The shell can be flattened if it thins at the poles

A uniform spherical shell pulls nothing inside it. The proof is a pair of cones: through any interior point, a thin double cone cuts the shell in two patches whose areas grow as the squares of their distances from the point, so the masses in them grow as the squares of the distances while their pulls fall as the squares, and the two pulls cancel exactly. Every cone through every interior point balances, and the interior is empty of force. It is one of the two results that make spherical bodies calculable; the other is that the same shell pulls outside as though its mass were at the centre.

A flat ring does not share the property. Inside a ring the pull points outward: the cone argument fails in two dimensions, the nearer part of the ring wins, and a body placed off-centre inside a rigid ring is pulled towards the rim. Between a sphere and a ring lies every degree of flattening, and the obvious guess is that the interior field grows smoothly from nothing to the ring’s outward pull as a shell is squashed. The guess is wrong for one family of flattened shells, and that family is the reason the interior of a flattened body can be calculated at all.

A shell with the right thickness #

Newton found the family and proved the result in the Principia. Take two ellipsoids, one a scaled copy of the other — the same axis ratios, the same centre, one slightly larger in every direction by the same factor. The region between them is a homoeoid. Because every dimension grows by the same factor, it is thicker where the ellipsoid is long and thinner where it is short: a flattened homoeoid is fat at its equator and thin at its poles, in proportion to the local radius.

The figure computes the field directly, by placing twenty thousand equal masses on a homoeoid of axis ratio one half — far flatter than any planet; Saturn, the flattest, has an axis ratio of about nine-tenths — and adding their pulls at points along its two axes. Inside the shell the sum is zero to the precision of the sampling, along the long axis and the short one alike. The flattened shell is as empty of force inside as the spherical one. Outside, it is nothing like a point: along the long axis, just beyond the shell, the pull is half as strong again as a point of the same mass would exert, because the mass concentrated in the thick equatorial rim is close by; along the short axis it is weaker, because the thin polar caps are close and the heavy rim is far.

The masses in the calculation are placed by a trick that is itself the reason the theorem holds. A thin spherical shell, uniformly covered, is stretched along two axes and compressed along the third; the stretch carries every piece of the sphere to a piece of the homoeoid and carries equal masses to equal masses. The homoeoid is a sphere seen through a linear transformation, and a linear transformation preserves the property the cone argument needs.

Newton needed the result for a practical question: how flattened is the Earth? Treating it as a homogeneous fluid spinning once a day, and using the theorem to compute the gravity inside a flattened body of uniform density, he found that its equator should bulge by one part in 230 — the first prediction of the Earth’s shape from its rotation, made before any measurement had shown that it was not a sphere. The measured value, established by the geodetic expeditions of the following century, is one part in 298. The difference is not an error in the theorem but in the assumption of uniformity: the Earth is denser towards its centre, its inner layers are less flattened than its surface, and a centrally concentrated planet bulges less for the same spin. The shortfall of Newton’s figure was the first evidence, from the shape alone, that the Earth has a dense interior.

Why the cones still balance #

The cone argument needs one thing: that a thin cone through the interior point cuts equal thicknesses of shell at its two ends. For a spherical shell that is obvious by symmetry. For a homoeoid it follows from a property of similar, concentric ellipses. Any straight line meets the inner ellipse in two points and the outer in two, and the midpoints of the two chords coincide — the midpoint of a chord of an ellipse lies on the diameter conjugate to its direction, which is the same line for similar concentric ellipses. The two segments of shell between the inner and outer crossings are therefore equal in length on both sides of the point, whatever its position and whatever the line’s direction. The figure checks four lines and finds the segments equal to the precision of the arithmetic.

With equal thicknesses at both ends of every cone, the rest of the argument is the sphere’s. The masses cut out at the two ends are proportional to the squares of their distances from the point, their pulls are inversely proportional to those squares, and they cancel. Every cone balances, and the interior is empty — however extreme the flattening.

The same surface with the wrong thickness #

The theorem depends on the thickness following the homoeoid’s rule, and the simplest way to see how much is to break the rule.

Spread the same total mass evenly over the same ellipsoidal surface — a shell of uniform thickness rather than a homoeoid — and the interior field returns, and is not small: a fifth of the field a point of the same mass would exert at the shell’s long radius. Compared with a homoeoid, a uniform-thickness shell has too much mass near its poles and too little in its equatorial rim, and the polar excess pulls: a body inside is drawn towards the poles along the short axis and pushed towards the centre along the long one. The empty interior is not a property of flattened shells, or of ellipsoids. It is a property of one precise way of distributing mass on them, and a slight departure from it leaves a field inside.

Every ellipsoid is a nest of homoeoids #

The theorem’s power is that any body whose surfaces of equal density are similar, concentric ellipsoids can be built from homoeoids, like an onion. A uniform ellipsoid is one; a flattened planet whose density rises inwards on similar surfaces is another; so is a galaxy’s dark halo if its density contours are similar ellipsoids. For a point inside such a body, every homoeoid outside the point contributes nothing, by the theorem, and only those inside it pull. The field at any point depends only on the mass interior to the ellipsoidal surface through it — exactly as the field inside a spherical body depends only on the mass inside the sphere through the point, which is what makes a straight tunnel through a uniform sphere take the same time whatever its length.

For a uniform ellipsoid the result is especially simple: the field inside is linear in position, with a different constant along each axis. For a uniform oblate spheroid the pull towards the axis is proportional to the distance from it, and the pull towards the equatorial plane is proportional to the height above it, with coefficients that depend only on the flattening. That is the calculation behind the shapes of rotating fluid planets and every model of the interior of a flattened body; without the homoeoid theorem each would need a separate, much harder integral over the whole body.

A planet’s layers are not quite similar #

The theorem’s condition — layers that are similar ellipsoids — is met exactly by a uniform body and only approximately by anything else, and the most familiar case shows how. A rotating fluid planet flattens because its spin throws its outer layers outward at the equator, and each level surface inside it takes up the shape that balances gravity against rotation at that depth. Deep inside, where the density is high and the enclosed mass pulls strongly, a layer is flattened less than the surface is; near the surface, where the pull is weaker relative to the spin, more. Clairaut worked out the relation in the eighteenth century: the flattening of the layers increases outwards, by an amount that depends on how centrally concentrated the planet is. A planet with a dense core has nearly spherical inner layers and a flattened outer skin.

Such a planet is not a nest of homoeoids — its layers change shape with depth — and the simple statement that only the interior layers pull no longer holds exactly. The interior field depends slightly on the layers outside, through the difference between their shape and that of the layers inside. The departure is small for the Earth, whose flattening changes by a factor of about two from core to surface, and it is where the information about the interior lives: a moment of inertia that says whether the heavy material sank and the flattening of the surface are tied together through exactly this variation of shape with depth, and measuring both constrains how the density rises inward.

It is also why the part of a planet’s shape that its spin cannot explain is so informative. A fluid body in equilibrium has layers whose shapes follow from its rotation and its density profile alone; any excess flattening, or any departure from axial symmetry, is either a relic of a faster spin frozen into a rigid body or a sign of mass that is not in equilibrium, and the homoeoid theorem is the baseline against which that excess is measured.

The exterior, which the theorem does not simplify #

The theorem makes the inside of a homoeoid trivial and says nothing simple about the outside, and the first figure shows why that matters for measurement. Outside a flattened body the field departs from a point’s most strongly near the body and along its axes, and the departure is expanded in a series of terms of increasing order: the quadrupole that the Earth’s bulge produces, and the smaller terms after it. Each term is an integral over the interior density weighted by a different power of the radius, which is how a planet’s gravity field, measured from outside by a spacecraft’s orbit, becomes a coarse profile of its interior. A body built of homoeoids has a particularly simple series, with every term determined by the shapes and masses of its layers; a body whose layers are not homoeoids has additional terms, and those are the ones that reveal the non-uniformities.

A faster bounce than an orbit #

The two coefficients differ, and the difference has a consequence for any star moving inside a flattened body.

A star on a nearly circular orbit, tilted slightly out of the equatorial plane, goes round at the orbital frequency and bobs above and below the plane at the vertical frequency. Inside a sphere the two are equal, and the tilted orbit is simply a circle in a tilted plane, closing on itself. Inside a flattened body the mass is concentrated towards the equatorial plane, the pull back towards the plane is stronger than the pull towards the axis, and the star bobs faster than it goes round: fourteen per cent faster for an axis ratio of 0.8, half as fast again for 0.5. It is the same effect that a satellite feels from the Earth’s equatorial bulge, seen from inside the bulge rather than outside it.

The plane that turns #

A star that bobs faster than it orbits crosses the equatorial plane a little earlier each time round, so its ascending node — the point where it rises through the plane — slips backward along the orbit. The orbit’s plane turns steadily about the symmetry axis. Inside a homogeneous spheroid with a ten per cent flattening, the node slips by more than twenty degrees per revolution; after a dozen revolutions the plane has turned all the way round. In a real galaxy halo, whose density falls with radius, the effect is smaller at a given flattening than in the homogeneous case drawn, but it is of the same kind and accumulates in the same way.

That turning is the most direct handle there is on the shape of a dark halo. A satellite galaxy torn apart by the Milky Way’s tides leaves a stream of stars spread along its orbit, and a stream several orbits long records the orbit’s plane at several epochs. If the halo is spherical, the stream lies in a single plane; if it is flattened, the older parts of the stream lie in planes turned from the younger, and the amount of turning measures the flattening. Streams stretching across much of the sky have been used exactly this way, and so have polar-ring galaxies, whose rings of gas and stars orbit over the poles of their host discs and would be twisted by any flattening of the host’s halo.

What the streams say, and do not #

The answers have not been consistent, and the homoeoid theorem is part of why. The Sagittarius stream, the longest and best-studied, was fitted for years by halo shapes that disagreed with each other — flattened, elongated, or triaxial with an orientation no formation model predicted — depending on which part of the stream was fitted and which other constraints were imposed. Part of the difficulty was that the Milky Way’s largest satellite, the Large Magellanic Cloud, is massive enough to drag the halo’s centre and distort the streams that pass near it, a perturbation the static, ellipsoidal halo of the theorem does not contain. Models that include it describe the stream much better with a halo closer to spherical. Thinner streams from globular clusters, closer to the disc, suggest a halo nearly round in the inner tens of kiloparsecs, with limits on the flattening of a few tenths.

The theorem underlies all of these analyses because it is what makes an ellipsoidal halo’s field calculable in the first place: a halo whose density contours are similar ellipsoids has a field computable from one-dimensional integrals, and every halo model fitted to a stream uses that structure. A halo whose contours are not similar ellipsoids — twisted with radius, or lopsided — has to be computed another way, and one lesson of the Sagittarius results is that the real halo may be one of those.

What the pictures do not show #

The field figures use a thin homoeoid sampled at twenty thousand points; the tiny residual inside is the sampling’s, and a finer sampling shrinks it without limit. The shell is infinitesimally thin in the argument and thin in the calculation, and its own field on the shell is singular, which is why the curves are broken there. The frequency and precession figures use a homogeneous spheroid, whose interior field is exactly linear; real haloes and planets are not homogeneous, and for them the ratio of frequencies depends on radius, is smaller at a given flattening, and needs the theorem applied layer by layer. And the exterior field of a homoeoid, which the first figure shows departing from a point’s, has its own beautiful property the figures leave out: two thin homoeoids whose surfaces are confocal rather than similar — sharing the same foci — have the same exterior field if they have the same mass. The interior of a homoeoid is simple because of similarity; the exterior is simple because of confocality.

Still open: whether the halo’s layers are similar #

Everything the theorem gives assumes that the density’s contours are similar ellipsoids — the same shape at every radius, aligned, concentric. Simulated dark haloes are not like that: they are rounder in the middle, where the galaxy’s own gravity has pulled them into shape, and more elongated outside, with axes that twist with radius as the halo’s assembly history varied. A halo whose layers change shape is not a nest of homoeoids, its interior field depends on more than the mass inside each surface, and the planes of streams at different radii can turn at rates no single flattening explains. Whether the Milky Way’s halo is round inside and flattened outside, as simulations suggest, or round throughout, as some streams suggest, is being measured stream by stream, radius by radius — and the answer will say how far the theorem that makes haloes calculable applies to the one galaxy that can be mapped in this much detail.