Dynamically Relevant Local Coordinates for Halo Orbits

Abstract

A local coordinate system based on the eigenstructure of the Halo orbit is proposed. We show that one only needs to keep track of six intuitive scalars to easily understand the qualitative dynamic evolution of spacecraft states near a halo orbit. Special attention is given to the center subspace and the space associated with the unity eigenvalues of the monodromy matrix. Examples are given for halo orbits in the Hill Three-Body Problem. I. Introduction Halo orbits have shown great usefulness in space missions. The Wilkinson Microwave Anisotropty Probe (WMAP), near the Earth-Sun L2 point is studying the cosmic microwave background radiation and the origins of the universe while the Solar and Heliospheric Observatory (SOHO) is currently studying the Sun from an orbit about L1. Dynamical Systems Theory, when applied to halo orbits, has allowed missions such as Genesis to use very little fuel while still accomplishing complicated objectives. Our goal is to make the analysis of trajectories near halo orbits more intuitive. When one numerically integrates a halo orbit, the simulation is typically carried out in a Lagrangian or Hamiltonian coordinate frame. When studying trajectories close to the periodic orbit, the dynamics may be linearized using the same frame in which the full nonlinear simulation was computed. This frame does not provide clear insight on how the spacecraft moves when it is in close proximity to a halo orbit. We propose a more dynamically intuitive local coordinate system based on the eigenstructure of the linearization. This method provides a simple way to compute and understand spacecraft motion near a halo orbit in terms of the local dynamical manifolds. The monodromy matrix associated with a particular halo orbit is defined as the state transition matrix (STM) evaluated after one period of the orbit (T ), denoted by F(T; 0). The interpretation of F(T; 0) is that it maps the local state forward by one orbit period. Therefore, the eigenstructure of F(T; 0) describes how spacecrafts behave near the nominal halo orbit. For all Hamiltonian systems, the STM is symplectic, and in the cases we study, F(T; 0) has the following six eigenvalues: