Periodic orbits in the concentric circular restricted four-body problem and their invariant manifolds

Abstract

We give numerical calculations of periodic orbits in the planar concentric restricted four-body problem. It is assumed that the motion of a massless body is governed by three primaries m1, m2 and m3. We suppose that m1≫m2,m3 and that, in an m1 centered inertial reference frame, m2 and m3 move in different circles about m1 and m1 is fixed. Although the motion of the primaries m1,m2,m3 does not satisfy Newton’s equations of motion, this approximation is a good to model, for instance, the Jupiter–Europa–Ganymede–spacecraft system. We compute periodic orbits in both the m1–m2 and m1–m3 rotating frames. Periodic orbits that orbit around one of the primaries are found. Using a method that is based on the well-known Laplace resonance we also find unstable periodic orbits about the collinear libration points near m2 and m3. Since the periodic orbits near the collinear libration points are unstable they have stable/unstable manifolds, which we compute. We observe a lack of intersection of the stable and unstable manifolds of different periodic orbits.