Characterizing Accuracy of Normal Forms to Study Trajectories in Cislunar Space

Qualitative understanding of cislunar trajectories is increasingly important as lunar missions become more commonplace. Researchers have used the Lie series method to reduce the Circular Restricted Three-Body Problem (CR3BP) to its normal form up to some approximation order and in the vicinity of the five libration points. This approximation allows for analytical propagation in proximity to the libration points by defining action-angle variables. These variables are such that, up to the approximation order, the actions are constant and the angles are linear in time. An objective of this work is to examine how the normal form coordinates characterize trajectories in the vicinity of the libration points and maintain accuracy of propagation. Normal form coordinates qualitatively separate periodic, quasiperiodic, transit, and reflective trajectories. Another objective of this work is to examine the accuracy of the approximate normal form centered at L₁ and L₂ at various approximation orders, distances, and energy levels. At higher approximation orders, the normal form is able to accurately propagate trajectories in a ball around the libration point of origin. Finally, two example applications of this method are then examined, including maneuver characterization and Halo orbit identification.