Impossibility of the capture of retrograde satellites in the restricted three-body problem

Impossibility of the capture of retrograde satellites is proved in the framework of the circular planar restricted three-body problem.Motion of the third body is described in Birkhoff's variables. First, various kinds of retrograde orbits around one of the primaries are defined. Among them are captured retrograde orbits and retrograde satellite orbits. Second, the surface of section and surface transformation are defined in the three dimensional manifold, in which the motion of the third body is possible for a fixed value of the Jacobi constant. Third, the surface transformation is shown to be homeomorphic and measure-preserving. Finally, with the use of these properties of surface transformation, impossibility of the capture is shown.