The lightest body leaves

In 1913 the Danish astronomer Carl Burrau posed a problem that looks as simple as a three-body problem can: three masses of 3, 4 and 5 units, placed at the corners of a right-angled triangle with sides 3, 4 and 5 — each mass opposite the side of its own length — and released from rest. Burrau could follow them by hand for only a short time. In 1967 Victor Szebehely and Frederick Peters followed them by computer to the end, and the end was the same kind of end that every non-hierarchical triple reaches: after a long sequence of close encounters, one body was ejected on an escape orbit and the other two left as a binary. The body ejected was the lightest.

That outcome is the general one, and this essay is about the two things that make it interesting. The first is that no individual outcome can be predicted without integrating the orbits all the way — a prediction for a chaotic system has an expiry date, and for a democratic triple it expires within a few crossing times. The second is that the outcomes, taken together, are predictable after all: they follow a statistical law in which the mass of the escaper matters enormously and the details of the start matter not at all.

Why a democratic triple cannot last #

A triple is democratic when no pair is much tighter than the others — when the three bodies share one region of space and one timescale. The stability boundary that governs hierarchical triples says that a third body inside about three separations of a binary destroys it; a democratic triple is one in which every body is inside that distance of every other pair, so every pair is being destroyed by the third body all the time.

What follows is a sequence of close two-body encounters. Two bodies approach, swing round each other, and exchange energy with the third, which is sent out on a wide excursion and falls back. At each encounter the energy is redistributed, and the redistribution depends sensitively on the geometry of the approach. Sooner or later an encounter gives one body enough energy to leave for good. Because the total energy of a triple released from rest is negative, the system cannot simply fly apart; the energy the escaper carries off has to come from somewhere, and it comes from the pair left behind, which is bound more tightly than any pair was at the start.

The end state is the only simple thing about the run. During the democratic phase the separations wander across two decades with no pattern, and a tiny change in any initial position would produce a completely different sequence of encounters and possibly a different escaper. Where the chaos comes from is the same here as anywhere in the three-body problem: each close encounter amplifies small differences in the approach, and the amplifications multiply.

Why the light one #

The escaper is usually the lightest body, and the reason is conservation of momentum.

When one body leaves, the binary left behind recoils. The two carry equal and opposite momenta in the centre-of-mass frame, $m_e v_e = M_b v_b$, so their speeds are in inverse proportion to their masses, and their kinetic energies are too: the light escaper carries off most of the energy released, and the heavy binary’s recoil takes little. For a given amount of energy made available by the pair tightening, a light body is much more easily sent over the escape threshold than a heavy one, because almost all of that energy goes into it.

That is the mechanism, and it gives the direction. The strength of the preference — the power of the escaper’s mass in the outcome probability — needs the statistical theory. Its central assumption is that during the democratic phase the chaos explores all the states consistent with the conserved energy and angular momentum more or less uniformly, so that the probability of a given outcome is proportional to the volume of the states that lead to it. That is the ergodic hypothesis of statistical mechanics, applied to three bodies instead of $10^{23}$. Its predictions, worked out by Monaghan in the 1970s and refined by Valtonen, Karttunen and others, include the distribution of the escaper’s speed, the eccentricity of the binary left behind, and the escape probability, which at leading order falls as the cube of the escaper’s mass.

The integrations drawn here are a direct test of that assumption. Sixty triples of masses 1, 2 and 3 give the lightest body 81 per cent of the escapes against a prediction of 86, and the other two 11 and 9 per cent against 11 and 3. The heaviest body escapes more often than the leading-order law allows — the sample is small, and the law is a leading-order law that omits the angular-momentum and geometry factors that favour heavy escapers slightly — but the ordering and the dominance of the lightest are as predicted. In 2019 Stone and Leigh derived a more complete statistical solution along these lines and compared it with a large ensemble of integrations, finding agreement in the distributions of outcome, and the three-body problem, famously without a solution, turned out to have a statistical one.

The binary left behind #

The escaper takes the headlines; the binary it leaves behind is the more lasting product, and its properties are predicted by the same statistics.

Its binding energy is fixed by bookkeeping. The triple started with a negative total energy; the escaper leaves with a positive one; so the binary’s energy is more negative than the triple’s was, by the escaper’s kinetic energy at infinity. The binary is therefore always harder than the triple it came from — typically bound two or three times more tightly than the original configuration — and it is the binary that holds almost all of the system’s binding energy. That is the basic process by which binaries heat the star clusters they live in: every encounter that ejects a star or speeds one up is paid for by a binary becoming tighter, and the energy goes into the motions of the other stars.

Its eccentricity is where the ergodic assumption shows most plainly. If the democratic phase samples its states uniformly, then the orbits available to the final binary at a fixed energy are sampled uniformly too, and for a fixed energy the number of orbits with eccentricity less than $e$ is proportional to $e^2$ — the “thermal” distribution, in which the probability density rises as $2e$. The binaries produced by break-ups are therefore typically highly eccentric, with a median eccentricity of about 0.7. Wide binaries in the solar neighbourhood, measured with Gaia, have an eccentricity distribution close to thermal, and even slightly beyond it; close binaries, which have been circularised by tides, do not. The eccentricities of the widest binaries in the sky look like the output of a statistical process, which is what they would be if many of them had been through one.

When the masses are equal #

The same experiment with three equal masses removes the mechanism and leaves the statistics.

Equal masses also break up more slowly. Only 38 of 60 triples broke up within the time integrated, against 57 of 60 for masses of 1, 2 and 3. A strongly unequal triple has an obvious candidate for ejection and ejects it; an equal-mass triple has to wait for a fluctuation large enough to throw out a body that is no easier to throw out than the others.

That consequence runs through the demography of stars. Binaries formed in small multiple systems that dissolve are preferentially made of the heavier members; the lightest stars and brown dwarfs are preferentially the escapers. Among the explanations offered for why brown dwarfs are rarely found in wide binaries with Sun-like stars is that many of them were born in small groups and ejected early, before they could accrete enough to become stars — a population of ejected embryos whose low masses are a consequence of their ejection as much as its cause.

A process with no memory #

The time a democratic triple takes to break up is as unpredictable in a single case as the identity of the escaper, and just as regular across an ensemble.

That is the same law as radioactive decay, and for the same reason. A nucleus has a fixed probability per unit time of tunnelling out, whatever its history; a democratic triple has a roughly fixed probability per crossing time that its next close encounter will be the one that ejects a body, because the chaos has erased any memory of where it started. The half-life is a property of the ensemble, measurable to any precision with enough triples, and meaningless for any single one.

The long tail beyond the exponential is the one place where memory survives. Some triples fall, during their democratic phase, into configurations in which one body is sent on a long, nearly unbound excursion while the other two form a temporary tight pair. The excursion can last many crossing times, and while it lasts nothing much happens; then the body falls back and the democratic phase resumes. Those long excursions produce a tail of survival times longer than the exponential predicts — a heavy tail of the kind statistical physicists call Lévy flights — and in a large ensemble a few triples last hundreds of times longer than the half-life. Chaos erases memory except during the intervals when there is almost no dynamics to do the erasing.

Runaway stars from an encounter that has already happened #

The strongest evidence that democratic break-ups happen in the sky comes from stars that are moving away from where one happened.

Among the massive stars of the Milky Way a few per cent are runaways — O and B stars moving through space at 40 kilometres a second or more, far faster than the stars around them and too fast to have been born where they are. Two runaway stars, AE Aurigae and mu Columbae, are moving in almost exactly opposite directions at about a hundred kilometres a second each, and when their motions are traced back, they meet about two and a half million years ago in the region of the Orion Nebula Cluster — at the position of the massive binary iota Orionis, which has a high eccentricity and a peculiar motion of its own consistent with recoil.

That is a break-up reconstructed after the fact: a close encounter between two binaries, or a binary and a single star, in the dense young cluster, which ejected two of the stars and left a hard, eccentric binary behind. The speeds, the directions and the momentum balance all fit. Proper motions measured two and a half million years after an encounter have told which stars were in it, and the ejected stars were the lighter members, as the statistics say they should be.

A second channel for runaways — a supernova in a binary, which releases the companion at its orbital speed — leaves a surviving companion moving fast from the explosion site instead. The two channels predict different things about the runaway’s companions, its rotation and its chemistry, and the Gaia proper motions have made it possible to trace large numbers of runaways back to their clusters and ask which channel produced each. Both occur, in comparable numbers.

The same break-up at the centre of the Galaxy #

A three-body break-up does not need three stars. It needs three bodies and no hierarchy between the relevant energies, and one of the most dramatic versions of it happens at the centre of the Milky Way.

A binary star that passes close to the four-million-solar-mass black hole at the Galactic Centre meets a third body far heavier than itself. At closest approach the black hole’s tide exceeds the binary’s own binding, and the binary is separated: one star is captured into a tight orbit round the black hole, and the other is flung away. The escaper leaves with a speed of order the binary’s orbital speed times the cube root of the ratio of the black hole’s mass to the binary’s — for a binary of Sun-like stars with a separation of a tenth of an astronomical unit, well over a thousand kilometres a second. Jack Hills predicted in 1988 that the Milky Way should therefore be ejecting stars at speeds greater than its own escape speed, and in 2005 the first of them was found, a B star moving away from the Galactic Centre at more than seven hundred kilometres a second.

Several dozen such hypervelocity stars are now known, and their companions — the stars captured by the black hole — may be the young, massive stars on eccentric orbits a few thousand astronomical units from it, which are hard to explain by any other route because the black hole’s tide would have prevented them from forming there. The Hills mechanism is the light body leaving, taken to its extreme: the third body is so heavy that the escaper is always one of the binary’s stars, and the only question is which.

Black holes that merge on eccentric orbits #

In the dense cores of globular clusters the heaviest objects are stellar black holes, which sink to the centre and interact among themselves. Their encounters are democratic triples of the kind drawn here, with one important addition: a very close encounter between two black holes can radiate enough energy in gravitational waves to capture them into a binary on the spot.

Most pairs of black holes that merge do so after a long, slow inspiral that circularises their orbits long before they enter the frequency band the ground-based detectors hear. But a pair captured during a chaotic triple encounter can be formed so tight and so eccentric that it merges within hours, still eccentric when its signal arrives. Integrations of cluster cores predict that a few per cent of the mergers from clusters should carry measurable eccentricity, and none from isolated binaries should. An eccentric merger would be a fossil of a three-body encounter, and at least one detected event has been argued to show one, though the measurement of eccentricity from a short signal is difficult and the claim is disputed. If confirmed, it would make the statistics of democratic break-ups directly observable in gravitational waves, for objects that emit no light at all.

What was actually integrated #

The triples are planar and start from rest. Zero initial angular momentum and zero initial kinetic energy make the democratic phase as violent as possible, and the break-ups fast; triples with angular momentum live longer, and the statistical theory predicts how the outcome shares depend on it. The runs here test the theory in its simplest case.

Sixty runs is a small sample. The shares quoted carry statistical uncertainties of several per cent for the common outcomes and are uncertain by factors of about two for the rare ones — the heaviest body’s nine per cent against a predicted three is within that range.

Escape is declared by a criterion: a body counts as escaped when it is more than eight units from the other two and its energy relative to their centre of mass is positive. A body far away and marginally bound is counted as still in the system, and a very long excursion near the end of a run may be miscounted.

And the integrator struggles at the closest encounters. The step shrinks with the separation of the closest pair, and in rare near-collisions the energy error reaches a few parts in ten thousand. That is small compared with the energy exchanged at an encounter, so it does not change which body escapes, but it is a reminder that the arithmetic of an integration has its own errors, and for chaotic orbits they are amplified along with everything else.

Still open: where the ergodic assumption fails #

The statistical theory assumes that the democratic phase explores its available states uniformly, and the success of its predictions says that, on average, it does. But the phase space of three bodies is not uniformly chaotic. It contains islands of regular motion — periodic orbits like the figure-eight, and quasi-periodic orbits around them — that a chaotic trajectory can approach and stick to for a long time without entering. Those islands are one source of the heavy tail in the survival times, and they are a systematic departure from the ergodic assumption whose size depends on how much of the phase space they occupy at a given energy and angular momentum. How large that fraction is, how it depends on the masses, and whether the stickiness near the islands biases which body escapes are open questions at the point where the three-body problem meets statistical mechanics — and where a problem long described as having no solution turns out to have two, one for a single system and one for all of them, which do not quite agree.