Optimal low-thrust Earth escape spiral trajectories from geosynchronous orbit

Minimum time low thrust Earth escape spiral trajectories from geosynchronous orbit were calculated. The two point boundary value problem was solved with very good convergence by one of the direct methods (two-dimensional search conjugated gradient method with penalty function), using the digital computer FACOM 203-75. Starting from geosynchronous orbit, the spacecraft increases both its orbital altitude and its total energy E (kinetic energy plus potential energy in Earth's gravitational field), rotating round the Earth along the spiral trajectory, and then attains the parabolic velocity (E=O). The thrust vector oscillates around the velocity vector, i.e., the tangential direction with the period almost equal to the rotational period of the orbit. The synchronization is clearer in the first portion of the trajectory and for a smaller initial acceleration. The amplitude of the oscillation increases gradually in the first portion, but begins to decrease in the final portion of the trajectory, and reaches zero at the escape point.