Regularized phase-space volume for the three-body problem

TL;DR

This paper introduces a regularized method to compute the phase-space volume for the three-body problem, addressing divergence issues and enabling better understanding of disintegration times within statistical theory.

Contribution

It defines a regularized phase-volume by subtracting a hierarchical reference, reducing the problem to a shape sphere, and provides numerical evaluations for planar and 3D systems.

Findings

Regularized phase-volume can be computed and visualized for three-body systems.

In the test mass limit, the phase-volume becomes negative, indicating theory breakdown.

The method paves the way for comparing theoretical disintegration times with simulations.

Abstract

The micro-canonical phase-space volume for the three-body problem is an elementary quantity of intrinsic interest, and within the flux-based statistical theory, it sets the scale of the disintegration time. While the bare phase-volume diverges, we show that a regularized version can be defined by subtracting a reference phase-volume, which is associated with hierarchical configurations. The reference quantity, also known as a counter-term, can be chosen from a 1-parameter class. The regularized phase-volume of a given (negative) total energy, $\bar\sigma(E)$, is evaluated. First, it is reduced to a function of the masses only, which is sensitive to the choice of a regularization scheme only through an additive constant. Then, analytic integration is used to reduce the integration to a sphere, known as shape sphere. Finally, the remaining integral is evaluated numerically, and presented by a contour plot in parameter space. Regularized phase-volumes are presented for both the planar three-body system and the full 3d system. In the test mass limit, the regularized phase-volume is found to become negative, thereby signalling the breakdown of the non-hierarchical statistical theory. This work opens the road to the evaluation of $\bar\sigma(E,L)$, where $L$ is the total angular momentum, and it turn, to comparison with simulation determined disintegration times.