Stationary Configurations for Co-orbital Satellites with Small Arbitrary Masses

Abstract

We derive general results on the existence of stationary configurations for N co-orbital satellites with small but otherwise arbitrary masses mi , revolving on circular and planar orbits around a massive primary. The existence of stationary configurations depends on the parity of N.I fN is odd, then for any arbitrary angular separation between the satellites, there always exists a set of masses (positive or negative) which achieves stationarity. However, physically acceptable solutions (mi > 0 for all i) restrict this existence to sub-domains of angular separations. If N is even, then for given angular separations of the satellites, there is in general no set of masses which achieves stationarity. The case N = 3 is treated completely for small arbitrary satellite masses, giving all the possible solutions and their stability, to within our approximations. Configurations with a massive central body surrounded by small co-orbital satel- lites are found in several instances in the Solar System. Examples are found in the Saturnian system: a satellite (Helene) librates near the L4 point of Dione, and two satellites, Telesto and Calypso, librate near the L4 and L5 of Tethys, respectively, while the co-orbital satellites Janus and Epimetheus oscillate in horseshoe orbits around their mutual L3 point. In another context, the four co-orbital Neptune ring arcs might be explained, at least partly, by the presence of several hypothetical co-orbital satellites which would confine the observed dusty ring material (Renner and Sicardy, in prepara- tion). More generally, a ring close to the Roche zone of its planet might evolve, through accretion, into a collection of N co-orbital satellites which gather most of the mass of the ring material. Our aim in this paper is to derive some general results on stationary planar configurations for N co-orbital satellites orbiting a much more massive central planet (planar 1 + N body problem). These stationary configurations are some- times called relative equilibria, that is, special configurations of masses of the N body problem which rotate rigidly, with a constant angular velocity about their center of mass, if given the correct initial momenta. In rotating coordinates these special solutions become fixed points, hence the name relative equilibria. The