The four-body linear equations of planar relative motion

Abstract

The very restricted four-body problem is used to derive linear equations of relative motion that take into account the perturbing effects of a secondary gravitational source. The outcome is a system of linear differential equations that has an analytical solution. The results of this solution are compared to the outcomes of the very restricted four-body problem and the well known Hill–Clohessy–Wiltshire equations for various cases in the Sun–Ceres system. The results demonstrate that the Benavides–Spencer formulation's accuracy exceeds that of the Hill–Clohessy–Wiltshire equations when compared to the real-life outcomes returned by the numerical integration of the very restricted four-body problem.