Analytical conditions for bounded mean inter-satellite distances in the J₂ problem

Finding satellite relative orbits that are resilient to differential gravitational perturbations has received much attention in the literature. In particular, detecting "invariant" relative orbits under the J₂ perturbation was considered a solved problem. These "invariance" conditions result in constraints on the differential mean semimajor axis, inclination, and eccentricity. In this paper, it is shown that alternative conditions can be used to further reduce the drift among J₂-perturbed satellites. These alternative conditions are found by investigating the secular part of the averaged intersatellite distance. Averaging is performed with respect to the mean anomaly and argument of perigee. A closed-form expression for the first-order J₂-perturbed mean distanceis obtained. It is found that the mean relative distance squared drifts as a quadratic function of time. Four conditions for mean-distance boundedness are derived. Using simulations, it isshown that the new conditions can improve previously obtainedresults, inthe senseofreducing the residual intersatellite distance drift.