## A Statistical Solution to the Chaotic, Non-Hierarchical Three-Body Problem

I’m a bit late passing this on but I think some of my readers might find this interesting, as I did when I came across it a week or so ago. There’s a paper on the arXiv by Nicholas Stone and Nathan Leigh with the title *A Statistical Solution to the Chaotic, Non-Hierarchical Three-Body Problem* and the following abstract:

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 full paper can be downloaded here. The abstract is very clear but you might want to read the wikipedia entry for the three-body problem for general background. Here’s a fun figure from the paper:

Let me just add a note of explanation of the word `hierarchical’ as applied here: it means when the mass of one body is very different from the other two, or that two of the bodies have a much smaller separation from each other than they do from the third.

This paper does not present an analytic solution of the unrestricted three-body problem (which is known to be intractable) but does provide some very useful statistical insights into the long-term evolution of three-body systems, for example confirming the generally held opinion that most such systems evolve into a state in which one body is ejected and the other two form a tight binary.

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February 26, 2020 at 2:32 pm

Excellent work!

March 6, 2020 at 3:19 pm

I’d be interested in whether the two long but temporary excursions of the black body, extending below the bottom of the figure, were consecutive and immediately before the final separation of the black body from the other two, when it departs off the top of the figure.

March 6, 2020 at 3:25 pm

I’d guess so. I imagine the particle has almost enough energy to reach escape velocity from the other two and the last encounter just tips it over.

March 6, 2020 at 5:49 pm

Wanna bet on it? Obviously it is the first time it reaches escape energy/velocity, but that it gets there monotonically is far from given in a chaotic system…

March 6, 2020 at 6:41 pm

You know I’m not a gambling man.

March 7, 2020 at 6:25 am

Videos of these simulations would be nice.

March 9, 2020 at 9:45 am

Also, if expulsion takes place to give a 2-body system and a lone body, is the latter always the same body (ie, the lightest, or the middle mass, or the heaviest)?