Arbitrary Angular-Separation Relative Motion in Near-Circular Orbits with Similar Periods

Abstract

The time-explicit analytical description of the arbitrary angular-separation relative motion in near-circular orbits with similar periods is investigated. Using Lagrange’s generalized expansion and neglecting the higher-order terms, an approximate dynamics equation of relative motion is obtained. The analytical solution is obtained by using an iterative substitution method. For the convenience of analyzing the characteristics of the relative motion, the analytical solution is described in a new form by converting the Cartesian coordinates to six relative parameters. Based on this expression, it is found that the relative trajectory in the along-track and radial directions in the local vertical local horizontal frame of the periodic relative motion is a combination of circular, elliptical, and rectilinear motions. Seven different orbital configuration simulations are provided to analyze the periodical orbits. Both small and large angular-separation relative-motion simulations are conducted to examine the accuracy of the analytical solutions.