Lunar capture trajectories and homoclinic connections through isomorphic mapping

Analysis and design of low-energy transfers to the Moon has been a subject of great interest for many decades. This paper is concerned with a topological study of such transfers, with emphasis to trajectories that allow performing lunar capture and those that exhibit homoclinic connections, in the context of the circular restricted three-body problem. A fundamental theorem stated by Conley locates capture trajectories in the phase space and can be condensed in a sentence: "if a crossing asymptotic orbit exists then near any such there is a capture orbit". In this work this fundamental theoretical assertion is used together with an original cylindrical isomorphic mapping of the phase space associated with the third body dynamics. For a given energy level, the stable and unstable invariant manifolds of the periodic Lyapunov orbit around the collinear interior Lagrange point are computed and represented in cylindrical coordinates as tubes that emanate from the transformed periodic orbit. These tubes exhibit complex geometrical features. Their intersections correspond to homoclinic orbits and determine the topological separation of long-term lunar capture orbits from short-duration capture trajectories. The isomorphic mapping is proven to allow a deep insight on the chaotic motion that characterizes the dynamics of the circular restricted three-body, and suggests an interesting interpretation, and together corroboration, of Conley's assertion on the topological location of lunar capture orbits. Moreover, an alternative three-dimensional representation of the phase space is profitably employed to identify convenient lunar periodic orbits that can be entered with modest propellant consumption, starting from the Lyapunov orbit.