Optimal Interplanetary Orbital Transfers via Electrical Engines

The Hohmann transfer theory, developed in the 19th century, is the kernel of orbital transfer with minimum propellant mass by means of chemical engines. The success of the Deep Space 1 spacecraft has paved the way toward using advanced electrical engines in space. While chemical engines are characterized by high thrust and low specific impulse, electrical engines are characterized by low thrust and hight specific impulse. In this paper, we focus on four issues of optimal interplanetary transfer for a spacecraft powered by an electrical engine controlled via the thrust direction and thrust setting: (a) trajectories of compromise between transfer time and propellant mass, (b) trajectories of minimum time, (c) trajectories of minimum propellant mass, and (d) relations with the Hohmann transfer trajectory. The resulting fundamental properties are as follows: (a) Flight Time/Propellant Mass Compromise. For interplanetary orbital transfer (orbital period of order year), an important objective of trajectory optimization is a compromise between flight time and propellant mass. The resulting trajectories have a three-subarc thrust profile: the first and third subarcs are characterized by maximum thrust; the second subarc is characterized by zero thrust (coasting flight); for the first subarc, the normal component of the thrust is opposite to that of the third subarc. When the compromise factor shifts from transfer time (C=0) toward propellant mass (C=1), the average magnitude of the thrust direction for the first and third subarcs decreases, while the flight time of the second subarc (coasting) increases; this results into propellant mass decrease and flight time increase. (b) Minimum Time. The minimum transfer time trajectory is achieved when the compromise factor is totally shifted toward the transfer time (C=0). The resulting trajectory is characterized by a two-subarc thrust profile. In both subarcs, maximum thrust setting is employed and the thrust direction is transversal to the velocity direction. In the first subarc, the normal component of the thrust vector is directed upward for ascending transfer and downward for descending transfer. In the second subarc, the normal component of the thrust vector is directed downward for ascending transfer and upward for descending transfer. (c) Minimum Propellant Mass. The minimum propellant mass trajectory is achieved when the compromise factor is totally shifted toward propellant mass (C=1). The resulting trajectory is characterized by a three-subarc (bang-zero-bang) thrust profile, with the thrust direction tangent to the flight path at all times. (d) Relations with the Hohmann Transfer. The Hohmann transfer trajectory can be regarded as the asymptotic limit of the minimum propellant mass trajectory as the thrust magnitude tends to infinity. The Hohmann transfer trajectory provides lower bounds for the propellant mass, flight time, and phase angle travel of the minimum propellant mass trajectory. The above properties are verified computationally for two cases (a) ascending transfer from Earth orbit to Mars orbit; and (b) descending transfer from Earth orbit to Venus orbit. The results are obtained using the sequential gradient- restoration algorithm in either single-subarc form or multiple-subarc form.