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We consider a modification of the restricted three-body problem where the primary (more massive body) is a triaxial rigid body and the secondary (less massive body) is an oblate spheroid and study periodic motions around the collinear equilibrium points. The locations of these points are first determined for 10 combinations of the parameters of the problem. In all 10 cases, the collinear equilibrium points are found to be unstable, as in the classical problem, and the Lyapunov periodic orbits around them have been computed accurately by applying known corrector–predictor algorithms. An extensive study on the families of three-dimensional periodic orbits emanating from these points has also been done. To find suitable starting points, for all the computed families, semianalytical solutions have been obtained, for both two- and three-dimensional cases, around the collinear equilibrium points using the Lindstedt–Poincaré method. Finally, the stability of all computed periodic orbits has been studied.

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An investigation of three-dimensional periodic orbits and their stability emanating from the collinear equilibrium points of the restricted three-body problem with oblate and radiating primaries is presented. A simulation is done by using five binary systems: Sirius, Procyon, Luhman 16, α-Centuari and Luyten 726-8. Firstly, based on the topological degree theory, the total number of the collinear equilibrium points for the five binary systems were obtained and then, their positions were determined numerically. The linear stability of these equilibrium points was also examined and found to be unstable in the Lyapunov sense. An analytical approximation of three-dimensional periodic solutions around them was established via the Lindstedt-Poincaré local analysis. Finally, using the analytical solution to obtain starting orbits, the families of three-dimensional periodic orbits emanating from these equilibria have been continued numerically.