Saturn's Rings

Numerical solutions of the restricted three-body problem are obtained in order to investigate the behavior of orbits of asteroids with periods at or near n/(n+1) times that of Jupiter. The four cases n= 1, 2, 3, and 4 are examined and the results confirm and extend earlier analytical studies for the case in which the perturber moves in a circular orbit. For the class of orbits defined by the condition that at t =0 an asteroid lies either at perihelion or aphelion at inferior conjunction with Jupiter, the resulting period and certain osculating parameters for periodic orbits are given as a function of the asteroid's initial radius vector. Also indicated are regions in which no periodic orbits are possible. Our investigation shows, for example, a complete absence of orbits with semimajor axes between 3.28 and 3.36 a.u. and a clustering of semimajor axes at 3.97 a.u., both in accord with observation.