Astronomy & Celestial Mechanics
the sky, computed
Series

The three-body problem — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Three bodies, integrated. Three equal masses integrated forward under mutual gravity: the figure-eight choreography — all three bodies on one closed curve. Every point is a step of the equations of motion, and the total energy is conserved to 2.5e-11 across the run.

    Three bodies, and what "no solution" actually means

    The three-body problem is routinely called unsolvable. Trajectories are computed for it every day, exact periodic solutions are known, and both statements are true — the word is doing more work than it looks.

    part 1 · gravitation
  2. Zero-velocity curves, mass fraction 0.15. Level sets of the Jacobi constant in the frame that rotates with the two bodies. A particle with a given value of C is confined to the side of its own curve where the kinetic energy would be positive; as C falls the curves open, first at the inner point between the bodies, then behind the smaller one, and finally around the whole system.

    The curve that says where a body cannot go

    The restricted three-body problem has no solution and one conserved quantity. That quantity is enough to draw a boundary the body can never cross — without integrating anything, without knowing where it started, and for all time.

    part 2 · gravitation
  3. Which hierarchical triples survive, against the outer orbit's pericentre and eccentricity. Each cell is one integration of a triple: an equal-mass inner binary on a circular orbit of unit separation, and a third body of 1 times the binary's mass on an outer orbit of the given eccentricity with its pericentre at the given distance from the binary's centre of mass, all in one plane and going the same way round. Cells coloured as survivors kept the binary intact and ejected nothing for 30 outer orbits — 37 of 50; the others broke up, shaded darker the longer they lasted. The curve is the Mardling–Aarseth stability criterion, a pericentre of 2.8 inner separations times ((1 + q)(1 + e)/√(1 − e))^0.4, derived from where the resonances between the outer orbit and the inner one begin to overlap, and meant for survival over very many orbits. Every triple outside it survived. Inside it, 17 of 30 also survived this run; the outermost break-up is at a pericentre of 3.0 for a circular outer orbit and 2.6 at e = 0.6. The boundary drawn from a finite integration is therefore a lifetime rather than a line — triples between the break-ups and the curve survived a run of this length and are not thereby shown to be stable; the criterion is a statement about how the resonances are arranged, and a finite integration can only fail to refute it.

    A third body kept at a safe distance

    Most triple stars last for billions of years because they are hierarchical — a close pair and a distant third. How distant is distant enough has a formula, the formula comes from overlapping resonances, and an integration of any finite length can only fail to refute it.

    part 3 · gravitation
  4. Which of three bodies is thrown out. Of 60 integrations of three bodies of mass 1, 2, 3 released from rest in random triangles, 57 broke up within the time integrated, each leaving one body unbound and a binary behind. The escaper was the body of mass 1 in 46 runs (81 per cent), 2 in 6 runs (11 per cent), 3 in 5 runs (9 per cent). The bars beside them are the shares the statistical theory of break-up predicts to leading order, in proportion to the escaper's mass to the power −3: 86, 11, 3 per cent. Every run is deterministic and none of them could be predicted without integrating it, but the ensemble is: the lightest body leaves most often when the masses differ, because the energy of the binary left behind is shared between the escaper and the binary's recoil in inverse proportion to their masses, and the lightest body gets the most speed. Energy is conserved in every run to better than 3·10⁻⁴.

    The lightest body leaves

    Three stars close together, with no hierarchy between them, cannot stay together. One of them is thrown out, which one is unpredictable in any single case, and across many cases it is almost always the lightest — a statistical law emerging from a deterministic chaos that cannot be integrated far enough to forecast.

    part 4 · gravitation
  5. The figure-eight with equal masses, and with one mass changed. The paths of three bodies in the figure-eight choreography over 100 periods, with equal masses (left) and with the third body's mass changed by 0.0001 (right). With equal masses the three paths lie on one another: each body runs round the same figure-of-eight, a third of a period behind the next. With the mass changed the three bodies stay bound but leave the single curve, filling a broad band round where it was. The choreography is an exact solution only for exactly equal masses, and the island of stable orbits near it is narrow.

    A periodic orbit that the sky cannot keep

    Three equal masses can chase one another for ever round a single figure of eight. The orbit exists, it is stable, and it will almost certainly never be seen — because its island of stability is a hundredth of a per cent wide in the masses, and nothing in nature makes three stars equal to that precision.

    part 5 · gravitation

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