Astronomy & Celestial Mechanics
the sky, computed
The observed sky

Four numbers that weigh a planet's core

Mercury rocks about its mean rotation by thirty-eight arcseconds, which is more than twice what a planet turning in one piece could manage. The excess says that most of the planet's moment of inertia is not following the mantle on an eighty-eight-day timescale, and the only thing that does not follow a mantle is a liquid.

Assumes Libration and Moment of inertia.

15 min read 8 figures The sky as a clockMeasuring the unreachable

The Moon’s face is not quite fixed: it rocks by a few degrees a month, for two unrelated geometric reasons, and the rocking has shown fifty-nine per cent of its surface to people who never left the ground. That essay was about the optical libration, which is a matter of where the observer is standing and how the Moon’s orbit is shaped, and not about the Moon at all.

There is a second kind of rocking, much smaller, which is about the body. A planet on an eccentric orbit is torqued unevenly through that orbit, and it responds by physically oscillating about its mean rotation. The amplitude is set by how much moment of inertia has to be swung — and if part of the interior does not participate, the amplitude is larger than it should be.

Mercury’s is more than twice as large as it should be.

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. 1 The forced libration amplitude against the fraction of the planet’s polar moment that rocks with the mantle. The curve is a rectangular hyperbola, because the same torque applied to less moment produces proportionally more angle. A Mercury turning rigidly in one piece would librate by 16.2 arcseconds; the measured value is 38.5 ± 1.6, so only 42 per cent of the moment is being swung. The other 58 per cent is not following the mantle on an 88-day timescale, and the only interior that does not follow its own mantle is a liquid one.

Why the torque is uneven

A body with an equatorial bulge feels a torque from whatever it orbits, and the torque tries to align the bulge’s long axis with the line to the primary. For a body on a circular orbit rotating synchronously, the alignment is exact and the torque is zero.

Mercury is neither. Its orbit has an eccentricity of 0.2056, which is the largest of any planet, and it rotates not synchronously but in a 3:2 spin–orbit resonance: three rotations for every two orbits. So through each orbit the angular velocity of the line to the Sun varies enormously — by Kepler’s second law, the planet sweeps through perihelion far faster than the rotation carries it round — while the spin rate is nearly constant.

The bulge therefore leads and lags alternately, and the restoring torque swings the planet back and forth. The amplitude of that swing depends on the size of the bulge, which is measured, and on the moment of inertia being swung, which is not.

The sub-Earth point over 400 days. Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats.
Fig. 2 The other libration, for comparison. Optical libration is a change in what an observer sees rather than in what the body does: the Moon’s constant rotation against its variable orbital angular velocity, plus the tilt of its equator, plus the observer’s own displacement across the Earth’s radius. It is degrees; the physical libration this essay is about is tens of arcseconds, four orders of magnitude smaller, and it is the only one that says anything about an interior.

There is a subtlety in the 3:2 resonance itself that makes the whole thing possible. A body captured into synchronous rotation feels a torque that vanishes on a circular orbit, so its forced libration is driven only by the eccentricity and is small. A body in the 3:2 state is being forced much harder, because the mismatch between spin rate and orbital angular velocity is large through most of the orbit. Mercury’s libration is measurable because Mercury is in the wrong resonance for a quiet life — and it is in that resonance because its eccentricity made capture into 3:2 far more likely than into 1:1.

Four observables, two unknowns

The measurement is usually called Peale’s experiment, after the 1976 paper that laid it out before any of the four numbers could be measured. The proposal was that a particular combination of four observables would determine both the whole planet’s moment of inertia and the mantle’s separately — and that the ratio would say whether the core was liquid.

The four are:

J2J_2 and C22C_{22}, the two lowest gravity coefficients, which give the differences between the planet’s three principal moments of inertia in units of MR2MR^{2}. These come from tracking a spacecraft in orbit.

The obliquity, the tilt of the spin axis from the orbit normal. Mercury occupies a Cassini state, meaning its spin axis, its orbit normal and the normal to the invariable plane stay coplanar as all three precess together. In such a state the obliquity is not free: it is fixed by the planet’s moment of inertia and by the known precession rate of the orbit. So measuring the obliquity gives C/MR2C/MR^{2} for the whole planet.

The libration amplitude, which gives Cm/MR2C_m/MR^{2} for the part that rocks.

Two of the four give the shape of the mass distribution, the third gives the whole body’s moment and the fourth gives the mantle’s. The ratio Cm/CC_m/C follows, and it is 0.421 ± 0.021.

Measured moment-of-inertia factors, from the Sun's 0.3307 to the Moon's 0.3931. Eleven bodies whose interiors have never been sampled, arranged by the one interior quantity that has been measured for all of them. C/MR² is 2/5 for a uniform sphere and falls as mass is concentrated toward the centre, and the values here span from 0.3307 to 0.3931. That spread is the content. The Moon at 0.3931 is barely differentiated — whatever iron core it has is a few per cent of its radius, which is why the Moon is the one large body in the inner solar system without a magnetic field of its own. Mercury at 0.346 is nearly as low as the Earth despite being an eighth of its mass, and for a body that small the only way to get there is an iron core filling most of the radius. The Sun at 0.07 is off the scale of anything a two-layer model describes; a star is not a planet with a bigger core but a body whose density falls by five orders of magnitude between centre and surface. The faint curves behind are the two-layer relation at a few density contrasts, drawn to show what kind of interior each value is consistent with — and the horizontal placement of each body on them is an illustration rather than a result, since one factor never fixes one core. Every number here 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.
Fig. 3 Where the answer puts Mercury. Its whole-planet factor is 0.346, almost as low as the Earth’s for a body an eighteenth of the Earth’s mass — which by itself already requires an iron core filling most of the radius. The libration then adds that most of that core is liquid, which no static measurement could have said.

The elegance of the scheme is that the two moments are obtained from measurements of different kinds. The obliquity is a secular, geometric property of an equilibrium that took millions of years to establish; the libration is an oscillation at the orbital period. Nothing about either measurement depends on the other, and the ratio of the two moments — which is the answer — is therefore not a difference of two nearly equal numbers from one experiment but a quotient of two independent ones.

It is also worth noticing what the scheme does not need. It needs no seismometer, no lander, no sample, and no assumption about the planet’s composition. The four observables are angles and timings, and the two moments come out of them by rigid-body mechanics. The composition enters only afterwards, when somebody wants a core radius in kilometres.

What the Cassini state contributes

The obliquity half of the experiment deserves a paragraph of its own, because it is the least obvious.

A planet’s spin axis precesses under the torque from its primary, at a rate proportional to its own oblateness and inversely proportional to its polar moment. Its orbit’s normal precesses too, under the perturbations of the other planets. If the two rates are commensurate, the system can settle into a state in which the spin axis, the orbit normal and a fixed reference normal remain in one plane, all precessing together — a Cassini state.

Mercury is in one, and the equilibrium obliquity of that state depends on the moment of inertia. So a single angle — measured at two arcminutes, which is 0.034 degrees — determines C/MR2C/MR^{2}.

The measurement is delicate for a reason worth stating: the obliquity is small, about two arcminutes, so the geometry is nearly degenerate and small errors in the spin-axis direction are large fractional errors in the obliquity. Two independent techniques were used, Earth-based radar speckle tracking and spacecraft altimetry, and their agreement is the reason the result is believed.

What was actually measured

The libration amplitude was measured from the Earth, by radar, before any spacecraft orbited Mercury.

The technique is speckle tracking. Radar illuminates the planet; the reflected signal, received at two widely separated antennas, carries a speckle pattern produced by interference from the rough surface. That pattern sweeps across the Earth as the planet turns, and the time offset between its arrival at two antennas measures the instantaneous rotation rate. Repeating over years builds up the rotation as a function of time, and the periodic departure from uniform rotation at the orbital period is the libration.

The precision is remarkable given what is being done: the amplitude of 38.5 arcseconds corresponds to a displacement of about 450 metres at Mercury’s equator, measured across a hundred million kilometres.

Two of the four numbers came later, from a spacecraft in orbit measuring the gravity field, and one of the two — the obliquity — was measured both ways. The chain also has an internal check that is worth stating, because it is what turns four measurements of two unknowns into a test rather than a fit. Four observables and two unknowns is one more constraint than is needed. The redundancy was used: the whole-planet moment obtained from the obliquity had to be consistent with the moment implied by the gravity coefficients under the assumption of hydrostatic equilibrium, and it was not — Mercury’s J2J_2 is larger than a hydrostatic body spinning at its rate would have. That failure is itself informative: it says the planet retains a non-hydrostatic figure, frozen in from a time when it spun faster, and it is the reason the Radau–Darwin shortcut cannot be used here.

The two things this does not say

It does not give a core radius. Cm/C=0.42C_m/C = 0.42 constrains the combination of the core’s size and density, and inverting that needs a density profile — which comes from a mean density, an assumed mantle composition and a laboratory equation of state for iron alloys. The published core radius of about 2,020 kilometres is a model output, and its uncertainty is dominated by the light-element content of the core, which nothing measures.

It does not say the whole core is liquid. What the libration shows is that 58 per cent of the moment is decoupled on an eighty-eight-day timescale. A solid inner core inside a liquid outer one would rock with the mantle only if it were coupled to it, and on that timescale it would not be — so the measurement bounds the liquid layer’s extent from one side and leaves an inner core’s existence open. Other arguments, from the magnetic field and from the planet’s thermal history, favour a solid inner core, and the libration cannot see it.

Why a small planet has a liquid core at all

This is the part that made the result interesting rather than merely precise.

A body’s interior cools by conduction and convection, and a small body cools faster because it has more surface per unit volume. Mercury is a twentieth of the Earth’s volume, so by the simplest argument its core should have frozen long ago — and the standard expectation before the measurement was that it had.

Two things save it. The first is sulphur: an iron core with a few per cent of sulphur has a melting point hundreds of kelvin lower than pure iron, and a few per cent is what cosmochemistry allows. The second is the planet’s own history: a core that is freezing releases latent heat and drives compositional convection, which is also the standard explanation for Mercury’s magnetic field, and the field is observed.

So a rocking measured by radar constrains a sulphur abundance, which constrains where in the disc the planet’s material came from. That is a longer chain than most in this collection and every link of it is checkable.

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 — 3 gives 0.317 at 69 per cent of the radius, 5 gives 0.279 at 65 per cent of the radius, 8 gives 0.246 at 61 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. 4 The relation the core radius is extracted from. A measured factor is a horizontal line crossing each curve twice, so a moment of inertia alone never gives a core size — the density contrast has to be assumed, and the assumption is a cosmochemical one. Mercury’s low factor puts it toward the bottom of this diagram, where the curves are steep and the inference is comparatively well constrained, which is a piece of luck.
Measured moment-of-inertia factors, from the Sun's 0.3115 to the Moon's 0.3931. Eleven bodies whose interiors have never been sampled, arranged by the one interior quantity that has been measured for all of them. C/MR² is 2/5 for a uniform sphere and falls as mass is concentrated toward the centre, and the values here span from 0.3115 to 0.3931. That spread is the content. The Moon at 0.3931 is barely differentiated — whatever iron core it has is a few per cent of its radius, which is why the Moon is the one large body in the inner solar system without a magnetic field of its own. Mercury at 0.346 is nearly as low as the Earth despite being an eighth of its mass, and for a body that small the only way to get there is an iron core filling most of the radius. The Sun at 0.07 is off the scale of anything a two-layer model describes; a star is not a planet with a bigger core but a body whose density falls by five orders of magnitude between centre and surface. The faint curves behind are the two-layer relation at a few density contrasts, drawn to show what kind of interior each value is consistent with — and the horizontal placement of each body on them is an illustration rather than a result, since one factor never fixes one core. Every number here 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.
Fig. 5 The same four numbers for bodies where they were measured by spacecraft rather than by radar. A moment-of-inertia factor of 0.4 is a uniform sphere and anything below it is a body with its heavy material inside — Io at 0.378 has a core, Callisto at 0.355 is only partly differentiated, and Ganymede at 0.311 is more centrally condensed than the Earth. Each of those is a gravity flyby and a rotation state, which is the identical measurement this essay makes for Mercury by a different route.

The same experiment elsewhere

Peale’s argument is general, and it has been run or proposed on several other bodies.

The Moon. Laser ranging to the retroreflector arrays measures the physical libration to milliarcseconds, and the residuals require dissipation at a core–mantle boundary — evidence for a small fluid core of about 350 kilometres, later supported by a reanalysis of the Apollo seismic data.

Enceladus. The libration amplitude of a tenth of a degree is four times what a body frozen through could show, and it is one of the two independent arguments for a global subsurface ocean, the other being a tidal Love number far too large for a solid.

Titan. The same experiment is harder because the libration is small and the atmosphere exchanges angular momentum with the surface seasonally, which produces a rotation variation of its own that has to be separated from the tidal one.

Mars. A lander’s radio link measures the planet’s rotation and nutation, and the amplitude of the nutation is resonantly amplified by the fluid core’s own free oscillation — so the same idea, applied to a different periodicity, gives a core radius. The answer, 1,830 kilometres, was later confirmed by a seismometer.

Why the resonance is three to two

Everything above depends on Mercury being in the 3:2 spin–orbit state rather than the 1:1 that tides normally produce, and the question of how it got there is worth following, because the answer changed within the last twenty years.

Tides despin a body. A planet born rotating fast is slowed by the torque on its own tidal bulge until it approaches a rate commensurate with its orbit, and as it passes each commensurability there is a chance of capture. The chance depends on the strength of the resonance and on how fast the spin is changing, in exactly the way capture into an orbital resonance depends on the drift rate: a slow approach is captured and a fast one passes through.

The 3:2 resonance’s strength depends on the eccentricity, and Mercury’s is large, so the resonance is strong enough to capture. The standard calculation of the 1960s gave a capture probability of about seven per cent — enough to be possible and small enough to be uncomfortable, since it means the planet’s present state is a one-in-fourteen accident.

Two additions have since raised it. The first is friction at the core–mantle boundary. A liquid core does not despin with the mantle, so there is a torque between them, and that torque changes the rate at which the mantle’s spin drifts through the resonance. Including it raises the capture probability substantially — and the ingredient it requires is a liquid core, which is what the libration measured.

The second is that the eccentricity has not been constant. Mercury’s orbit is chaotic on billion-year timescales, and integrations show its eccentricity wandering between near zero and above 0.4. A body that passed the resonance at a moment of low eccentricity would not have been captured; one that passed at high eccentricity would. Averaging over the possible histories rather than assuming the present eccentricity raises the probability again, to values near certainty in some treatments.

The argument closes on itself in a satisfying way. The libration is measurable because the planet is in the 3:2 state; the planet is probably in the 3:2 state because its core is liquid; and the liquid core is what the libration measured. None of the three steps was known when Peale proposed the experiment.

The comparison across bodies is the check on all of it, and it is worth restricting to the three whose interiors are measured by three different techniques.

Measured moment-of-inertia factors, from the Sun's 0.346 to the Moon's 0.3931. Eleven bodies whose interiors have never been sampled, arranged by the one interior quantity that has been measured for all of them. C/MR² is 2/5 for a uniform sphere and falls as mass is concentrated toward the centre, and the values here span from 0.346 to 0.3931. That spread is the content. The Moon at 0.3931 is barely differentiated — whatever iron core it has is a few per cent of its radius, which is why the Moon is the one large body in the inner solar system without a magnetic field of its own. Mercury at 0.346 is nearly as low as the Earth despite being an eighth of its mass, and for a body that small the only way to get there is an iron core filling most of the radius. The Sun at 0.07 is off the scale of anything a two-layer model describes; a star is not a planet with a bigger core but a body whose density falls by five orders of magnitude between centre and surface. The faint curves behind are the two-layer relation at a few density contrasts, drawn to show what kind of interior each value is consistent with — and the horizontal placement of each body on them is an illustration rather than a result, since one factor never fixes one core. Every number here 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.
Fig. 6 The measured factors for Mercury, Mars and the Moon. Mercury’s comes from a forced libration, Mars’s from a precession rate, the Moon’s from laser ranging, and the three agree in placing all of them below the uniform-sphere value — three techniques, one conclusion, and no shared assumption between them.

What the picture assumes

The hyperbola in the hero figure is exact under three assumptions, and each is worth naming because each has been questioned.

The mantle is rigid. The relation treats the librating shell as a solid body swinging as one. A real mantle is elastic and deforms slightly under the same torque, which reduces the effective moment being swung by a per cent or two. The correction is included in the published analyses and is small for a rocky mantle; for an ice shell floating on an ocean it is not small at all, which is why the same experiment at Enceladus needs more care.

The coupling to the core is negligible. A liquid core is not perfectly decoupled: viscous and magnetic torques at the boundary drag it along a little, and topography on the boundary couples it mechanically. Estimates put the effect below the measurement error for Mercury, which is a statement about a core–mantle boundary nobody has seen.

The forcing is the orbital one. Other periodicities exist — a long-period term from the planetary perturbations to Mercury’s orbit, and a free libration if anything has excited one — and they have to be separated in the fit rather than assumed absent. It is worth noticing that the capture argument is the only part of this chain that is not a measurement, and that it is the part most likely to be revised.

The four observables reduce to two unknowns through a chain of models, and each link in that chain is worth drawing at a second setting.

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, 3 gives 0.317 at 69 per cent of the radius, 6 gives 0.266 at 63 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. 7 The moment-of-inertia factor against core size for three density contrasts. A measured factor picks out a curve on this plot and not a point, so the core radius that comes out depends on an assumed contrast — which is supplied by cosmochemistry rather than by any measurement of the planet.
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. 8 The Radau–Darwin approximation against the measured factors. It is good to a per cent for most bodies and fails by twelve per cent for Mars, because Mars is not in hydrostatic equilibrium — and the failure is itself the measurement of how far from equilibrium it is.

Where the ladder goes

The first rung of this anchor was about a libration that reveals nothing about an interior — the optical one, which is geometry. This one is about the physical libration, which reveals almost nothing else.

The next steps go toward the periodicities. A body has a free libration as well as a forced one, at a period set by its own moment distribution rather than by the orbit, and detecting a free libration would say that something excited it recently — which for a body assumed geologically quiet is a strong statement. Mercury’s free libration has been searched for and not found, and the upper limit is itself a constraint on how much internal activity there has been.

And there is the thread this whole phase runs on. What a libration measures is the fraction of a body’s moment of inertia that is participating — which is exactly what a pulsar’s glitches measure about a neutron star’s superfluid, and what the moment-of-inertia factor is about in the first place. A planet whose core does not follow its mantle and a neutron star whose superfluid does not follow its crust are the same measurement at twenty orders of magnitude’s difference in density.

Where this goes next

The sub-Earth point over 400 days. Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats. The face that is not quite fixed The Moon keeps one face turned toward the Earth, and the sentence is exactly true only of a fictitious Moon on a circular orbit. The real one rocks by a few degrees each month, in two directions and for two unrelated reasons, and the rocking has shown 59 per cent of its surface to people who never left the ground. Measured moment-of-inertia factors, from the Sun's 0.07 to the Moon's 0.3931. Eleven bodies whose interiors have never been sampled, arranged by the one interior quantity that has been measured for all of them. C/MR² is 2/5 for a uniform sphere and falls as mass is concentrated toward the centre, and the values here span from 0.07 to 0.3931. That spread is the content. The Moon at 0.3931 is barely differentiated — whatever iron core it has is a few per cent of its radius, which is why the Moon is the one large body in the inner solar system without a magnetic field of its own. Mercury at 0.346 is nearly as low as the Earth despite being an eighth of its mass, and for a body that small the only way to get there is an iron core filling most of the radius. The Sun at 0.07 is off the scale of anything a two-layer model describes; a star is not a planet with a bigger core but a body whose density falls by five orders of magnitude between centre and surface. The faint curves behind are the two-layer relation at a few density contrasts, drawn to show what kind of interior each value is consistent with — and the horizontal placement of each body on them is an illustration rather than a result, since one factor never fixes one core. Every number here 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. 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. The Kirkwood gaps. Asteroid numbers against semi-major axis, with the resonant radii marked. Each gap sits where the orbital period is a simple fraction of Jupiter's, and each of those radii is computed from the harmonic law rather than placed by eye. Resonance clears a gap in one place and locks a moon in another When two orbital periods are in a simple ratio, small tugs stop averaging away and start accumulating. Sometimes that empties a region entirely. Sometimes it holds three moons together for the age of the solar system.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 10 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Cassini stateCore mantle boundaryForced eccentricityGravity scienceLibrationMoment of inertia factorObliquityRadar astronomySpin orbit resonanceSynchronous rotationZonal harmonic