Optimal trajectories and mirror properties for round-trip Mars missions

Abstract

This paper deals with optimal trajectories for round-trip Mars missions, starting from LEO (low Earth orbit), arriving to LMO (low Mars orbit), and then returning to LEO after a waiting period in LMO. The optimization criterion is the minimization of the total characteristic velocity. The assumed physical model is the restricted four-body model. The optimization problem is solved via the sequential gradient-restoration algorithm employed in conjunction with a variable-stepsize integration technique to overcome numerical difficulties due to large changes in the gravity field near Earth and near Mars. The results indicate that, if the Earth and Mars orbits around Sun are assumed to be circular and coplanar, the total characteristic velocity for a round-trip Mars mission is 11.30 km/s and the total mission time is 970 days. One important relation between the LEO-to-LMO and LMO-to-LEO optimal transfers is a kind of image property or mirror property: the return trajectory can be obtained from the outgoing trajectory via a sequential procedure of rotation, reflection, and inversion. In an optimal round-trip Mars mission, the mirror properties are present in many areas, for example: (i) the characteristic velocities of the outgoing and return trajectories are the same, 5.65 km/s; (ii) the flight times of the outgoing and return trajectories are the same, 258 days; (iii) the Mars/Earth inertial phase angle differences on arrival to LMO (−75 deg) and at departure from LMO (+75 deg) are the same in modulus, albeit opposite in sign; (iv) to achieve the transition from optimal arrival to LMO to optimal departure from LMO, the Mars/Earth inertial phase angle difference must change by 210 deg, corresponding to 454 days, since the Earth/Mars angular velocity difference is 0.462 deg/day.