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 binary3,4, 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 ergodicity7–9, 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 mergers11–13, but the rates and properties14,15 of the resulting gravitational waves depend on the distribution of post-disintegration eccentricities.
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Acknowledgements We acknowledge discussions with D. Heggie, P. Hut, R. Sari and S. Portegies-Zwart, as well as feedback from E. Michaely and O. C. Winter. N.C.S. received financial support from NASA, through Einstein Postdoctoral Fellowship Award number PF5-160145 and the NASA Astrophysics Theory Research Program (grant NNX17AK43G; Principal Investigator, B. Metzger). N.C.S. also thanks the Aspen Center for Physics for its hospitality during early stages of this work. N.W.C.L. acknowledges support by Fondecyt Iniciacion grant number 11180005. We thank the Chinese Academy of Sciences for hosting us as we completed our efforts. We thank M. Valtonen and H. Karttunen, whose book on the three-body problem motivated much of this work.
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