A Statistical Solution to the Chaotic, Non-Hierarchical Three-Body Problem. (arXiv:1909.05272v1 [astro-ph.GA])

<a href="http://arxiv.org/find/astro-ph/1/au:+Stone_N/0/1/0/all/0/1">Nicholas C. Stone</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Leigh_N/0/1/0/all/0/1">Nathan W.C. Leigh</a>

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 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 ergodicity, suggesting that the distribution of outcomes is

uniform across the accessible phase volume. Here, we use the ergodic hypothesis

to 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

non-hierarchical triple. The generally super-thermal distributions of survivor

binary eccentricity that we predict have notable applications to many

astrophysical scenarios. For example, non-hierarchical triples produced

dynamically in globular clusters are a primary formation channel for black hole

mergers, but the rates and properties of the resulting gravitational waves

depend on the distribution of post-disintegration eccentricities.

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 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 ergodicity, suggesting that the distribution of outcomes is

uniform across the accessible phase volume. Here, we use the ergodic hypothesis

to 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

non-hierarchical triple. The generally super-thermal distributions of survivor

binary eccentricity that we predict have notable applications to many

astrophysical scenarios. For example, non-hierarchical triples produced

dynamically in globular clusters are a primary formation channel for black hole

mergers, but the rates and properties of the resulting gravitational waves

depend on the distribution of post-disintegration eccentricities.

http://arxiv.org/icons/sfx.gif