Searching for periodic orbits of the spatial elliptic restricted three-body problem by double averaging

We calculate new families of periodic orbits of the spatial elliptic restricted three-body problem. The original system has three degrees of freedom and depends on time explicitly, but surprisingly after doubly averaging the Hamiltonian over the time and the mean anomaly the system gets reduced to one degree of freedom. This fact allows us to determine families of periodic orbits of the original system. We compute the relative equilibria, study their linear and Lyapunov stability and reconstruct the corresponding invariant manifolds in the original Hamiltonian. Additionally, we determine six families of periodic orbits with large semimajor axes having any eccentricity. Making use of adequate variables, we easily see that four families lie on a plane perpendicular to the plane of the primaries, two of them being stable and the other two unstable. The other two families are stable families of closed orbits coplanar with the primaries.