Numerical and analytical study of the gravitational capture in the bicircular problem

The objective of the present paper is to study the ballistic gravitational capture in a dynamical model that has the presence of four bodies. In particular, the Earth-Moon-Sun-Spacecraft system is considered. This phenomenon is explained in terms of the integration of the perturbing forces with respect to time. Analytical equations are derived to estimate their magnitude and to show the best directions of approaching for the spacecraft. Using those equations, an analytical estimate of the fourth body effect valid for any system of primaries is derived. Numerical simulations are also performed to show the differences between the models with three and four bodies in the trajectories of the spacecraft.