)

IAU 2000/2006 Changes

What IAU 2000 Changed

IAU 2000 Resolution B1.6 adopted:

• The Celestial Intermediate Pole (CIP) to replace the Celestial Ephemeris Pole
• The Celestial Intermediate Origin (CIO) to replace the equinox as the origin of right ascension
• The Earth Rotation Angle (ERA) as the fundamental measure of Earth rotation (replacing sidereal time)
• The IAU 2000A precession-nutation model (MHB2000) as the standard for the non-rigid Earth

What IAU 2006 Changed

IAU 2006 Resolution B1 adopted:

• Improved precession polynomial expressions (5th degree in $t$) based on dynamical theory
• The $J_2$ secular rate effect incorporated into precession
• The mean obliquity value $\epsilon_0 = 84381.406''$ (vs. IAU 2000’s $84381.448''$)
• Small adjustments to the IAU 2000A nutation to maintain consistency with the new precession

Two Equivalent Transformation Methods

IERS 2010 Chapter 5 describes two methods that agree to microarcsecond precision:

Method 1: CIO-Based (Recommended)

$$[\text{GCRS}] \xrightarrow{C(X,Y,s)} [\text{CIRS}] \xrightarrow{R_3(-ERA)} [\text{TIRS}] \xrightarrow{W(x_p, y_p, s')} [\text{ITRS}]$$

• Uses CIP coordinates $(X, Y)$ directly (no separate precession/nutation angles)
• Uses Earth Rotation Angle (linear in UT1)
• Conceptually cleaner: ERA measures pure rotation, precession-nutation is in $Q(t)$

Method 2: Equinox-Based (Classical)

$$[\text{GCRS}] \xrightarrow{N \cdot P \cdot B} [\text{True equinox}] \xrightarrow{R_3(-GAST)} [\text{TIRS}] \xrightarrow{W(x_p, y_p, s')} [\text{ITRS}]$$

• Uses classical nutation angles ($\Delta\psi$, $\Delta\epsilon$) with precession
• Uses Greenwich Apparent Sidereal Time
• More familiar to older software and textbooks

SOFA Library

The IAU Standards Of Fundamental Astronomy (SOFA) library provides reference implementations in Fortran and C. Key routine families:

Routine	Purpose
XY06 / XYS06A	CIP coordinates from IAU 2006/2000A series
ERA00	Earth Rotation Angle
NUT06A	Nutation components
PN06 / PNM06A	Precession-nutation matrices
C2TCIO / C2TEQX	Complete celestial-to-terrestrial matrices

How Skyfield Uses This

Skyfield uses a combination approach:

• Downloads IERS finals2000A data for UT1-UTC and polar motion
• Uses the de421.bsp (or de440.bsp) JPL ephemeris for planetary positions
• Implements the full IAU precession-nutation via precomputed tables
• For SGP4 satellites, correctly handles the TEME intermediate frame by using the truncated nutation for the TEME-to-PEF step and the full model for subsequent frame conversions

Practical Note for Craft

Craft uses Skyfield on the backend for all celestial computations. Skyfield correctly handles the TEME-to-GCRS transformation internally. On the frontend, satellite.js outputs TEME directly — the client-side code converts to geodetic coordinates (latitude/longitude/altitude) using the simplified GMST approach, which is adequate for visualization at CesiumJS rendering precision.

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Appendix C: TEME Coordinate System The Rev-1 paper's detailed TEME frame definition and worked examples

WGS-84: Modern Geodetic Datum The datum used for observer coordinates and display

Kelso (2007): SGP4 Validation Empirical accuracy bounds for the SGP4/TEME pipeline

SGP4 Theory Overview Index of all theory archive documents