Regularized phase-space volume for the three-body problem
arXiv:2205.04294 · doi:10.1007/s10569-022-10108-1
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, , 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 , where is the total angular momentum, and it turn, to comparison with simulation determined disintegration times.
arXiv comment27 pages, 6 figures
References in corpus (5)
- A Statistical Solution to the Chaotic, Non-Hierarchical Three-Body Problem
- An Analytical, Statistical Approximate Solution for Dissipative and non-Dissipative Binary-Single Stellar Encounters
- Chaos and Lévy Flights in the Three-Body Problem
- Flux-based statistical prediction of three-body outcomes
- Testing the Flux-based statistical prediction of the Three-Body Problem