EXACT SOLUTIONS OF EQUATIONS OF THE TWO-PRIMARY-BODY PROBLEM IN THE RESTRICTED THREE-BODY PROBLEM WITH VARIABLE MASSES

Abstract

This work examines the translational-rotational motion of a non-stationary, axisymmetric body of variable mass in the Newtonian gravitational field of two primary spherical bodies with variable masses, formulated within the framework of the restricted three-body problem with variable masses in a barycentric coordinate system. The masses of the bodies change isotropically over time, so no reactive forces or moments arise. The small axisymmetric body may change its size and shape while remaining axially symmetric throughout the process. The restricted formulation implies that the small body does not influence the motion of the two primary spherical bodies with variable masses. The study focuses on the secular perturbations of translational-rotational motion in the considered three-body system. Since the exact solutions for the translational-rotational motion of the two primary spherical bodies with variable masses in the barycentric coordinate system are unknown, the differential equations of the two-body problem and those of the non-stationary small body are investigated jointly. Due to the complexity of the problem, the translational-rotational motion of the three-body system is studied using perturbation theory in analogues of Delaunay-Andoyer variables. Exact analytical solutions of the differential equations for the secular perturbations of the translational-rotational motion in the problem of two primary spherical bodies in terms of Delaunay-Andoyer variable analogues are obtained. These exact solutions open the possibility of further investigating the secular perturbations of the translational-rotational motion of a non-stationary, axisymmetric body within the restricted three-body problem with variable masses.