Symplectic geometry and space mission design

TL;DR

This paper advances symplectic geometry methods for analyzing periodic orbits in space mission design, providing new tools and numerical studies for orbit stability and bifurcations in planetary systems.

Contribution

It introduces an algorithm for computing Conley--Zehnder indices and applies symplectic geometry techniques to study orbit families relevant to space missions.

Findings

Refined Broucke stability diagram.

Numerical computation of Conley--Zehnder indices.

Identification of orbits near Enceladus suitable for missions.

Abstract

Using methods from symplectic geometry, the second and fifth authors have provided theoretical groundwork and tools aimed at analyzing periodic orbits, their stability and their bifurcations in families, for the purpose of space mission design. The Broucke stability diagram was refined, and the "Floer numerical invariants" where considered, as numbers which stay invariant before and after a bifurcation, and therefore serve as tests for the algorithms used. These tools were later employed for numerical studies. In this article, we will further illustrate these methods with numerical studies of families of orbits for the Jupiter-Europa and Saturn-Enceladus systems, with emphasis on planar-to-spatial bifurcations, from deformation of the families in Hill's lunar problem studied by the first author. We will also provide an algorithm for the numerical computation of Conley--Zehnder indices, which are instrumental in practice for determining which families of orbits connect to which. As an application, we use our tools to study a family of periodic orbits that approaches Enceladus at an altitude of 29km, and therefore may be used in future space missions to visit the water plumes.