In the framework of multi-body dynamics, successive encounters with a third body, even if well outside of its
sphere of influence, can noticeably alter the trajectory of a spacecraft. Examples of these effects have already been
exploited by past missions such as SMART-1, as well as are proposed to benefit future missions to Jupiter, Saturn or
Neptune, and disposal strategies from Earth’s High Eccentric or Libration Point Orbits. This paper revises three
totally different descriptions of the effects of the third body gravitational perturbation. These are the averaged
dynamics of the classical third body perturbing function, the Öpik’s close encounter theory and the Keplerian map
approach. The first two techniques have respectively been applied to the cases of a spacecraft either always
remaining very far or occasionally experiencing extremely close approaches to the third body. However, the paper
also seeks solutions for trajectories that undergo one or more close approaches at distances in the order of the sphere
of influence of the third body. The paper attempts to gain insight into the accuracy of these different perturbative
techniques into each of these scenarios, as compared with the motion in the Circular Restricted Three Body Problem