Fuel- or Time-Optimal Transfers Between Coplanar, Coaxial Ellipses Using Lambert’s Theorem

Undoubtedly, minimum-fuel and-time orbit transfer are the two major goals of the op timal orbit maneuver. This paper considers two coplanar elliptic orbits when the apsidal l ines coincide. We analytically find the conditions for the two-impulse minimum-time tra nsfer orbit using Lambert’s theorem. In the minimum-time transfer, the transfer time is a decreasing function of a variable related to the transfer orbit’s semimajor axis. Conseque ntly there exists no unique minimum-time solution. Thus for the minimum-time case, th ere is a limiting solution only, however, there exists a unique solution in the case of mini mum-fuel transfer, for which we find the necessary and sufficient conditions. Furthermor e, as a special case, we consider when the transfer angle is one hundred and eighty degre es. In this case we show that we obtain the fuel-optimal Hohmann transfer orbit. We als o derive the Hohmann transfer time and delta-velocity equations from more general equa tions, which are also obtained using Lambert’s theorem. There is a tradeoff between mini mum-time and-fuel transfer. Finally, we propose an optimal coplanar orbit maneuver alg orithm for trading off the minimum-time goal against the minimum-fuel goal