Earth-to-moon transfer with a limited power engine

Abstract

This paper presents a power-limited low-thrust optimal guidance law for transfer in a central gravitational field. This guidance law is obtained analytically as a particular solution to the case of transfer with free final time. The optimal thrust acceleration is shown to be colinear with the vehicle velocity vector multiplied by a time-varying gain that depends on a constant-guidance parameter. An analysis is performed of the system trajectories as obtained when this guidance law is applied. This guidance law is applied to the case of Earth-to-moon transfer. The initial condition is a low circular Earth orbit. For the final condition two cases are considered: 1) impact on the moon and 2) moon orbit injection. The solution for both cases is obtained by considering the gravity field of one body at a time. The initial Earth-centered trajectory consists of a number of revolutions during which the trajectory spirals away gradually from Earth. In the impact case, by adequately choosing the guidance parameter, it is possible to achieve, at a prescribed range, a moon-relative velocity such that the moon gravitational field will capture the spacecraft. For moon orbit injection, the trajectory is obtained by matching the outward Earth-spiral trajectory with a moon inward spiral. Both spirals are generated by the guidance law with the guidance parameter positive for the Earth outward spiral and negative for the moon inward one.