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Scattering in the Positive Energy Isosceles Three-Body Problem

Richard Moeckel

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 $s_+$.

Scattering in the Positive Energy Isosceles Three-Body Problem

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 . Furthermore, the triangles have well-defined asymptotic shapes . The scattering problems asks which asymptotic shape can occur for a given choice of . 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 scatters to essentially all possible isosceles shapes .
Paper Structure (10 sections, 25 theorems, 89 equations, 19 figures)

This paper contains 10 sections, 25 theorems, 89 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Blown-up Jacobi Coordinates
  3. Regularization of binary collisions
  4. Triple collision
  5. Estimates in the natural timescale
  6. The Infinity Manifold and the scattering relation
  7. Flow at Infinity
  8. Triple collision to Infinity
  9. Scattering theorems
  10. Scattering experiments

Key Result

Proposition 1

The energy equations (eq_renergy) and (eq_senergy) define smooth three-dimensional submanifolds of phase space and the solutions $\gamma(\tau)$ of equations (eq_regode) and (eq_sode) exist for $\tau\in\mathbb{R}$.

Figures (19)

  • Figure 1: Isosceles shapes.
  • Figure 2: Scattering from the equilateral triangle shape in the equal mass isosceles three-body problem.
  • Figure 3: One parameter family of final shapes $\theta_+$ with $\theta_-$ fixed at the equilateral triangle as in Figure \ref{['fig_scatterm31thetalag']}. The discontinuity occurs at a triple collision orbit. In this case $\theta_+$ varies over $[-\frac{\pi}{2},\frac{\pi}{2}]\setminus\{-\theta_-\}$.
  • Figure 4: Scattering at infinity with and without binary collisions. On the left we have the isosceles problem with binary collision. On the right, a possible free particle scattering in the plane without collisions. The final shapes are different and even not rotationally equivalent in the plane.
  • Figure 5: The graph of the function $\theta(u)$ relating the two shape variables. As $u$ varies over an interval of length $2\pi$, the shape parameter $\theta$ is double-covered except at the binary collisions.
  • ...and 14 more figures

Theorems & Definitions (40)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5: Simo
  • Proposition 6
  • ...and 30 more