The chaotic four-body problem in Newtonian gravity I: Identical point-particles

TL;DR

This paper analyzes the chaotic four-body problem in Newtonian gravity with identical point particles, developing an analytic formalism for outcome probabilities and velocity distributions, and comparing these with numerical simulations.

Contribution

It introduces a formalism for calculating outcome probabilities and velocity distributions in four-body encounters, extending previous three-body models and validated through simulations.

Findings

Distinct velocity distributions for each encounter outcome.

Analytic predictions match simulations for stable triples at low virial ratios.

Single star escape velocities follow a predictable distribution similar to the three-body case.

Abstract

In this paper, we study the chaotic four-body problem in Newtonian gravity. Assuming point particles and total encounter energies $\le$ 0, the problem has three possible outcomes. We describe each outcome as a series of discrete transformations in energy space, using the diagrams first presented in Leigh \& Geller (2012; see the Appendix). Furthermore, we develop a formalism for calculating probabilities for these outcomes to occur, expressed using the density of escape configurations per unit energy, and based on the Monaghan description originally developed for the three-body problem. We compare this analytic formalism to results from a series of binary-binary encounters with identical point particles, simulated using the \texttt{FEWBODY} code. Each of our three encounter outcomes produces a unique velocity distribution for the escaping star(s). Thus, these distributions can potentially be used to constrain the origins of dynamically-formed populations, via a direct comparison between the predicted and observed velocity distributions. Finally, we show that, for encounters that form stable triples, the simulated single star escape velocity distributions are the same as for the three-body problem. This is also the case for the other two encounter outcomes, but only at low virial ratios. This suggests that single and binary stars processed via single-binary and binary-binary encounters in dense star clusters should have a unique velocity distribution relative to the underlying Maxwellian distribution (provided the relaxation time is sufficiently long), which can be calculated analytically.