Numerical study of the time required for the gravitational capture in the bi-circular four-body problem

A gravitational capture occurs when a spacecraft (or any particle with negligible mass) changes from a hyperbolic orbit with a small positive energy around a celestial body into an elliptic orbit with a small negative energy without the use of any propulsive system. The force responsible for this modification in the orbit of the spacecraft is the gravitational force of the third and the fourth bodies involved in the dynamics. In this way, those forces are used as a zero cost control, equivalent to a continuous thrust applied in the spacecraft. One of the most important applications of this property is the construction of trajectories to the Moon to minimize fuel consumption. The concept of gravitational capture is used, together with the basic ideas of the gravity-assisted maneuver and the bi-elliptic transfer orbit, to generate a trajectory that requires fuel consumption smaller than the one required by the Hohmann transfer. The objective of the present paper is to study the time required for 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.