Eight-shaped Lissajous orbits in the Earth-Moon system

Euler and Lagrange proved the existence of five equilibrium points in the circular restricted three-body problem. These equilibrium points are known as Lagrange points (Euler points or libration points) L1 , . . . , L5 . The existence of families of periodic and quasi-periodic orbits around those points is well known (see [15, 16, 31]). Among them, halo orbits are 3-dimensional periodic orbits diffeomorphic to circles. They are the first type of so-called Lissa jous orbits. In this article we focus on Lissa jous orbits of the second type, which are almost vertical and have the shape of an eight, and that we call Eight Lissa jous orbits. In the Earth-Moon system, we first compute numerically a family of such orbits, based on Linsdtedt Poincar´e's method combined with a continuation method on the excursion parameter. Then, we study their specific properties. In particular, we put in evidence, using local Lyapunov exponents, that their invariant manifolds share nice global stability properties, which make them of interest in space mission design. More precisely, we show numerically that invariant manifolds of Eight Lissa jous orbits keep in large time a structure of eight-shaped tubes. This property is compared with halo orbits, the invariant manifolds of which do not share such global stability properties. Finally, we show that the invariant manifolds of Eight Lissa jous orbits can be used to visit almost all the Moon surface in the Earth-Moon system.