Fast and accurate estimation of fuel-optimal trajectories to Near-Earth Asteroids

This paper proposes an improved method for the preliminary evaluation of minimum-propellant trajectories to Near-Earth Asteroids (NEAs). The method applies to missions from Earth to asteroids with small eccentricity and inclination. A planar and a plane-change problem can be distinguished. In the planar problem, the solution assumes that multiple burn arcs are performed in correspondence of the apsides of the target asteroid in order to change the initial spacecraft orbit (i.e., Earth’s orbit) into the target one. The number of arcs is established once the time of flight is given (1 burn at each apsis per revolution, 1 revolution per year can be assumed). The length and propellant consumption of each arc to attain the required changes of semi-major axis and eccentricity are computed by a procedure based on Edelbaum’s approximation, which is well-suited to the problem at hand, as eccentricity changes are expected to be small for feasible missions. No numerical integration is required, but only the numerical solution of a three-unknown algebraic system is needed, making the procedure extremely fast. Plane change is taken into account assuming a constant out-of-plane thrust angle during each burn. A previous simple formulation used an averaged thrust effect over one revolution and neglected the fact that plane changes are more effective at the nodes. Several improvements are here introduced, which greatly increase the method accuracy. The influences of the eccentricity change, the angle between the asteroid line of nodes and line of apsides, and the expected length of the arc are considered: In fact, when the eccentricity is small, the thrust arc can be performed at the nodes where the inclination is efficiently changed, with little penalty in the planar maneuver. An efficient plane change is also performed when the angle between the asteroid line of nodes and line of apsides is small and/or the length of the arc is large, because, in this case, the node is comprised in the apsidal burn. A simple corrective formula accounts for this effect. The new method shows remarkable accuracy. The results comparison with solutions obtained with an indirect optimization method for a set of more than 60 NEAs shows a 0.95 correlation coefficient in the propellant masses. The estimation error is below 10% for 75% of the targets, below 15% for 95% of the targets, and always below 20%.