Some periodic solutions of a four-body problem☆

Abstract A four-body problem which is a model for the motion of a particle or vehicle in the Sun-Earth-Moon system is obtained by rotating the barycenter B of the two massive bodies in a restricted three-body configuration about a third massive body. It is shown that this problem can be regarded as a perturbation of the restricted three-body problem where the perturbation parameter is the mass of the third body divided by the cubed distance from the third body to B . It is proved that under certain conditions, especially with a restriction on the magnitude of the perturbation parameter, the particle of negligible mass has a periodic motion near each of the libration points of the restricted three-body problem; a method for estimating the restriction on the magnitude of the perturbation parameter is described.