Bounded relative motions near Keplerian orbits in spherical coordinates

Abstract

The problem of relative motion for arbitrary Keplerian orbits is modeled in a curvilinear coordinate system. A set of spherical coordinates rather than the classical Cartesian coordinates is used. The techniques such as linearization and normalization are performed to simplify the relative equations of motion as a linear form that has a similar structure as the Tschauner-Hempel (T-H) equations but expressed in spherical coordinates. The analytical solution in spherical coordinates is obtained by using the same technique as shown in the classical T-H equations. Based on this solution, bounded relative motions and the boundedness conditions are recognized for elliptical, parabolic and hyperbolic orbits, respectively. The geometry of bounded relative motions near parabolic and hyperbolic orbits are revealed, which has not been reported in literatures yet. It shows a strong connection with the “leader-follower” formation near elliptical orbits. Numerical simulation demonstrates the improved accuracy of the newly-developed solution and shows the existence of bounded relative trajectories for unbounded reference orbits.