Illustrating chaos: A schematic discretization of the general three-body problem in Newtonian gravity

TL;DR

This paper introduces a formalism using schematic diagrams to represent chaotic three-body interactions in Newtonian gravity, capturing all relevant dynamics and potentially reducing computational costs.

Contribution

The authors develop a novel schematic diagram approach for depicting three-body interactions, simplifying complex dynamics into discrete transformations in energy and angular momentum space.

Findings

Diagrams fully characterize bound and unbound interactions.

Prolonged particle excursions are reduced to single transformations.

Method applicable to evolving binary populations in dense environments.

Abstract

We present a formalism for constructing schematic diagrams to depict chaotic three-body interactions in Newtonian gravity. This is done by decomposing each interaction in to a series of discrete transformations in energy- and angular momentum-space. Each time a transformation is applied, the system changes state as the particles re-distribute their energy and angular momenta. These diagrams have the virtue of containing all of the quantitative information needed to fully characterize most bound or unbound interactions through time and space, including the total duration of the interaction, the initial and final stable states in addition to every intervening temporary meta-stable state. As shown via an illustrative example for the bound case, prolonged excursions of one of the particles, which by far dominates the computational cost of the simulations, are reduced to a single discrete transformation in energy- and angular momentum-space, thereby potentially mitigating any computational expense. We further generalize our formalism to sequences of (unbound) three-body interactions, as occur in dense stellar environments during binary hardening. Finally, we provide a method for dynamically evolving entire populations of binaries via three-body scattering interactions, using a purely analytic formalism. In principle, the techniques presented here are adaptable to other three-body problems that conserve energy and angular momentum.