Relative equilibria and periodic orbits in a Circular Planar (2+2)-Body Problem

We present a planar four-body model, the Circular Planar (2+2)-Body Problem, for the motion of two asteroids (having small but positive masses) moving under the gravitational attraction of each other and under the gravitational attraction of two primaries (with masses much larger than the two smaller mass bodies) moving in uniform circular motion about their center of mass. We show the Circular Planar (2+2)-Body Problem has (at least) 6 relative equilibria and (at least) 10 one-parameter families of periodic orbits, two of which are of Hill-type. The existence of six relative equilibria and eight one-parameter families of periodic orbits is obtained by a reduction of the Circular Planar (2+2)-Body Problem in which the primaries have equal mass, the asteroids have equal mass, and the positions of the asteroids are symmetric with respect to the origin. The remaining two one-parameter families of periodic orbits, which are of comet-type, are obtained directly in the Circular Planar (2+2)-Body Problem.