The wobble inside the wobble

The Earth’s axis traces a circle of radius $23.44°$ on the sky once every 25,772 years, and that motion has a cause: the Sun and the Moon pull harder on the near side of the equatorial bulge than on the far side, and the resulting torque, applied to a spinning body, moves the axis sideways rather than tipping it over.

That is the pole star’s shelf life, and it is the smooth part of the answer. The torque is not smooth. It depends on where the Moon is, and the Moon’s orbit is inclined to the ecliptic by $5.15°$ with a line of nodes that regresses once in 18.61 years — so part of the pull oscillates with that period instead of accumulating. The axis therefore does not trace a circle. It traces a circle with a small loop on it, executed a thousand and four hundred times per circuit.

Why one torque produces two motions #

The torque on an oblate spinning Earth from a body of mass $m$ at distance $r$ and declination $\delta$ goes as

$$N \propto \frac{Gm}{r^3}\,(C-A)\sin\delta\cos\delta,$$

with $C-A$ the difference between the polar and equatorial moments of inertia. The important factor is $\sin\delta\cos\delta$, which is $\tfrac12\sin 2\delta$: it vanishes when the perturbing body is over the equator and again when it is over a pole, and peaks at $45°$.

So the torque varies as the Sun and Moon move in declination. Average it over a year and the Sun’s contribution leaves a steady term — the solar part of precession. Average the Moon’s over a month and the same happens. But the Moon’s monthly range of declination is not fixed: it is the obliquity plus or minus the $5.15°$ inclination of the lunar orbit, and which of those applies depends on where the node is. Over 18.61 years the Moon’s declination range swings between $18.3°$ and $28.6°$, and the average torque swings with it.

That is nutation. It is not a separate effect with a separate cause; it is the part of the same torque that does not average to a constant, because the geometry supplying the torque is itself turning.

The Moon’s node, which is the clock #

The Moon’s orbit is inclined at $5.15°$ and its line of nodes regresses steadily — westward along the ecliptic, completing a circuit in 6,798 days. That coincidence is worth dwelling on, because it makes nutation checkable in an unexpected way. The lunar node’s period is measurable from eclipse records going back millennia, entirely independently of any observation of the Earth’s axis. The nutation series’ principal period must equal it, and does.

The circle it is drawn on #

The radius of that circle is the obliquity, and the obliquity is not a constant either: it oscillates by about a degree and a half over forty-one thousand years, so the circle the pole traces is slowly breathing while the pole runs around it.

The relative rates are the cleanest way to hold the two apart. Precession carries the pole 50.29 arcseconds along its circle per year, so in one node cycle it moves 936 arcseconds — 47 times the height of the loop drawn on it. The pole therefore never returns to a point it has visited: the loop is open, and the path is a shallow scallop rather than a series of closed curls.

The size of it, and why the ratio matters #

The two components of the principal term are

$$\Delta\psi = -17.20''\sin\Omega,\qquad \Delta\varepsilon = +9.20''\cos\Omega,$$

with $\Omega$ the longitude of the ascending node. The first is a motion along the ecliptic — a displacement in celestial longitude, which shifts the equinox — and the second is a change in the tilt itself.

On the sky, a displacement in longitude is foreshortened by $\sin\varepsilon = 0.398$. So the pole’s actual path is an ellipse with semi-axes $17.20\times0.398 = 6.84''$ and $9.20''$: the obliquity axis is the larger, by a factor of 1.35, and the loop is elongated towards and away from the ecliptic pole.

Nearly every textbook figure of nutation draws it the other way round, because $17.2$ is a bigger number than $9.2$ and the foreshortening is forgotten. The generator behind the opening figure refuses to draw it if the ratio comes out the wrong way, which is the kind of assertion this site exists to make: the figure cannot be drawn in the shape everybody remembers.

What it does to a catalogue #

Twenty arcseconds is a large angle in astrometry. Modern positions are quoted to tens of microarcseconds, so nutation is a hundred thousand times the measurement error, and no position means anything until it is stated whether the frame it is in includes nutation or not.

The convention is a division of the pole into two. The mean pole moves only with precession — the smooth part — and the true pole includes nutation. Coordinates referred to the mean equinox and equator of a date are “mean” coordinates; those referred to the instantaneous ones are “apparent”. A star catalogue quotes mean positions at a standard epoch, and an observer converts to apparent coordinates before pointing anything.

Why it took so long to find #

Bradley found nutation in 1748, twenty years after he found aberration, and he found it the same way — by observing $\gamma$ Draconis, which passes near the zenith at Greenwich and is therefore free of refraction, for long enough.

Twenty years is not incidental. It is a little over one node period, and that is the shortest run of observations from which the effect can be separated from everything else. Aberration announced itself within a year because its period is a year; nutation required Bradley to observe continuously from 1727 to 1747 and then to notice that the residuals after removing aberration and precession were themselves periodic, with a period matching the lunar node’s.

This is the same separability argument that governs any fit of several parameters to one moving point: what makes a signal recoverable is not its size but its having a period nothing else has. Nutation’s period is not shared by aberration (one year), by parallax (one year), by precession (secular), or by proper motion (secular). It was recoverable the moment somebody had 18.6 years of data.

The series, and how long it is #

The two terms above are the largest. The standard model in use since 2003 has 1,365 of them: 678 lunisolar terms indexed by combinations of five fundamental arguments, and 687 planetary terms. The largest planetary term is 0.1 milliarcseconds and the smallest kept are a tenth of a microarcsecond.

That is an unusual sort of theory, and worth naming as such. It is not a differential equation being solved; it is a Fourier series whose frequencies are known exactly from the orbits and whose amplitudes are computed from a model of the Earth’s interior — because a rigid Earth and a real one respond differently, and the difference is measurable. The amplitude of the principal term in a rigid-Earth model is about one per cent away from the observed one, and that one per cent is a measurement of the fluid core.

Nutation is therefore a geophysical instrument. The free core nutation — a resonance of the fluid outer core inside the mantle, with a period of about 430 days in the terrestrial frame — shows up as an amplification of the nutation terms whose frequencies approach it, and its period and damping are known almost entirely from that amplification. A pattern in the sky is being used to measure something 3,000 km underground. The component that carries the largest consequence for anybody who is not an astrometrist is the one that shifts celestial longitude, and it is worth following through.

The component that shifts celestial longitude — the $17.20''$ term — moves the equinox, which is the origin of right ascension and also the moment the Sun crosses the equator. So nutation displaces the equinox by up to 17 arcseconds along the ecliptic, and the Sun covers that in about seven minutes.

The equinox as an instant therefore has a nutation term in it of a few minutes, on top of the much larger effects of the tropical year not being a whole number of days. That is not a curiosity: the ecclesiastical calculation of Easter depends on a fictitious equinox fixed at 21 March precisely so that the astronomical one’s wandering does not have to be tracked, and the two can differ by more than a day.

What is actually measured #

Nothing on this page is inferred from a picture of the pole, because the pole cannot be observed. What is observed is the direction to something distant, and the pole is a parameter.

Since the 1980s that has meant very-long-baseline interferometry: the delay between the arrival of a radio wavefront at two antennas thousands of kilometres apart, measured against extragalactic sources with no measurable proper motion. The delay depends on the baseline’s orientation in the inertial frame, which depends on the Earth’s orientation, which is what is being solved for. The technique returns all five Earth-orientation parameters at once — two for polar motion, one for rotation angle, two for nutation — and it does so to about 50 microarcseconds per session.

The residuals after the standard model is removed are what keeps the subject alive. They are not zero: there is a slowly varying offset of order a milliarcsecond, attributed to the free core nutation being excited by something atmospheric, and it is monitored and published rather than modelled.

What the ladder has established #

The first rung of this anchor was the circle: a torque, a spinning body, and 25,772 years. This one says that the circle is the average of something that is not smooth, and that the departure is set by a third body’s orbital plane rather than by anything about the Earth.

The general shape of that is worth carrying. A perturbation with a period long compared with the driving motions leaves a secular term; one with a period comparable to them leaves an oscillation. Which is which depends on the ratio of periods and not on the strength of anything, and the same division separates the elements of an orbit that drift from those that oscillate. Precession and nutation are the two halves of one expansion, and the boundary between them is a choice about what counts as fast.

The wobble nothing is driving #

Every motion in this essay is forced: a torque with a period, producing a response at the same period. There is one that is not, and it was found in the residuals after all the forced terms had been removed.

A spinning body whose rotation axis is not exactly aligned with its axis of maximum moment of inertia wobbles freely, with no torque required. For a rigid Earth the period of that free wobble follows from the flattening and comes out at about 305 days.

The observed period is 433.

The discrepancy is the Earth’s failure to be rigid. An elastic Earth deforms as the axis wanders, which changes the moments of inertia and lengthens the period; the oceans, which are free to redistribute, lengthen it further. Reconciling 305 with 433 was the first quantitative measurement of the Earth’s elastic response as a whole body, and it was made in the 1890s from a hundred years of latitude observations.

The wobble’s amplitude is a few tenths of an arcsecond, which is a few metres of motion of the pole across the Earth’s surface — enough to change a station’s latitude measurably, which is how it was found.

The unsettled part is why it is still going. Internal friction should damp a free wobble in a few decades, and this one has been observed for well over a century without dying away, so something is continuously re-exciting it. The leading candidate is the atmosphere and the ocean — pressure fluctuations at the sea floor and mass redistribution in the air, acting as a broadband noise source that keeps ringing the resonance. The excitation is at about the right level and the case is not closed.

The resonance in the core #

There is a second free motion, and it is the reason the forced nutations of this essay cannot be computed for a solid Earth.

The Earth’s fluid outer core sits in an ellipsoidal cavity, and it need not rotate about exactly the same axis as the mantle around it. If it does not, the pressure of the fluid on the sloping boundary provides a restoring torque, and the result is a free mode — the free core nutation — with a period, seen from space, of about 430 days retrograde.

Nothing drives that mode directly, but its existence changes how the Earth responds to everything that does. A forced nutation whose period lies near the free one is amplified, in the same way any driven oscillator is amplified near resonance — and one of the forced terms, the retrograde annual nutation, lies close enough that its observed amplitude is roughly a third larger than a solid Earth would give.

That amplification is a measurement of the core. The free mode’s period depends on the flattening of the core–mantle boundary, so fitting the observed nutation amplitudes returns that flattening — and it comes out about five per cent larger than hydrostatic equilibrium predicts, implying that the boundary has topography held up by mantle convection.

A discrepancy of a few milliarcseconds in the wobble of the Earth’s axis is therefore a statement about the shape of a surface three thousand kilometres down, and it is one of very few ways that surface can be reached at all.

Both of the free motions in this section share a property worth naming: neither is a response to anything in the sky, so neither can be predicted from an ephemeris, and both have to be measured continuously and published as observed quantities. That is why the service that maintains the Earth’s orientation parameters issues them weekly rather than tabulating them in advance, and why a position computed from a formula alone is wrong at the level of tens of milliarcseconds.

The forced terms in this essay, by contrast, are predictable centuries ahead, because their causes are two orbits whose geometry is known — which is the cleanest possible illustration of the difference between a driven response and a free one.

It also explains the division of labour in the standards. The forced nutation is published as a model — a series with fixed coefficients, valid indefinitely — while polar motion and the rotation angle are published as measurements with a short prediction attached, and the two are combined at the point of use.

That division also decides what a user has to fetch. A prediction of where a star will be next year needs only the model; a reduction of an observation made last Tuesday needs the measured parameters for that day, which is why every serious astrometric pipeline has a network dependency and a stale-data failure mode.

The forced series is also the part that can be improved by theory alone. Every refinement of the Earth’s internal structure changes the predicted coefficients, and comparing a new prediction against the same accumulated observations is a test that costs no observing time at all — which is why the nutation model has been revised several times without any new instrument being built.

It is one of the few places in observational astronomy where a century-old data set is not merely still useful but is the primary constraint, because what is being tested is a prediction about a slow motion and the length of the record is the whole of the sensitivity.

Where the ladder goes next #

The rung after this is polar motion — which sounds like the same subject and is not. Precession and nutation move the axis in space while the Earth stays put on it; polar motion moves the axis within the body, so the geographic pole wanders by about fifteen metres in a fourteen-month Chandler wobble plus an annual term. One is astronomy and the other is geophysics, and telling them apart requires exactly the frame distinction this essay has been about.