Numerical study about natural escape and capture routes by the Moon via Lagrangian points L1 and L2

The lunar sphere of influence, whose radius is some 66,300 km, has regions of stable orbits around the Moon and also regions that contain trajectories which, after spending some time around the Moon, escape and are later recaptured by lunar gravity. Both the escape and the capture occur along the Lagrangian equilibrium points L1 and L2. In this study, we mapped out the region of lunar influence considering the restricted three-body Earth–Moon–particle problem and the four-body Sun–Earth–Moon–particle (probe) problem. We identified the stable trajectories, and the escape and capture trajectories through the L1 and L2 in plots of the eccentricity versus the semi-major axis as a function of the time that the energy of the osculating lunar trajectory in the two-body Moon–particle problem remains negative. We also investigated the properties of these routes, giving special attention to the fact that they supply a natural mechanism for performing low-energy transfers between the Earth and the Moon, and can thus be useful on a great number of future missions.