The field outside that cannot find the core

Counting what comes out is the whole of Gauss’s law: the flux of a field through any closed surface depends on what the surface encloses and on nothing else. Applied forwards, it is one of the most powerful tools in physics, because it says the field outside a symmetric body can be written down knowing only the body’s total charge or mass. The pull that grows on the way down used it to compute gravity inside the Earth from a density profile that seismology had supplied.

Read backwards, the same law is a statement about what cannot be known. If the field outside depends only on what is enclosed, then everything about how the enclosed mass is arranged, beyond the quantities the field happens to depend on, is invisible from outside. That is not a limit of instruments. No measurement of gravity outside a body, however precise and however complete, can supply what the field does not contain. What it does contain turns out to be a specific and short list, and the most important thing about a planet’s interior is not on it.

One field outside, three bodies inside #

The cleanest case is a spherically symmetric body. Outside it, the field is that of a point with the same mass, whatever the distribution of that mass with radius — the result Newton needed a page of geometry for and Gauss’s law gives in a line.

The claim in the figure is not quoted from the theorem. At a distance of 1.7 radii the pull of each body was computed by adding up the pull of every one of its mass elements, and all three agree with the pull of a point of the same mass to better than a part in five hundred. Outside, the three bodies are the same body.

Inside they could hardly be less alike. The uniform ball’s gravity falls in a straight line to the centre. The layered body’s rises through its light mantle to 1.24 times its surface value at the top of the dense core, then falls. The hollow shell has no gravity at all in its cavity. And the number that best summarises each arrangement, the moment-of-inertia factor $C/MR^2$ — how hard the body is to spin, relative to a point mass at its surface — is 0.400, 0.330 and 0.551. The Earth’s value is 0.3307. Gravity measured anywhere outside the Earth is the same for all three models, and only one of them has the Earth’s moment of inertia.

The electrical version of the statement is just as strict. The potential outside a charged sphere is the same whether the charge sits on its surface, as on a conductor, or is spread through a ball, or is concentrated at the centre. An electrode array wrapped around any volume sees only what the charge inside it looks like from outside, and Hermann von Helmholtz proved in 1853 that the currents inside a conducting body cannot be reconstructed uniquely from the potentials they produce on its surface. It is why the electrical activity of the brain, measured on the scalp, has no unique source inside the head, and why every reconstruction of one is a model added to the data.

Every exterior field has a skin that makes it #

There is a sharper way to state how little the outside knows, and it comes from the uniqueness of solutions to Laplace’s equation. In empty space outside a body, the potential is fixed completely by its values on any surface enclosing the body — its value at a point is an average over what surrounds it — so any two sources that produce the same potential on that surface produce the same potential everywhere outside it.

It follows that every exterior field, however complicated the body inside, can be reproduced exactly by a thin layer of mass or charge spread over the body’s own surface with a suitable varying density. George Green showed this in 1828. A planet with a molten iron core, a hollow shell and a painted skin of the right pattern are indistinguishable from outside. The skin is not a physical proposal; it is the demonstration that the exterior field carries at most the information of a function on a surface, and a function on a surface cannot encode a whole volume.

What the field does contain #

Remove the spherical symmetry and the field outside acquires more structure, and that structure is a list of numbers. The potential outside any body — a single number at every point, fixed by its values on any enclosing surface — can be expanded in a series of moments — the total mass first, then terms that fall off faster with distance and describe progressively finer features of the shape of the field. For a planet spinning about an axis the leading departure from a sphere is $J_2$, which measures how much the field is flattened, followed by $J_4$, $J_6$ and the rest.

Each coefficient is an integral over the body: the density, weighted by the distance from the centre raised to the power of the degree and by a polynomial in the direction. So each is one number obtained by averaging the interior in a particular way, and two very different interiors can give the same one.

A uniform body flattened by 0.34 per cent, about the Earth’s flattening, has $J_2 = 0.001339$. A perfect sphere with a thin ring around its equator holding 0.268 per cent of its mass has exactly the same $J_2$, and at the level of the flattening of their fields the two cannot be told apart. At the next order they part completely: $J_4$ is $-3.85 \times 10^{-6}$ for the flattened body and $-0.001$ for the ring, a factor of 260, and $J_6$ differs by a factor of seventy thousand. Each coefficient was computed by integrating over the body rather than taken from a formula.

So the higher coefficients carry real information about shape, and a well-measured field rules out a great many interiors. What no finite set of them does is fix where along each radius the mass lies. Every coefficient of degree $l$ weights the density by $r^l$, and a redistribution of mass that preserves every one of those weighted averages — which is always possible, in infinitely many ways — leaves the whole exterior field unchanged.

The number the field leaves out #

The most important instance of that freedom is the moment of inertia, and it can be stated exactly. The moment of inertia about the spin axis is an integral of the density weighted by the squared distance from the axis. Part of that integral is the flattening: the difference between the polar and equatorial moments, $C - A$, is $MR^2J_2$, which gravity fixes. The rest is the trace, the density weighted by the squared distance from the centre in all directions equally, and that part produces no field outside at all — it is precisely the spherically symmetric redistribution that the first figure showed to be invisible.

So gravity alone fixes $C - A$ and not $C$. Something that responds to $C$ itself is needed, and the Earth supplies one in the motion of its spin axis. The Sun and Moon pull on the equatorial bulge and make the axis sweep round a cone, and the rate of that precession depends on the torque, which is set by $C - A$, divided by the spin angular momentum, which is set by $C$.

Every point on the line has the Earth’s gravity outside. Moving along it changes only how the mass is arranged with depth, and the precession period changes in proportion to $C/MR^2$. The measured precession, once every 25,772 years, gives the ratio $(C-A)/C = 3.27379 \times 10^{-3}$, and dividing the field’s $J_2$ by it gives $C/MR^2 = 0.3307$. A uniform Earth with the same gravity outside would precess once every 31,173 years. The difference between those two numbers, astronomical in origin, is the evidence that the Earth’s mass is concentrated towards its centre — and it was known in the nineteenth century, long before seismic waves located the core.

The same combination weighs other worlds. Mars’s moment-of-inertia factor, 0.364, comes from tracking its precession with landers on its surface; the Moon’s, 0.393, from its slow rocking librations, tracked by laser ranging to reflectors left on its surface. Where no precession has been measured — for Jupiter and Saturn — the factor is inferred by assuming the planet is a fluid in equilibrium, which the next section returns to.

Two numbers, and still a family #

The mass and the moment of inertia together rule out the uniform ball and the hollow shell. They do not pick out one planet. A body built from just two uniform layers, a core and a mantle, has three free numbers — the core’s radius and the two densities — and two constraints, so every core radius has its own pair of densities that satisfies both.

A core a third of the planet’s radius would have to be more than five times the mean density, with a mantle only a little lighter than average. A core three quarters of the radius would be barely denser than the mantle wrapped round it. Every combination on the two curves was checked by integrating the moment of inertia over the layered body, and every one produces the same gravity outside, the same precession and the same total mass. Gravity and precession between them have reduced an infinite freedom to a one-parameter family, and no further measurement of either will reduce it more.

The seismic radius of the core, 3,480 kilometres, closes the family. At that radius the two layers must be 4.15 and 12.5 grams per cubic centimetre, which is close to the averages of the Earth’s real mantle and core — neither of which is uniform, since both are compressed more strongly with depth. The agreement is a genuine test rather than an input: the core’s radius was found from the timing of waves that pass through it, which knows nothing about mass, and the densities it implies have to be consistent with rock and iron under the pressures the Earth’s interior imposes. They are.

This is also why a planet large enough to pull itself round is easier to model than it has any right to be. A body in hydrostatic equilibrium has a shape and a density profile linked by its own gravity and spin — the flattening that its rotation’s outward term produces depends on how centrally its mass is concentrated — and assuming equilibrium supplies a relation between $J_2$ and $C/MR^2$ without any measurement of precession. That assumption, known as the Radau–Darwin approximation, is how moment-of-inertia factors are quoted for Jupiter and Saturn, and it is exactly as good as the equilibrium it assumes.

Why depth is so hard to see #

The last figure explains why even a perfectly measured field cannot settle how deep a mass lies, and it is the practical form of the whole argument.

A lump of mass at depth $d$ contributes to the degree-$l$ coefficient in proportion to $((R-d)/R)^l$, which was checked for one depth and degree by projecting the potential of a buried point mass directly. Degree 60 resolves features a few hundred kilometres across. A lump 30 kilometres down shows at three quarters of its surface strength there; 670 kilometres down, at a thousandth; at the top of the core, 2,891 kilometres down, at $2 \times 10^{-16}$. Fine detail in the field is a map of the crust. The deep interior shows only in the broadest terms.

That fading is also a trade. At any single degree, a small mass near the surface and a larger mass deep down produce the same coefficient, and the ratio of masses is just the attenuation factor. Satellites such as GRACE, which measured the Earth’s field month by month from 2002 by tracking the changing distance between two spacecraft, can see a season’s groundwater arrive in a river basin; they can do so because the water is at the surface, where its signal is undamped, and because the question asked is about a known depth. Asked where along a radius an unknown anomaly sits, the data have no answer without another kind of measurement.

What the idealised bodies leave out #

The bodies are rigid and static. A real planet responds to the tides that make it precess, its fluid core does not rotate with the mantle, and its precession includes nutations whose details depend on the elasticity and the core. Those effects are measured, and they add information about the interior rather than removing it — which is how the core’s flattening and the coupling between core and mantle are estimated.

The field is taken as the body’s alone. A spinning planet’s measured gravity includes the centrifugal term of its rotation, and the tides raised by the Sun and Moon deform it and add time-varying coefficients of their own. Both are removed before the coefficients are interpreted, and the tidal response is itself a measurement of the interior: how far the solid planet yields to the Moon depends on how rigid it is and on whether its core is liquid.

The coefficient figures use idealised shapes. A uniform flattened body and a sphere with a ring are chosen because their coefficients are easy to compute and very different at the next order, not because either resembles a planet. A real planet’s $J_4$ is small and negative, close to what a flattened fluid body gives, and that closeness is itself evidence of near-equilibrium.

The attenuation is for a single point mass. A spread-out anomaly has its own shape, which further suppresses its high-degree signal, and the curves are the most favourable case for seeing a buried mass rather than a typical one.

An infinite family behind three drawings #

The first figure shows three bodies, and the family it stands for is infinite: every rearrangement of mass along the radius that keeps the total fixed produces the same exterior curve. The drawing cannot show the size of that family, only three members of it, and it is the size that makes the inverse problem ill-posed rather than merely difficult.

Nor do the figures show what does resolve the ambiguity, because it is not gravity. Seismic waves travel through the interior and their speeds depend on the material; the free oscillations of the whole planet after a large earthquake, each a mode with its own count of nodes, depend on its density at every depth; the magnetic field constrains the conducting core. The interior profile that the pull that grows on the way down used is a joint inversion of all of them, and gravity’s role in it is to fix the mass and the moments while the waves supply what the moments cannot.

Still open: how much of the deep interior can be seen at all #

Some deep structure is visible in the broadest features of the field. Two vast regions of unusually slow seismic speeds sit at the base of the mantle beneath Africa and the Pacific, each thousands of kilometres across, and whether they are denser or lighter than their surroundings — whether they are ancient piles of dense material or hot, buoyant upwellings — changes the long-wavelength field, the tides of the solid Earth and the way the planet responds to the Moon’s pull. Measurements of all three have been used to argue that the regions are dense, and other analyses find that the evidence is compatible with either. The answer is a question about how much the broadest few coefficients of the field, combined with everything else, can say about mass three thousand kilometres down — which is the inverse problem at its hardest.

The habit worth carrying away is to read a conservation law in both directions. A law that says the field depends only on the total is also a law that says the field cannot see anything but the total, and the useful question about any such law is which other measurement responds to what it hides. For the Earth’s core the answer was its wobble; for the brain’s currents it is a model of the head; and for the deep mantle it is still being worked out.