Precession, Nutation and Equinox-Based Frames

Precession and nutation models, sidereal time, and the MOD, TOD and TEME frames

Giza, around 2560 BC

The Great Pyramid's descending passage runs straight and narrow into the bedrock, angled at about 26° and aimed almost exactly at true north. Stand at the bottom and look up the shaft, and you see a small patch of the northern sky, framed in stone. In 1839 the astronomer John Herschel worked out which star would have sat in that patch when the pyramid was built. It was not Polaris. Polaris was more than twenty degrees from the pole and wheeled round the sky like any other star. The pole star of the pyramid builders was Thuban, a faint star in Draco — and over the following millennia it drifted away from the pole, and Polaris drifted in. Nothing happened to either star. The thing that moved was the axis of the Earth. And because every coordinate system tied to Earth's equator inherits that motion, a satellite's "inclination" or "right ascension" means nothing until you say: measured from the equator of which date?

Watch first · presented by Shoaib Aziz

The numbers in this film are the labs' own: your precession matrix matches SOFA to 1e-16; Polaris comes closest to the pole in 2102, at 0.46°; nutation swings reach 18.9″ in longitude and 10.0″ in obliquity with an 18.6-year period (about 2 km at GEO); and treating a TEME position as J2000 puts it 41.9 km off, while the full chain lands within a metre of the modern frame.

The cause is the equatorial bulge you will meet again in Unit 7. Earth is about 21 km wider at the equator than pole to pole, and that bulge is tilted 23.44° to the ecliptic. The Sun and the Moon pull harder on the near side of the bulge than on the far side. The net effect is a torque that tries to pull the equator into the ecliptic plane — to stand the axis up straight.

A spinning body does not respond to a torque by tipping over. It responds by precessing: the spin axis sweeps sideways, at right angles to the torque, exactly as a spinning top leans but does not fall. Earth's axis traces a cone of half-angle 23.4° about the ecliptic pole, once every ~25,800 years. The Moon supplies about two-thirds of the torque; the Sun the rest.

The equinox — the line where equator meets ecliptic — is dragged around with it. It slides westward along the ecliptic at 50.29″ per year, which is why it has left Aries, sits in Pisces today, and will cross into Aquarius around AD 2600.

💡Two motions, one top

Precession is the slow, smooth sweep: 50.29″/yr in longitude, a period of ~25,800 years. It comes from the average torque. Nutation is the nod on top of it. The torque is not constant: the Sun's pull peaks twice a year (at the solstices, when it is furthest from the equator), the Moon's twice a month — and the Moon's orbit itself is tilted 5° to the ecliptic and swivels round once every 18.6 years. That last cycle dominates. The true pole traces a small ellipse around the mean pole, about ±17.2″ in longitude and ±9.2″ in obliquity. So there are two poles of date, and two equinoxes of date: Mean — precession only. The smooth, averaged axis. True — precession plus nutation. The axis Earth actually spins about (to within a few milliarcseconds).

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Those two motions define the classical frames of date, and the chain that connects them to J2000. Each step is one rotation.

MOD — Mean of Date. Mean equator and mean equinox at the instant of interest. Reached from J2000 by the precession matrix P. TOD — True of Date. True equator and true equinox. Reached from MOD by the nutation matrix N.

The IAU-1976 precession model (Lieske) writes P as three rotations through angles that grow almost linearly with time:

\mathbf{P} = \mathbf{R}_3(-z)\,\mathbf{R}_2(\theta)\,\mathbf{R}_3(-\zeta), \qquad \begin{aligned} \zeta &= 2306.2181^{\prime\prime}\,T + 0.30188^{\prime\prime}\,T^2 + 0.017998^{\prime\prime}\,T^3 \\ z &= 2306.2181^{\prime\prime}\,T + 1.09468^{\prime\prime}\,T^2 + 0.018203^{\prime\prime}\,T^3 \\ \theta &= 2004.3109^{\prime\prime}\,T - 0.42665^{\prime\prime}\,T^2 - 0.041833^{\prime\prime}\,T^3 \end{aligned}

T is Julian centuries of TT since J2000.0. In 2026 the total rotation is ~0.37°. Lab 1 builds this matrix and matches SOFA to 10⁻¹⁶.

\mathbf{N} = \mathbf{R}_1(-\varepsilon-\Delta\varepsilon)\,\mathbf{R}_3(-\Delta\psi)\,\mathbf{R}_1(\varepsilon)

Rotate the mean equator down onto the ecliptic (ε), slide along it by the nutation in longitude Δψ, and tip back up by the true obliquity ε + Δε.

💡Generations of the model

The matrices keep their shape; the series behind them improve. IAU-1976 precession + IAU-1980 nutation — 106 nutation terms, errors of tens of milliarcseconds. The basis of FK5, TEME, SGP4 and an enormous amount of flight software still in service. IAU-2000A nutation — 1,365 terms (678 luni-solar, 687 planetary), accurate to ~0.2 mas. IAU-2000B is a 77-term abridgement good to 1 mas. IAU-2006 precession (P03) — rebuilt on a dynamically consistent model of Earth's slowly changing shape. The practical rule: never mix generations. A matrix from one family and a sidereal time from another can disagree by more than the effect you are trying to model.

Earth's spin enters through sidereal time: the angle from the equinox to the Greenwich meridian. There are two, matching the two equinoxes:

GMST — Greenwich Mean Sidereal Time, measured from the mean equinox. GAST — Greenwich Apparent Sidereal Time, measured from the true equinox.

The difference is the equation of the equinoxes, essentially the nutation in longitude projected onto the equator. It never exceeds about 1.15 seconds of time — 17 arcseconds of rotation, half a kilometre at GEO.

\mathrm{GAST} = \mathrm{GMST} + \underbrace{\Delta\psi\cos\varepsilon}_{\text{equation of the equinoxes}}, \qquad \vec{r}_{\mathrm{PEF}} = \mathbf{R}_3(\mathrm{GAST})\,\vec{r}_{\mathrm{TOD}}

PEF (Pseudo Earth-Fixed) is TOD spun round by Earth’s rotation. Add polar motion and you reach the true Earth-fixed frame — lesson 3.

💡TEME and SGP4

True Equator, Mean Equinox. Its z-axis is the true pole of date; its x-axis is the mean equinox, projected onto the true equator. It is a hybrid, and it exists for one reason: the SGP4 propagator. SGP4 (Spacetrack Report #3, 1980) turns a satellite's position into Earth-fixed coordinates by rotating through GMST alone. Whatever frame makes that one rotation correct is the frame SGP4 is working in — and that frame is TEME: true pole, but measured from the mean equinox, so that GMST rather than GAST is the right angle. Every Two-Line Element set ever published is therefore a TEME product. Every TLE user must convert, and the conversion is two short steps:

\vec{r}_{\mathrm{TOD}} = \mathbf{R}_3(-\mathrm{Eq}_E)\,\vec{r}_{\mathrm{TEME}}, \qquad \vec{r}_{\mathrm{J2000}} = \mathbf{P}^{\mathsf T}\,\mathbf{N}^{\mathsf T}\,\vec{r}_{\mathrm{TOD}}

TEME → TOD → MOD → J2000. Going backwards along an arrow means using the transpose, because every one of these matrices is a rotation.

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Frame	z-axis	x-axis	From the previous frame by	Size in 2026
GCRF	ICRS pole	ICRS origin	—	—
J2000 / EME2000	J2000 mean pole	J2000 mean equinox	Frame bias B	23 mas
MOD	Mean pole of date	Mean equinox of date	Precession P	~0.37°
TOD	True pole of date	True equinox of date	Nutation N	~20″
TEME	True pole of date	Mean equinox (projected)	R₃(Eq_E) from TOD	≤ 17″
PEF	True pole of date	Greenwich meridian	R₃(GAST) from TOD, or R₃(GMST) from TEME	a full turn per day
ITRF	Conventional terrestrial pole	Greenwich (IERS)	Polar motion W	~0.3″

⚠TEME "of date" or "of epoch"?

A TLE has an epoch. SGP4 propagates it to some later time. Is the output in the TEME of the epoch or of the output time? It is TEME of date — the frame at the instant of each output. Older software sometimes converted everything with the epoch's matrices, an error that grows with the propagation span. Vallado, Crawford, Hujsak and Kelso settled the convention in "Revisiting Spacetrack Report #3" (2006). If your TLE-derived positions drift away from precise ephemerides by a steadily growing angle, check this first.

🕰Twenty years of watching one star

James Bradley spent from 1727 to 1747 measuring the positions of stars near the zenith with a fixed telescope, looking for parallax. He found something else: a yearly wobble (the aberration of light, his first great discovery) and, underneath it, a slower nod that did not repeat with the year. It took him nearly two decades to see that it repeated with the 18.6-year cycle of the Moon's orbital node. He announced the nutation of Earth's axis in 1748. A year later d'Alembert explained it from Newton's mechanics — the same torque on the bulge as precession, varying as the Moon's orbit swivels.

🛰TLEs, TEME and a real collision

On 10 February 2009 the active Iridium 33 and the dead Kosmos-2251 collided at 11.7 km/s, 790 km above Siberia, creating thousands of trackable fragments. CelesTrak's SOCRATES service, screening conjunctions from public TLEs, had predicted a close approach of 584 m that day — and it was not even among the top 150 predicted conjunctions. TLEs are only good to about a kilometre, which is why conjunction screening now uses higher-accuracy ephemerides and covariance (lesson 4). But the frame matters on top of that: as the lab shows, treating a TEME vector as J2000 adds an error of ~40 km for a LEO satellite in 2026 — dozens of times worse than the TLE itself.

↓But there is a problem

Look again at GMST. It is a single angle that tries to do two jobs at once: measure how far Earth has spun, and track where the moving equinox has got to. The polynomial for it has a T² term, which a spinning planet does not have — that term is precession, smuggled in. In 2000 the IAU separated the two. The new chain measures Earth's spin from a point on the equator that does not slide — and with spin finally isolated, it has to face the last moving part: the crust, which wanders on the axis, drifts on plates, and is the frame every GPS receiver reports in.