Numerical study and analytic estimation of forces acting in ballistic gravitational capture

The objective of the present paper is to study in some detail the physical reasons of the ballistic gravitational capture. A numerical study is performed to show the magnitude of the forces involved in the ballistic gravitational capture as a function of time. It shows that the resultant force is near zero all of the time. Then, this phenomenon is explained in terms of the integration of the perturbing forces with respect to time. The relation between those integrals and the reduction of the two-body energy with respect to the moon is derived and applied to a large set of trajectories. Analytical equations for those forces are derived to estimate their magnitude and to show the best directions of approach for the ballistic gravitational capture. Using those equations, an analytical estimate of the effects for different values of the mass parameter of the couple of primaries is derived.