Scattering in the Positive Energy Isosceles Three-Body Problem

TL;DR

This paper investigates how solutions in the positive energy isosceles three-body problem evolve asymptotically, showing that initial shapes can scatter into a wide range of final shapes, including near-collision scenarios.

Contribution

It extends scattering analysis to include binary and near-triple collisions in the isosceles three-body problem, revealing broad asymptotic shape possibilities.

Findings

Initial shapes can scatter into almost all possible asymptotic shapes.

Binary and near-collision effects significantly influence scattering outcomes.

Results depend on mass parameters but show general scattering behavior.

Abstract

In the three-body problem with positive energy, solutions which avoid triple collision have the property that the size of the triangle formed by the bodies tends to infinity as $t\rightarrow \pm\infty$. Furthermore, the triangles have well-defined asymptotic shapes $s_\pm$. The scattering problems asks which asymptotic shape $s_+$ can occur for a given choice of $s_-$. Previous work shows that this can be viewed as the problem of finding heteroclinic orbits connecting equilibrium points on a boundary manifold ``at infinity'' and some results were obtained for solutions which avoid collisions. The goal of this paper is to study the scattering effect of binary and near-triple collisions in a simple setting -- the isosceles three-body problem. The details depend on the mass parameters but in many cases, a fixed isosceles initial shape $s_-$ scatters to essentially all possible isosceles shapes $s_+$.