Optimal ballistically captured Earth–Moon transfers

The optimality of a low-energy Earth–Moon transfer terminating in ballistic capture is examined for the first time using primer vector theory. An optimal control problem is formed with the following free variables: the location, time, and magnitude of the transfer insertion burn, and the transfer time. A constraint is placed on the initial state of the spacecraft to bind it to a given initial orbit around a first body, and on the final state of the spacecraft to limit its Keplerian energy with respect to a second body. Optimal transfers in the system are shown to meet certain conditions placed on the primer vector and its time derivative. A two point boundary value problem containing these necessary conditions is created for use in targeting optimal transfers. The two point boundary value problem is then applied to the ballistic lunar capture problem, and an optimal trajectory is shown. Additionally, the problem is then modified to fix the time of transfer, allowing for optimal multi-impulse transfers. The tradeoff between transfer time and fuel cost is shown for Earth–Moon ballistic lunar capture transfers.