Construction of the local orbits near the $L_2$ libration point of the Sun – Earth system

We describe the way to construct spacecraft orbits, that keep near the $L_2$ libration point of the Sun – Earth system for a long time. We construct them in two stages. The first stage consists in constructing the orbit as a solution of the restricted three-body problem. On the second stage, that orbit is transformed to the solution of the restricted four-body problem with real orbits of the Sun, the Earth, and the Moon. A prototype of the first stage orbit is formed by a Lyapunov periodic solution that exists near the $L_2$ point and lies in the orbital plane of the large bodies and a second order system solution describing small deviations of the spacecraft from this plane along the periodic solution. The desired orbit is looked for as a solution of the three-body problem, providing the best approximation of the prototype in the least squares sense. That orbit is an analogous prototype for the second stage. In both stages the approximating solutions are constructed by the continuation on the length of the approximation interval.