  We present a planar four-body model, called the Binary Asteroid 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 asteroids) moving in
uniform circular motion about their center of mass. We show the Binary Asteroid
Model 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 8 one-parameter families of periodic orbits is
obtained by a reduction of the Binary Asteroid 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 Binary Asteroid Problem.
