Trajectory Optimization Operations for Satellites in Elliptic Orbits

Minimum-fuel and -time orbit transfer are two major goals of the satellite trajectory optimization. In this paper, we consider satellites in two coplanar elliptic orbits when the apsidal lines coincide, and analytically find the conditions for the two-impulse minimum-time transfer orbit using Lambert's theorem. The transfer time is a decreasing function of a variable related to the transfer orbit's semimajor axis in the minimum-time case. In the minimum-time case, there is no unique minimum-time solution, but there is a limiting solution. However, there exists a unique solution in the case of minimum-fuel transfer, fur which we find analytically the necessary and sufficient conditions. As a special case, we consider when the transfer angle is one hundred and eighty degrees. In this case, we show that we obtain the classical fuel-optimal Hohmann transfer orbit. We also derive the Hohmann transfer rime and delta-velocity equations from more general equations, which are obtained using Lambert's theorem. We note the tradeoff between minimum-time and - fuel transfer. An optimal coplanar orbit maneuver algorithm to trade off the minimum-time goal against the minimum-fuel goal is proposed. Finally, the numerical simulation results are given to demonstrate the derived theory and the algorithm.