Optimal Free-Return Trajectories for Moon Missions and Mars Missions

Abstract

Free-return trajectories for Moon or Mars missions depart from a low Earth orbit (LEO), pass near the Moon or Mars, and then return to LEO; propulsive maneuvers are employed only at the departure from LEO and return to LEO. The assumed physical model is the restricted three-body model for Moon missions and the restricted four-body model for Mars missions. Trajectory optimization is carried out via the sequential gradient-restoration algorithm in mathematical programming format. Moon Missions. In a rotating coordinate system, the return trajectory Moon/Earth is the mirror image of the outgoing trajectory Earth/Moon with respect to the Earth-Moon axis; in an inertial coordinate system, the mirror image property still holds with respect to the Earth-Moon axis at Moon flyby. The return flight time is the same as the outgoing flight time, and the total flight time of the optimal free-return trajectory is considerably smaller than that of the minimum energy trajectory. Should a decision be made at the apex of the free-return trajectory to circularize the motion in the low Moon orbit, the total characteristic velocity of the free-return trajectory would be slightly larger than that of the minimum energy trajectory. To sum up, for Moon missions, the free-return trajectory provides an ideal protection for safe return with the added advantage of a smaller flight time, albeit at the expense of an acceptable increase in total characteristic velocity. Mars Missions. The mirror image property noted for free-return Moon missions does not apply to free-return Mars missions: indeed, the return flight time is more than three times the outgoing flight time, corresponding to the fact that the return angular travel around the Sun (more than 1.5 loops) is more than three times the outgoing angular travel (less than 0.5 loops). There is an interesting interplay between total characteristic velocity and Mars flyby altitude. Should the total characteristic velocity be optimized with respect to the Mars flyby altitude, the resulting altitude would be too high for useful observation of the Mars. On the other hand, as the Mars flyby altitude is lowered, the total characteristic velocity increases and might reach unacceptable values. To sum up, for Mars missions, the free-return trajectory is not as useful as for Moon missions; indeed, compared with the minimum energy trajectory, the free-return trajectory has the twin disadvantage of larger flight time and larger characteristic velocity.