Three Body

Math · chapter 12 of 13

What usually happens

Throw three bodies together and the usual ending is a fight two of them win: one gets thrown out, the other two are left closer than they started. You cannot predict which — but you can price it.

Give up on predicting the path and ask a different question: out of many random triples, what fraction end which way? That question has answers, and they are stable, and they are useful.

Start a pair in orbit and send a third body in. Three things can happen. It can fly past, leaving the pair a little changed — a flyby. It can trade places with one of the pair and leave the other one out in the cold — an exchange. Or all three can mill about for a while in a resonance, drawing loops around each other, and then one leaves for good — an ejection. Given enough time, and with the energy to allow it, ejection is what almost always happens.

The bookkeeping is exact even though the path is not. Energy is conserved, so the body that leaves takes kinetic energy with it, and the pair left behind must go down by the same amount — which for a gravitational pair means closer together and faster. This is how a binary tightens: by throwing things out. It is also where a slingshot comes from. Voyager 2 did not gain energy from nothing; Jupiter lost a slice, and Jupiter can afford it.

In 2019 Nicholas Stone and Nathan Leigh derived the outcome distribution — a closed-form answer for the statistics of the chaotic problem, with the escaper's energy and the leftover pair's shape as probabilities. The individual path stays unpredictable; the odds are now written down.

The stable arrangements are the hierarchical ones: a close pair with a distant third, each level looking like a two-body orbit with a small correction. Sun, Earth, Moon. Every triple star that has lasted. Even those have a slow knife: if the outer orbit is tilted more than about 39.2° from the inner one, the Kozai–Lidov mechanism trades the tilt for stretch, and the inner pair's orbit swings between round and cigar-shaped over thousands of orbits. That mechanism is now standard equipment in explanations of hot Jupiters and of black-hole pairs that merge.

The demo below runs a few hundred random triples in your browser and counts what they did. The percentages on the page are whatever the run produced, computed there and then.

Epair, after = Epair, before − Eescaper

Reading it: Whatever energy the ejected body leaves with comes out of the pair that stays, which pulls them closer together.

A tighter pair is a lower-energy pair, because the energy is negative and grows more so. This is the engine behind hard binaries in star clusters getting harder.

aafter = abefore1 + abeforeΔ  ·  cos icrit = ±√(3/5), i ≈ 39.2°

Reading it: How much the pair tightens, and the tilt beyond which a distant third body starts stretching the inner orbit.

The second is the Kozai–Lidov threshold. Above it, the inner pair's roundness and its tilt trade back and forth on a long cycle.

teject — heavy tailed

Reading it: How long the milling-about lasts has no typical value worth quoting: mostly quick, with a long tail of triples that hang on.

Which is why the demo shows the spread rather than an average. An average over a heavy-tailed distribution is a number that describes nothing.

Try it. Run the demo once, then again. The individual endings differ and the percentages barely move. That gap — unpredictable one at a time, dependable in bulk — is the whole reason stellar dynamics is a science.

This one runs in the browser, with JavaScript on. The words above it stand on their own.

0 of them left between 0 and 20 of them left between 2 and 41 of them left between 4 and 611 of them left between 6 and 815 of them left between 8 and 1015 of them left between 10 and 129 of them left between 12 and 1420 of them left between 14 and 1617 of them left between 16 and 1829 of them left between 18 and 2025 of them left between 20 and 2227 of them left between 22 and 2421 of them left between 24 and 2621 of them left between 26 and 2817 of them left between 28 and 3028 of them left between 30 and 3216 of them left between 32 and 3416 of them left between 34 and 3611 of them left between 36 and 3817 of them left between 38 and 4016 of them left between 40 and 4222 of them left between 42 and 4417 of them left between 44 and 4617 of them left between 46 and 4812 of them left between 48 and 508 of them left between 50 and 527 of them left between 52 and 546 of them left between 54 and 567 of them left between 56 and 587 of them left between 58 and 600102030405060how many left, per slice of timetime units

643 random triples of equal weight, run for 60 time units while this page was being built. 435 of them — 67.7% — threw a body clear out; 208 were still together at the end. Half the ejections had happened by t = 29.0, a tenth before 12.5, and a tenth not until after 48.4.

The audit on that figure. The bottom of each well is rounded off at 0.05, because at a fixed step a near pass throws a body across the sky and the answer you get is the stepper's rather than the physics'. The median energy error over the run is 2e-05. Then the first 150 of the same starts were run again at a third of the step: the share that ejected came out at 64.0% against 70.0% — the same answer — while only 69% of the individual triples ended the same way. The statistics hold and the trajectories do not, which is the finding of this whole page.

Sources

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