Motion in the ER3BP Under a Heterogeneous Triaxial Primary and a Modified Newtonian Force of the Secondary

This paper investigates the dynamical behaviour of an infinitesimal mass body moving under the gravitational influence of a heterogeneous triaxial primary body and a secondary body that produces a modified Newtonian gravitational force in the elliptic restricted three-body problem. In this model, the two primary bodies move around their common center of mass in elliptical orbits, while the mass of the third body is assumed to be sufficiently small so that its effect on the motion of the primary bodies can be neglected. The equations of motion of the infinitesimal body are formulated, and the corresponding mean motion of the system is determined by taking into account the triaxial structure and mass heterogeneity of the primary body as well as the modified gravitational field of the secondary. The equilibrium points of the system are then investigated. Both collinear equilibrium points, which lie along the line joining the two primary bodies, and non-collinear equilibrium points, which are located away from this line, are determined analytically. Their linear stability is examined by studying the characteristic equations associated with small perturbations around these points. In addition, the dynamical characteristics of the system are illustrated numerically through potential surfaces, the locations of equilibrium points, permissible regions of motion, periodic orbits, and basins of attraction. These numerical results provide a clear understanding of how the eccentricity of the motion, triaxiality of the primary body, and the modified gravitational force of the secondary influence the motion of the infinitesimal body. The results may be useful in understanding the dynamics of celestial bodies.