On the bifurcation and continuation of periodic orbits in the three-body problem

TL;DR

This paper investigates the existence and continuation of asymmetric periodic orbits in the three-body problem, revealing their presence beyond known resonances and illustrating how these orbits connect across different problem variants.

Contribution

It demonstrates the existence of asymmetric periodic orbits in the elliptic restricted three-body problem and shows how these orbits connect with those in the circular case, extending prior understanding.

Findings

Asymmetric periodic orbits exist in the elliptic restricted problem.

Families of periodic orbits connect smoothly across problem variants.

The results verify the scenario proposed by Bozis and Hadjidemetriou (1976).

Abstract

We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only in particular resonances called "asymmetric resonances". However, numerical studies showed that in the general three body problem asymmetric orbits may exist not only for asymmetric resonances, but for other kinds, too. In this work, we show the existence of asymmetric periodic orbits in the elliptic restricted problem. These orbits are continued and clarify the origin of many asymmetric periodic orbits in the general problem. Also, we illustrate how the families of periodic orbits of the restricted circular problem and those of the elliptic one join smoothly and form families in the general problem, verifying in this way the scenario described firstly by Bozis and Hadjidemetriou (1976).