Three bodies, and what "no solution" actually means

Two bodies under gravity are solved completely and forever. The orbit is a conic, the timing comes from one equation, and the position at any future time is a finite calculation.

Add a third body and the standard statement is that the problem becomes unsolvable. That statement is repeated so often that it has acquired a meaning it does not have, and unpicking it is worth doing, because three completely different things are true at once: no general formula exists, particular exact solutions do exist, and trajectories are computed routinely to whatever precision anybody wants.

What Poincaré actually showed #

The result usually cited is Poincaré’s, from 1890, and the popular version of it — “he proved the three-body problem cannot be solved” — is not what he proved.

What is missing is not a solution but a reduction. The two-body problem is tractable because it has enough conserved quantities to eat all its degrees of freedom: energy, angular momentum, the motion of the centre of mass, and the extra vector that keeps the orbit from precessing. Count them against the number of coordinates and the system reduces to a single equation in one variable.

The three-body problem has energy, angular momentum and the centre-of-mass motion too. It has far more degrees of freedom, and what Poincaré showed is that no further independent algebraic conserved quantities exist. There is nothing more to find, so the reduction cannot be completed, and the system is said to be non-integrable.

That is a statement about a particular kind of solution: a closed-form expression, built from a finite number of standard operations, giving positions as functions of time. No such expression exists. It is emphatically not a statement that the motion is unknowable.

The proof also produced something Poincaré had not gone looking for. Working on a prize essay for the King of Sweden, he initially submitted a version containing an error; correcting it, he found that the stable and unstable trajectories emanating from certain unstable orbits do not simply miss each other or join smoothly, but intersect infinitely often, weaving a structure of unimaginable complexity. He wrote that he would not even attempt to draw it. That structure is a homoclinic tangle, and it is the mechanism of chaos, discovered seventy years before the word.

Sensitive dependence, measured #

The practical content of non-integrability is not the absence of a formula. It is that neighbouring trajectories separate exponentially, so a prediction’s accuracy decays at a rate set by the system rather than by the effort put into it.

That figure is Burrau’s problem — masses 3, 4 and 5 released from rest at the corners of a 3-4-5 triangle, posed in 1913 and computed properly only in 1967. It runs through a long sequence of close approaches and eventually ejects the lightest body altogether, leaving a binary behind. That ending is the generic fate of a three-body system: one body leaves, and the other two pair up about their common centre.

The rate of separation defines a timescale. If a difference grows as $e^{t/\tau}$, then $\tau$ — the Lyapunov time — is how long it takes an error to grow by a factor of $e$, and beyond a few dozen $\tau$ any initial uncertainty has filled the accessible space. For the inner solar system that timescale is about five million years — comfortably shorter than the intervals over which the planets’ orbital elements are quoted.

The consequence is worth stating in the form that makes it concrete. Improving the initial data by a factor of ten buys $\tau \ln 10$ of extra prediction — about eleven million years for the inner planets. Measuring the planets’ positions a thousand times better than they are known now would extend the horizon of prediction by about thirty-four million years, and no further. The limit is not instrumental. It is the dynamics.

Solutions that do exist #

Against that background, the exact solutions are genuinely surprising, and there are more of them than the “unsolvable” framing suggests.

Euler, in 1767, found the three collinear solutions: three bodies in a line, rotating rigidly, with the spacings fixed by a quintic. Lagrange, in 1772, found the equilateral one — three bodies at the corners of an equilateral triangle, rotating together, for any mass ratio whatsoever. Those five configurations, applied to the restricted case where the third body is massless, are the Lagrange points, and they are occupied.

Then, for two hundred and twenty years, nothing new.

In 1993 Christopher Moore found numerically, and in 2000 Alain Chenciner and Richard Montgomery proved rigorously, that three equal masses can chase each other around a single figure-eight curve, each a third of a period behind the next. It is the hero figure above. Nothing about it was suspected; the curve had been sitting in the equations since 1687.

The figure-eight opened a search. Several thousand families of periodic three-body orbits are now known, most of them found in the last fifteen years by systematic numerical scanning of initial conditions. They are all unstable — a perturbation destroys them — and they are all exact.

The moral is that “no general solution” and “no solutions” are different claims. The set of exact solutions is infinite and richly structured. What is missing is a formula that covers the arbitrary case.

Restricting until something works #

The route that produced most of the subject’s usable results is to weaken the problem until it becomes tractable, and to be explicit about what was given up.

The circular restricted three-body problem makes the third body massless, so it does not disturb the other two, and puts the two massive bodies on circles. In the frame rotating with them, both are stationary, and the problem acquires one conserved quantity — the Jacobi integral — which is not an energy but behaves like one.

That single conserved quantity is worth more than it looks. Fixing its value bounds the region a body can reach — the zero-velocity surfaces, outside which the kinetic energy would have to be negative. A body inside the surface around one primary cannot escape to the other without a change of Jacobi constant, which gravity alone cannot supply. That is a genuine prediction about a chaotic system: not where a body will be, but where it cannot be, forever.

The elliptic restricted problem, where the primaries move on ellipses, loses even the Jacobi integral, and the equilibrium points become periodic paths rather than places. Almost every real system is this one.

What was actually computed #

The solar system’s own stability is the question this all points at, and it has an answer with an interesting shape: probabilistic, obtained numerically, and unavailable by any other route.

The calculation is a direct integration of the planets’ equations of motion over billions of years, with the relativistic corrections included, using symplectic integrators designed to conserve the system’s structural properties rather than merely its accuracy. Jacques Laskar and colleagues have run thousands of such integrations, differing only in a starting position perturbed by a few metres.

The results diverge, as they must, after about a hundred million years. What survives is a statistics of outcomes. In roughly 99% of runs the solar system remains recognisably as it is for the next five billion years. In about 1%, Mercury’s eccentricity — pumped by a resonance with Jupiter’s apsidal precession — grows large enough that the planet’s orbit crosses Venus’s, and the ending is a collision, an ejection, or in a small number of runs a destabilisation that reaches the Earth.

Two things about that result are worth separating. The first is that a 1% chance over five billion years is not a prediction about anything anybody will see. The second is that the number could not have been obtained any other way: there is no perturbation series that converges over that span, and no analytic criterion that decides it. The stability of the solar system is a numerical experiment.

The measurement that constrains it is more modest and much more precise. Planetary ephemerides are fitted to decades of spacecraft ranging and radar, and they reproduce the observed positions to metres. Those ephemerides are integrations of exactly this kind, and their agreement with observation over the fifty years for which the data exist is what licenses running the same machinery for five billion. The same integrations supply the positions every mission is navigated against.

The three-body problem as a machine for making things #

Non-integrability sounds like a defect. In one respect it is an engine, and the astrophysical systems that matter most are the ones that exploit it.

The mechanism is that a chaotic three-body system redistributes energy and angular momentum in ways a two-body system cannot. Two bodies are stuck with the orbit they have. Three can trade, and the trade is what produces almost every compact-object binary anyone has detected.

Ejection and hardening. A generic triple ends with one body leaving and the remaining two tightened. The escaper carries away positive energy, so the survivors must be more tightly bound than the original configuration was; the binary “hardens”. In a dense star cluster this happens repeatedly, and it is the standard route to producing binaries tight enough to merge under gravitational radiation within the age of the universe.

The Kozai–Lidov mechanism. A close pair with a distant third body on a steeply inclined orbit does something entirely counterintuitive: the inner pair’s eccentricity and the mutual inclination oscillate into each other, with the eccentricity climbing to values near 1 while the semi-major axis stays fixed. The inner orbit’s periapsis can be driven down until tides or gravitational radiation take over. Discovered independently in 1961 and 1962, in work on asteroid and satellite orbits respectively, it now appears in explanations of hot Jupiters, of black-hole mergers, and of why so many close binaries have a distant companion.

Resonant capture. A resonance is a three-body phenomenon by construction — two orbits plus the primary — and capture into one is possible only because the dynamics are not integrable. A system with a formula could not have been captured into anything.

None of these has a closed-form description. All of them are computed, and all of them are observed.

The sphere where one body wins #

The restricted problem produces one more piece of practical machinery, and it is what makes the whole patched-conic approach to mission design defensible.

Around each body there is a region within which its gravity dominates the perturbation from the primary. Its radius is the Hill radius,

$$r_H \approx a\left(\frac{m}{3M}\right)^{1/3},$$

with $a$ the body’s orbital radius, $m$ its mass and $M$ the primary’s. For the Earth it is about 1.5 million kilometres — four times the Moon’s distance, which is why the Moon is bound to the Earth at all. For Jupiter it is 53 million kilometres, and for a low-mass asteroid it is a few hundred metres, which is why almost nothing has a satellite.

The Hill radius is a statement about the restricted problem, and it is derived from the same effective potential the Lagrange points come from — L₁ and L₂ sit at roughly one Hill radius either side of the secondary, and the Hill sphere is essentially the region those two points enclose. A body inside it can be treated as orbiting the secondary with the primary as a perturbation; outside, the reverse.

That division is what licenses splitting a trajectory into two-body arcs and stitching them together. It has no rigorous justification, it is a chaotic system being carved into integrable pieces, and every interplanetary mission ever flown was planned this way.

Easier again with a million #

There is a pattern in the difficulty that is worth naming, because it is not monotonic in the number of bodies.

Two bodies are solved exactly. Three are not, and neither are any number between three and something very large. But take the number to a million and the problem becomes tractable again — not exactly, but in a way that yields real answers.

The reason is that a system of enough bodies can be described by a distribution rather than by a list. Instead of following each star, one follows the density of stars in position and velocity, evolving under the smooth gravitational field that the density itself produces. Individual encounters are neglected, which is legitimate when there are enough bodies that no single one dominates any other’s motion, and the resulting equations describe a galaxy well enough to predict its structure.

So the difficulty peaks in the middle. Few enough bodies to need exact treatment and too many to have one is the hard region, and it happens to be where planetary systems, stellar triples and small clusters all live — which is why numerical integration was invented for exactly those problems and why the largest systems and the smallest were both understood first.

Where the model stops #

Point masses. Real bodies have tides and oblateness, and both feed back into the orbits. Over the age of the solar system the Earth–Moon tidal exchange has moved substantial angular momentum around.

Newtonian gravity. Mercury’s relativistic precession is 43 arcseconds per century, which is small and is not negligible for stability: leaving it out changes the frequency of the resonance that destabilises the planet, and the computed instability probability rises by more than an order of magnitude. A correction discovered as a curiosity turns out to control the outcome.

Three bodies. The solar system has eight planets, and the mechanisms that matter are resonances between three or more of them at once. The three-body problem is where the phenomena appear; it is not where the solar system lives.

Finite precision. Every integration is done in floating-point arithmetic, and the round-off is itself a perturbation that grows exponentially. Long integrations use extended precision and symplectic methods, and their results are trusted statistically rather than trajectory by trajectory.

The figures share a limitation that is intrinsic to what they show. Each of them is a trajectory plot — the path traced in space — which is exactly the wrong object for a chaotic system, because it discards the timing. Two three-body systems can trace visually similar paths while being at completely different points along them, and the divergence figure only works because the two runs are stopped at the same instant. The real object of study is a trajectory in a twelve-dimensional phase space, and no projection of it onto a page is faithful.

One further consequence of non-integrability deserves recording, because it inverts what the word suggests. A system with a closed-form solution is predictable and therefore, in a precise sense, contains no information that was not in its initial conditions. A chaotic one generates information: two states indistinguishable at the start become distinguishable later, so the trajectory is continuously revealing digits of the initial condition that no measurement had access to. That is why a chaotic system cannot be predicted and can be simulated — and why the statistics of the solar system’s possible futures are a meaningful result while any single one of them is not.

One more integration follows a periodic solution twice as far as its own period.

The ladder from here #

Later rungs on this anchor: the Jacobi integral derived, and the zero-velocity surfaces it bounds. Hill’s problem and the Hill sphere. Poincaré sections, and reading a chaotic system through them. The homoclinic tangle. Lyapunov exponents, and how one is measured from a numerical trajectory. KAM theory, which says which orbits survive a small perturbation and which do not. The figure-eight and the catalogue of periodic families. Symplectic integrators. Laskar’s stability integrations in detail. Hierarchical triples in stellar dynamics, and the Kozai–Lidov mechanism that turns an inclination into an eccentricity. And the gravitational-wave sources produced by three-body encounters in dense clusters, which is where the problem is currently doing the most work.

Poincaré’s prize essay was awarded, printed, and then recalled at his own expense when he found the error — which cost him more than the prize was worth, and led directly to the discovery of chaos.