Astronomy & Celestial Mechanics
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Gravitation

An ocean is detected and its depth is not

A tidal response thirty-six times too large for any solid body proves a moon has a liquid layer under its crust. It does not say how deep the liquid is, how thick the crust above it is, or what the core beneath it is made of — because the same one number is produced by a whole surface of interiors.

Assumes Love numbers and Moment of inertia.

16 min read 8 figures Measuring the unreachableScales that defeat intuition

The argument that Titan has a global ocean is one of the cleanest in planetary science and it runs in one line. A solid Titan, at the rigidity a silicate and high-pressure-ice interior must have, gives a tidal Love number of about 0.017. Cassini measured 0.616. Nothing solid of that size reaches that value at any rigidity a material has, so the shell is not attached to what is underneath it.

That argument is about an inequality, and inequalities are robust. The temptation is to read the same number as a measurement — to convert 0.616 into an ocean depth, or a shell thickness, or a core radius. It does not convert. The number is one integral of a whole interior against a fixed weighting, and inverting one integral for a function is not a problem with a solution.

What this essay is about is exactly how much the number does say, which is more than nothing and much less than an interior.

The ceiling a measured response has to be read against. The fluid Love number of a layered body — the value it would have if its outer shell offered no resistance whatever — against the radius of its core, for cores 1×, 1.6×, 2.4×, 3.6× the density of the material outside. Every curve starts at 3/2, because a core of no size is a uniform body, and falls as the core grows: central condensation stiffens a fluid body without giving it any strength, since the tidal forcing is strongest out where there is then very little mass to move. A body 0.6 of the way core at 3.6× density has a fluid ceiling of 0.787. A measurement above the relevant curve is impossible for any interior of that layering; a measurement below it says only that something is resisting, and does not say what.
Fig. 1 The ceiling a measured response has to be read against. The fluid Love number of a layered body — what it would be if the outer shell offered no resistance at all — against the size of its core, for cores at several densities relative to the material outside. Every curve starts at 3/2 because a core of no size is a uniform body, and falls as the core grows: central condensation stiffens a fluid body without giving it any strength, because the tidal forcing is strongest out where there is then very little mass left to move. Titan’s 0.616 is drawn across them.

The ceiling is not three halves

The first thing a measured Love number has to be compared with is the largest value the body could possibly have, and for a layered body that is not the uniform 3/2.

The Radau integration gives the fluid response for any run of density, and what it says about a rock-and-ice body is that the response falls steeply as the rock is gathered into the middle. A body that is six-tenths core at three and a half times the density of what surrounds it has a fluid ceiling of 0.79 — half the uniform value — and it has that ceiling without any strength anywhere in it.

So a measurement of 0.616 against a fluid ceiling of 0.79 is not a body deforming almost as freely as a fluid. It is a body deforming to about four-fifths of its own fluid limit, and the remaining fifth is what the ice shell is doing.

That reframing matters for what comes next. The quantity the measurement constrains is the ratio of the observed response to the fluid one, and the fluid one depends on the core — on how much rock there is and how tightly it is gathered — while the deficit depends on the shell. One measurement, two unknowns, and they enter at opposite ends of the body.

Titan's measured k₂ of 0.616, against the 0.017 a solid body allows. The tidal Love number of a small solid body, against the rigidity of the material it is made of, both axes logarithmic. For a homogeneous elastic sphere k₂ = (3/2)/(1 + 19μ/2ρgR), and the group in the denominator is the ratio of the material's strength to the pressure its own weight can generate. That ratio decides everything. A body has to be large before its own gravity can overwhelm the strength of rock or ice, so for a moon the denominator is in the hundreds and k₂ collapses: a solid Titan at its interior's own rigidity of 60 gigapascals would have k₂ = 0.0170, drawn where the heavy curve crosses that rigidity. The fainter curves are the same relation for other bodies, and they are ordered by size — the Earth sits far above the moons because ρgR is a hundred times larger. The measured value is the horizontal line at 0.616 ± 0.067, 36 times the solid prediction and outside anything the curve reaches at any rigidity a solid supports. That is the argument, and its shape matters: not that the interior is soft, but that no rigidity whatever makes a solid body of this size respond this much. What responds is a shell floating on a global liquid layer, mechanically decoupled from whatever is beneath it and free to deform almost as a fluid would. The picture assumes homogeneity throughout, which no icy moon has; a layered body is stiffer or softer than this by a factor of a few, and never by the factor of a hundred the argument turns on.
Fig. 2 The inequality that makes the detection secure, drawn on a different axis. The elastic response of a homogeneous body of Titan’s size and density, against the rigidity of whatever it is made of, swept across four decades. The measurement is thirty-six times the value at any rigidity a solid supports. The argument’s strength is that it is an exclusion rather than a fit: a formula with one free parameter, swept across every value that parameter can have, never reaches the observation.

What the shell contributes, and why it is a product

An ice shell floating on a global ocean resists the tide the way a stretched membrane resists a push. Its contribution to the stiffness is not its rigidity alone and not its thickness alone; to a good approximation it is the product, divided by the pressure the body’s own weight generates.

That product is the whole difficulty. A shell twenty kilometres thick at the rigidity of cold ice and a shell sixty kilometres thick at the rigidity of ice warmed to near its melting point produce nearly the same stiffness, and therefore nearly the same Love number. The measurement cannot separate them, and the two describe completely different moons: one with a thin brittle lid over a deep ocean, one with a thick convecting shell.

And the rigidity of ice is not a constant. It falls steeply with temperature near the melting point, and the base of any conducting ice shell is at the melting point by construction, so the effective rigidity of a shell is an average over a profile that the shell’s own thickness sets. The unknown and the thing it is being solved for are the same quantity.

The ocean’s depth barely enters at all. A layer of water between a rigid core and an elastic shell decouples the two whether it is ten kilometres deep or two hundred, because what the tide needs from it is only that it cannot support a shear stress. Doubling the ocean changes the response by a per cent or two, which is inside the measurement error of every determination there is.

That is the sharpest statement this essay has: the quantity most people want from the measurement is the one it is least sensitive to.

The ceiling a measured response has to be read against. The fluid Love number of a layered body — the value it would have if its outer shell offered no resistance whatever — against the radius of its core, for cores 2×, 3×, 4.2× the density of the material outside. Every curve starts at 3/2, because a core of no size is a uniform body, and falls as the core grows: central condensation stiffens a fluid body without giving it any strength, since the tidal forcing is strongest out where there is then very little mass to move. A body 0.6 of the way core at 4.2× density has a fluid ceiling of 0.713. A measurement above the relevant curve is impossible for any interior of that layering; a measurement below it says only that something is resisting, and does not say what.
Fig. 3 The same construction for a rockier body, with Europa’s predicted response marked. Europa is half again as dense as Titan and its rock fraction is much larger, so its fluid ceiling is lower throughout and the curves fall faster. The predicted value with an ocean is near 0.25 and without one is near 0.015 — a factor of seventeen, which is the size of the signal a mission is designed to detect. The gap between the curves at a given core size is what makes the measurement worth making; the width of the band of interiors consistent with any one value is what stops it being an inversion.

What a single number cannot do, stated as a count

It is worth doing the accounting explicitly, because “degenerate” is a word that gets used where “underdetermined” is meant, and the distinction is about numbers.

A three-layer model of an icy moon has, at minimum: a core radius, a core density, an ocean depth, a shell thickness, and a shell rigidity. That is five parameters. Two of them are constrained by the body’s bulk density and its total radius, both of which are known precisely, leaving three.

A tidal Love number is one number. It reduces three unknowns to a two-dimensional surface of interiors, every point of which reproduces the measurement exactly.

The surface is not featureless, which is why the measurement is worth having. Every point on it has a decoupling layer, because the solid corner of the parameter space is excluded outright. And the surface has edges: a shell cannot be thinner than the thermal calculation allows, a core cannot be denser than the densest plausible rock, an ocean cannot be deeper than the body’s mass permits. Within those bounds the answer is a region rather than a point, and quoting a shell thickness from a Love number alone means choosing a point in that region by assumption.

The Love number a solid Europa could have, across four decades of rigidity. The tidal Love number of a small solid body, against the rigidity of the material it is made of, both axes logarithmic. For a homogeneous elastic sphere k₂ = (3/2)/(1 + 19μ/2ρgR), and the group in the denominator is the ratio of the material's strength to the pressure its own weight can generate. That ratio decides everything. A body has to be large before its own gravity can overwhelm the strength of rock or ice, so for a moon the denominator is in the hundreds and k₂ collapses: a solid Europa at its interior's own rigidity of 50 gigapascals would have k₂ = 0.0193, drawn where the heavy curve crosses that rigidity. The fainter curves are the same relation for other bodies, and they are ordered by size — the Earth sits far above the moons because ρgR is a hundred times larger. Nothing is marked as measured here because this body's interior is read from a different observable — predicted near 0.25 with an ocean, near 0.015 without — and the point of drawing it is the size of the number a solid model predicts. The picture assumes homogeneity throughout, which no icy moon has; a layered body is stiffer or softer than this by a factor of a few, and never by the factor of a hundred the argument turns on.
Fig. 4 Europa’s version of the exclusion, which has not yet been tested. The predicted response with a global ocean is more than an order of magnitude above anything a solid body of that size and density reaches, so the measurement is as decisive as Titan’s was — and the case for the ocean currently rests on the induced magnetic field, which is a different physics with a different set of assumptions in it. Two independent arguments for the same conclusion is the position this subject wants to be in and is not yet.

The second number, and what it has to be

Breaking a degeneracy takes a measurement whose dependence on the unknowns has a different shape, not merely another measurement.

The moment of inertia is the obvious candidate and it is the one that has actually been used. It is a different integral of the same density profile, weighted by the square of the distance from the axis rather than by the response to a quadrupole, so it emphasises the outer parts differently. A body with a small dense core and a large light mantle has a small moment-of-inertia factor and a low fluid Love number; the two are correlated, and they are not the same function, so measuring both narrows the region considerably.

The difficulty is that the moment of inertia is itself not directly measured. It comes from the degree-2 gravity coefficients through the Radau–Darwin relation, which assumes hydrostatic equilibrium — and a body with a rigid shell that has frozen in some of its past shape is not hydrostatic. So the second number arrives with an assumption the first did not need.

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. 5 The complementary constraint, and it is complementary rather than redundant. The moment-of-inertia factor also falls as mass is gathered inward, so the two quantities are correlated — but C/MR2C/MR^2 weights the interior by the square of the distance from the spin axis and k2k_2 weights it by the response to a quadrupole. Measuring both is worth far more than measuring either twice, which is the whole argument for a mission that does gravity science at several orbital phases rather than one.

The phase of the response is the second candidate and it carries different information again. The tidal bulge lags the forcing, the lag is set by how the material creeps rather than by how the mass is arranged, and the amplitude and the phase therefore separate structure from rheology. Measuring the phase requires resolving a fraction of a degree in a signal that is already at the edge of detectability.

The obliquity is the third and it is the cleverest. A synchronously rotating moon settles into a state where its spin axis, its orbit normal and the Laplace plane are coplanar, and the angle it settles at depends on its moment of inertia — so measuring a pole position measures an interior. For Titan this gave an independent constraint, and for Enceladus the analogous measurement is a libration: the moon’s rotation oscillates about the synchronous rate by an amount that depends on how much of it is rigidly coupled to the surface. Enceladus’s libration is four times what a body frozen through could show, which is a detection of a decoupling layer by a method with nothing in common with gravity science.

Three of the four routes measure the same interior through different integrals, and it took all of them to establish what is now stated as a fact in one sentence.

The ceiling a measured response has to be read against. The fluid Love number of a layered body — the value it would have if its outer shell offered no resistance whatever — against the radius of its core, for cores 1.4×, 2.2×, 3.2× the density of the material outside. Every curve starts at 3/2, because a core of no size is a uniform body, and falls as the core grows: central condensation stiffens a fluid body without giving it any strength, since the tidal forcing is strongest out where there is then very little mass to move. A body 0.6 of the way core at 3.2× density has a fluid ceiling of 0.847. A measurement above the relevant curve is impossible for any interior of that layering; a measurement below it says only that something is resisting, and does not say what.
Fig. 6 And the case the method is calibrated on, where the answer is known by other means. The Moon’s measured Love number is 0.0242, from laser ranging, which sits far below every fluid curve — as it must, since the Moon is solid throughout apart from a small partially molten layer at the base of its mantle. A body that behaved as a fluid would read a value forty times higher. The distance between a measurement and the fluid ceiling is a measurement of strength, and for the Moon it is nearly the whole distance.

What was actually measured at Titan

The chain from the observation to the number is long enough to be worth setting out, because the error on the Love number is what decides how large the allowed region is.

What Cassini recorded was Doppler shifts of its radio carrier during six close flybys. A spacecraft accelerating toward or away from the Earth shifts the frequency of the signal it returns, and the residual after every known acceleration is removed is the gravity field of the body being flown past.

Titan’s orbit is eccentric, so the tide it feels varies through its orbital period, and the quadrupole it acquires varies with it. The six flybys were timed to sample different orbital phases, so the varying part of the degree-2 gravity could be separated from the static part. That varying part is the tidal response.

The difficulty is that the tidal term is small and correlated with everything else in the fit — the spacecraft’s own state, the higher-order static field, the ephemeris of the moon. The published uncertainty of ±0.067\pm 0.067 on 0.616 is eleven per cent, and it is dominated by those correlations rather than by the noise on any single measurement.

Eleven per cent is enough to exclude a solid body by a factor of thirty. It is not enough to distinguish a twenty-kilometre shell from a fifty-kilometre one, and it is nowhere near enough to say anything about the ocean.

Enceladus, where three observations agree and none of them is a Love number

The smallest body in this argument is the one where the case is strongest, and it was built without a tidal Love number at all.

Enceladus is 252 kilometres in radius, a tenth of Titan’s, so the group ρgR\rho g R that decides an elastic response is a hundred times smaller and a solid Enceladus would have a k2k_2 of about two thousandths. Nothing can measure that. The moon is too small for gravity science to separate a tidal term at the required precision, and the detection had to come from elsewhere.

It came from three places.

The plume is the obvious one: water vapour and ice grains leaving the south polar region at a rate of a couple of hundred kilograms a second, driven by heat a moon makes by not being allowed to relax, with salts in it at concentrations that indicate contact with rock. That is direct evidence of liquid water, and on its own it is evidence of a local reservoir rather than a global ocean.

The libration is the decisive one. A synchronously rotating moon on an eccentric orbit does not rotate uniformly — the tidal torque speeds it up and slows it down through each orbit, and it oscillates about the mean rate by an amplitude that depends on how much of the body is rigidly coupled to the surface. Enceladus’s libration, measured from the positions of surface features across seven years of Cassini images, is 0.120 degrees. A body frozen through, with the shell rigidly attached to the core, could librate by at most a third of that. The shell is not attached.

The gravity and topography together are the third. The degree-2 gravity field implies a mass distribution; the shape implies a different one; and the mismatch is what a shell floating in isostatic balance on a denser liquid produces. Fitting the two jointly gives a shell thickness averaging twenty to twenty-five kilometres and thinning to a few at the south pole.

Three measurements with nothing in common — a mass flux, an angle, and a mismatch between two fields — and all three require a global liquid layer. That is the standard the Titan result is held to and it is why the single-number version of the argument is worth being careful about: a factor of thirty is persuasive, and three independent factors are an established fact.

The surprise in the number itself

There is a detail of the Titan measurement that is usually passed over and is the most interesting thing about it.

The fluid ceiling for Titan’s layering is around 0.7 to 0.8. The measurement is 0.616. So the body is responding at four-fifths of the freedom a completely fluid version of itself would have — which means the ice shell is contributing almost nothing to the stiffness.

That is not what was expected. A shell of cold ice a hundred kilometres thick would have reduced the response substantially, and pre-Cassini models with such shells predicted values near 0.2. Recovering 0.616 requires either a shell much thinner than those models, or a shell much warmer and therefore much softer, or a shell that is decoupled from itself as well as from the ocean — a layer of warm ductile ice at its base that flows on tidal timescales.

The measurement is therefore a statement about the shell’s temperature as much as about the ocean’s existence, and it pushes toward a warm, thin, or partially mobile crust. Which of those three it is remains open, and it is the same three-way ambiguity the product of thickness and rigidity always produces.

Where the picture stops

The shell has structure the model does not. A conducting ice shell has a temperature profile, a rigidity that varies through it by more than an order of magnitude, and quite possibly a convecting lower half whose effective behaviour over a tidal period is not that of an elastic solid at all. The single effective rigidity used in every formula here is an average over that profile with a weighting nobody has derived.

An ocean need not be global. The whole argument assumes a continuous liquid layer, because that is what decouples the shell everywhere. A partial ocean, or a layer of warm ductile ice that behaves as a liquid on tidal timescales and as a solid on geological ones, produces an intermediate response the model has no parameter for.

And the high-pressure ice under the ocean is not rock. Titan’s interior below the ocean is ice at pressures where it takes a denser crystalline form, and whether that layer is rigid on tidal timescales or itself deforms is unsettled. If it deforms, the “core” the fluid ceiling depends on is smaller than the rock fraction suggests, and every ceiling in the first figure moves.

The Love number against central condensation: 3/2 for a uniform body, 2.4e-3 at n = 4. How willingly a body deforms, drawn against how concentrated it is. The horizontal axis is the polytropic index, which is a proxy for the run of density inside — n = 0 is uniform, n = 1.5 is a non-relativistic degenerate gas, n = 3 is a radiative star like the Sun — and the vertical axis is the fluid Love number k₂ on a logarithmic scale. The curve is the Radau equation integrated over each polytrope's own density profile, and its two ends are exact rather than fitted: a uniform incompressible body has k₂ = 3/2 exactly, and a body with all its mass at the centre has k₂ = 0, because a point mass has no quadrupole to offer. Everything real lies between. The fall is steep — four orders of magnitude across the family — which is what makes the number diagnostic: k₂ is not a mild function of structure, it is a sensitive one, and measuring it to ten per cent constrains the interior far better than measuring a mean density to the same precision. The Sun, at n ≈ 3, sits near 2.9e-2. Two conventions collide here and the figure uses one of them: the planetary literature's k₂, for which a uniform body gives 3/2. The stellar literature's apsidal-motion constant is half of this at every point, so a uniform body gives 0.75, and the two are the same quantity. What the picture cannot show is rigidity — every body on it is a fluid, and for anything smaller than a planet that assumption fails badly.
Fig. 7 The same quantity for the other family of layered bodies, where the layering is continuous rather than in steps. A polytrope’s index is a measure of how centrally condensed it is, and the Love number falls from 3/2 to two thousandths across the family — the same steep sensitivity the layered curves show, arrived at with no interfaces at all. What k2k_2 measures is central condensation, and whether that condensation is achieved by a discrete core or by a smooth run of density is invisible to it.

The shape of the argument, and where else it appears

The structure here recurs wherever a single integral of an unknown function is measured, and it is worth naming.

An exclusion is worth more than a fit. The Titan result is secure because a model with one free parameter, swept across its whole range, could not reach the observation. That is a different kind of claim from a best-fit interior, and it survives changes to everything the fit would have depended on.

A degeneracy is broken by a different weighting, not by a better measurement. Improving the Love number from eleven per cent to one per cent shrinks the allowed region along one direction and leaves it just as extended along the others. A moment of inertia, a libration amplitude or a phase lag each cut it a different way.

The same pattern governs how many compositions share one bulk density, and it governs the reading of a stellar spectrum, where a single line strength constrains a product of abundance, temperature and pressure and only lines of different character separate them. In every case the useful question is not how precisely the number is known but what else has a different dependence on the same unknowns.

the Moon's measured k₂ of 0.02422, against the 0.023 a solid body allows. The tidal Love number of a small solid body, against the rigidity of the material it is made of, both axes logarithmic. For a homogeneous elastic sphere k₂ = (3/2)/(1 + 19μ/2ρgR), and the group in the denominator is the ratio of the material's strength to the pressure its own weight can generate. That ratio decides everything. A body has to be large before its own gravity can overwhelm the strength of rock or ice, so for a moon the denominator is in the hundreds and k₂ collapses: a solid the Moon at its interior's own rigidity of 65 gigapascals would have k₂ = 0.0226, drawn where the heavy curve crosses that rigidity. The fainter curves are the same relation for other bodies, and they are ordered by size — the Earth sits far above the moons because ρgR is a hundred times larger. The measured value is the horizontal line at 0.02422 ± 0.00022, and the solid curve passes through it — 1.07 times the prediction. That agreement is the control on the whole method. A relation that returned "no solid model fits" for every body would be evidence of nothing; this one returns a fit where a fit is expected, from GRAIL, from the degree-2 tidal field. The picture assumes homogeneity throughout, which no icy moon has; a layered body is stiffer or softer than this by a factor of a few, and never by the factor of a hundred the argument turns on.
Fig. 8 And the control the whole method rests on. The Moon’s Love number is measured by laser ranging to a few parts in a thousand, and the homogeneous elastic formula at a lunar bulk rigidity reproduces it. A formula that returned “no solid model fits” for every body would be evidence of nothing whatever, and the reason the Titan exclusion is believed is that the same formula, on a body known to be solid, gives the right answer.

Still open: a thickness nobody has measured

Every published ice-shell thickness for Titan, Europa or Enceladus is the output of a thermal model rather than of a measurement. The models balance the heat generated by tidal dissipation against the heat conducted or convected out, and they return thicknesses from a few kilometres to over a hundred depending on assumptions about the rheology, the heat source and the history.

The measurement that would settle it is a radar sounding: a pulse that penetrates the ice and returns from the water beneath, giving a two-way travel time and therefore a thickness directly. It is the one observation in this subject that inverts to a number rather than to a region, and it has not yet been made at any icy moon.

Whether it will work is itself uncertain, because the attenuation of radio waves in ice depends on the ice’s temperature and impurity content, and a warm salty shell may be opaque at every useful frequency. Until that is tried, the thickness of the crust over the best-established ocean beyond the Earth remains a modelled quantity with a factor-of-five range, next to a tidal response measured to eleven per cent.

Where this goes next

The Love number against central condensation: 3/2 for a uniform body, 2.4e-3 at n = 4. How willingly a body deforms, drawn against how concentrated it is. The horizontal axis is the polytropic index, which is a proxy for the run of density inside — n = 0 is uniform, n = 1.5 is a non-relativistic degenerate gas, n = 3 is a radiative star like the Sun — and the vertical axis is the fluid Love number k₂ on a logarithmic scale. The curve is the Radau equation integrated over each polytrope's own density profile, and its two ends are exact rather than fitted: a uniform incompressible body has k₂ = 3/2 exactly, and a body with all its mass at the centre has k₂ = 0, because a point mass has no quadrupole to offer. Everything real lies between. The fall is steep — four orders of magnitude across the family — which is what makes the number diagnostic: k₂ is not a mild function of structure, it is a sensitive one, and measuring it to ten per cent constrains the interior far better than measuring a mean density to the same precision. The Sun, at n ≈ 3, sits near 2.9e-2. Two conventions collide here and the figure uses one of them: the planetary literature's k₂, for which a uniform body gives 3/2. The stellar literature's apsidal-motion constant is half of this at every point, so a uniform body gives 0.75, and the two are the same quantity. What the picture cannot show is rigidity — every body on it is a fluid, and for anything smaller than a planet that assumption fails badly. How much a world gives A body pulled on from one side deforms, and how much it deforms is a single dimensionless number. That number is three halves for a uniform fluid, three hundredths for the Sun, and two thousandths for a moon made of ice — so measuring it is a measurement of what is inside. 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. One mass and one radius, and every composition that gives them. A planet of 5 Earth masses and 1.6 Earth radii, and the compositions consistent with it. The horizontal axis is the fraction of the planet's mass in an iron core and the vertical axis the fraction in a water layer outside the rock; the heavy curve is every pair that reproduces the measurement exactly, and the band around it is what the 0.05 Earth-radius uncertainty allows. The answer is a curve, not a point, and that is not a failure of precision. Two numbers cannot determine three components: a planet can be made denser by adding iron or lighter by adding water, and along this locus the two changes cancel exactly. The ends of it are not small variations on one planet. At the left is a body with no iron at all and 0 per cent of its mass in water; at the right, one with an iron core like Mercury's and 23 per cent water. Those have different formation histories, different interiors, different everything, and the same mass and radius to the precision anybody can measure them. Breaking the degeneracy needs an observation that is neither a mass nor a radius. The usual one is a transmission spectrum, which measures the atmosphere's scale height and so its mean molecular weight — a hydrogen envelope and a steam envelope differ by a factor of nine in that, and the corresponding factor in the size of the spectral features. What the picture assumes is that the planet is differentiated into clean layers, which is the standard assumption and is false in detail: water dissolves into silicate melt at these pressures, and a mixed interior sits at neither end of this curve. One density, and every planet that has it A mass and a radius are two numbers, and a differentiated planet has at least three components. The set of compositions matching a measurement is therefore a curve rather than a point — and its two ends are a body with no iron and half its mass in water, and a body with a Mercury-like core.

About the same objects

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

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Central condensationDegeneracyGravity scienceIce shellLayered interiorLove numberMoment of inertiaRadau darwin relationRigiditySubsurface ocean