Designing the low-energy lunar transfers trajectories which pass in the vicinity of the libration points of the Earth - Moon system. Part 1. Theory and method

A method for designing low-energy trajectory of transfer to the Moon with the insertion of a spacecraft into a low circumlunar orbit is proposed. The analysis of this trajectory is based on the solution of a boundary value problem for the system of differential equations of the restricted four-body problem. The trajectory of the low-energy flight passes through a region of space where the gravitational attraction of the Earth, the Moon, and the Sun tend to cancel. The trajectory turns out to be very sensitive to the initial conditions of the spacecraft motion. Difficulties arise in solving the boundary value problem. Weak stability boundary issue appears. An additional difficulty in designing the trajectory of a low-energy transfer of a spacecraft is related to the multi-extremality of the optimization problem under consideration. The authors assume that the transfer trajectory passes in the vicinity of the libration point L1 or L2 of the Earth - Moon system and introduces some restrictions on the velocity vector of the spacecraft at the moment the spacecraft passes the vicinity of the libration point. This assumption and the use of enumeration in space of the two main parameters of the transfer pattern allows to find an initial approximation for the low-energy transfer trajectory.