The four-body linear equations of planar relative motion

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.