On the co-orbital motion of two planets in quasi-circular orbits

TL;DR

This paper develops an analytical Hamiltonian model to study the co-orbital motion of two planets, identifying key configurations and bifurcations in their quasi-circular orbits.

Contribution

It introduces a comprehensive Hamiltonian formalism for co-orbital planets, revealing relations between fixed points, special orbits, and bifurcation phenomena.

Findings

Identification of fixed points linking to special orbits

Analysis of fundamental frequencies near periodic solutions

Proof of bifurcation of key configurations from a common fixed point

Abstract

We develop an analytical Hamiltonian formalism adapted to the study of the motion of two planets in co-orbital resonance. The Hamiltonian, averaged over one of the planetary mean longitude, is expanded in power series of eccentricities and inclinations. The model, which is valid in the entire co-orbital region, possesses an integrable approximation modeling the planar and quasi-circular motions. First, focusing on the fixed points of this approximation, we highlight relations linking the eigenvectors of the associated linearized differential system and the existence of certain remarkable orbits like the elliptic Eulerian Lagrangian configurations, the Anti-Lagrange (Giuppone et al., 2010) orbits and some second sort orbits discovered by Poincar\'e. Then, the variational equation is studied in the vicinity of any quasi-circular periodic solution. The fundamental frequencies of the trajectory are deduced and possible occurrence of low order resonances are discussed. Finally, with the help of the construction of a Birkhoff normal form, we prove that the elliptic Lagrangian equilateral configurations and the Anti-Lagrange orbits bifurcate from the same fixed point L4.