Mapping Two-Body and Cislunar Impulsive Maneuvers into Fuel-Optimal Low-Thrust Trajectories

A fundamental astrodynamics problem is to map impulsive trajectories into finite- and low-thrust ones. We propose a thrust-continuation based mapping between minimum- Δ v Δ v impulsive and finite-thrust trajectories. Two notable features of the proposed method are: a) guessing the non-intuitive costates is completely circumvented, and b) multiple-impulse trajectories can be converted into multiple-revolution finite- and low-thrust fuel-optimal trajectories. A novel derivation is proposed to estimate the mass costate from extremal impulsive solutions, based on a high-thrust assumption. The method is tested on three trajectory design problems under two-body dynamics and one problem under the circular restricted three-body problem (CR3BP) dynamics of the Earth-Moon system. A remarkable result, under two-body dynamics, is the uniqueness of the fuel-optimal solutions for the specified total number of revolutions, when limited to a single orbital revolution on each potential phasing orbit. Results show the method applies successfully to extremal multi-impulsive trajectories under both two-body and CR3BP dynamics. If we consider incipient minimum- Δ v Δ v impulsive solutions, results indicate that impulsive maneuvers are mapped into local fuel-optimal trajectories. A fascinating heretofore unknown finding is the existence of a critical thrust value (for the same initial mass and specific impulse values) at which two completely different fuel-optimal trajectories result in the same final mass for the same total time of flight and number of orbital revolutions.