Periodic Solutions Around the Out-of-Plane Equilibrium Points in the Restricted Three-Body Problem with Radiation and Angular Velocity Variation

In the present work, we study the motion of an infinitesimal body near the out-of-plane equilibrium points of the restricted three-body problem in which the angular velocity of the two primary bodies is considered in the case where both of them are sources of radiation. Firstly, these equilibria are determined numerically, and then the influence of the system parameters on their positions is examined. Due to the symmetry of the problem, these points appear in pairs and, depending on the parameter values, their number may be zero, two, or four. The linear stability of the out-of-plane equilibrium points is also studied, and it is found that there are cases where they can be stable. In addition, periodic motion around them is investigated both analytically and numerically. Specifically, the Lindstedt–Poincaré method is used in order to obtain a second order analytical solution, while the families emanating from the out-of-plane equilibrium points are finally computed numerically either in case where the corresponding equilibrium points are stable or unstable. For the numerical computation of a three-dimensional periodic orbit, we apply known unconstrained optimization methods to an objective function that is formed by the respective periodicity conditions that have to be fulfilled.