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.

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.

This study explores the orbital behaviour surrounding out-of-plane equilibrium points (OEPs) within the circular restricted three-body problem (CR3BP) framework, with a particular emphasis on binary star systems where the primary stars are represented as oblate and radiating entities. The research centres on the stability (Lyapunov-wise) of two pairs of OEPs, and, respectively, which are influenced by the oblateness and radiation pressure coefficients of the primary stars. By applying the theoretical framework to five specific binary systems—Lalande 21258, BD+195116, Ross 614, 70 Ophiuchi, and 61 Cygni—we assess the stability properties of these equilibrium points. Our findings indicate that the OEPs exhibit instability across all five systems, as evidenced by the positive real parts of the complex roots linked to their perturbations. This instability implies that any perturbations will amplify over time, resulting in significant deviations from the equilibrium states. The implications of this research are significant for the design of satellite constellations and the planning of space missions, as a thorough understanding of the stability of these equilibrium points is essential for successful mission execution and orbital insertion strategies. This work contributes to the wider domain of celestial mechanics by deepening our comprehension of dynamical behaviours in intricate binary systems.

In this study, the generalized restricted three-body problem (R3BP) is applied to the EQ Pegasi binary system to investigate the effects of zonal harmonics on the motion of a satellite. Building on previous research, it enhances the standard potential function to represent the oblateness of the primaries by adding higher-order zonal harmonic terms and for both the primary and secondary bodies. After establishing the equations regulating motion, the collinear equilibrium points (CEPs) are found numerically using the Newton-Raphson technique. The Lyapunov stability theorem is used to assess their stability. According to the results, the addition of zonal harmonics somewhat changes the locations of the CEPs, with the equatorial bulge having the most significant impact. The Jacobian constants indicate the energy levels at these moments, showing slightvariations across different scenarios. Even after adding higher-order zonal harmonics, stability evaluations show that the CEPs are still unstable, which aligns with classical R3BP findings. The necessity of active station-keeping in satellite mission planning, particularly close to these locations, is highlighted by this ongoing volatility. The study highlights the importance of considering higher-order gravitational effects while designing missions and analyzing stability, particularly for Nigerian spacecraft operating in complicated gravitational fields. The findings underscore the importance of advanced modeling in addressing the complex dynamics of binary star systems, thereby supporting earlier studies and offering valuable insights for future space missions.