A Closed Form Vectorial Solution to the Relative Orbital Motion

Time-explicit solutions for relative Keplerian dynamics are presented for the general case in which the reference trajectory is elliptic, parabolic or hyperbolic. The non-linear differential equation modeling the motion is solved by means of tensorial and vectorial regularization methods. This regularizing procedure extends the approaches of Levi-Civita and Kustaanheimo-Stiefel in the case of Kepler problem in rotating reference frames. The solution generalizes the approximate results obtained from Hill-Clohessy-Wiltshire, Lawden and Tschauner-Hempel equations. The approximate solutions derived from the liniarized models are in fact the first linear approximation of the exact solution the authors have found.