The spinning fluid that becomes a cigar

In the Principia, Newton worked out the shape of a rotating Earth by treating it as a uniform fluid and balancing gravity against the spin at the poles and the equator. He found it should be flattened, its polar radius shorter than its equatorial by one part in 230. French astronomers, who measured the Earth’s shape by surveying long arcs of meridian, at first found the opposite, an Earth elongated towards the poles, and the dispute between Newton’s followers and Cassini’s school was settled only in the 1730s by expeditions to Lapland and Peru, which found the Earth flattened as Newton had said — though by less than his uniform model predicted.

Colin Maclaurin worked out the exact shape in 1742: a uniform, self-gravitating fluid spinning rigidly can be in equilibrium as an oblate spheroid, a flattened ellipsoid symmetric about its axis, for any degree of flattening up to a limit. For nearly a century everyone assumed those spheroids were the only possibility. Then, in 1834, Carl Jacobi showed that they were not.

How much spin a given flattening needs #

A spinning fluid is shaped by a balance between its gravity, which pulls it towards a sphere, and the outward push of its rotation, which is strongest at the equator. The surface a spin decides found the free surface of water in a spinning bucket curving into a paraboloid, where the bucket’s gravity is a fixed field from below; for a fluid held together by its own gravity, the field depends on the shape, and the calculation is harder. Maclaurin’s result is that a spheroid of meridian eccentricity $e$ is an exact equilibrium if

$$\frac{\Omega^2}{\pi G\rho} = \frac{2\sqrt{1-e^2}\,(3-2e^2)\,\arcsin e}{e^3} - \frac{6(1-e^2)}{e^2},$$

where $\Omega$ is the spin rate and $\rho$ the density. The combination $\Omega^2/\pi G\rho$ is the natural measure of spin for a self-gravitating body: how fast it turns compared with how strongly it holds itself together.

For slow spin the flattening is small and grows as the square of the spin. The Earth, turning once a day with a mean density of 5.5 grams per cubic centimetre, sits far down at the left of the curve, and a uniform Earth at that spin would have an eccentricity of 0.093 — Newton’s 1/230. The real Earth’s eccentricity is 0.082, a flattening of 1/298, and the difference is not an error: the Earth is not uniform. Its iron core is three times as dense as its outer rocks, so much of its mass sits near the centre, where it contributes to the inward pull but is not itself flattened much. Jupiter and Saturn, spinning in ten hours and of low average density, sit much further up the curve and are visibly flattened — Saturn by almost a tenth — but again less than uniform bodies would be, because their mass too is concentrated towards their centres. The gap between a planet’s dot and the curve is a measure of how centrally condensed it is.

A limit on how fast a fluid can spin #

The curve does not rise for ever. It peaks at $\Omega^2/\pi G\rho = 0.4493$, at an eccentricity of 0.930, and beyond that it falls: a flatter spheroid needs a slower spin. That is not as strange as it looks. A flatter spheroid of the same volume is wider, so its angular momentum, which grows as the square of its radius, can keep growing even as its rate of spin falls; the curve against angular momentum, below, rises smoothly through the peak. But it does mean that no uniform fluid body, of any shape in this family, can spin faster than $\Omega^2 = 0.449\,\pi G\rho$ and hold together. For a body of the Earth’s mean density that is a rotation period of about two hours and twenty-five minutes; for a body as dense as water, about five and three-quarter hours.

The size at which a body becomes round found that bodies larger than a few hundred kilometres are pulled into a near-sphere by their gravity, overcoming the strength of rock. The Maclaurin limit is the corresponding statement about spin: a self-gravitating body larger than a few hundred kilometres, behaving as a fluid, has a maximum spin set only by its density. Asteroids smaller than about 150 metres, held together by their strength rather than their gravity, can spin faster than this “spin barrier” — some in minutes — while larger ones, which are rubble piles held by gravity, almost never spin faster than about 2.2 hours. The barrier seen in surveys of asteroid rotation is a version of this limit.

The shape nobody looked for #

Lagrange and Laplace had assumed that a spinning fluid in equilibrium must be symmetric about its axis of rotation, as the Maclaurin spheroids are. Jacobi asked whether an ellipsoid with three unequal axes, rotating about its shortest one, could also be an equilibrium, and found that it could — but only for a definite relationship between the three axes and the spin, and only beyond a certain degree of flattening.

The Jacobi ellipsoid rotates about its shortest axis, but its two equatorial axes differ: seen from above, it is an ellipse rather than a circle, and it rotates like a rugby ball spun end over end. The conditions for equilibrium are two equations among the three axes and the spin, involving integrals over the ellipsoid’s shape that have no closed form; the figures were computed by evaluating those integrals numerically, and the same numerical method, applied to a spheroid, reproduces Maclaurin’s closed formula, which is how the calculation checks itself.

The Jacobi family begins where it meets the Maclaurin family. At an eccentricity of 0.8127 — a spheroid with its polar axis 58 per cent of its equatorial — the two equatorial axes of the Jacobi ellipsoid become equal and the two shapes coincide. From there the Jacobi ellipsoids stretch out along one equatorial axis and shrink along the other two, becoming longer and thinner as their angular momentum grows.

Jacobi announced the result in a short note in 1834, and it was greeted with surprise, because it contradicted a belief so general that nobody had thought to test it: that a symmetric cause — a fluid spinning about an axis, with nothing outside to prefer one direction in its equator over another — must have a symmetric effect. It does not have to. The equations allow both shapes, and the symmetric one is not always the one nature takes. Joseph Liouville and Charles Meyer soon showed where on Maclaurin’s sequence the new branch begins, and the question of which branch a real body follows, and why, occupied mathematicians from Poincaré to Chandrasekhar for the next hundred and thirty years. It is one of the first and cleanest examples of what is now called spontaneous symmetry breaking: a system with a symmetry settling into a state that lacks it.

Two shapes for one angular momentum #

The natural way to compare the two families is by the quantity a spinning body conserves: its angular momentum. The quantity that survives a change of shape found a skater’s angular momentum unchanged as arms are drawn in and the spin speeds up; a body that changes shape slowly under its own internal forces keeps its angular momentum, and the question is which shape it settles into.

Below an angular momentum of 0.304 in these units only the spheroid exists. Above it there are two possible shapes for each angular momentum: the spheroid, flatter and spinning faster, and the ellipsoid, elongated and spinning more slowly. The computed bifurcation agrees with the value Chandrasekhar gives in his monograph on the subject to the third decimal place. Beyond it the Jacobi branch turns downward in spin: as an ellipsoid gains angular momentum it lengthens, and its moment of inertia grows faster than its angular momentum, so it actually slows.

Why the stretched shape wins #

Which of the two shapes a real body takes depends on how it loses energy.

At the same angular momentum the ellipsoid always has less energy than the spheroid. A real fluid has some viscosity, which turns the energy of any internal motion into heat while leaving the total angular momentum unchanged. So a spheroid past the bifurcation, given a small disturbance that stretches it along one equatorial axis, will slowly lose energy to friction as that disturbance grows, and drift onto the Jacobi branch. This is a secular instability, one that needs dissipation to act and proceeds at the rate dissipation allows. The axis a leak of energy chooses found the same principle in a spinning satellite: at fixed angular momentum, dissipation drives a body towards the state of least energy, which for a rigid body is rotation about its axis of greatest inertia, and here is the stretched ellipsoid.

Without any viscosity the spheroid survives past the bifurcation, as an equilibrium that small disturbances only rock. It becomes dynamically unstable, with oscillations that grow even without friction, only at an eccentricity of 0.9529, close to the top of its curve. The region between the two thresholds is where a perfectly frictionless spinning fluid could remain a spheroid while any real fluid would not.

Stretching towards fission #

The Jacobi sequence does not go on for ever either.

As the angular momentum rises the ellipsoid lengthens, its long axis growing to nearly three times the radius of a sphere of the same volume while its other two shrink. Henri Poincaré showed in 1885 that at an axis ratio of about 0.43 between the two equatorial axes the Jacobi family is itself crossed by another, a family of pear-shaped figures, one end fatter than the other. George Darwin, the son of Charles, followed the pears in the hope that they would pinch off into two bodies, and proposed that the Moon had formed by fission from a rapidly spinning early Earth in exactly this way. Later work by Lyapunov and others showed the pear-shaped sequence to be unstable from the start, and the Moon’s origin is now attributed to a giant impact; but the idea that a spinning fluid can divide is not wrong, only more complicated than a smooth march along a sequence of shapes. The thread that cannot stay a thread found a liquid cylinder breaking into drops when its surface energy favoured them; a spinning, self-gravitating fluid can break for the analogous reason, when two bodies in orbit carry its angular momentum with less energy than one.

The same shapes, squeezed by a neighbour #

Spin is not the only thing that can stretch a self-gravitating fluid. A moon close to its planet is pulled harder on its near side than its far side, and in its own rotating frame it settles into an ellipsoid elongated towards the planet — a Roche ellipsoid, the tidal cousin of Jacobi’s, computed with the same integrals over the same kind of shape. The distance that forgets the moon found the Roche limit, inside which a fluid moon is pulled apart, and the factor 2.44 in it comes from exactly this calculation: the sequence of Roche ellipsoids, like the Maclaurin sequence, has a last member, and a fluid satellite closer to its planet than that has no equilibrium shape at all.

Io, kept molten in places by the flexing the moon kept molten by its neighbours described, is measurably elongated towards Jupiter, by a few kilometres; the inner moons of Saturn are elongated far more. Every such body is a point somewhere on one of these families of fluid ellipsoids, displaced from it by its strength and its internal structure, and the size of the displacement is one of the few measures of a distant moon’s interior that can be read from its outline alone.

A flattening that sets a wobble #

The Earth’s modest flattening has a consequence that has been measured to great precision. A spinning body whose moment of inertia about its spin axis exceeds that about an equatorial axis can wobble freely, its spin axis tracing a small circle in the body, with a period set by the ratio of those moments — by how flattened it is. For a rigid Earth with its measured flattening the period would be about 305 days; the observed wobble, discovered by Seth Chandler in 1891, takes about 433, because the Earth is not rigid and its oceans and mantle yield to the wobble, as the wobble that is slow because the Earth gives found. The flattening that Newton first estimated from a uniform spinning fluid is the same quantity that, measured from the orbits of satellites, now gives the Earth’s moments of inertia to several significant figures, and through them the concentration of its mass towards the core.

Where real bodies sit #

Most planets and stars spin far too slowly to approach either threshold. A few do not. The dwarf planet Haumea, beyond Neptune, spins once every 3.9 hours, one of the fastest rotations of any large body in the solar system, and its brightness varies as it turns in a way that only an elongated shape explains; an occultation of a star in 2017 measured its axes as roughly 2,300, 1,700 and 1,000 kilometres. Whether its shape is exactly a Jacobi ellipsoid depends on how fluid its interior has been, which is debated, but its elongation is the Jacobi branch’s signature: a body too fast-spinning to stay axially symmetric.

Newborn neutron stars may spin near their limits too. A neutron star formed with a few millisecond period is far past the Maclaurin bifurcation, and its internal friction, or the emission of gravitational waves, can drive it onto a bar-like shape. A spinning bar radiates gravitational waves strongly, the shape the wave that stretches one way and squeezes the other found such waves making, and the possibility that young neutron stars pass through such a phase is one of the sources gravitational-wave detectors search for. In stars spinning less violently, deformations are limited to small mountains, the subject of the mountain a spinning star is allowed.

A uniform, incompressible fluid turning as one piece #

Every figure is for a uniform, incompressible fluid. Real planets and stars are compressible and centrally condensed, and both effects move the thresholds: a centrally condensed body flattens less at a given spin, as the planets on the first figure do, and its bifurcation to non-axisymmetric shapes occurs at a different ratio of rotational to gravitational energy — for compressible stars the instability to a bar depends on that ratio, which is about 0.14 for the secular instability and 0.27 for the dynamical one, rather than on an eccentricity. The rotation is taken as rigid, where real bodies rotate faster in some places than others, and the energies use the uniform-density formulas for an ellipsoid’s self-gravity.

The secular instability’s rate depends entirely on the dissipation, which is not modelled: for a planet with a viscous mantle the drift onto the Jacobi branch could take longer than the age of the solar system, and for a star whose internal friction is small, gravitational radiation may drive a different branch of instability altogether. The planet dots use measured flattenings and densities, typed in; Haumea’s axes are quoted, not drawn.

Still open: how a spinning fluid divides #

What happens to a spinning, self-gravitating fluid that has too much angular momentum to stay in one piece is still not fully known. Numerical simulations show bar-shaped stars shedding spiral arms, rapidly spinning stellar cores splitting into fragments as they collapse, and binary stars forming from rotating gas clouds, but the clean sequences of equilibrium shapes — Maclaurin, Jacobi, Poincaré’s pears — give only the starting points of those processes, not their outcomes. Whether some binary stars form by the fission of a single rapidly rotating star, rather than by the fragmentation of the cloud they formed from, has been argued for over a century and is not settled.

The general lesson is the one Jacobi found against the expectations of his time. A symmetric equilibrium is not always the preferred one: past a definite angular momentum, a spinning fluid has a less symmetric shape available with less energy, and any loss of energy at fixed angular momentum will carry it there. A spinning ball of fluid flattens as it spins faster, but past an eccentricity of 0.81 it can lower its energy by giving up its roundness in the equator and becoming a tumbling cigar.