The part of a shape the spin cannot explain

Spin a fluid and it flattens. How much it flattens depends on how fast it spins and on how its density is arranged, and both of those are things about the body rather than about its history. Given a rotation rate and an internal density profile, the equilibrium shape is not a matter of opinion; it is computed, and it has one answer.

Real planets do not have that shape. They have that shape plus something, and the something is the subject of this essay. For the Earth the excess is a few parts in a hundred thousand of the total bulge and is nevertheless the largest signal in the field after the bulge itself. For the Moon the excess is the bulge: essentially all of it.

The distinction matters because the two parts are read for completely different things. The equilibrium bulge is a statement about the whole interior at once — one integral of the density, and the reason a moment of inertia can be inferred from an outline at all. The residue is a statement about a place: a load, a frozen epoch, a region of mantle that will not flow. One is an average and the other is a feature, and separating them is the price of using either.

Subtracting the shape the spin is entitled to #

The equilibrium figure of a rotating body is fixed to first order by two numbers: the rotation parameter $q = \omega^2R^3/GM$, which says how hard the spin is pulling material outward relative to gravity, and the internal density distribution, which decides how much of that pull the body concedes.

Given a moment of inertia, the flattening follows. Given a flattening, the moment of inertia follows — that is the Darwin–Radau relation, and it is how almost every quoted interior in the solar system was obtained. Both directions assume the body is a fluid that has had time to relax.

The non-hydrostatic figure is what is left when the equilibrium part is taken away:

$$J_2^{\rm nh} = J_2^{\rm obs} - J_2^{\rm eq}(q, C/MR^2).$$

There is a circularity in that line and it has to be confronted rather than hidden. The equilibrium term needs the moment of inertia, and the usual route to the moment of inertia is the observed $J_2$ under the assumption that the whole of it is equilibrium. Subtracting requires already knowing the answer.

That is why this essay is a rung above the last one rather than a footnote to it. Breaking the circle needs a second measurement, and the second measurement is a body’s own rotation rather than its satellite’s.

The Moon, where the excess is the whole of it #

The Moon rotates once every 27.3 days. Its rotation parameter is $7.6\times10^{-6}$, so the flattening its present spin can support is about four parts in a million. The observed flattening is three parts in ten thousand — seventy-five times larger.

The standard explanation is a fossil bulge. The Moon was once much closer to the Earth, spinning much faster in synchronous lock, and warm enough to relax to the equilibrium shape appropriate to that era. Then it cooled, its outer layers became rigid, and it has been carrying an obsolete shape ever since.

The size of the excess can be put in more physical terms. The Moon’s polar and equatorial radii differ by about two kilometres, and the amount of that difference the present rotation can hold up is under thirty metres. Nineteen hundred and seventy metres of lunar shape is unsupported by anything happening now, which is the largest anomaly of its kind anywhere in the solar system and is carried by a body small enough that its lithosphere is a substantial fraction of its radius.

That turns the residue into a clock. The equilibrium figure of a synchronously rotating satellite has a known dependence on distance, so the observed excess corresponds to a particular Earth–Moon separation at the moment of freezing — around twenty-four Earth radii, against the sixty of today. Combined with the rate at which the Moon is receding, that is a date.

The date does not quite work, and the disagreement is instructive. The bulge implies freezing at a distance the Moon reached early — within the first few hundred million years — while the lunar crust’s ages and the thermal models both suggest the outer layers stayed warm considerably longer. The leading resolution is that the frozen shape is not the equilibrium figure of a circular synchronous orbit but of an eccentric one, which raises a larger bulge for the same distance, and that the Moon’s orbit was substantially eccentric at the time. Whether that is right is open. What is not open is that the shape is a record of something, because it cannot be a record of the present.

There is a second, quieter consequence of the Moon’s frozen shape, and it is the reason the fossil is not merely a curiosity. A synchronously locked satellite with a permanent equatorial bulge pointing at its primary is locked far more firmly than one whose bulge is raised freshly each orbit, because the restoring torque is proportional to the permanent asymmetry. The Moon’s fossil is what makes its libration small and its lock stable, and it is therefore part of the explanation of why the same face has been presented for four billion years. A record of the past is also a constraint on the present.

Venus, where there is nothing to subtract #

Venus makes the argument in its cleanest form. Its rotation is so slow that the equilibrium bulge is unmeasurably small, so the observed quadrupole is entirely non-hydrostatic, and no subtraction is needed at all.

What is left is a gravity field that correlates strongly with the topography — much more strongly than the Earth’s does. That correlation is measured as an admittance, the ratio of gravity to topography as a function of wavelength, and its size says how the topography is held up. A mountain floating on a weak layer is compensated at shallow depth and produces little gravity; a mountain held up by something deep produces a lot.

It is worth noticing what makes this measurement possible at all, because it is a piece of luck rather than a piece of design. On the Earth the equilibrium bulge is a hundred times the largest non-hydrostatic term, so isolating the interesting part means subtracting two large numbers and keeping their difference — an operation that propagates every error in the larger one. On Venus there is no larger one. The slow rotation that makes Venus strange in every other respect is what makes its interior easiest to read, and the whole of its degree-two field is signal.

Venus’s admittance is high at long wavelengths, which says its topography is supported deep — hundreds of kilometres — rather than by a shallow crustal root. The standard reading is that Venus has no asthenosphere: no weak, low-viscosity layer beneath the lithosphere for the crust to float on, which is also the reason it has no plate tectonics. That conclusion comes entirely from a quantity that on any other planet would be a small correction to be subtracted before the interesting work began.

Mars, where the excess is a mountain #

Mars sits in between and is the cautionary case, because its non-hydrostatic figure is large enough to matter and small enough to be missed.

The size of the excess is a mass. Converting the non-hydrostatic $J_2$ into the load that produces it gives something around $10^{21}$ kilograms for Tharsis — roughly a three-thousandth of the planet, or a layer three kilometres thick spread over a quarter of its surface. That is a measurement of a geological province made from orbit, without a single image.

It also has a dynamical consequence that is easy to state and startling to notice. A body’s rotation axis migrates to align with its largest moment of inertia, so a load of that size, emplaced away from the equator, would have reoriented the entire planet beneath it. The observed position of Tharsis, almost exactly on the equator, is what that reorientation predicts, and the fossil shorelines and valley networks around it are cited as a record of the pole having moved. True polar wander is a consequence of a non-hydrostatic figure and nothing else, and it is a case where the residue does not merely record history — it makes it.

The Earth’s own residue is worth a sentence for contrast, because it is the case where the subtraction is hardest and the answer is best known. After removing the equilibrium figure the largest remaining term is a degree-two anomaly of about $10^{-5}$ of $J_2$, and it is not static: post-glacial rebound is still returning mass towards the poles from the last ice age, so the Earth’s oblateness has been measured to be decreasing by satellite laser ranging since the 1970s. A shape that is relaxing on a measurable timescale is the direct observation of the process every other body in this essay is assumed to have finished.

What was actually measured #

Three instruments, three bodies, three completely different techniques, and all three measure the same kind of quantity.

Lunar laser ranging. Corner cubes left by Apollo and Lunokhod return laser pulses fired from Earth, timed to a few picoseconds, giving the Earth–Moon distance to millimetres. The Moon’s physical libration — its rocking about the mean synchronous rotation — is read directly out of that time series, and it depends on the ratios of the three principal moments. Combined with the gravity field from orbiters, it gives $C/MR^2 = 0.3931$ with no hydrostatic assumption in it, which is what allows the fossil bulge to be quantified at all.

Radar on Mercury. Two radio telescopes observing the same radar echo can measure the instantaneous spin rate to a part in $10^5$. The campaign that did so found Mercury’s forced libration to be more than twice the amplitude a solid planet would show.

Landers on Mars. Tracking radio transponders on the surface across twenty years — Viking, then Pathfinder, then the rovers — gives the precession rate and hence the dynamical ellipticity. That is what produced the 0.3644 that disagrees with the hydrostatic value, and the disagreement is the Tharsis measurement.

What the residue can and cannot say #

A non-hydrostatic figure is a single number per harmonic degree, and a mass distribution is a function. The residue bounds a family of loads rather than picking one out.

What breaks the ambiguity is the same thing that broke it before: more coefficients. A load at one depth and a load at another produce the same $J_2$ and different $J_4$, so a gravity field measured to high degree distinguishes them, and the admittance spectrum is exactly that comparison carried out wavelength by wavelength. The residue at degree two is where the argument starts rather than where it ends.

Where the picture stops #

There are three, and the first is the awkward one.

The subtraction needs the answer. Computing the equilibrium figure requires a moment of inertia, and outside the four bodies with a measured precession there is no model-independent one. For the Galilean satellites, for Titan, for every trans-Neptunian body, the quoted moment of inertia assumes the figure is hydrostatic, and the assumption cannot be checked with the data in hand. Those numbers are not wrong so much as conditional, and the condition is rarely printed.

A fossil is not dated by its size alone. Reading a frozen bulge as a distance assumes the body froze in one event, at one shape, and has not relaxed since. Partial relaxation over billions of years would leave a shape intermediate between the frozen one and the current equilibrium, and would be indistinguishable from a fossil of a smaller original bulge.

And a slowly rotating body has no defined hydrostatic reference at all. For Venus the equilibrium term is smaller than the measurement error on anything, so “non-hydrostatic figure” and “figure” are the same quantity. That is convenient here and is a warning elsewhere: the decomposition is only meaningful when the two parts are comparable, and for most of the small bodies in the solar system they are not.

There is also a fourth limit that belongs with those three and is about the word rather than the measurement. “Non-hydrostatic” describes what the residue is not, and the three cases above show it can be at least three different things: an elastic memory of a former equilibrium, a load supported by lithospheric strength, and a dynamic anomaly maintained by ongoing convection. The three have different timescales, different depths and different implications, and a single degree-two coefficient does not distinguish them. Naming the residue by its negation is a way of admitting that.

Why a residue is worth more than the quantity it was subtracted from #

The bulge itself is nearly uninformative. It is a smooth, predictable response to rotation that any body of roughly the right density would show, and its size says one thing — how centrally condensed the body is — which is an average over the whole interior.

The residue is the opposite. It is localised, it is not predicted by anything, and it records a specific event or a specific load: a mountain, a frozen epoch, a mantle that will not flow. Every one of the three cases above turns a small leftover into a statement no bulk measurement could make.

That is a pattern this collection meets in every field. The forty-three arcseconds Mercury has left over after every known planet’s pull is subtracted; the comet that arrives a day early after gravity is accounted for; the residual acceleration of a spacecraft after the model has done its work. In each the subtracted part is enormous, well understood, and of no further interest, and the remainder is where the physics is.

The difference in this case is that the subtraction is the difficult step rather than the routine one. Nobody has to wonder what the Sun’s gravity does to Mercury. Working out what shape a planet would have if it were a fluid — and knowing whether it is — is the whole of the problem.

One last point about where this fits. The residue is measured from orbit, and measuring it well means measuring the gravity field well, which is the same node regression and the same Doppler tracking that produced the bulge in the first place. There is no separate instrument for the interesting half. The signal and the thing it has to be separated from arrive in the same data, differing only in how they depend on the rotation rate — which is why a body with an unusual spin is so much more informative than a body with an ordinary one, and why Mercury’s spin state had to be measured before its interior could be.

End on the sign convention, because it is the thing most often got wrong when comparing bodies. The observed flattening is compared against the hydrostatic value computed for the body’s present spin, so a residual is positive when the body is more flattened than it should be and negative when it is less. A positive residue means an excess of mass near the equator or a shape frozen from a faster past rotation; a negative one means the interior is denser towards the centre than the hydrostatic calculation assumed. The two cases point at completely different interiors, and quoting only the magnitude of the residue — which happens — discards precisely the half of the measurement that says which.

Where the ladder goes next #

The obvious next rung is the admittance itself: the ratio of gravity to topography as a function of wavelength, and how its shape distinguishes a mountain floating on a weak layer from one held up by a strong one. The rung after that is the tidal response, which asks the body how much it gives rather than what shape it holds, and which separates a liquid interior from a solid one carrying the same residue.