The minimum time control of the Kepler equation is considered. The typical application is the transfer of a satellite from an orbit around the Earth to another one, both orbits being elliptic. We recall the standard model to represent the system. Its Lie algebraic structure is first analyzed, and controllability is established for two different single-input subsystems, the control being oriented by the velocity or by the orthoradial direction. In both cases, a preliminary analysis of singular and regular extremals is also given, using the usual concept of order to classify the contacts. Moreover, the singularity of the multi-input model---which is a particular case of a subriemannian system with drift---is resolved, and the related nilpotent model is given. Finally, second order optimality conditions are recalled, for smooth regular and singular extremals. For both, the algorithms to compute conjugate points are detailed and applied to check numerically the time optimality of orbit transfers.