Trajectory Design for Escape from the Secondary body in a Non-autonomous Model

This study aims to characterize the solution space for escape trajectories from the vicinity of a secondary body in the Bi-Circular Restricted Four-Body Problem (BCR4BP). The behavior of spacecraft near the secondary body, particularly with respect to escape, has traditionally been analyzed within the framework of the Circular Restricted Three-Body Problem (CR3BP), where escape trajectories are organized using lobe and tube structures associated with invariant manifolds. However, these dynamical structures are no longer time-invariant in non-autonomous models such as the BCR4BP. They depend explicitly on time and exist only as time-dependent structures. To address this limitation, Lagrangian Coherent Structures (LCS) are employed to define the escape lobe, that is, the set of initial conditions from which a spacecraft immediately escapes the vicinity of the secondary body without completing a full revolution about it. The perilune lobes are then constructed by backward propagation of trajectories from the escape lobe and by collecting the previous perilune states along each trajectory. In this way, the escape solution space in the BCR4BP is represented in terms of the escape lobe and the perilune lobes. It is shown that transitions among the lobes exhibit a structure analogous to that found in the CR3BP. Their shapes and areas depend strongly on the initial solar phase angle and the Jacobi constant, and the number of available escape trajectories changes accordingly. The resulting lobe structures are further applied to the design of one-impulse escapes from a Low Lunar Orbit (LLO), demonstrating their utility for escape trajectory design.