An Exact Solution to Satellite Relative Orbital Motion

Abstract

The paper gives an exact solution to the motion of a celestial body related to another, both orbiting around the same attraction center under the influence of a Keplerian force. The result is a replica for Lawden’s equations of motion, and the approach is completely different. We do not try to linearize the equation of motion by approximating, but solve it and give an exact solution. This is possible due to a vectorial differential regularization introduced, that transforms the non-linear Kepler’s problem into an ordinary second order linear differential equation with constant coefficients. The relative orbital motion is described by exact equations; the only implicit function used being the eccentric anomaly related to a keplerian motion. The subject of this paper is of great interest in space engineering, as results from several orbital mechanics papers and books written in the past years. We propose an alternative to the actual approaches that leads to simple and elegant solutions.