Comparisons between the circular restricted three-body and bi-circular four body problems for transfers between the two smaller primaries

Important properties of the dynamics of a spacecraft can be obtained from the Circular Restricted Three Body Problem and the Bi-Circular Bi-planar Four Body Problem. In this work, both systems are compared under the perspective of the costs involved in a transfer between the smaller primaries. An analytical approach shows several properties of the perturbation due to the gravity of the Sun and the motion of the smaller primaries around it over a spacecraft in the region of interest, like its behavior at and around the barycenter or at any point in a circle around the Sun. The costs involved in transfers between the smaller primaries are numerically evaluated and analyzed using the newly developed Theory of Functional Connections. The results show that the influence of this perturbation over the costs is significant for systems like the Sun–Earth–Moon or Sun–Mars–Phobos. On the other hand, it is also shown that this influence may be negligible for other very different systems, like the Sun–Saturn–Titan or Sun–Ida–Dactyl. Maps of perturbation are drawn in the region of interest, which can be used for mission designers. Finally, a new approach to describe the influence of the Sun over the tides of the smaller primaries is proposed under the Four Body Problem model.