Periodic orbits for 3 and 4 co-orbital bodies

Abstract: We investigate the natural families of periodic orbits associated with the equilibrium configurations of the the planar restricted $1+n$ body problem for the case $2\leq n \leq 4$ equal mass satellites. Such periodic orbits can be used to model both trojan exoplanetary systems and parking orbits for captured asteroids within the solar system. For $n=2$ there are two families of periodic orbits associated with the equilibria of the system: the well known horseshoe and tadpole orbits. For $n=3$ there are three families that emanate from the equilibrium configurations of the satellites, while for $n=4$ there are six such families as well as numerous additional connecting families. The families of periodic orbits are all of the horseshoe or tadpole type, and several have regions of neutral linear stability.