The properties of the optimal and sub-optimal solutions to multiple-pass aeroassisted plane change are studied in terms of the trajectory variables in Refs.  and . The solutions show the strong orbital nature. Then, it is proposed to obtain the variational equations of the orbital elements. We shall use these equations and the approximate control derived in Ref.  to calculate the trajectories. In this respect, the approximate control law and the transversality condition are transformed in terms of the orbital elements. Folllowing the above results, we can reduce the computational task by further simplification. with W and Q being small and returning to the value of zero after each revolution, we neglect the equations ~or ~, an~ Q. Also, since ~ 0, that is c~ ~ f, we can neglect the eqnation for the ~ and have only three state equations for the integration. Still the computation over several revolutions is long since it is performed using the eccentric anomaly along the osculating orbit as the independant variable. Here, we shall use the method of averaging as applied to the problem of orbit contraction5~6 to solve the problem of optimal plane change. This will Iead to the integration of a reduced set of two nonlinear equations.
Advances in the Astronautical Sciences Series 82, pp.1139-1154