A statistical solution to the chaotic, non-hierarchical three-body problem

The three-body problem is arguably the oldest open question in astrophysics and has resisted a general analytic solution for centuries. Various implementations of perturbation theory provide solutions in portions of parameter space, but only where hierarchies of masses or separations exist. Numerical integrations¹ show that bound, non-hierarchical triple systems of Newtonian point particles will almost² always disintegrate into a single escaping star and a stable bound binary^3,4, but the chaotic nature of the three-body problem⁵ prevents the derivation of tractable⁶ analytic formulae that deterministically map initial conditions to final outcomes. Chaos, however, also motivates the assumption of ergodicity^7-9, implying that the distribution of outcomes is uniform across the accessible phase volume. Here we report a statistical solution to the non-hierarchical three-body problem that is derived using the ergodic hypothesis and that provides closed-form distributions of outcomes (for example, binary orbital elements) when given the conserved integrals of motion. We compare our outcome distributions to large ensembles of numerical three-body integrations and find good agreement, so long as we restrict ourselves to 'resonant' encounters¹⁰ (the roughly 50 per cent of scatterings that undergo chaotic evolution). In analysing our scattering experiments, we identify 'scrambles' (periods of time in which no pairwise binaries exist) as the key dynamical state that ergodicizes a non-hierarchical triple system. The generally super-thermal distributions of survivor binary eccentricity that we predict have notable applications to many astrophysical scenarios. For example, non-hierarchical triple systems produced dynamically in globular clusters are a primary formation channel for black-hole mergers^11-13, but the rates and properties^14,15 of the resulting gravitational waves depend on the distribution of post-disintegration eccentricities.