A method based on Jacobi Integral variational equation for computing Earth-Moon trajectories in the four-body problem

Abstract

This work proposes an alternative method for solving the two-point boundary value problem concerning to Earth-Moon bi-impulsive trajectories in the dynamics of the planar bi-circular restricted four-body problem, which describes the motion of a space vehicle subjected to the gravitational attraction of Earth, Moon and Sun. Initially, the space vehicle is at a circular low Earth orbit (LEO) with prescribed altitude. After applying the first impulsive velocity increment, the space vehicle is inserted into a transfer trajectory. The second velocity increment is applied to decelerate and circularize the movement of the space vehicle at a circular low Moon orbit (LMO) with prescribed altitude. To solve this problem, a new two-point boundary value problem (TPBVP) is formulated, which includes an unknown value of the Jacobi integral at the departure time, and, a prescribed value at the arrival time. Since the Jacobi integral is not a first integral for the four-body problem, it is taken as additional state variable, and, its variational equation is added to the system of differential equation in the description of the dynamics of the space vehicle. Taking into account the boundary conditions, expressions for the velocity increments are deduced from the Jacobi integral computed at the initial and final times. Based on this new TPBVP, a numerical procedure is proposed to obtain different families of Earth-Moon trajectories with decreasingly fuel consumption.