Astronomy & Celestial Mechanics
the sky, computed
Concept

Three-body problem — where it appears

The motion of three gravitating bodies, which has no general solution in closed form and is chaotic over most of its parameter space. Its unsolvability is a statement about closed-form expressions rather than about computation, and the trajectories are computable for as long as their Lyapunov time allows.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

The five Lagrange points at mass fraction 0.12. The five points at which a small body can keep station with two larger ones. The three on the line of centres are roots of a quintic and are unstable; the two forming equilateral triangles are stable for a sufficiently lopsided mass ratio.

Five places that keep station, in a problem with no solution

Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles.

gravitation · Lagrange points 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.

gravitation · The three-body problem The Hill radius against the mass ratio, and where four planets keep their outermost moon. The Hill radius R_H = a(μ/3)^⅓ and the exact L₁ distance from the same quintic, both in units of the planet's own semi-major axis, against the mass ratio μ = m/(M+m) from 10⁻⁸ to 0.12. The two are the same line to the width of the stroke across the whole planetary range — the cube-root formula runs 0.33% short of the exact L₁ distance at the Earth's mass ratio and 10.0% short at the right-hand edge — so the departure between them is drawn on its own percentage scale at the right, which is the only place it can be seen. Below the curves are the numerically determined stability limits, 0.5 R_H for a prograde satellite and 0.7 R_H for a retrograde one, and the outermost known satellite of each of four planets, each at a Hill fraction computed from that planet's own mass and orbit: Moon 0.257, Sinope 0.451, Phoebe 0.198, Neso 0.424. Of the four, three are retrograde — Sinope, Phoebe, Neso — and they reach 0.451 of their planet's Hill radius against 0.257 for the one prograde satellite: 1.8 times as far out, in a diagram whose two stability limits stand in the ratio 0.7 to 0.5. No equilibrium argument predicts that, and it is what the drawing makes plain.

The region a planet may keep a moon in

A satellite is not held by orbiting but by orbiting inside the radius at which the Sun's tidal field would take it away. That radius is the same distance the innermost Lagrange point sits at, asked a different question — and the boundary the sky actually respects is smaller, by a factor that depends on which way the moon is going round.

gravitation · Hill sphere Energy error over 240 revolutions, at one step size. The relative error in total energy against revolution number, for three integrators run on the same Kepler orbit at e = 0.5 with the same step of 200 per revolution. The exact energy is a constant, so every curve here is the method rather than the problem. Euler climbs steadily: its energy at the end is 106.5% wrong, and the orbit it draws has spiralled outwards. Runge–Kutta 4 begins 4.7e+2 times more accurate than leapfrog and ends at 2.22e-4, having grown by a factor of 10 across the run: the error is SECULAR. Leapfrog oscillates inside a band and stays there — worst error 2.62e-3, and the second half of the run is no worse than the first, which is measured here rather than claimed. That is the property that decides whether a five-billion-year integration means anything, and it is not accuracy: a symplectic method is the exact solution of a Hamiltonian a step-size away from the intended one, so its energy cannot wander, while a more accurate non-symplectic method has no such constraint and eventually wanders further.

Wrong about where, and right about how much

Runge–Kutta is the more accurate method and loses energy steadily; leapfrog is cruder and its energy error never leaves a band. Over five billion years only one of those properties survives — and neither method knows where the planet is.

gravitation · Numerical integration A separation of 10⁻¹⁰ becomes 0.0285. The distance between two copies of Burrau's problem, started 10⁻¹⁰ apart in one coordinate of one body and then stepped in lockstep — one loop, one step size, both states advanced by it — for 60 time units. The vertical axis is logarithmic, so the straight stretch is exponential growth, and its slope is the Lyapunov exponent: 0.3413 per time unit, fitted over the 604 samples lying between ten times the starting offset and a tenth of the system's own size. That is an e-folding every 2.93 time units, so the separation multiplies by ten every 6.75 and by 7.8·10⁸ across the whole run. It has not saturated within the drawn interval: the growth rate over the last tenth of the run is still 0.6002 per time unit, 176% of the fitted exponent, and the separation has reached only 0.90% of the system's own size. The exponent is what sets a prediction horizon, and this run is drawn short of it on purpose: the straight line is the measurement, and the flat part that follows is arithmetic about how far apart two bounded systems can get.

A prediction with an expiry date

The inner solar system's Lyapunov time is about five million years, so a centimetre of error becomes an orbit in a hundred million. The ephemeris dies while the system survives — because the elements stay bounded when the phase does not, and only one of those is what stability means.

gravitation · Chaos A transit that will not keep time. Transit times of a planet of 8 Earth masses on a 3-day orbit, minus the best straight line through them, from a three-body integration in which the second planet at 4.62 days is the only thing perturbing it. The residual swings by ±1.7 minutes and repeats over 58 days — the super-period 1/|3/P₂ − 2/P₁|, which is long precisely because the pair is near the 3:2 resonance and gets longer the nearer it is. The amplitude is proportional to the perturbing planet's mass, so fitting this curve weighs a planet that may never transit at all. The short jagged component is not noise: it is the chopping signal, one kick per conjunction at the 8.6-day synodic period, sampled once every 3 days by the transits themselves and so barely above the rate at which it can be followed. The integration conserved energy to 1.6e-10.

Planets found by a transit running late

A planet on a fixed orbit transits like a clock. A second planet pulling on it makes the clock run fast and slow by minutes — and fitting that wander weighs a planet that may never cross the star at all.

exoplanets · Transit timing An ephemeris fitted to 120 days, 43 minutes wrong within a year against a band of ±5.1. Transit times of a 6 Earth-mass planet on a 10-day orbit, perturbed by a 14 Earth-mass planet at 15.24 days, integrated for 1460 days and compared with a straight-line ephemeris fitted only to the transits in the first 120 days — the shaded window. Inside the window the line fits to 2.1 minutes. Outside it the pair's 317-day super-period carries the transits away from the line, and within a year of the window closing the prediction is 42.8 minutes early of the observed transit, 347 days after the last fitted one. The narrow band is the formal three-sigma uncertainty of the same line for a timing precision of 0.5 minutes per transit, which at that date is ±5.1 minutes: the error is 8.4 times the band. A statistical uncertainty assumes the residuals are noise, and these are a signal, so the band describes a planet that does not exist.

A forecast that fails on a schedule

A transiting planet perturbed near a resonance keeps a clock that wanders, and a straight-line ephemeris fitted to part of the wander predicts the next transit with a confidence the wander does not deserve. The error is not noise and does not average down; it grows on the pair's super-period, it is many times the formal uncertainty within a year, and how soon it appears depends on which stretch of the wander happened to be observed. When a model that includes the known perturber still fails, the failure has a period, and the period is a planet.

exoplanets · Transit timing A 10 Earth-mass Trojan started 10° from its point, seen in the planet's transits. A Jupiter-mass planet on a 4-day orbit about a 1 solar-mass star, with a 10 Earth-mass companion started 10 degrees beyond the leading Lagrange point of the same orbit, integrated for 160 days. Above, the angle between the companion and the planet as seen from the star: it swings about 60 degrees, between 51.1 and 70.3, with a period of 48.5 days measured off the curve, against 49.8 from the small-amplitude formula P/√(27μ/4). Below, the planet's own transit times minus a straight line: they swing by ±300.0 seconds at the same period, because the planet and the companion orbit their common centre of mass and the companion's libration moves that centre along the orbit. The companion shares the planet's period exactly, so it produces no timing signal at any orbital period of its own and would not transit on a schedule distinct from the planet's; the libration period is the only clock it has.

A companion on the same orbit, seen in the planet's clock

A body sharing a planet's orbit at one of its Lagrange points has the planet's period exactly, so no search for periodic dips or wobbles can find it at a period of its own. It shows instead in two ways the planet's own signals carry — a slow swing of the transit times at the companion's libration period, and a fixed offset between the planet's transit and its star's velocity curve.

exoplanets · Transit timing The quantity a capture has to change. The Jacobi constant along three trajectories through the same encounter with the secondary, integrated in the rotating frame at mass fraction 0.0009543. The undissipated one holds its value to 3e-11 over the whole passage — that is not an approximation but an identity of the equations of motion, and the residual is the integrator's. A body that enters through the neck at L₁ with the neck open leaves through it, because C has not changed and the neck is still open; a purely gravitational two-body encounter is time-reversible and cannot end in a bound orbit. The other two runs remove energy in the rotating frame. A drag proportional to velocity raises C continuously at a rate 2k v², and a single impulse against the motion raises it in a step. Both finish above C(L₁) = 3.039, which is the line at which the neck shuts and the body is trapped in the secondary's lobe. Which mechanism operated for a given satellite is not recoverable from its orbit today — but it is partly recoverable from the population, because the three leave different distributions behind.

A moon that two bodies could not have caught

A body that falls into a planet's Hill sphere under gravity alone leaves it again, because the quantity the rotating frame conserves is the same on the way out as on the way in. Every captured moon is therefore a record of something that removed energy — and which of the three candidates did it is written in the inclinations rather than in any one orbit.

gravitation · Hill sphere 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.

gravitation · The three-body problem 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.

gravitation · The three-body problem 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.

gravitation · The three-body problem

Named alongside it

The objects these essays reach for when they reach for this one.

Lagrange pointsNumerical integrationChaosMass ratioN-body integrationRotating frameTransit timing variationAngular momentumChoreographyEccentricityEffective potentialEphemeris

All concepts