Finite energy foliations in the restricted three-body problem

TL;DR

This paper uses pseudo-holomorphic curves to establish finite energy foliations in the restricted three-body problem, explaining homoclinic orbits and proving Birkhoff's retrograde orbit conjecture near a critical mass ratio.

Contribution

It introduces a novel application of finite energy foliations to the restricted three-body problem, linking convexity conditions to the existence of global surfaces of section.

Findings

Finite energy foliations exist for certain mass ratios and energies.

Homoclinic orbits to Lyapunov orbits are explained by the foliation.

Birkhoff's retrograde orbit conjecture is proved in the specified regime.

Abstract

This paper is about using pseudo-holomorphic curves to study the circular planar restricted three-body problem. The main result states that for mass ratios sufficiently close to $\frac{1}{2}$ and energies slightly above the first Lagrange value, the flow on the regularized component $\mathbb R P^3 \# \mathbb R P^3$ of the energy surface admits a finite energy foliation with three binding orbits, namely two retrograde orbits around the primaries and the Lyapunov orbit in the neck region about the first Lagrange point. This foliation explains the numerically observed homoclinic orbits to the Lyapunov orbits. The critical energy surface is proved to satisfy the strict convexity condition in regularizing elliptic coordinates. This allows for the application of a general abstract result for Reeb vector fields on holed lens spaces, concerning the existence of finite energy foliations with prescribed binding orbits. As a by-product of the convex analysis, Birkhoff's retrograde orbit conjecture is proved for mass ratios sufficiently close to $\frac{1}{2}$ and all energies below the first Lagrange value. This conjecture states that the retrograde orbit bounds a disk-like global surface of section on each regularized component $\mathbb R P^3$ of the energy surface.