On the period of the periodic orbits of the restricted three body problem

TL;DR

This paper establishes a relationship between the period of closed orbits in the restricted three-body problem and the region they enclose, revealing a Keplerian-like property and analyzing orbit directions near Lagrange points.

Contribution

It introduces a new formula linking orbit period to enclosed region and shows that near L4 and L5, periodic solutions must move clockwise.

Findings

Period depends on enclosed region and a specific integral.

Periodic solutions near L4 and L5 are clockwise.

The formula generalizes Keplerian relations to the three-body context.

Abstract

We will show that the period $T$ of a closed orbit of the planar circular restricted three-body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, $2 T=k\pi+\int_\Omega g$ where $k$ is an integer, $\Omega$ is the region enclosed by the periodic orbit and $g:\mathbb{R}^2\to \mathbb{R}$ is a function that only depends on the constant $C$ known as the Jacobian integral; it does not depend on $\Omega$. This theorem has a Keplerian flavor in the sense that it relates the period with the space "swept" by the orbit. As an application, we prove that there is a neighborhood around $L_4$ such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for $L_5$.