Gravitation

Whether the heavy material sank

A moment of inertia is 0.4 of MR² for a uniform sphere and less for everything that has differentiated, and it is measured by watching a body wobble. Mercury's 0.346 says most of the planet is iron core; the Moon's 0.393 says almost none of it is.

Assumes Oblateness and Precession.

16 min read 8 figures Measuring the unreachable Scales that defeat intuition

Nothing has ever been retrieved from the inside of a planet. The deepest hole ever drilled reached twelve kilometres, which on the Earth is a fifth of the way through the crust and a five-hundredth of the way to the centre. Every statement about what a planetary interior is made of rests on inference from outside, and the two quantities that carry most of the weight are a mean density and a moment of inertia.

A mean density is easy and weak. It is a mass over a volume, and a mass comes from anything that orbits while a volume comes from a photograph. It says how much heavy material there is and nothing about where it is: a well-mixed body and a strongly layered one of the same composition have the same mean density.

The moment of inertia is the second number, and it is the one that says where. Because it weights each parcel of mass by the square of its distance from the axis, it is not a volume average but a lever-arm average, and moving mass inward lowers it. That single sentence is the whole method.

Fig. 1 Eleven bodies, arranged by the one interior quantity measured for all of them. C/MR² is exactly 2/5 for a uniform sphere and falls as mass is concentrated inward, and the values here run from the Sun’s 0.070 to the Moon’s 0.3931. Every one of them was obtained by watching the body turn — a precession rate, a libration amplitude, a gravity field sampled on a flyby, or in the Sun’s case the frequencies of its own oscillations.

Two-fifths, and why both ends return to it

For a uniform sphere the integral is elementary and gives $C = \tfrac{2}{5}MR^{2}$ . What makes the quantity useful is what happens when the sphere is not uniform, and the two-layer case can be done exactly:

\frac{C}{MR^{2}} = \frac{2}{5}\cdot\frac{\rho_m + (\rho_c-\rho_m)x^{5}}{\rho_m + (\rho_c-\rho_m)x^{3}},

with $x$ the core’s fractional radius. The fifth power upstairs and the third power downstairs are the moment and the mass, and their difference is the whole effect.

The moment-of-inertia factor against core size, for five density contrasts. What a moment of inertia can say. The vertical axis is C/MR², the polar moment divided by what a hoop of the same mass and radius would have, and for a uniform sphere it is exactly 2/5 — the value both ends of every curve return to, because a body with no core and a body that is entirely core are both uniform. In between, the ratio dips: a moment weights mass by the square of its distance from the axis, so moving density inward lowers it, and the deeper the dip the more differentiated the body. The five curves are five core-to-mantle density ratios, and the minimum moves down and inward as that ratio grows — a denser core reaches its greatest effect at a smaller radius, because beyond that the core is so much of the body that the whole thing looks uniform again — 1.5 gives 0.370 at 75 per cent of the radius, 2 gives 0.348 at 72 per cent of the radius, 3 gives 0.317 at 69 per cent of the radius, 5 gives 0.279 at 65 per cent of the radius, 10 gives 0.231 at 59 per cent of the radius. Two things the figure makes visible are worth more than the numbers. The relation is not invertible: one measured factor is met by two core sizes on each curve and by a whole family of curves, so a moment of inertia alone never gives a core radius — it gives a constraint that a second measurement has to be combined with. And the whole diagram lives between 0.4 and about 0.15, which is a narrow range for so much physics; distinguishing a large core from a small one means measuring C/MR² to a per cent or two, and every technique for doing so is a way of watching the body turn. — Fig. 2 The relation for five core-to-mantle density ratios. Both ends of every curve return to 2/5, because a body with no core and a body that is entirely core are both uniform. In between there is a minimum, and it moves down and inward as the contrast grows — a denser core reaches its greatest effect at a smaller radius, because beyond that the core is so much of the body that the whole thing looks uniform again. The intuition that a denser core needs to be bigger to concentrate the mass is the natural one and it is wrong.

The figure also shows the method’s central limitation, which is that it is not invertible. One measured factor is met by two core radii on each curve and by a whole family of curves, so a moment of inertia never gives a core radius. It gives one constraint, which some second measurement has to be combined with — a mean density, a Love number, a seismic travel time, or a composition assumed from meteorites.

And the whole diagram lives between 0.4 and about 0.15. That narrowness is the practical difficulty: telling a large core from a small one means measuring the factor to a per cent or two, and every technique for doing so is a way of watching a body turn very carefully indeed.

There is one further reason the quantity is worth as much as it is, and it is not about planets at all. $C/MR^{2}$ is dimensionless. It carries no mass, no length and no time, so it can be compared across objects that have nothing else in common — a moon two hundred kilometres across and a star seven hundred thousand — and the comparison means something. Very few quantities in this collection have that property. A density does not: comparing the Earth’s five and a half grams per cubic centimetre with the Sun’s one and a half says something about composition and almost nothing about structure.

Four ways of watching a body turn

Precession of the spin axis. A body with an equatorial bulge, tilted to its orbit, is torqued by whatever it orbits, and it precesses. The rate depends on the ratio $(C-A)/C$ : the bulge supplies the torque and the polar moment supplies the resistance. Since $(C-A)/MR^{2}$ is $J_2$ , which is measured from the external gravity field, dividing one by the other gives $C/MR^{2}$ . This is how the Earth’s 0.3307 was known long before anything was in orbit — the precession of the equinoxes has been measured since Hipparchus, and $J_2$ came from the shape. Forced libration. A body on an eccentric orbit is torqued unevenly through its orbit and rocks about its mean rotation. The amplitude is inversely proportional to the moment that has to be rocked, which makes it a direct measurement — and, crucially, a measurement of the moment of only the part that is rigidly participating. Mercury’s case is the worked example, and the discrepancy between the whole body’s moment and the rocking part’s is how its core was shown to be liquid.

Flyby gravity. A spacecraft passing a moon is deflected by its quadrupole as well as by its mass, and $J_2$ and $C_{22}$ come out of the Doppler residual. That gives the bulge but not the moment, so the conversion needs a further assumption — normally hydrostatic equilibrium, through the relation below. Every moment of inertia quoted for a Galilean satellite is a hydrostatic one, and the assumption is load-bearing.

Seismology and oscillation. The Earth’s is now known to six figures from free oscillations after great earthquakes; the Sun’s 0.070 comes from inverting its acoustic spectrum. These are the only two bodies where the interior is measured rather than constrained, and both required an instrument on or in the object — a seismometer network in the first case, and in the second a spectrum resolved to parts in a million. The four are not equally trustworthy, and the ordering is not the one their precisions suggest. Flyby gravity gives the smallest formal errors and rests on the largest assumption. Precession gives a number with no interior model in it at all but needs a body whose spin axis has been watched for a long time. Libration is the only method that can distinguish a part of a body from the whole of it. And seismology is the only one that does not have to assume the body is in equilibrium.

A moment of inertia from a photograph

For a body that is a fluid in hydrostatic equilibrium, its shape is a response to its own rotation, and the response contains the same information about the interior. The Radau–Darwin relation makes that precise:

\frac{C}{MR^{2}} = \frac{2}{3}\left[1 - \frac{2}{5}\sqrt{\frac{5q}{2f}-1}\right],

with $f$ the flattening and $q = \omega^{2}R^{3}/GM$ . Nothing about the interior appears in it. A photograph gives $f$ , a light curve or a radio rotation gives $\omega$ , and an orbiting moon gives $GM$ — so the interior of a body nobody has visited follows from three things any telescope can supply.

Radau–Darwin against the measured factor, and the 12 per cent Tharsis costs. A moment of inertia from a shape. The Radau–Darwin relation, C/MR² = (2/3)[1 − (2/5)√(5q/2f − 1)], connects the polar moment to two quantities measurable from a great distance: the flattening f, which is a photograph, and q = ω²R³/GM, which is a rotation period and a mass. Nothing about the interior appears in it. The diagonal is where prediction equals measurement, and the Earth sits on it to 0.2 per cent — a moment of inertia obtained from the planet's outline and its day. Jupiter and Saturn sit a few per cent below the line, and the sign is not an accident: the relation is first order in the flattening, and a body flattened by a tenth has second-order terms of exactly that size. Mars is drawn twice, and the pair is the point of the figure. Fed the flattening Mars actually has, the relation returns 0.408 against a measured 0.364 — 12 per cent high, worse than anything else here. Fed the flattening a fluid Mars of the same spin would have, it returns 0.369 and is back within 1.3 per cent. The difference between those two shapes is Tharsis: a volcanic province a quarter of the planet across, held up by a lithosphere strong enough not to relax, and therefore invisible to a relation that assumes the body is a fluid. What the figure cannot supply is the judgement it depends on. Nothing in the method says which bodies are hydrostatic, and where that judgement has been made wrongly the published interior was wrong with it. — Fig. 3 How well it works, and where it fails. The Earth is on the diagonal to two parts in a thousand — a moment of inertia obtained from an outline and a day. Jupiter and Saturn sit a few per cent low, and the sign is not accidental: the relation is first order in the flattening, and a body flattened by a tenth has second-order terms of exactly that size. Mars is drawn twice, and the pair is the point. Fed the shape Mars actually has, the relation is twelve per cent high; fed the shape a fluid Mars of the same spin would have, it is within one per cent.

The twelve per cent is Tharsis: a volcanic province a quarter of the planet across, standing several kilometres above the datum and held up by a lithosphere strong enough not to relax. It contributes to the observed flattening and has nothing to do with the density distribution the relation is trying to measure. The method cannot tell a hydrostatic body from a non-hydrostatic one — that judgement has to be imported — and where the judgement has been made wrongly, the published interior was wrong with it.

Vesta is the case where this went furthest. It is large enough to have differentiated and small enough to retain a substantial non-hydrostatic figure, so a shape-derived moment of inertia for it is meaningless; the actual value came from a spacecraft in orbit measuring the gravity field directly, which is the only method that does not have to assume anything about the shape.

What the numbers say

The Moon, 0.3931. Very nearly uniform. Whatever iron core it has is a few per cent of its radius and under two per cent of its mass, which is a strong constraint on how it formed and the reason it has no dynamo of its own today. The number comes from laser ranging to the retroreflector arrays, which measures the physical libration to milliarcseconds.

Mercury, 0.346. Almost as low as the Earth’s, for a planet an eighteenth of the Earth’s mass. There is only one way to get there at that size: an iron core filling about four-fifths of the radius and three-fifths of the mass. Nothing else in the solar system is built like that, and every proposed explanation involves losing a mantle.

Mars, 0.3644. Between the two, which is what a body with a core of about half its radius gives. It was inferred from precession for decades and confirmed by tracking a lander, and later by a seismometer that measured the core radius directly at 1,830 kilometres — one of the few cases where an inference from a wobble was checked against a wave.

Jupiter, 0.2756, and Saturn, 0.22. Strongly condensed, which is what a hydrogen-dominated body under its own weight has to be, and consistent with heavy-element cores of some tens of Earth masses. The values are model-dependent in a way the terrestrial ones are not: what is measured is the gravity field, and reading a core out of a series of harmonics requires an equation of state for hydrogen at fifty million atmospheres.

The Sun, 0.070. Off the scale of anything a two-layer model describes. A star is not a planet with a bigger core; its density falls by five orders of magnitude from centre to surface, and the polytropic index that fits it is around 3, where a mass becomes independent of a radius.

The one number and the several unknowns

It is worth writing out what a moment of inertia is actually being asked to do in a typical planetary paper, because the arithmetic is less impressive than the confidence with which the results are quoted.

An interior model of a terrestrial planet has, at minimum, a core radius, a core density, a mantle density, and a light-element fraction in the core that ties the last two together. That is three or four unknowns. The measurements available are a mass, a radius and a moment of inertia: three numbers, one of which — the mass — is nearly used up fixing the overall scale.

So the system is underdetermined, and it is closed by importing chemistry: the mantle is assumed to be a silicate assemblage with a composition taken from meteorites, and the core an iron alloy whose equation of state is measured in a diamond anvil cell. The published core radius is then a number that depends on a laboratory experiment on a milligram of iron at a pressure held for a microsecond. This is not a criticism — there is no other way to do it — but it does mean that the error bar quoted on a core radius is a propagation of the moment of inertia’s error and not of the assumption’s.

Why a low number is a history

Differentiation is not a state a body is born in. A body assembled from planetesimals starts mixed, and the heavy material sinks only if the interior is hot enough to melt, which requires either accretional heating, or short-lived radioactivity, or both. So a moment-of-inertia factor is a statement about the first few million years of a body’s existence, read four and a half billion years later.

That is why the Moon’s 0.3931 is interesting rather than dull. The Moon is large enough that it should have differentiated, and it did — there is a crust, and the crust is anorthositic, which requires a magma ocean — but the iron that separated amounts to almost nothing. The most economical reading is that the material the Moon was made from had already had its iron removed, which is one of the strongest constraints on the giant-impact hypothesis and is a piece of evidence obtained entirely by bouncing lasers off a mirror.

Mercury’s number is the same argument run the other way and is harder to accommodate. A body that formed where Mercury is should be perhaps a third iron by mass; Mercury is around seventy per cent. Something removed most of a mantle, and the candidates — a giant impact that stripped it, evaporation in an unusually hot inner disc, or drag stripping in a dense early nebula — make different predictions for the surface composition, which is why the flyby that measured the libration also carried a gamma-ray spectrometer. The surface turned out to be volatile-rich, which an impact or an evaporation should have removed, and the question is open. The same trade drawn over a wider range of density contrast shows how weakly the factor constrains any one of the quantities in it.

The bodies where the number is measured differently

The technique in this essay uses a spin, and there are bodies whose interiors are read from their gravity fields instead — with far more resolution and a different set of assumptions.

A rotating fluid planet is flattened, and its external gravitational field departs from that of a point mass in a way that depends on how the mass is distributed. Expanding the field in spherical harmonics gives a sequence of coefficients — $J_2$ , $J_4$ , $J_6$ and onward — of which the first is dominated by the flattening and the later ones become progressively more sensitive to the deep interior.

A spacecraft in a close polar orbit measures those coefficients by the Doppler shift of its radio link, and the recent missions to the giant planets have measured them to remarkable precision: for Jupiter, the even harmonics are known to better than a part in a million, and the odd ones — which vanish for a body symmetric about its equator and do not vanish for one with deep winds — have been measured at all for the first time.

The results have not been what was assumed. A giant planet was expected to have a compact core of heavy elements surrounded by a hydrogen envelope, with a sharp boundary between them. The measured harmonics do not fit that: they require a dilute core, in which the heavy elements are spread over a substantial fraction of the planet’s radius rather than concentrated at the centre.

That is a statement about the formation history as direct as any moment-of-inertia factor. A sharp core is what accretion of a solid body followed by gas capture would leave; a dilute one requires either that the heavy material was mixed outward afterwards, or that a giant impact disrupted the core early on.

The same question — did the heavy material sink, and how far — is therefore answered by a spin for a solid body and by a gravity field for a fluid one, and the fluid case now has by far the better data.

The comparison also shows what each method cannot reach. A gravity field is a statement about the exterior, so it constrains the density distribution only through a small number of integrals and is blind to any rearrangement that leaves those integrals unchanged; a moment-of-inertia factor is one such integral by itself. Neither method locates a boundary — both constrain the distribution’s moments, and the interpretation as a core of a particular size comes from a model that has to be assumed and then checked for consistency.

That is the standing situation for planetary interiors: a handful of integral constraints, a model with more parameters than constraints, and a conclusion that is secure about the gross arrangement and negotiable about everything below it.

It is worth saying that this is not a defect of the measurements, which are excellent, but of what an exterior observation can constrain about an interior — and no improvement in the data changes the number of integrals available.

Two more measurements of the same quantity, made in entirely different ways.

Mercury's forced libration: 38.5″ against the 16.2″ a solid body would give. The measurement that finds a liquid core from a distance. A body on an eccentric orbit does not feel a steady torque: the pull on its equatorial bulge swings back and forth through the orbit, and the body rocks about its mean rotation by a small angle. How small depends on how much moment of inertia has to be rocked, and on nothing else — every other factor in the problem belongs to the orbit or to the body's own measured gravity field. The horizontal axis is therefore the fraction of the total polar moment that participates, one if the whole body turns rigidly together, and the vertical axis is the resulting amplitude. The curve is a rectangular hyperbola, because the same torque applied to less moment produces proportionally more angle. A Mercury turning in one piece would librate by 16.2 arcseconds. Radar measurements of the actual rocking give 38.5 ± 1.6, which is 2.38 times larger and many standard deviations away, so only 42 per cent of the moment is being rocked at all. The other 58 per cent is not following the mantle on an eighty-eight-day timescale, and the only way for an interior not to follow its own mantle is for the two to be separated by a liquid. That is how a planet nobody has landed on was shown to have a molten core — by watching, from Earth, the tiny irregularity of its turning. The figure treats the librating shell as rigid, which is right for a rocky mantle and wrong for an ice shell floating on an ocean, where the shell's own elasticity enters at the same level as the effect. — Fig. 7 Mercury’s forced libration: the rocking its orbit imposes on it once per revolution, measured by radar at 38.5 arcseconds against the 16.2 a solid body would show. The factor of two is not a refinement of the moment of inertia; it says that only the outer 42 per cent of the planet is rocking, and therefore that the core is liquid.

Fig. 8 The measured factors for the five bodies whose interiors are best known, without the giant planets compressing the scale. All five sit below the uniform-sphere value of 0.4 and all five sit above 0.3, which is a narrow range for objects differing by a factor of twenty in mass.

Where the ladder goes

This anchor’s later rungs are about the ways the single number is escaped. One is to measure more moments: the whole zonal series of a gravity field is a set of differently weighted integrals over the same density, and a spacecraft in a close polar orbit can measure a dozen of them. Another is to measure the response rather than the state, which is the Love number. A third is to put an instrument on the surface and listen, which has now been done on Mars and gave a core radius directly.

And there is a thread out of planetary science entirely. The moment of inertia of a star’s convection zone against its radiative interior is what sets how much angular momentum has to be moved to keep them turning together; the moment of inertia of a neutron star’s crustal superfluid against the rest of it is what a pulsar glitch measures. The same integral, over objects fifteen orders of magnitude apart in density, doing the same job: saying what fraction of a body is participating.