Periodic orbits in the restricted four-body problem

One solution to the restricted three-body problem is that of three-dimensional, periodic halo orbits. These orbits emanate from the three collinear libration points and exist at all possible mass ratios of the primaries, with infinite possible trajectories. Their existence motivated this study of the effect of an additional gravitational influence on the motion. The approach is to seek a solution in the restricted four-body problem using halo solutions as an initial approximation. The method first solves for periodic, coplanar motion of the three primaries under their mutual gravitational attractions and represents them as trigonometric series. Then, under the modified gravity force model, a three-dimensional solution is obtained in the problem of four bodies, periodic with respect to the synodic system of the three primaries. The additional primary remains relatively far removed and acts as a perturbing influence on the original motion. Some shape and stability characteristics are presented for three such solutions.