Lunar Capture Trajectories and Homoclinic Connections Through Isomorphic Mapping

The determination of low-energy trajectories to the Moon has attracted a long interest in the last decades for space mission analysis. Outer and inner transfers, based on the transit through the regions where the collinear Lagrange points L1 and L2 are located, have been studied for a long time and some space missions have already taken advantage of the results of these studies. Two examples of this kind of missions are represented by the European Smart-1 mission and by the Japanese Hiten mission. This paper is concerned with a topological study of the transfer trajectories to the Moon, with particular interest to the low energy trajectories that allow performing (long-term) lunar captures and to trajectories that exhibit homoclinic connections. In general, the third body trajectory oscillates, in the sense that the spacecraft passes continuously from the neighborhood of the Earth to the region in proximity of the Moon (and viceversa). A fundamental topological theorem stated by Conley is focused on the location of capture trajectories in the phase space, which includes all the dynamical information on the spacecraft state (defined by its position and velocity). Conley’s theorem 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 three-dimensional representation of trajectories. This cylindrical representation is associated with an isomorphism between actual coordinates (of position and velocity) and transformed coordinates. For a given energy level, the (stable and unstable) manifolds associated with the Lyapunov orbit around L1 are computed and represented in cylindrical coordinates as tubes that emanate from the (transformed) periodic Lyapunov orbit. Sections of these tubes are considered. A relevant number of points lying on these sections are selected, and the corresponding positions and velocities are assumed as initial states and then numerically propagated, to find how long each trajectory remains in the neighborhood of the Moon, with the final intent of evaluating the capture duration as a function of the initial state. The cylindrical representation is then employed to detect homoclinic connections. The analysis included in this paper, based on isomorphic mapping, allows finding the geometrical locus that corresponds to these connections. In fact, each manifold is associated with a tube in the transformed space. As a result, the geometrical locus associated with homoclinic connections is represented as the intersection between distinct tubes. The existence of these connections suggests an interesting interpretation, and together corroboration, of Conley’s assertion on the topological location of lunar capture orbits.