A periodic orbit that the sky cannot keep

For three bodies of comparable mass that are not arranged in a hierarchy, the outcome is almost always a break-up: one body is thrown out and the other two leave as a binary. “Almost always” leaves room for exceptions, and the most beautiful of them was found in 1993 by Cristopher Moore, searching numerically for orbits in which three bodies share a single path. Three equal masses, set up correctly, chase one another round a figure of eight — each a third of a period behind the one in front, crossing at the centre, swinging round the two lobes — and they do it for ever. In 2000 Alain Chenciner and Richard Montgomery proved that the orbit exists: not an approximation to something periodic, but an exact periodic solution of Newton’s equations for three bodies.

It is also stable, in the strict sense that small disturbances to it stay small. And it will almost certainly never be seen in the sky. That combination — an exact, stable solution with essentially no chance of being realised — is what this essay is about, because it says something precise about what “stable” means in a problem that has no general solution.

A solution found by looking for it #

Before the figure-eight, the known periodic solutions of the general three-body problem were few. Lagrange’s equilateral triangle and Euler’s collinear configurations, from the eighteenth century, rotate rigidly and are unstable for most mass ratios. A handful of families found numerically in the 1970s by Roger Broucke and Michel Hénon extended the list. The figure-eight was of a new kind: a choreography, in which all the bodies travel one closed curve.

Chenciner and Montgomery’s proof did not solve the equations. It minimised a quantity — the action, the integral over one period of kinetic minus potential energy — over all closed paths with the figure-eight’s symmetry, and showed that the minimum exists and does not involve a collision. Any path that minimises the action is a solution of Newton’s equations, by the principle of least action; the difficulty is always to show that the minimising path does not pass through a collision, where the equations break down, and the symmetry was what made that possible. The existence of the orbit is a theorem about a minimum, and the orbit itself is known only numerically, to as many digits as anyone cares to compute.

The method opened a floodgate. Carles Simó found hundreds of choreographies for three and more bodies within a few years. In 2013 Milovan Šuvakov and Veljko Dmitrašinović found thirteen new families of periodic three-body orbits by a systematic numerical search, and in 2017 Xiaoming Li and Shijun Liao, using high-precision arithmetic on a supercomputer, reported more than six hundred. The three-body problem, famous for having no solution, turned out to have an enormous number of exact periodic ones.

Why the symmetry does the work #

The figure-eight is special because equal masses make the three bodies interchangeable, and the orbit uses that interchangeability as fully as a planar orbit can.

Its path has two mirror symmetries — it is unchanged by reflection in either axis of the eight — and the motion is unchanged by reversing time. More strikingly, the configuration of the three bodies a third of a period later is the same configuration with the bodies relabelled: body 1 is where body 2 was, body 2 where body 3 was, body 3 where body 1 was. That cyclic relabelling is only a symmetry of Newton’s equations if the three masses are equal, because swapping two bodies of different mass changes the forces. The existence proof works within the class of paths that have all these symmetries, and the symmetries are what exclude the collisions that would otherwise spoil the minimum.

So the orbit is built from the equality of the masses, not merely compatible with it. When one mass changes, the cyclic symmetry is gone, and there is no longer any reason for the three bodies to share a path; the nearest periodic orbit, if there is one, is a different curve for each body. The choreography does not deform smoothly into a slightly lopsided choreography as the mass ratio moves. It breaks — the periodic orbit continues as a non-choreographic periodic orbit for a little way, and its stability, which depended on a delicate balance between the three bodies’ motions, is lost within about a ten-thousandth.

Stable, in the strict sense #

Most of those periodic orbits are unstable: a disturbance of any size grows exponentially along the orbit, and a real system started near one of them would drift away and, eventually, break up. The figure-eight is among the few that are not. Its linear stability — the behaviour of infinitesimal disturbances — was computed by Simó shortly after its discovery, and all the disturbances oscillate rather than grow.

Linear stability is not enough for a system with more than one degree of freedom; a disturbance that does not grow at first order can grow through the nonlinear terms. What saves the figure-eight is KAM theory, the result of Kolmogorov, Arnold and Moser that says a stable periodic orbit of a Hamiltonian system is typically surrounded by invariant tori — surfaces in phase space on which nearby orbits move quasi-periodically and from which they cannot escape. Such tori act as barriers, and a region of phase space enclosed by them is an island of stability: orbits that start inside stay inside for ever.

The figure-eight sits in the middle of such an island. An orbit started close enough to it stays close to it for all time, not because anything damps the difference, but because the tori round it enclose the motion. The question that decides whether the orbit could exist in nature is how large the island is.

How large the island is #

The island has a size in every direction of the space of initial conditions — positions, velocities, angular momentum — and in the masses. The masses are the decisive direction, because they are the one thing a real system cannot adjust after it forms: three stars with masses that differ at the level of one per cent will differ at that level for their whole lives.

A continuation study of the orbit’s stability made in 2002 by Galán, Muñoz-Almaraz, Freire, Doedel and Vanderbauwhede, following the figure-eight as one mass is varied, found stability over an interval of the same order, about a hundred-thousandth to a ten-thousandth on either side of equality. The integrations drawn here agree: the orbit tolerates a change in one mass of a few parts in a hundred thousand and does not tolerate one part in ten thousand.

That second figure is the same lesson as the stability boundary of hierarchical triples, seen from inside an island rather than outside a boundary. An orbit just outside the island is not violently unstable; it diffuses away from the choreography on a timescale that lengthens as the mass change shrinks, and a finite integration catches only the diffusion fast enough to finish within it. The true edge of the island is where the diffusion time becomes infinite, and integrations of any finite length locate it only from outside.

A proof of stability made by computer #

That the figure-eight is stable in the full nonlinear sense, and not only to first order, is itself a theorem, and a theorem of an unusual kind.

KAM theory guarantees invariant tori round a linearly stable periodic orbit only if certain conditions hold: the frequencies of the small oscillations round the orbit must not be in low-order resonance with the orbit’s own frequency, and a quantity called the twist must be non-zero. Checking those conditions for the figure-eight requires the orbit and its linearisation to be known to a precision no analytic method can reach. In 2007 Tomasz Kapela and Carles Simó checked them with a computer-assisted proof: a numerical integration carried out in interval arithmetic, in which every number is an interval guaranteed to contain the true value and every operation widens the interval to account for rounding. At the end, the intervals were narrow enough to establish the conditions rigorously, and the stability of the figure-eight became a theorem proved by a machine.

That is a different kind of knowledge from the integrations drawn here. A numerical integration is wrong about where and right about how much: its trajectory drifts from the true one, but its energy and its statistics stay honest. An interval integration is right about where, within a bound it computes for itself, and it is what is needed when the question is not “what usually happens” but “is this so”. The two meet in the island’s width: the proof establishes that an island exists; the integrations show how small it is.

Leaving the island #

Once outside, the orbit’s fate depends on how far outside it is. A mass change of a thousandth or a hundredth produces an orbit that wanders over a broad region round the figure-eight for as long as it was integrated, with no escape: the three bodies stay close, their paths tangle, and the choreography is gone. At a tenth, a body escapes.

The intermediate regime — bound but no longer a choreography — is a chaotic region that the figure-eight’s island sits inside. Orbits there are democratic triples with an unusually long life, because they begin near a structure that holds them, and they end, in the long run, as every democratic triple ends: with the lightest body leaving. The figure-eight is an island in a sea whose shore is a break-up, and the island is small.

How many figure-eights the Galaxy might hold #

An orbit this finely tuned could form only by accident, in an encounter that happened to deliver three nearly equal stars to nearly the right positions with nearly the right velocities. Douglas Heggie asked in 2000 how often that might happen, by computing how often encounters between pairs of binaries in star clusters end in a figure-eight triple, and combining that with the rate of such encounters in the Galaxy and with the mass tolerance.

The estimate is dominated by the tolerance. Three stars equal in mass to one part in ten thousand is not something any star-forming process produces, except by chance; the masses of stars in a cluster are spread across a factor of a hundred, and three drawn from that spread match to one part in ten thousand very rarely. His answer was famously wide: somewhere between about one figure-eight in each galaxy and one in the whole observable universe. An exact, stable, periodic solution of the laws of gravity is expected to be realised a handful of times in the cosmos, if at all.

That is not a failure of the solution. It is a measure of how much of the stability of the Solar System and of stable triple stars comes from hierarchy rather than from symmetry. Hierarchical systems are stable over wide ranges of masses and orbits because their stability comes from separated timescales, which do not care about the details. The figure-eight is stable because of an exact symmetry between equal masses, and a symmetry is broken by anything that is not exactly symmetric.

The other kind of three-body stability #

The figure-eight’s fragility is sharpest set beside the one exact configuration of three bodies that nature builds all the time.

Lagrange’s equilateral solution places three bodies at the corners of a rotating equilateral triangle, and it is a solution for any masses. It is stable only if one mass dominates: Routh showed in 1875 that the triangle is linearly stable when $27(m_1m_2 + m_2m_3 + m_3m_1) \lt (m_1 + m_2 + m_3)^2$, which requires the largest body to hold about 96 per cent of the total mass. The Sun, Jupiter and a Trojan asteroid satisfy it by a huge margin, and several thousand Trojan asteroids sit in Jupiter’s two Lagrange points; Saturn’s moons and Neptune have Trojans too, and the Earth has at least two.

The contrast is exact. The Lagrange triangle is stable because of hierarchy — one body so heavy that the others are test particles in its field — and it tolerates any mass for the light bodies, which is why nature has filled its islands. The figure-eight is stable because of symmetry, among three bodies of equal weight, and it tolerates almost no departure from that symmetry, which is why nature has probably never filled its island at all. Stability that comes from one body dominating is robust; stability that comes from three bodies balancing is not, and that is most of the reason planetary systems, with their dominant stars, are the stable configurations of the gravitational many-body problem.

Where other choreographies stand #

The hundreds of periodic orbits found since 2000 are mostly unstable, and the few stable ones share the figure-eight’s problem to varying degrees. Some families remain stable over wider ranges of mass ratio, and the search for periodic orbits that are stable at unequal masses — which would be candidates for actually existing in the sky — has become a programme of its own. A periodic orbit that survives a mass ratio of two or three would have a chance of being assembled in a cluster encounter; one that needs equal masses to one part in ten thousand does not.

The periodic orbits matter even if none is ever observed, because they are the skeleton of the phase space. Unstable periodic orbits organise the chaotic sea: a chaotic orbit spends time near one, leaves along its unstable direction, and approaches another — which is how the chaos of the three-body problem can be described in terms of the periodic orbits it shadows. And stable ones, with their islands, are where chaotic orbits stick for long intervals, producing the long tails of survival times in ensembles of democratic triples. The periodic orbits are rarely where the systems are; they are what shapes the paths of the systems that are not on them.

What the integrations are and are not #

Every orbit is planar with zero total angular momentum, the figure-eight’s own. Changing the angular momentum is a second direction in which the island has a width, and the figures scan only the mass.

Straying is measured from the equal-mass curve, as the largest distance from any sampled body position to the nearest point of the curve. Measuring against the curve rather than against the unperturbed positions at the same times separates a change of shape from a change of period, which a mass change also produces and which would otherwise look like a departure.

The integrator is a fourth-order Runge–Kutta scheme with an adaptive step, not the symplectic or high-precision methods used in the published stability studies. It conserves the energy of the equal-mass orbit to parts in a hundred billion over the runs drawn, far below the scale of the departures measured, but it would not be adequate for integrations of millions of periods, where rounding errors accumulate like a random walk, and an adaptive step has costs of its own for long integrations.

And “the island’s width” is read at a finite time. The two scans differ by a factor of ten because of it; the true width is the limit of such scans as the time goes to infinity, which the published continuation studies approach through linear stability analysis rather than through integration.

Still open: stable orbits the sky could build #

The figure-eight showed that exact, stable periodic orbits of three comparable masses exist, and the searches since have found them by the hundred. What is not known is whether any of them is stable across a range of unequal masses wide enough for nature to assemble it — a range of a factor of two, say, rather than one part in ten thousand. If such an orbit exists, dense star clusters, where binary–binary encounters happen routinely, should occasionally produce it, and it would appear as a triple of comparable stars in a compact non-hierarchical configuration that persists, which is a configuration the statistics of break-up say should not last. None has been identified. Finding one would be the first periodic three-body orbit found by looking at the sky rather than at a computer, and its existence would say that the island round some choreography is large enough for chance to find.