  We develop a theoretical framework for the calculation of orbits for a system
consisting of a spherical object and a non-spherical body, which is then
specialized to a prolate ellipsoid. Particular trajectories are presented that
illustrate a drastic contrast between the familiar elliptical orbits of
spherical binary systems and the trajectories around the prolate spheroid. We
also show here, and in a media video representation of the computed orbits, how
the spherical satellite instantaneous orbital plane and eccentricity evolve. We
also explicitly verify the conservation of the total angular momentum and
energy of the system, prolate plus satellite, while the intrinsic rotational
angular momentum and energy of the prolate changes with time at the expense of
the orbital energy and angular momentum of the sphere. We then consider a
particular orbit where an initially bound satellite gains sufficient orbital
energy and eventually escapes, with its total energy now positive. The inverse
process, where a satellite is captured by a prolate, is also considered, and we
determine the probability of this event occurring, as a function of the initial
relative velocity and parameter of impact of the system. We end with a
discussion of a plausible scenario where an escaping satellite in the Oort
cloud could wind up with a new heliocentric Earth`s crossing orbit. In the
Appendices we develop the necessary equations for the application of the above
formalism to orbits around a general homogeneous ellipsoid.
