Design of Moon Missions

Abstract

In this paper, a systematic study of the optimization of trajectories for Earth-Moon flight is presented. The optimization criterion is the total characteristic velocity and the parameters to be optimized are: the initial phase angle of the spacecraft with respect to Earth, flight time, and velocity impulses at departure and arrival. The problem is formulated using a simplified version of the restricted three-body model and is solved using the sequential gradient-restoration algorithm for mathematical programming problems.

For given initial conditions, corresponding to a counterclockwise circular low Earth orbit at Space Station altitude, the optimization problem is solved for given final conditions, corresponding to either a clockwise or counterclockwise circular low Moon orbit at different altitudes. Then, the same problem is studied for the Moon-Earth return flight with the same boundary conditions.

The results show that the flight time obtained for the optimal trajectories (about 4.5 days) is larger than that of the Apollo missions (2.5 to 3.2 days). In light of these results, a further parametric study is performed. For given initial and final conditions, the transfer problem is solved again for fixed flight time smaller or larger than the optimal time.

The results show that, if the prescribed flight time is within one day of the optimal time, the penalty in characteristic velocity is relatively small. For larger time deviations, the penalty in characteristic velocity becomes more severe. In particular, if the flight time is greater than the optimal time by more than two days, no feasible trajectory exists for the given boundary conditions.

The most interesting finding is that the optimal Earth-Moon and Moon-Earth trajectories are mirror images of one another with respect to the Earth-Moon axis. This result extends to optimal trajectories the theorem of image trajectories formulated by Miele for feasible trajectories in 1960.