On the analytical construction of halo orbits and halo tubes in the elliptic restricted three-body problem

TL;DR

This paper develops an analytical method using resonant normal forms to construct and analyze halo orbits and their manifolds in the elliptic restricted three-body problem, extending previous circular problem models.

Contribution

It introduces a non-linear Floquet-Birkhoff resonant normal form approach to generalize halo orbits for the elliptic three-body problem, including error analysis.

Findings

Provides a large order approximation of halo orbits in the elliptic problem.

Defines manifold tubes associated with these halo orbits.

Includes an error analysis comparing the method to actual elliptic three-body dynamics.

Abstract

The halo orbits of the spatial circular restricted three-body problem are largely considered in space-flight dynamics to design low-energy transfers between celestial bodies. A very efficient analytical method for the computation of halo orbits, and the related transfers, has been obtained from the high-order resonant Birkhoff normal forms defined at the Lagrangian points L1-L2. In this paper, by implementing a non-linear Floquet-Birkhoff resonant normal form, we provide the definition of orbits, as well as their manifold tubes, which exist in a large order approximation of the elliptic three-body problem and generalize the halo orbits of the circular problem. Since the libration amplitude of such halo orbits is large (comparable to the distance of L1-L2 to the secondary body), and the Birkhoff normal forms are obtained through series expansions at the Lagrangian points, we provide also an error analysis of the method with respect to the orbits of the genuine elliptic restricted three-body problem.