Reachable Set of Periapsis Parameters for Invariant Manifolds from a Periodic Orbit

This study investigates the reachable set of periapsis along invariant manifolds from periodic orbits in the three-body system. Given a nominal stable or unstable manifold, the mapping between the periapsis parameters and the initial state deviation is first derived using state-transition tensors (STTs). A distance-based STT is proposed to mitigate the impact of the duration from the periodic orbit to the smaller primary, thus enhancing the mapping accuracy. Then, the reachable set of periapsis radius and inclination is determined within a feasible region of initial states, of which the boundary is defined by the extremum of periapsis radius and the extremum of inclination with fixed periapsis radius. These extrema are determined through a series of optimization problems with the derived optimal necessary and sufficient conditions. Finally, the proposed method is applied to calculate the reachable set from Halo orbits. Numerical simulations show that the computation for a reachable set is efficient. The relative errors of the periapsis radius and the inclination for a single reachable set are within 3.64% and within 1% for the reachable set from a Halo orbit.