Flux-based statistical prediction of three-body outcomes

Abstract

The gravitational three-body problem is a rich open problem, dating back to Newton. It serves as a prototypical example of a chaotic system and has numerous applications in astrophysics. Generically, the motion is non-integrable and susceptible to disintegration, and for negative total energy the decay outcome is a free body flying apart from a binary. Since Poincar\'e, the problem is known to be chaotic and is believed to lack a general deterministic solution. Instead, decades ago a statistical solution was marked as a goal. Yet, despite considerable progress, all extant approaches display two flaws. First, probability was equated with phase space volume, thereby ignoring the fact that significant regions of phase space describe regular motion, including post-decay motion. Secondly and relatedly, an adjustable parameter, the strong interaction region, which is a sort of cutoff, was a central ingredient of the theory. This paper introduces remedies and presents for the first time a statistical prediction of decay rates, in addition to outcomes. Based on an analogy with a particle moving within a leaky container, the statistical distribution is presented in an exactly factorized form. One factor is the flux of phase-space volume, rather than the volume itself, and it is given in a cutoff-independent closed-form. The other factors are the chaotic absorptivity and the regularized phase space volume. The situation is analogous to Kirchhoff's law of thermal radiation, also known as greybody radiation. In addition, an equation system for the time evolution of the statistical distribution is introduced; it describes the decay rate statistics while accounting for sub-escape excursions. Early numerical tests indicate a leap in accuracy.