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Symplectic geometry and space mission design

Cengiz Aydin, Urs Frauenfelder, Otto van Koert, Dayung Koh, Agustin Moreno

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.

Symplectic geometry and space mission design

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.
Paper Structure (18 sections, 1 theorem, 37 equations, 23 figures, 17 tables)

This paper contains 18 sections, 1 theorem, 37 equations, 23 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notions
  4. B-signs
  5. Conley--Zehnder index
  6. Floer numerical invariants
  7. Global topological methods
  8. Circular restricted three-body problem
  9. Hill's lunar problem
  10. Numerical work
  11. Result I. Planar direct/prograde orbits
  12. Result II. Bifurcation graphs with the same topology
  13. Result III. Bifurcation graphs between prograde and retrograde orbits
  14. Result IV. Bifurcation graph for Saturn--Enceladus
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem A

The characteristic polynomial of a symplectic matrix $A \in Sp(2n)$ is palindromic, i.e. there are $a_0,\ldots,a_n$ such that Furthermore, if $\lambda$ is an eigenvalue of $A$, then so are $\lambda^{-1}$, $\bar{\lambda}$ and $\bar{\lambda}^{-1}$.

Figures (23)

  • Figure 1: $\mu_{CZ}$ jumps by $\pm 1$ when crossing $1$, according to direction of bifurcation, as shown. If it stays elliptic, the jump is by $\pm 2$. This is determined by the $B$-sign.
  • Figure 2: A sketch of a bifurcation at a degenerate orbit, with the before/after orbits determined by the deformation parameter (the energy), each branch with its own CZ-index. The Floer number is a signed count of orbits which stays invariant.
  • Figure 3: The 2D GIT sequence. One obtains more refined information for symmetric orbits.
  • Figure 4: The 3D Broucke stability diagram. Here, $\Gamma_{\pm 1}$ corresponds to eigenvalue $\pm 1$, $\Gamma_d$ to double eigenvalue, $\mathcal{E}^2$ to doubly elliptic (stable region), and so on FM.
  • Figure 5: The branches (represented as lines) are two-dimensional, and come together at the 1-dimensional "branching locus" (represented as points), where we cross from one region to another of the Broucke diagram.
  • ...and 18 more figures

Theorems & Definitions (3)

  • Theorem A
  • Example A.1
  • Remark A.2