Impact of Triaxiality and Mass Variations on Motion around Triangular Equilibrium Points of the Restricted Three-body Problem

Abstract

This paper investigates the impact of triaxiality and mass variations of the primaries on motion around the triangular equilibrium points (TEPs) of the restricted three-body problem (R3BP). The motion of the primaries takes place within the framework of the Gylden–Mestschersky problem while their mass variation occur in accordance with the unified Mestschersky law and the bigger primary is a triaxial body. The dynamical equations of the time-dependent system are derived and transformed to a system of equations with constant coefficients. The solutions of the equations with constant coefficients were explored, and it was seen that there exists analytically a pair of TEPs. The stability of the TEPs is investigated and it is seen that the points can be stable and unstable, depending on the variable mass parameters, critical mass parameters, triaxiality of the bigger primary and the constant \(\kappa \) of mass variation. The region of stability increases and decreases depending on the stabilizing or destabilizing behaviors of the triaxiality of the bigger primary and the variable mass parameter. The stability of the TEPs of the time-dependent system are unstable as solutions do not converge but tend to infinity with time. Numerically, we explore the study for the general case of a particle in the gravitational field of the bigger triaxial primary and a smaller spherical body using the mass parameters which covers for most astronomical systems to demonstrate the applications of our problem in astronomy. It is seen that there can be finitely many pairs of TEPs for variable mass parameters, and the region of stability of the TEPs for different mass parameters was explored. Further, the zero velocity curves unveiled more interesting scenarios that could not have been observed analytically. It is observed that an increasing mass parameter of the binary gives a larger region where motion of the particle is allowed and vice versa, while the departure of the bigger primary to a triaxial body always reduces region of allowed motion and the variation constant can increase or decrease region where motion is allowed. The study can be applied to provide insights into the dynamics of an infinitesimal mass around variable mass primaries coupled with variations in shape.