Construction of the flights from low Earth orbits to local orbits near the $L_2$ libration point in the Sun – Earth system

We describe the way to construct spacecraft orbits for the flight from a low Earth orbit to a halo-orbit near the $L_2$ libration point of the Sun – Earth system. 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 solution lying in the orbital plane of the large bodies and a solution of the second order system that describes small deviations of the spacecraft from this plane along the plane solution. The plane solution begins near the Earth and tends with time to a Lyapunov solution that exists in the vicinity the $L_2$ libration point. The desired orbit is looked for as a solution of three-body problem that provides the best approximation of the prototype in the least squares sense. That orbit is an analogous prototype for the second stage. The approximating solutions are constructed in both stages by the continuation on the length of the approximation interval. Initial conditions of approximating solutions correspond to a spacecraft start from a low Earth orbit with prescribed values of the perigee distance and the inclination.