Orbits and Relative Motion in the Gravitational Field of an Oblate Body

This paper studies the orbits of satellites in the noncentral gravitational field of an oblate body that includes the J 2 term. A discussion of the angular momenta and equations of motion of objects in the equatorial, near-equatorial, and polar orbits is presented. Some progress toward the solution and approximate solution of the equations of motion is presented for these cases. The paper also derives the relative-motion equations that describe a spacecraft in the vicinity of a satellite in equatorial, near-equatorial or polar orbit in this gravitational field. We show that the rendezvous equations can be simplified to resemble the corresponding equations for a central-force field. In particular, equations similar to the Tschauner-Hempel equations appear for orbits in the equatorial plane. For circular orbits in this plane, these become modified Clohessy-Wiltshire equations and possess closed-form solutions. Various types of powered optimal rendezvous problems can be formulated from these equations.