REVIEW OF THE SOLUTIONS TO THE TSCHAUNER-HEMPEL EQUATIONS FOR SATELLITE RELATIVE MOTION

The Tschauner-Hempel equations model the motion of a deputy satellite relative to a chief satellite with arbitrary eccentricity. They are linear non-autonomous differential equations with the chiefs true anomaly as the independent variable. Since they first appeared, numerous analytical solutions have been presented. This paper provides a focused review of some of these solutions: highlighting how they are related and their singularities. The fundamental solutions of the Tschauner-Hempel equations can be interpreted geometrically as generalizations of the drifting two-by-one ellipse that describes relative motion in circular orbits. General solutions are formed by taking linear combinations of these fundamental solutions.