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The Chaotic Three-Body Problem in Newtonian Gravity

Published onJun 01, 2020
The Chaotic Three-Body Problem in Newtonian Gravity

This paper derives a general statistical solution to what is arguably the oldest open question in both physics and astrophysics: the three-body problem. The three-body problem 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 integrations show that bound, non-hierarchical triples of Newtonian point particles will almost always disintegrate into a single escaping star and a stable, bound binary, but the chaotic nature of the three-body problem prevents the derivation of tractable analytic formulae deterministically mapping initial conditions to final outcomes. However, chaos also motivates the assumption of thermodynamic ergodicity. Using this assumption, we derive a complete statistical solution to the non-hierarchical three-body problem, one which provides closed-form distributions of outcomes (e.g. binary orbital elements) 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” encounters (the ∼50% of scatterings that undergo chaotic evolution). In analyzing our scattering experiments, we identify “scrambles” (periods in time where no pairwise binaries exist) as the key dynamical state that ergodicizes a three-body system. The generally super-thermal distributions of survivor binary eccentricity that we predict have notable applications to many astrophysical scenarios, such as the formation of gravitational wave sources in globular clusters. The last effects are further quantified by the development of a Boltzmann equation, to dynamically evolve a population of binaries in a cluster due to single-binary scatterings.

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