Local Orbital Elements for the Circular Restricted Three-Body Problem

Abstract

The Keplerian orbital elements make up a set of parameters that uniquely describe a two-body trajectory. Once perturbations are imposed on two-body dynamics, one often studies the time evolution of the orbital elements. Moving in complexity beyond two-body perturbations [that is, studying the dynamics in the circular restricted three-body problem (CR3BP)], the Keplerian orbital elements are no longer well defined in certain regions of phase space, especially when the gravitational attractions of both the primary and secondary bodies have similar magnitudes, as occurs in the vicinity of the libration points. In this work, we define a generalization of orbit elements that can be applied in these regions and others. Specifically, we define a set of semi-analytical action-angle orbital elements that are defined locally about any bounded special solution in the CR3BP: equilibria, periodic orbits, and quasi-periodic orbits. Local action-angle orbital elements are defined using action-angle coordinates in the Birkhoff–Gustavson normal form about the bounded invariant manifold. We include detailed examples around the five equilibria in the Earth–Moon CR3BP.