Dynamical equations of the restricted three-body problem with variable masses and zonal harmonics

Abstract

In this paper, the derivations of the equations of motion of a massless particle in the frame of the restricted three-body problem (R3BP) with variable masses and zonal harmonics, is explored. The motion and mass variations of the primaries are governed by the Gylden-Mestschersky problem (GMP) and the unified Mestschersky law (UML), respectively, with further assumptions that the shapes of the primaries are varying oblateness with zonal harmonics coefficients up to J₄ terms. The non-autonomous differential equations of the particle in a rotating frame of reference are derived by adopting the Lagrangian method and transmuted to a system with constants coefficients using the Mestschersky transformation, the UML, the particular integral and solutions of the GMP and additionally, a transformation we deduced for the varying oblateness with zonal harmonics up to J₄ term. The autonomized system depends on the oblateness of the primaries with zonal harmonics coefficients up to J₄, the mass parameter and the mass variation parameter. The derivations of the dynamical equations will pave way for more explorations of the dynamical predictions of a test particle in the gravitational environment of oblate spheroidal binary under zonal harmonics coefficients up to J₄. This will in no doubt expand the knowledge base of celestial mechanics and space missions.