A third body kept at a safe distance

Three bodies have no general solution, and three bodies of comparable mass started anywhere near each other do not stay together: sooner or later one of them is thrown out and the other two leave as a binary. And yet the sky is full of triple stars that have lasted for billions of years. Alpha Centauri is one — a pair of Sun-like stars twenty-three astronomical units apart, and Proxima, a red dwarf thirteen thousand astronomical units away. Polaris is another. Roughly one star system in ten in the solar neighbourhood has three or more members.

The ones that last are hierarchical: a close inner binary and a third body far enough away that, from out there, the binary looks almost like a single mass, and from inside the binary, the third body looks almost like a slowly varying tide. Each orbit is nearly a Kepler ellipse, and the problem is nearly two separate two-body problems. The word “nearly” is where all the difficulty lives, and the question this essay asks is how far away the third body has to be for “nearly” to hold forever.

What goes wrong when the third body is close #

A third body passing the inner binary exerts a tidal force on it: it pulls the nearer star slightly harder than the farther one. At each passage that force does work on the binary, adding to or removing a little of its orbital energy and changing its eccentricity, by an amount that depends on where the two inner stars are in their orbit at that moment.

If the outer orbit is wide, the passages are gentle and the binary goes round many times during each one. The kicks from the different phases of the inner orbit average away within a passage, and what remains is a slow, secular change in the shape of the inner orbit — the kind of exchange of eccentricity and inclination that is large but perfectly regular. Nothing accumulates in the energy, and the triple lasts.

If the outer orbit is close, each passage is short compared with the inner period. The binary receives a kick that depends on its phase, and the phase at the next passage depends on everything that happened in between. The kicks no longer average out. They add like steps of a random walk, and after enough of them the binary’s energy has wandered far enough that it is either broken up or so changed that the third body can capture one of its members. The transition between the two regimes is the stability boundary, and it is where the outer period stops being long enough, compared with the inner one, for the averaging to work.

Where the boundary comes from #

The averaging fails when the outer orbit resonates with the inner one. A resonance is a ratio of small whole numbers between the two periods; at an exact resonance the third body’s passages always meet the binary at the same phase, and their effects add coherently instead of averaging. Resonances clear gaps and lock moons throughout the solar system for exactly this reason.

Each resonance has a width — a range of period ratios around it within which the coherent forcing dominates — and the width grows as the perturbation grows, which is to say as the third body comes closer or becomes heavier. Far from the binary the resonances are narrow and widely spaced, and an orbit between two of them is regular. As the third body comes closer, the resonances widen and crowd together, and at some point neighbouring resonances overlap. An orbit in the overlap region wanders from the influence of one resonance to the next and is chaotic. That is Chirikov’s resonance-overlap criterion, and it is the general route by which regular motion turns into chaos when a perturbation grows.

Rosemary Mardling applied it to hierarchical triples, and with Sverre Aarseth calibrated the result against integrations. For a coplanar prograde triple the critical outer pericentre, in units of the inner semi-major axis, is

$$\frac{R_p}{a_{\rm in}} = 2.8\left[\frac{(1+q)(1+e)}{\sqrt{1-e}}\right]^{2/5},$$

where $q$ is the outer body’s mass in units of the inner binary’s and $e$ is the outer orbit’s eccentricity. For an equal-mass outer body on a circular orbit that is 3.7 inner separations; at an outer eccentricity of 0.6 it is 5.4.

The shape of the formula carries its physics. The pericentre appears because that is where the tidal kick is given. The mass ratio appears to the two-fifths power because the resonance widths grow with the strength of the perturbation but the spacing between resonances grows too, and the ratio of the two scales gently. And the eccentricity appears because a more eccentric outer orbit delivers the same pericentre passage with a longer orbital period — which crowds more inner-orbit resonances into the same range — and with more of the outer orbit’s energy concentrated near pericentre, where the coupling is strongest.

A boundary measured by integrating #

The map at the top of the essay is the direct test. For each cell a triple was set up and integrated — thirty outer orbits, several thousand inner ones for the widest — and marked by whether it survived.

Every triple on the stable side of the criterion survived, which is what the criterion predicts. But on the unstable side the picture is not a clean edge. Close to the binary the triples break up within a few outer orbits; nearer the curve an increasing share of them survive the whole run. Seventeen of thirty cells inside the criterion survived thirty outer orbits, and the outermost break-up is at a pericentre of 3.0 separations, well inside the curve’s 3.7.

That gap is not a failure of the criterion. The criterion is a statement about where chaos sets in, not about how fast chaotic orbits break up, and a chaotic triple near the boundary can wander for a very long time before an unlucky sequence of passages ejects a body — the rate at which a chaotic orbit leaks through a partial barrier can be slow. A finite integration can show that a triple is unstable, by watching it break up. It cannot show that a triple is stable, only that it has not broken up yet. The boundary drawn from any integration of fixed length is therefore a lifetime contour, and the criterion is the line towards which the contours move as the integration is lengthened.

That last point is worth dwelling on, because it runs against intuition. A companion ten times the mass of the inner binary — a massive star with a close pair of light ones — needs to keep its pericentre only about twice as far from the pair as an equal-mass companion does. The resonances that drive the instability are between periods, and periods depend on masses only through their square roots; the widths depend on the mass ratio, but only weakly once it is large. Heavy companions are much less dangerous than their masses suggest, which is one reason why triples in which a massive star holds a close low-mass binary are common.

The triples that are actually seen #

The criterion predicts a gap in the population of real triple stars, and the gap is there.

Catalogues of multiple stars in the solar neighbourhood, assembled over decades from visual orbits, spectroscopic binaries and, more recently, Gaia astrometry, give the inner and outer periods of several hundred triples. The ratio of the two is the most direct observable: the criterion, rewritten in periods, says that a coplanar triple with a circular outer orbit needs its outer period to be at least about five times the inner one, and more for eccentric outer orbits. The observed distribution of period ratios in nearby triples runs from that neighbourhood upward over many decades, with no systems in the forbidden zone and a sharp edge close to where the criterion puts it. Andrei Tokovinin, who assembled much of that catalogue, found the edge in the period ratios before the criterion was calibrated, and the two agree.

The upper end of the distribution is set by something else entirely. A triple whose outer orbit is thousands of astronomical units across is loosely bound to begin with, and it is perturbed by passing stars and by the tide of the Galaxy. Beyond about ten thousand astronomical units those perturbations disrupt the outer orbit in a few billion years, and the widest triples are young or lucky. Alpha Centauri’s third member, Proxima, is at about thirteen thousand astronomical units from the inner pair, on an orbit that radial velocities and astrometry now show to be bound but eccentric, with a period of more than half a million years — close to the distance at which the Sun itself stops governing the orbits of bodies round it. The population of triples is bounded below by resonance overlap and above by the Galaxy, and between the two it spans a factor of about a million in period ratio.

The Earth, the Moon and the Sun #

The most familiar hierarchical triple sits so far on the stable side of the boundary that it is rarely thought of as one.

The Earth and the Moon are a binary with a separation of 384,000 kilometres, and the Sun is its third body, 390 separations away and 330,000 times as massive as the pair. Applied literally at a mass ratio of three hundred thousand, the criterion gives a critical pericentre of about 450 separations and puts the Sun slightly too close — which says only that a formula calibrated on stars of comparable mass should not be pushed five decades past its calibration. At extreme mass ratios the right boundary is the Hill criterion, which scales as the cube root of the mass ratio rather than its two-fifths power, and by it the Moon is safe: it sits at about a quarter of the Earth’s Hill radius, where prograde satellites are stable for the age of the Solar System. What keeps it safe is also that the Sun’s pull on the pair, though larger than the Earth’s pull on the Moon, is almost the same on both members and does no work on their relative orbit except through its difference across the pair.

That difference is the solar tide on the Earth–Moon orbit, and it is not small. It perturbs the Moon’s orbit by thousands of kilometres, in a set of inequalities that were the central problem of celestial mechanics for two centuries — the evection, with a period of about thirty-two days, which stretches the Moon’s orbit when its long axis points at the Sun; the variation, which distorts the orbit twice a month; and the annual equation, from the Earth’s own eccentric orbit round the Sun. Each is a kick from the third body that averages away over a longer cycle, exactly as the stable side of the boundary requires. The Moon’s orbit is the best-studied case there is of a hierarchical triple in which every perturbation is large, periodic and harmless.

Light companions, heavy companions #

The same map, redrawn for a light and for a heavy third body, shows how much room the mass ratio gives.

The comparison makes the lifetime reading concrete. For the light companion, the gap between the criterion and the observed break-ups is wide: its kicks are weak and chaotic orbits take a long time to break. For the heavy one it is narrow: the kicks are strong and chaos, once present, acts fast. The distance between the resonance-overlap line and the break-up line is a measure of how weakly the chaos is driven, and it is widest precisely where the third body matters least.

Eccentric companions #

An eccentric outer orbit raises the boundary, and the reason is visible in a single pair of runs.

That is why the criterion is written in terms of the pericentre rather than the semi-major axis, and why the eccentricity enters it only weakly once the pericentre is fixed. It is also why eccentric triples live dangerously when they are subject to a slow change of eccentricity from any other cause. The Kozai–Lidov mechanism drives a slow exchange of eccentricity and inclination in a hierarchical triple whose orbits are tilted, and can drive the outer or inner eccentricity high enough to move a triple from the stable side of the boundary to the unstable one — so that a triple which was stable when it formed becomes unstable a few million years later without anything external happening to it.

Planets round two suns #

The same criterion governs planets orbiting both stars of a close binary, and it is where the boundary has been tested most sharply by nature rather than by integration.

A circumbinary planet is a hierarchical triple in which the third body has a negligible mass ratio. Its critical orbital distance, from integrations of the restricted problem made by Holman and Wiegert, is between about two and four binary separations, depending on the binary’s eccentricity and mass ratio. The Kepler spacecraft found about a dozen such planets — Kepler-16b was the first, in 2011, orbiting a pair of stars in 229 days — and a remarkable number of them orbit just outside the critical distance, within twenty or thirty per cent of it.

The pile-up is not a coincidence of small numbers. Planets forming in a circumbinary disc migrate inward through the gas until they reach the disc’s inner edge, which is carved by the binary at roughly the stability boundary, because the gas inside it is itself on unstable orbits. The planets have been parked at the edge of chaos by the disc they formed in, and the edge is where the resonance overlap puts it. That is the cleanest demonstration anywhere that a stability boundary derived from overlapping resonances is a physical place, and not only a line in a parameter space.

When a cluster changes the answer #

A triple in the field is left alone once it forms, and its fate is settled by where it sits relative to the boundary. A triple in a dense star cluster is not left alone, and the boundary becomes a moving target.

Passing stars in a globular cluster core deliver kicks to the outer orbit much as the outer body delivers kicks to the inner one. A soft outer orbit — one whose binding energy is small compared with the kinetic energy of a typical passing star — is widened by them and eventually unbound; a hard one is tightened, the rule that makes binaries a cluster’s source of heat. A hierarchical triple in a cluster therefore drifts across the stability map over time: its outer orbit may be tightened until it crosses into the chaotic zone, at which point the triple breaks up in a few outer orbits, ejects one member and leaves a tighter binary behind.

That sequence — hardening, crossing the boundary, a democratic breakup — is one of the main channels by which dense clusters manufacture very tight binaries and eject stars at high speed. It is also a source of the merging black-hole binaries that gravitational-wave detectors observe: a triple of black holes in a cluster core, pushed across the boundary, produces a binary with a high eccentricity that radiates its orbital energy away far faster than a circular binary could. The boundary drawn at the top of this essay is where the triple stops being two problems and becomes one, and in a cluster that crossing is not an accident of initial conditions but a regular event.

What the integrations do not include #

The triples are coplanar and prograde. Mutual inclination changes the boundary in both directions: retrograde outer orbits are markedly more stable, because the resonant interactions are weaker when the bodies pass each other quickly, while inclined orbits bring in the Kozai–Lidov cycles, which the criterion’s calibrated inclination factor treats only approximately.

The inner binary starts circular. An eccentric inner binary presents a varying cross-section to the third body and is less stable; the criterion as written here assumes a circular one.

Each cell is a single integration at one phase. Near the boundary, survival depends on the relative phases of the orbits at the start, and a proper map would show, in each cell, the fraction of phases that survive. A single run per cell draws that fraction as zero or one.

And the integrator is not symplectic. It is a fourth-order Runge–Kutta scheme with a step set by the closest pair, and adapting the step breaks the conservation properties a symplectic integrator would have. Every surviving run reports its energy error, which is at the level of one part in ten million or better, far smaller than anything that decides survival; for integrations of millions of outer orbits a different integrator would be needed.

Still open: how long unstable is #

The criterion says where chaos begins, and it says nothing about how long a chaotic triple takes to break up. That lifetime varies by orders of magnitude across the chaotic zone — from a single outer orbit close to the binary to longer than the age of the universe just inside the boundary, particularly for light companions — and it decides which “unstable” triples should be seen in the sky. There is no formula for it of the same generality as the boundary. Statistical theories of the random walk in the binary’s energy give estimates in parts of the zone; large ensembles of integrations map it in others. For triples in dense star clusters, where passing stars add their own kicks, the question merges into the statistical theory of how clusters process their binaries. A hierarchical triple on the wrong side of the line by a small margin may outlive its own stars, and whether the sky contains many of them — metastable systems whose stability is a matter of time rather than of principle — is a question the criterion alone cannot answer.