On Robe’s restricted problem with a modified Newtonian potential

We analyze the existence of equilibrium points for a particle or dust grain in the framework of unperturbed and perturbed Robe’s motion. This particle is moving in a spherical nebula consisting of a homogeneous incompressible fluid, which is considered as the primary body. The second primary body creates the modified Newtonian potential. The perturbed mean motion and equations of motion are found. The equilibrium points (i.e. collinear, noncollinear and out–of–plane points), along with the required conditions of their existence are also analyzed. We emphasize that this analysis can be used to study the oscillations of the Earth’s core under the attraction of the Moon and it is also applicable to study the motion of underwater vehicles.