Satellite Orbits and Relative Motion in Levi-Civita Coordinates

Abstract

In this paper we consider satellite trajectories in central force field with quadratic drag using two formalisms. The first using polar coordinates in which the angular momentum plays a dominant role. The second is in Levi-Civita coordinates in which the energy plays a central role. We then unify these two formalisms by introducing polar coordinates in Levi-Civita space and derive a new equation for satellite orbits in which energy and and angular momentum are on equal footing {and thus characterize the orbit by its two invariants}. In the second part of the paper we derive in Levi-Civita coordinates a linearized equation for the relative motion of two satellites whose trajectories are in the same plane. We carry out also a numerical verification of these equations.