Effects of asteroid belt and oblateness on the stability and orbits near triangular Lagrange points

Abstract

This study investigates the dynamics of an infinitesimal body within the context of the restricted three-body problem, incorporating perturbations arising from the asteroid belt and the oblateness effects of the three bodies. The locations of the triangular Lagrange points are analytically derived, yielding results consistent with the classical case when perturbations are neglected. Furthermore, a detailed analysis of the relationship between various perturbations and the positions of the triangular Lagrange points is conducted, revealing that the horizontal positions are exclusively determined by the mass ratio $\mu$, whereas the vertical positions are influenced by all considered perturbations, including the asteroid belt and oblateness effects. Subsequently, the stability of the triangular Lagrange points is examined, with a focus on the critical mass ratio ${\mu }_c$, concluding that the triangular Lagrange points remain stable only under the condition $\mu <{\mu }_c$. Additionally, we calculate the second- and third-order periodic orbits near the triangular Lagrange point $L_4$ using the Lindstedt–Poincaré method. Finally, numerical simulations are performed to validate the analytical results, demonstrating that an increase in the mass of the asteroid belt, as well as a greater oblateness of the primaries and the third body, leads to a more noticeable divergence between the second- and third-order periodic orbits. These findings provide deeper insights into the complex dynamics of celestial systems under the influence of multiple perturbations.