The wobble that is slow because the Earth gives

The axis a leak of energy chooses added one ingredient to a rigid body — a way of losing energy — and found that it decides where the body ends up spinning, because at fixed angular momentum the energy is least about the axis of greatest inertia. That essay ended by pointing to the next ingredient, a body that is not merely leaky but deformable, whose shape and spin must be solved together. The Earth is the case that matters, and its free wobble is where the deformability shows up most cleanly: as a number of days that should have been three hundred and five and is four hundred and thirty-three.

The story is also a lesson in how a measurement can be hidden by assuming the answer. For more than a century astronomers looked for the Earth’s free wobble at the period a rigid Earth would have and found nothing, because it was not there. When it was found, at a different period, the difference turned out to be a measurement of the Earth’s interior made with a telescope pointed at the stars.

Euler’s ten months #

A body spinning about an axis that is not quite its axis of symmetry does not keep spinning about that axis. A spinning top nods because gravity torques it; a free body with no torque at all still wobbles, the spin axis circling the figure axis at a rate set by the body’s shape. For a flattened body spinning at rate Ω, with moments of inertia $C$ about its axis of symmetry and $A$ about an equatorial axis, the wobble completes one circuit, seen from the body, in

$$T_{\rm Euler} = \frac{A}{C - A}\cdot\frac{2\pi}{\Omega}.$$

Leonhard Euler derived this in 1765, and for the Earth, whose moments differ by about one part in three hundred, it comes to about three hundred and four days. The pole — the point where the rotation axis meets the surface — should trace a small circle round the figure axis every ten months, and the latitude of every observatory should rise and fall with that period.

The physical reason is the equatorial bulge. The Earth is fatter at the equator than through the poles by forty-three kilometres, a bulge raised by its own rotation. If the spin axis is displaced from the figure axis, the bulge is no longer symmetric about the spin, and in the rotating frame the centrifugal effect on the tilted bulge exerts a torque that swings the spin axis round the figure axis. The size of the bulge, measured by $C - A$, is how hard the swing is driven, and the period is inversely proportional to it.

The Earth spins about the axis of its greatest moment of inertia, the one a spinning body can hold stably — which is why a displaced axis wobbles round the figure rather than tumbling away from it. And the bulge that drives the wobble is itself a product of the spin: a fluid planet spinning at the Earth’s rate would settle into a flattened shape for the same reason the surface of a spinning bucket of water is curved, and the solid Earth, which behaves as a fluid over millions of years, has very nearly that shape. The wobble’s period is therefore a ratio of two consequences of one rotation: how fast it turns, and how much that turning has flattened the planet.

Fourteen months instead #

Through the nineteenth century astronomers compared latitudes measured over years and looked for a ten-month variation. They found variations of a few tenths of an arcsecond, which was at the edge of what could be measured, and nothing at ten months. In 1891 Seth Chandler, an American amateur who had built his own instruments and was working through the accumulated latitude observations of many observatories, found instead a variation with a period of about fourteen months — around four hundred and thirty days — plus a separate variation with a period of one year.

Simon Newcomb, the leading authority on the Earth’s motion, first doubted it and then, within months, supplied the explanation. Euler’s calculation assumes the Earth is rigid. If the Earth yields a little to the changing centrifugal forces as the axis wobbles, the period must lengthen, and a lengthening of forty per cent says how much.

A softer body wobbles more slowly #

The mechanism is not that a soft body swings more loosely. It is that a soft body’s bulge partly follows the spin axis. When the axis is displaced, the centrifugal potential that raised the bulge is now centred on a slightly different axis, and a body that yields deforms to fit it: part of the bulge re-forms round the new spin axis. That part is symmetric about the spin and exerts no torque. Only the part that stays with the body’s original figure drives the wobble, so the effective driving is weaker and the wobble slower.

How much of the bulge follows is measured by a Love number, $k$, the ratio of the deformation the body actually undergoes to a reference, and its value for a body that yields completely, like a fluid, is fixed by the body’s own shape: $k_s = 3G(C - A)/\Omega^2a^5$, about 0.94 for the Earth. The wobble period becomes

$$T = \frac{T_{\rm Euler}}{1 - k/k_s}.$$

The drawing plots it. A rigid body, $k = 0$, has Euler’s period. A fluid body, $k = k_s$, has an infinite period: its bulge follows any new spin axis entirely, nothing is left to push the axis back, and a displacement simply stays. The Earth’s 433 days needs $k/k_s$ close to 0.30 — the Earth yields to its own wobble about thirty per cent as much as a fluid would. That single number is a statement about the elasticity of the whole planet, and it was the first measurement of the Earth’s rigidity, made in the 1890s from observatory latitudes. It showed the Earth to be, on average, about as stiff as steel.

The pole’s path #

On the ground the wobble is small: the pole moves in a loop a few metres across, about the size of a room. Seen from above the North Pole it goes round anticlockwise, the same sense as the Earth’s rotation, and in a thousand days a rigid Earth’s pole would go round three and a third times and the real one goes round not quite two and a third. The size of the circle is set by whatever displaced the axis in the first place and says nothing about the Earth; the pace is the measurement.

The motion is measured now to a fraction of a millimetre. Networks of radio telescopes observing distant quasars, satellites carrying laser reflectors and the global positioning satellites together fix the pole’s position every day, and its track over a century and more is one of the best-known geophysical records there is. What the track shows, beneath the circles, is also a slow drift of the mean pole by about ten centimetres a year, as ice melted since the last glaciation lets the crust rebound and the mass distribution shifts — the body changing its figure, as a spinning body that changes its shape must, while its angular momentum stays fixed in space.

Two wobbles out of step #

Chandler’s discovery was hard for a reason the drawing shows. The pole does not only move at its free period. The seasons move air from continent to ocean and back, pile snow on the northern land masses in winter and melt it in summer, and shift water between hemispheres, and that redistribution of mass forces a wobble with a period of exactly one year. The two periods, fourteen months and twelve, are close enough that the combined motion beats: the two wobbles fall into step and add, then fall out of step and nearly cancel, every six and a half years.

An observer measuring latitude for two or three years, looking for a ten-month period, sees a motion whose amplitude is changing and whose period seems to wander, and can easily conclude there is nothing regular in it. It took a record long enough to span the beat before the two components could be separated. Chandler had the advantage of working with decades of other people’s data rather than a few years of his own.

A service built to watch one point #

The discovery produced an institution. From 1899 an International Latitude Service ran a chain of observatories on a single parallel, 39°08′ north — at Mizusawa in Japan, Carloforte in Sardinia, Gaithersburg and Ukiah in the United States, and others — each measuring its latitude against the same pairs of stars every clear night. Sitting on one parallel meant that the same stars could be used everywhere and errors in their catalogue positions cancelled; spreading the stations round the globe meant that the pole’s two coordinates could be separated from each station’s local errors. The service ran for most of the twentieth century, and its records are the backbone of the long series the drawings are modelled on.

It also produced a lesson in the difference between a signal and an artefact. Within a few years Hisashi Kimura at Mizusawa found an annual term that appeared equally at every station and so could not be a motion of the pole — the “z-term”, which turned out to come from errors in the star positions and from the observatories’ own seasonal effects on refraction. Separating what the Earth did from what the instruments did took decades, and it is why the modern record from quasars and satellites, which does not depend on catalogue stars, was such an improvement.

An annual wobble made larger by a nearby resonance #

The forced annual wobble has a feature worth noticing, because it is resonance at work. The seasonal forcing arrives at a period of twelve months, close to the free period of fourteen, and a forced oscillator driven near its natural frequency responds more strongly than the same push would move it far from resonance — the frequency that gets an answer, applied to a planet. The pole’s motion answers only the part of the seasonal forcing that turns the same way as the wobble, and for that part the response is one over one minus the ratio of the two periods: the forcing is faster than the free wobble, so the response is about five times what the same push would give far from resonance, and it lags the push by half a cycle. A rigid Earth, whose free period of ten months lies on the other side of twelve, would be amplified by a similar factor but would respond in step with the push. The timing of the annual wobble against the seasons is therefore also a check on which side of a year the free period lies, independent of Chandler’s own measurement.

Why it has not stopped #

A yielding Earth is also a lossy one. Each wobble flexes the mantle, which is not perfectly elastic, and sloshes the oceans, which have friction, and so the free wobble loses a little energy every cycle. The width that is a lifetime says how to read that loss: a resonance with quality factor $Q$ rings down in about $Q$ cycles divided by π. Estimates of the Chandler wobble’s $Q$ are around fifty to a hundred, so left alone it would fade to a third of its size in roughly thirty years and to almost nothing in a century.

It has not faded. It has been present, with an amplitude between about a tenth and a quarter of an arcsecond, for the whole of the record since the 1890s. Something is exciting it, and the drawing shows what that looks like: the same resonance, with the same damping, kicked continually by random forcing, holds an amplitude that wanders up and down over decades instead of decaying. For most of the twentieth century the source was a puzzle — earthquakes were proposed and are far too weak. Calculations and measurements since 2000 have found that the fluctuating pressure of the oceans on the sea floor, together with changes in atmospheric pressure over the continents, supplies enough excitation, with the oceans doing about two thirds of the work. The wobble is a resonance of the solid Earth, kept ringing by its fluid envelope.

The budget of the four months #

The single number $k/k_s = 0.30$ averages several things that can be separated with care. The elastic mantle accounts for most of the lengthening. The oceans add to it, because the ocean’s surface is an equipotential that tilts with the wobbling axis — the pole tide, a centimetre-high tide at the wobble period — and that water adds to the part of the bulge that follows the spin. The liquid outer core works the other way: it is not carried along by the mantle’s wobble, so the part of the planet that wobbles has a smaller moment of inertia than the whole Earth, which on its own would shorten the period. And the mantle’s slight anelasticity, its failure to be perfectly elastic at a period of fourteen months, lengthens it by a few more days. Put together from seismic models of the Earth’s interior, the budget comes out within a day or two of the observed period, which is a test of those models at a period far longer than any seismic wave.

The same softness shows up elsewhere as a separate measurement. The solid Earth rises and falls by tens of centimetres twice a day under the Moon’s tidal pull, and the size of that bodily tide is governed by the same Love number, measured at a period of hours rather than months. That the two agree, after the core and oceans are accounted for, is one of the checks that the Earth’s elasticity is understood. The same numbers, pushed to a moon squeezed hard and continuously by an eccentric orbit, are what heat Io from within, a moon kept molten by its neighbours.

Another planet with the same signature #

Mars has a free wobble too, and it was found in 2020 by tracking the orbits of spacecraft around it with enough precision to detect the planet’s rotation axis moving by about ten centimetres. Its period, about 207 days, is longer than the 191 days a rigid Mars of its shape would have, by about eight per cent. Mars yields less than the Earth does, relative to what a fluid body of its shape would do: it has no oceans to add a pole tide, and its cold, thick lithosphere is stiff. The lengthening is small enough that its interpretation depends on the size and state of Mars’s core, which the InSight lander’s seismometer has since measured independently, and the two approaches to the Martian interior can now be checked against each other.

Where the simple formula stops #

The formula $T_{\rm Euler}/(1 - k/k_s)$ treats the Earth as a single body with one Love number. The liquid core, which does not follow the mantle, needs its own equation, and a fluid core in a flattened mantle has a free mode of its own — a nearly diurnal free wobble of the core relative to the mantle, with a period of about 430 days as seen from space, discovered in the 1980s and 1990s in very-long-baseline measurements of the Earth’s nutation. Oceans that respond dynamically rather than as an equilibrium surface change the pole tide slightly. Anelasticity makes the Love number depend on period, so the $k$ that governs the fourteen-month wobble is not quite the $k$ of the twice-daily tide. Each correction is small; together they are what the budget above adds up.

What the pole’s circles do not show #

The drawings show the wobble as a clean circle with a single period. The real record is noisier and stranger. Around 2005 the wobble’s amplitude fell sharply, and around 2015 to 2020 it nearly vanished, to a few hundredths of an arcsecond, before recovering; there are also indications that its phase jumped around 1925. Such changes are consistent with a randomly excited, damped resonance, which can wander to small amplitudes by chance, but they are also what a change in the excitation — in how the oceans or atmosphere push — would produce, and the record is not long enough to tell the two apart with confidence.

Still open: what keeps the wobble going, in detail #

The oceanic and atmospheric excitation explains the wobble’s overall size, and modern models of ocean-floor pressure reproduce much of its variation over the last few decades. What they do not yet do reliably is predict when its amplitude will fall or recover, or explain the large excursions of the historical record, partly because the ocean-pressure models before the satellite era are poorly constrained. How much of the variability is chance, how much is a change in the oceans, and whether a changing climate will change the excitation are open, and the answers bear on how well the Earth’s orientation — which navigation satellites need to know to centimetres — can be predicted months ahead.

The habit worth carrying away is to ask what a rigid-body calculation assumes about the body. A free motion driven by a body’s shape is slowed if the shape can follow the motion, and the slowing measures how much of the shape follows — so a planet’s wobble, timed with a telescope, weighs its stiffness, and a fluid body with the same bulge would not wobble at all.