Theorem of Optimal Image Trajectories in the Restricted Problem of Three Bodies

The restricted three-body problem represents the dynamical framework employed for spacecraft mission analysis, in the presence of two attracting bodies, since the 1950s. In this context, orbital motion is often chaotic, although several special solutions (equilibrium points, periodic orbits, and quasiperiodic trajectories) exist, and can be—or have already been—profitably employed in space missions. The theorem of image trajectories, proven five decades ago by Miele, states that for a given path in the restricted problem of three bodies (with primaries in mutual circular orbits), there exists a mirror trajectory (in two dimensions) and three mirror paths (in three dimensions). This theorem regards feasible trajectories and proved extremely useful for investigating the natural dynamics in the restricted problem of three bodies, by identifying special solutions, such as symmetric periodic orbits and free return trajectories. This work extends the theorem of image trajectories to optimal paths, which minimize either the propellant consumption or the time of flight, by determining the relations between the optimal thrust sequence, magnitude, and direction of an outgoing path and a symmetrical returning trajectory. This means that while the theorem of image paths revealed extremely useful for investigating natural dynamics, the theorem of optimal image trajectories can be profitably employed for powered orbital motion, i.e., in the context of impulsive and finite-thrust orbit transfers and rendezvous, as well as for the purpose of analyzing artificial periodic orbits that use very low thrust propulsion.