Optimal two-impulse transfer between specified terminal states of Keplerian orbits

The integrals of motion for an inverse-square force field and a necessary condition for optimality are used in a transfer orbit co-ordinate system to formulate the solution of the optimal two-impulse transfer between fixed position and velocity vectors on Keplerian orbits. In this co-ordinate system, the equations reveal two asymptotes that are useful in analysing the solution. It is shown that there are only two real extremals (both minimums) which are separated by an asymptote. An approximate analytic solution is also obtained by ignoring a quadratic term whose coefficient is approximately zero for a large class of orbit transfer problems.