Special inclinations allowing minimal drift orbits for formation flying satellites

The possibility of obtaining a natural periodic relative motion of formation flying Earth satellites is investigated both numerically and analytically. The numerical algorithm is based on a genetic strategy, refined by means of nonlinear programming, that rewards periodic relative trajectories. First, we test our algorithm using a point mass gravitational model. In this case the period matching between the considered orbits is a necessary and sufficient condition to obtain invariant relative trajectories. Then, the J(2) perturbed case is considered. For this case, the conditions to obtain an invariant relative motion are known only in approximated closed forms which guarantee a minimal orbit drift, not a motion periodicity. Using the proposed numerical approach, we improved those results and found two couples of inclinations (63.4 and 116.6 deg, the critical inclinations, and 49 and 131 deg, two new "special" inclinations) that seemed to be favored by the dynamic system for obtaining periodic relative motion at small eccentricities.