Optimal Low-Thrust Maneuvers in Presence of Earth Shadow

The optimization of the very-low-thrust transfer between circular orbits in the presence of shadow is carried out by assuming that the spacecraft is on a circular orbit both on entering and exiting the shadow. The optimal steering law is found by extending Edelbaum’s approach to deal with an incomplete revolution around the Earth. The authors find an analytical solution, which does not show any appreciable difference when it is compared to the exact numerical solution that is obtained by using an indirect optimization method. The differences between the authors’ solution and other approximate solutions in the literature are highlighted and explained. The numerical analysis also demonstrates that no improvement can be achieved by removing the constraint of flying the shadow arc of a two-revolution maneuver on a circle; therefore, the proposed solution constitutes the optimal segment of a multirevolution transfer between circular orbits. The paper also analyzes the dual problem of maximizing the eccentricity change while maintaining the spacecraft energy, that is, the length of the ellipse major semiaxis. The rotation of the line of apsides during the maneuver, which is neglected in Edelbaum’s approach, is here considered. The optimal steering law is derived in the paper; the change in eccentricity is maximized when the major axes of the departure and arrival orbits have the same orientation and are perpendicular to the Earth-sun direction; therefore, the optimal position of the coast arc is symmetrical with respect to the minor axis.