Optimal transfers from an Earth orbit to a Mars orbit

Abstract

This paper deals with the optimal transfer of a spacecraft from a low Earth orbit (LEO) to a low Mars orbit (LMO). The transfer problem is formulated via a restricted four-body model in that the spacecraft is considered subject to the gravitational fields of Earth, Mars, and Sun along the entire trajectory. This is done to achieve increased accuracy with respect to the method of patched conics. The optimal transfer 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 gravitational field near Earth or near Mars. First, for given LEO and LMO radii, a basic optimization problem is considered: the minimization of the total characteristic velocity, the sum of the velocity impulses at LEO and LMO, assuming that the departure date is free, hence assuming that the planetary Mars/Earth phase angle difference at departure is free. At both departure and arrival, the optimal trajectory exhibits an asymptotic parallelism condition: at the end of near-Earth space, the spacecraft inertial velocity is parallel to the Earth inertial velocity; analogously, at the beginning of near-Mars space, the spacecraft inertial velocity is parallel to the Mars inertial velocity. The total characteristic velocity is Δ V=5.652 km/s, corresponding to a ratio of payload mass to initial mass of 0.224 for typical specific impulse and structural factor. Then, for given LEO and LMO radii, a departure window is generated by changing the departure date, hence changing the planetary Mars/Earth phase angle difference at departure, and then reoptimizing the transfer. This results into an one-parameter family of suboptimal transfers retaining the asymptotic parallelism condition at arrival, but not at departure. For the suboptimal transfers, the phase angle travel and transfer time decrease with late departure and increase with early departure. Also, a change in departure date of +32 days [−32 days] causes a characteristic velocity increase of 8% [5%], implying a payload mass decrease of 13% [8%].