Three-dimensional, periodic halo orbits in the restricted three-body problem

In the circular restricted three body problem, three dimensional, periodic orbits for the infinitesimal body exist that emanate from the general vicinity of the three collinear libration points. A numerical study was done to investigate the dependence of these orbits on mu, the mass ratio of the two primary bodies families containing stable three dimensional orbits were found to exist near the two exterior libration points for all values of mu, and that the size of these orbits increases with mu. Near the interior libration point, stable orbits exist only for mu .0573. Following a family toward the nearer primary, the orbits become thin and almost rectilinear, aligned essentially perpendicular to the plane of motion of the primaries. These near-collision orbits can be calculated and their stability analyzed by regularized numerical integration using a transformation of variables or by using 1st or 2nd order expansions in the small in-plane components of angular momentum, relative to the nearer primary, and of the osculating Laplace eccentricity vector.