The restricted P + 2 body problem

Abstract

In this paper we study a special case of the restricted n-body problem, called by us the restricted P + 2 body problem. The equilibrium configuration which the P + 1 bodies with mass form consists of one central mass encircled by a ring of P equally spaced particles of equal mass, the ring rotating at a specific angular velocity. We briefly discuss the stability of this configuration. We consider the dynamics of an infinitesimal mass under the influence of such a configuration. First the equilibrium points will be discussed, then the zero-velocity curves. We show that there are 3 P, 4 P or 5 P equilibrium points, depending on the ratio of the ring particle mass to the central body mass. Next motion about the equilibrium points is considered. We show that if the ring particle mass is small enough there will be P stable equilibrium points. Also if the number of particles, P, is large enough and the ratio of the ring particle mass to the central body mass is large enough there will be P different stable equilibrium points. Finally an analysis of the dynamics of the infinitesimal mass will be performed under the restriction that the particle does not cross or come close to the ring and lies in the plane of the ring. Under this restriction an approximate potential can be found which can be made arbitrarily close to the real potential under some circumstances. The dynamics of the particle under the approximate potential are integrable. We find a periodic orbit in this case with the Poincaré-Lindstedt method using the mass of the ring as a small parameter. The predictions from this approximate solution of the problem compare well with numerical integrations of the actual system.