Gravitation

A core weighed by something that never went in

The external gravity field of a planet is a series, and each term of it is an integral over the interior density weighted by a different power of radius. Measure enough terms and the series becomes a coarse depth profile — which is how Jupiter's core turned out to be smeared over half the planet rather than sitting at the middle.

Assumes Oblateness and Moment of inertia.

17 min read 8 figures Measuring the unreachable Small perturbations, large consequences

The Earth’s shape can be read off a satellite’s node, because a body that is not a sphere does not pull like a point and an orbit records the difference. That essay treated the departure from sphericity as one number, $J_2$ , and as a nuisance to be modelled.

It is not one number. The external potential of any body is an infinite series, and $J_2$ is its first term. Each subsequent term is an integral over the interior density weighted by a higher power of radius — so the series is not a refinement of a single measurement but a sequence of differently weighted measurements, and a spacecraft that never enters a planet can read a coarse depth profile out of it.

$Where each gravity harmonic gets its signal: J₂ from the bulk, J₁₀ from the outer 12 per cent. Why a spacecraft that never enters a planet can say something about its depth. Each zonal harmonic of the external field is an integral over the interior density weighted by r to the power of the degree plus two, and the curves here are those integrals accumulated outward: the fraction of each coefficient that has been contributed by the time the integration reaches a given fractional radius, for an interior of polytropic index 1. The weighting climbs steeply with degree, so the curves separate. Half of J₂ comes from inside 73 per cent of the radius, and half of J₁₀ from inside 88 per cent — the higher coefficients barely know the deep interior exists. That ordering is the whole basis of gravity science as a probe. A single coefficient is one number and constrains almost nothing; a series of them, each weighted differently, is a coarse depth profile, and it is how Jupiter's core turned out to be smeared over half the planet rather than sitting as a distinct sphere at the middle. Two limits are worth stating with it. The information falls off fast: by degree ten the kernel is concentrated in a shell so thin that measuring the coefficient says little about anything below it. And every curve here assumes north–south symmetry, under which the odd harmonics vanish identically — so a measured J₃ or J₅ is not a deeper probe of the same thing but a measurement of something else entirely, which at a giant planet is how fast and how deep the winds run.$

Fig. 1 Where each harmonic gets its signal. Each curve is the fraction of that coefficient contributed by the time the integration has reached a given fractional radius, for an interior of polytropic index 1. Half of J₂ comes from inside 51 per cent of the radius; half of J₁₀ from inside 88 per cent. The higher coefficients barely know the deep interior exists, and that ordering is the entire basis of gravity science as a probe.

Why a higher degree is a shallower probe

The external potential of an axisymmetric body is

V(r,\theta) = -\frac{GM}{r}\left[1 - \sum_{n\ge1} J_{2n}\left(\frac{R}{r}\right)^{2n}P_{2n}(\cos\theta)\right],

and each coefficient is an integral over the interior:

J_{2n} \propto \frac{1}{MR^{2n}}\int \rho(r)\,r^{2n+2}\,P_{2n}(\cos\theta)\,dV.

The weight is $r^{2n+2}$ . For $J_2$ that is $r^{4}$ ; for $J_{10}$ it is $r^{12}$ . A twelfth power is a very aggressive weighting: material at half the radius contributes $2^{-12}$ — a four-thousandth — of what the same material would contribute at the surface.

So the coefficients form a sequence of probes that march outward. In principle a long enough series inverts to a density profile; in practice the information falls off fast, and by degree ten the kernel is concentrated in a shell thin enough that the coefficient says almost nothing about anything beneath it.

There is a second reason the series is finite in practice. The coefficients themselves shrink rapidly — Jupiter’s $J_2$ is $1.47\times10^{-2}$ , $J_4$ is $-5.9\times10^{-4}$ , $J_6$ is $3.4\times10^{-5}$ , and by $J_{10}$ they are parts in $10^{7}$ — so measuring the tenth requires tracking precision that took until the 2010s to achieve.

Every degree but one falls on a line of slope −2. The root-mean-square size of the Earth's gravity-field coefficients at each spherical-harmonic degree, against the degree. The straight line is Kaula's rule, 10⁻⁵/ℓ², which is not a theory but a fit Kaula made in 1963 to the first satellite solutions and which every model since has stayed on. A slope of −2 says the planet's gravity has no preferred horizontal scale — the field looks the same statistically at a thousand kilometres and at a hundred. The exception is degree 2, which stands 194 times above the line, because J₂ is not a lump but the rotational flattening of the whole body and belongs to a different mechanism. That separation is the whole reason a satellite's node regression measures one number cleanly: everything else in the field is three orders of magnitude smaller and, above the atmosphere, smaller still. — Fig. 2 The general behaviour of a gravity field’s spectrum. Coefficient amplitude falls with degree along a characteristic line, which is a statement about how mass is distributed on all scales — and the departures from it are the interesting part. For a fluid planet the even zonal terms are the response to rotation and follow a steep sequence; everything else is small, and “everything else” is where the winds and the asymmetries live.

It is worth contrasting the two ends of the difficulty. Measuring $J_2$ is easy and interpreting it is hard, because $J_2$ is dominated by the bulk and every model reproduces it by construction. Measuring $J_{10}$ is hard and interpreting it is comparatively easy, because so few of the model’s parameters affect it — it is sensitive to a thin outer shell, and the shell’s density is close to something that can be measured at the cloud tops. The information about the deep interior therefore lives not in any one coefficient but in the pattern across them, and specifically in the residuals after the outer layers have been accounted for.

What the even harmonics said about Jupiter

Before a spacecraft flew a close polar orbit, Jupiter’s $J_2$ and $J_4$ were known and its $J_6$ approximately. Those three, with an equation of state for hydrogen and helium and an assumption of uniform rotation, gave interior models with a compact core of ten to twenty Earth masses at the centre and a smooth envelope above it.

A close polar orbit changed that by measuring the series out to $J_{10}$ and beyond, and to a precision several orders of magnitude better. What the higher coefficients required could not be reproduced by any model with a distinct, compact core. The models that fit have a dilute core: heavy elements not concentrated in a sphere at the middle but spread out to something like half the planet’s radius, in a composition gradient rather than a boundary.

That is a strong statement about how the planet formed, because a compact core is what core accretion produces and a smeared one is not. The leading explanations are that a giant impact during formation mixed a core outward, or that the core was never fully assembled and heavy material was dissolved into hot metallic hydrogen as the planet grew. The inference deserves scrutiny, because it is an inference about a region the harmonics barely reach. Nothing in the series carries much weight below half a radius, and a dilute core is precisely a statement about that region. What actually happened is subtler than “the harmonics saw the core”: the outer layers were pinned down so well by the high-degree coefficients that the low-degree ones, once those were subtracted, constrained the remainder tightly. The core is measured as a residual after everything above it has been fixed, which is why the precision of $J_8$ and $J_{10}$ mattered to a conclusion about the middle of the planet.

That is a common shape of argument in this collection and it is worth recognising: a quantity that no measurement is directly sensitive to becomes well determined because everything else is. It is also why the conclusion is fragile in a specific way — an error in the outer layers propagates entirely into the residual.

The odd harmonics, which should not exist

A body symmetric about its equator has no odd zonal harmonics. $J_3$ , $J_5$ and $J_7$ vanish identically for any north–south symmetric interior, whatever its radial structure.

Jupiter’s are not zero. They are small — parts in $10^{8}$ — and they were measured, and their existence is a measurement of an asymmetry rather than a deeper probe of a symmetric thing.

The asymmetry is the winds. Jupiter’s cloud-top zonal jets reach a hundred metres a second and are not symmetric between hemispheres; if those flows extend to depth, they carry mass with them and produce exactly this kind of signature. The magnitude of the odd harmonics then measures how deep the flows go, because a wind confined to a thin skin displaces very little mass and a wind reaching the core displaces a great deal.

The answer was about three thousand kilometres — a little under five per cent of the radius, and about one per cent of the mass. Below that the flows die away, and the reason they die is thought to be that hydrogen becomes metallic and conducting at those pressures, so a flow crossing field lines is braked magnetically.

That is a remarkable chain: an asymmetry in a gravity field, measured to parts in $10^{8}$ by tracking a spacecraft’s radio carrier, gives the depth at which hydrogen becomes a metal. Nothing about it involves seeing anything. There is a second asymmetry worth mentioning, because it shows how far the method has been pushed. The measured field also contains a small time-variable component, from the tide Jupiter’s moons raise on it and from the planet’s own dynamical response to that tide. Separating a tidal Love number from a static harmonic requires many passes at different moon phases, and the measurement that resulted disagreed slightly with the equilibrium value — a hint of a dynamical rather than a static response, which is a statement about the planet’s normal-mode frequencies.

A thousandth of the field, and all of the precession. Left: the Earth's figure against a sphere of the same equatorial radius, with the flattening drawn 26× its true value. The real difference between the equatorial and polar radii is 21.4 km on 6378 — 1 part in 298 — which at this size would be 0.5 pixels and invisible, so the drawing is a schematic and the number is here instead. Right: the two components of the J₂ perturbation at the surface, each as a fraction of the monopole μ/r², both differentiated from the potential rather than quoted. The radial one strengthens the inward pull by 1.62×10⁻³ over the equator, where the extra mass is, and weakens it by 3.25×10⁻³ over the poles, vanishing at ±35.26° where P₂ does. The transverse one is zero at the equator and at both poles and peaks at ±45°, at 1.62×10⁻³ — and that is the component that does the work. It pulls an inclined orbit back towards the equatorial plane, which is a torque about the line of nodes, and a torque applied to something already turning moves it sideways rather than back. Averaged over an orbit the pair leave a, e and i untouched and turn the whole plane instead, which is why a term a thousandth of the field is the largest single perturbation on almost every satellite ever flown. — Fig. 3 The shape the harmonics are a description of. A rotating fluid body is an oblate spheroid to first order and the flattening is set by the ratio of centrifugal to gravitational acceleration at the equator — so the even harmonics are, in the simplest reading, a measurement of how fast the body turns and how its mass is distributed, in a combination neither one alone resolves. Assuming the rotation is what turns this figure into a statement about the core.

The rotation that has to be assumed

There is an awkwardness at the heart of all of this and it deserves to be stated plainly.

The even harmonics are a response to rotation. A non-rotating fluid planet would be a sphere with no harmonics at all; the whole series exists because centrifugal force flattens the body, and the flattening’s shape depends on the interior’s response. So converting a measured $J_{2n}$ into a statement about density requires knowing the rotation rate.

For Jupiter that is easy: the magnetic field is tied to the deep interior and its rotation period is measured to milliseconds from the periodicity of radio emission. For Saturn it is not. Saturn’s magnetic field is very nearly axisymmetric, so it provides no clock, and the planet’s rotation period was uncertain by several minutes for decades. A rotation period uncertain by a few minutes is a several-per-cent uncertainty in the centrifugal term, which propagates directly into every interior model.

The eventual resolution came from an unrelated observable: waves in Saturn’s rings, driven by oscillations of the planet itself, whose frequencies encode the interior’s normal modes. That is seismology by proxy — the rings are a detector for the planet’s oscillations — and it gave a rotation period of about ten hours thirty-three minutes, and with it a set of interior models that had been waiting on the number. The Saturn episode is worth remembering as a caution about what counts as a measured quantity. For thirty years the planet’s rotation period was quoted from the periodicity of its kilometric radio emission — a real, repeatable measurement — and the period was later found to drift, to differ between hemispheres, and to be a property of the magnetosphere rather than of the interior. A number that had been treated as an input to every interior model turned out to be measuring something else entirely.

$Where each gravity harmonic gets its signal: J₂ from the bulk, J₈ from the outer 23 per cent. Why a spacecraft that never enters a planet can say something about its depth. Each zonal harmonic of the external field is an integral over the interior density weighted by r to the power of the degree plus two, and the curves here are those integrals accumulated outward: the fraction of each coefficient that has been contributed by the time the integration reaches a given fractional radius, for an interior of polytropic index 2. The weighting climbs steeply with degree, so the curves separate. Half of J₂ comes from inside 58 per cent of the radius, and half of J₈ from inside 77 per cent — the higher coefficients barely know the deep interior exists. That ordering is the whole basis of gravity science as a probe. A single coefficient is one number and constrains almost nothing; a series of them, each weighted differently, is a coarse depth profile, and it is how Jupiter's core turned out to be smeared over half the planet rather than sitting as a distinct sphere at the middle. Two limits are worth stating with it. The information falls off fast: by degree ten the kernel is concentrated in a shell so thin that measuring the coefficient says little about anything below it. And every curve here assumes north–south symmetry, under which the odd harmonics vanish identically — so a measured J₃ or J₅ is not a deeper probe of the same thing but a measurement of something else entirely, which at a giant planet is how fast and how deep the winds run.$

Fig. 4 The same expansion for a more centrally condensed interior. Each successive even harmonic samples a shallower shell, so the sequence

J_2, J_4, J_6\ldots

is a crude tomography of the density profile — and a body with more of its mass near the centre produces a sequence that dies away faster. Reading a core out of the sequence is reading how quickly the terms fall, which is why measuring

J_8

and

J_{10}

was worth a dedicated mission.

The same conclusion reached by a different instrument

Saturn’s interior was settled by a method that shares nothing with the one above, and the agreement is worth recording because the gravity inversion’s non-uniqueness makes an independent check unusually valuable.

The rings respond to the planet’s own oscillations. A normal mode of the interior produces a periodic gravitational perturbation at the mode’s frequency, and where that frequency matches the orbital frequency of ring material at some radius, the resonance launches a spiral density wave — a feature a few tens of kilometres wide, at a radius fixed entirely by the mode’s frequency. Reading the radii of those features off an occultation profile is therefore reading the planet’s oscillation spectrum, in the same way a stellar oscillation spectrum is read off a light curve.

Several dozen such waves have been identified and matched to modes. The frequencies they give require a stably stratified region extending from the centre out to something like sixty per cent of Saturn’s radius — a composition gradient rather than a boundary, which is to say a dilute core, reached without inverting any gravity harmonic at all.

That Jupiter’s gravity field and Saturn’s rings independently arrive at the same unexpected structure is the strongest reason to believe either. The two failure modes have nothing in common: the gravity inversion is limited by the equation of state and by the correlation between coefficients, while the ring measurement is limited by whether a wave has been correctly attributed to a mode. A conclusion that survives two methods with unrelated weaknesses is worth more than either method’s error bar suggests, which is the same argument the rest of this collection keeps making about masses, distances and ages, applied here to the inside of a planet.

There is a practical asymmetry between them that is worth keeping. The ring method needs a ring, so it applies to one planet in the solar system and to nothing outside it; the gravity method needs only a spacecraft and applies to every body one can be sent to. The check is available exactly once, and it is being used to certify a technique that will be applied where no check exists.

What has to be imported

A measured harmonic is a number. Turning it into a core mass requires an equation of state, and for a giant planet that means the behaviour of a hydrogen–helium mixture at pressures of tens of millions of atmospheres and temperatures of thousands of kelvin.

Those conditions are reached in the laboratory only fleetingly, by shock compression, and the measurements have moved substantially over the past two decades — including a revision to where hydrogen becomes metallic that changed published core masses by several Earth masses. Different equations of state applied to the same measured harmonics give core masses differing by more than their formal uncertainties.

There is also the question of helium. Helium is thought to be immiscible in metallic hydrogen at Saturn’s interior conditions, so it rains out — which redistributes mass, releases gravitational energy, and changes the composition of the envelope, in the same way and for the same reason that a freezing white dwarf separates its carbon from its oxygen. Saturn radiates more energy than it absorbs by considerably more than contraction alone can explain, and helium rain is the standard account of the excess. Every one of those effects enters the interpretation of the gravity field.

255 km sees degree 154; 35786 km sees degree 6. The same field spectrum, multiplied by the upward continuation factor (R/r)ˡ⁺¹ for orbits at 255 km, 450 km, 800 km, 35786 km. A harmonic of degree ℓ has ℓ bumps around the planet, so it falls off with height like a wave of that wavelength — fast, and faster the finer it is. The horizontal rule is a measurement floor; where each curve crosses it is the highest degree that orbit can feel at all, and the answer is 154 at 255 km, 101 at 450 km, 65 at 800 km, 6 at 35786 km, or 130 km, 198 km, 308 km, 3340 km of horizontal resolution on the ground. Nothing in that arithmetic is an instrument. It is why GOCE flew at 255 kilometres with drag compensation rather than at a comfortable altitude with a better gradiometer, why the Moon's mascons were not seen until something orbited low over them, and why a geostationary satellite's ephemeris needs a field with four terms in it. — Fig. 5 Why a close orbit is worth so much. A harmonic of degree ℓ falls off as the altitude to the power ℓ, so a spacecraft at a thousand kilometres senses degrees a satellite at geostationary altitude cannot detect at all. The measured degree limit is set almost entirely by the periapsis altitude, which is the reason a gravity mission flies a highly eccentric orbit and does its science in the few hours around each closest approach.

Two of the three quantities the method depends on can be read over wider ranges, and both show the same trade.

$Where each gravity harmonic gets its signal: J₂ from the bulk, J₁₂ from the outer 11 per cent. Why a spacecraft that never enters a planet can say something about its depth. Each zonal harmonic of the external field is an integral over the interior density weighted by r to the power of the degree plus two, and the curves here are those integrals accumulated outward: the fraction of each coefficient that has been contributed by the time the integration reaches a given fractional radius, for an interior of polytropic index 1. The weighting climbs steeply with degree, so the curves separate. Half of J₂ comes from inside 73 per cent of the radius, and half of J₁₂ from inside 89 per cent — the higher coefficients barely know the deep interior exists. That ordering is the whole basis of gravity science as a probe. A single coefficient is one number and constrains almost nothing; a series of them, each weighted differently, is a coarse depth profile, and it is how Jupiter's core turned out to be smeared over half the planet rather than sitting as a distinct sphere at the middle. Two limits are worth stating with it. The information falls off fast: by degree ten the kernel is concentrated in a shell so thin that measuring the coefficient says little about anything below it. And every curve here assumes north–south symmetry, under which the odd harmonics vanish identically — so a measured J₃ or J₅ is not a deeper probe of the same thing but a measurement of something else entirely, which at a giant planet is how fast and how deep the winds run.$

Fig. 6 Where degrees 2, 6 and 12 draw their signal from. The higher the degree, the shallower the shell that produces it, so a gravity field measured to high degree says a great deal about the outer layers and almost nothing about the centre.

The same method, elsewhere

The technique scales down and it scales sideways.

Icy moons. A flyby measures $J_2$ and $C_{22}$ , and if the body is assumed hydrostatic the two are in a fixed ratio and either gives the moment of inertia. Every published moment of inertia for a Galilean satellite comes from that assumption, and Io’s and Europa’s are consistent with it while Callisto’s is marginal — which is itself the evidence that Callisto is incompletely differentiated, and which sits alongside what the moments of inertia say about the rest of the solar system.

The Moon. A pair of spacecraft flying in formation measured the lunar field to degree 900 by tracking their separation to microns. That resolution reveals not a core but the crust: buried impact basins, dyke swarms, and a crustal thickness map with a resolution of tens of kilometres.

The Earth. Continuous measurement of the field’s changes now tracks water: ice sheets losing mass, aquifers being depleted, seasonal water storage. The instrument built to measure a static interior turned out to be a scale for the hydrosphere.

Asteroids. A small body’s field is measured by orbiting it closely, and the low-degree terms give the mass and the shape-implied density distribution — which, compared with the shape a stellar occultation gives, says whether the interior is uniform or has a denser core. For a body that is a pile of rubble the answer is usually “uniform, and half empty”.

What the picture cannot show

The hero figure draws kernels for a body of one polytropic index and a smooth density profile, and it makes the inference look cleaner than it is.

It does not show the correlation between coefficients, which is the largest practical difference between the figure and a real analysis. What a fit actually produces is not a set of independent measurements but a covariance matrix, and the higher harmonics are correlated with each other and with the rotation rate and with the wind profile. Quoting one coefficient with an error bar is a summary of something with a great deal more structure.

It does not show the non-uniqueness. The series is a set of integrals, and inverting a finite set of integrals for a function has infinitely many solutions. What the published core masses are is the range of core masses among models that a particular family of equations of state can produce, which is a narrower thing than the range consistent with the data.

And it assumes hydrostatic equilibrium and uniform rotation on cylinders, both of which are approximations whose failures are exactly the effects — winds, differential rotation, dynamical tides — that the odd harmonics and the highest even ones were measured to detect.

One more absence is worth naming. The figure is drawn for a rotating fluid body in equilibrium, and its kernels are the kernels of a static problem. A planet whose winds reach depth is not static in the relevant sense: part of the measured field comes from mass carried by a flow rather than from a density arrangement at rest, and the two are indistinguishable in the coefficients themselves. What separates them is the parity — the winds’ asymmetry between hemispheres puts their contribution into the odd harmonics, where a static symmetric interior has none. That is a fortunate accident of Jupiter’s particular flows rather than a general method, and for a planet whose winds happened to be symmetric there would be no way to tell. And the attenuation over the range of altitudes a real mission can occupy.

200 km sees degree 184; 20000 km sees degree 8. The same field spectrum, multiplied by the upward continuation factor (R/r)ˡ⁺¹ for orbits at 200 km, 600 km, 20000 km. A harmonic of degree ℓ has ℓ bumps around the planet, so it falls off with height like a wave of that wavelength — fast, and faster the finer it is. The horizontal rule is a measurement floor; where each curve crosses it is the highest degree that orbit can feel at all, and the answer is 184 at 200 km, 81 at 600 km, 8 at 20000 km, or 109 km, 247 km, 2505 km of horizontal resolution on the ground. Nothing in that arithmetic is an instrument. It is why GOCE flew at 200 kilometres with drag compensation rather than at a comfortable altitude with a better gradiometer, why the Moon's mascons were not seen until something orbited low over them, and why a geostationary satellite's ephemeris needs a field with four terms in it. — Fig. 8 What each altitude can resolve. Two hundred kilometres reaches degree 184 and twenty thousand reaches single figures, because every degree is attenuated by a further factor of the radius ratio — which is why a gravity mission is a low-altitude mission and why its lifetime is short.

Where the ladder goes

The earlier rungs of this anchor treated the oblateness as something an orbit feels: the shape read off a node, and which parts of the field a satellite is allowed to notice. This one turns the field into an instrument pointed inward.

The next steps are toward the two things the static field cannot supply. One is time dependence: the tidal response, which is a Love number and a completely independent constraint on the same interior. The other is normal modes — the ring seismology above, and its ambition of measuring a giant planet’s oscillation spectrum directly, which would do for Jupiter what helioseismology did for the Sun and turn one integral into a resolved profile.

About the same objects

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

Four numbers that weigh a planet's core gravity science · moment of inertia factor · zonal harmonic
A floor under the centre that assumes nothing equation of state · polytrope
An equation of state is already a star equation of state · polytrope
An ocean is detected and its depth is not gravity science · radau darwin relation
The mass a cold star cannot exceed equation of state · polytrope

What links here

Essays that link to this one from their own argument.

Whether the heavy material sank gravitation
An ocean found in a Doppler residual spaceflight

The objects this essay names

Each one links to every other essay that touches it.

Differential rotationDilute coreDoppler tracking Equation of state Gravity scienceMetallic hydrogenMoment of inertia factor Oblateness Polytrope Radau darwin relation Zonal harmonic

Why a higher degree is a shallower probe #

What the even harmonics said about Jupiter #

The odd harmonics, which should not exist #

The rotation that has to be assumed #

The same conclusion reached by a different instrument #

What has to be imported #

The same method, elsewhere #

What the picture cannot show #

Where the ladder goes #

Where this goes next