The dynamical equations of a test particle in the restricted three-body problem with a triaxial primary and variable masses

Abstract

The restricted three-body problem (R3BP) defines the motion of an infinitesimal mass moving under the gravitational attractions of two main masses, called primaries. In this paper, we examine the dynamical equations of a test body in the frame of the circular R3BP. Both primaries are assumed to vary their masses in accordance with the Unified Mestschersky law (UML) and their motion determined by the Gylden-Mestschersky problem (GMP). Further, the first primary is assumed to be a triaxial variable mass body. The potential between the primaries is deduced and in furtherance, the non-autonomous equations of motion of the model are derived. The derived equations are time varying and are thus transformed to the autonomized forms with constant coefficients using the Mestschersky transformation (MT), the UML, the particular integral and solutions of the GMP. We also use a transformation we introduced, which helps in converting the time dependent triaxiality of the bigger primary to one that is constant. The derived systems of equations with variable and constant coefficients can be used to model the long-term motion of satellites and planets in binary systems.