Design of Mars Missions

Abstract

This paper deals with the optimal design of round-trip Mars missions, starting from LEO (low Earth orbit), arriving to LMO (low Mars orbit), and then returning to LEO after a waiting time in LMO.The assumed physical model is the restricted four-body model, including Sun, Earth, Mars, and spacecraft. The optimization problem is formulated as a mathematical programming problem: the total characteristic velocity (the sum of the velocity impulses at LEO and LMO) is minimized, subject to the system equations and boundary conditions of the restricted four-body model. The mathematical programming problem is solved via the sequential gradient-restoration algorithm employed in conjunction with a variable-stepsize integration technique to overcome the numerical difficulties due to large changes in the gravity field near Earth and near Mars.The results lead to a baseline optimal trajectory computed under the assumption that the Earth and Mars orbits around Sun are circular and coplanar. The baseline optimal trajectory resembles a Hohmann transfer trajectory, but is not a Hohmann transfer trajectory, owing to the disturbing influence exerted by Earth/Mars on the terminal branches of the trajectory. For the baseline optimal trajectory, the total characteristic velocity of a round-trip Mars mission is 11.30 km/s (5.65 km/s each way) and the total mission time is 970 days (258 days each way plus 454 days waiting in LMO).An important property of the baseline optimal trajectory is the asymptotic parallelism property: For optimal transfer, the spacecraft inertial velocity must be parallel to the inertial velocity of the closest planet (Earth or Mars) at the entrance to and exit from deep interplanetary space. For both the outgoing and return trips, asymptotic parallelism occurs at the end of the first day and at the beginning of the last day. Another property of the baseline optimal trajectory is the near-mirror property. The return trajectory can be obtained from the outgoing trajectory via a sequential procedure of rotation, reflection, and inversion.Departure window trajectories are next-to-best trajectories. They are suboptimal trajectories obtained by changing the departure date, hence changing the Mars/Earth inertial phase angle difference at departure. For the departure window trajectories, the asymptotic parallelism property no longer holds in the departure branch, but still holds in the arrival branch. On the other hand, the near-mirror property no longer holds.Key WordsFlight mechanicsastrodynamicscelestial mechanicsEarth-to-Mars missionsround-trip Mars missionsmirror propertyasymptotic parallelism propertyoptimizationsequential gradient restoration algorithm