The Binary Returns!

TL;DR

This paper proves that in the spatial Newtonian three-body problem with fixed negative energy and angular momentum, all solutions pass through a bounded size, which has implications for understanding specific syzygy sequences.

Contribution

It establishes a bound on the size of solutions in the three-body problem at fixed energy and angular momentum, aiding in the study of syzygy sequence realizability.

Findings

Solutions pass through a bounded size I_0

Bound depends on energy, angular momentum, and masses

Supports impossibility results for certain syzygy sequences

Abstract

Consider the spatial Newtonian three body problem at fixed negative energy and fixed angular momentum. The moment of inertia $I$ provides a measure of the overall size of a three-body system. We will prove that there is a positive number $I_0$ depending on the energy and angular momentum levels as well as the masses such that every solution at these levels passes through $I\leq I_0$ at some instant of time. Motivation for this result comes from trying to prove the impossibility of realizing a certain syzygy sequence in the zero angular momentum problem.