The evolution of a satellite motion in the gravitational field of a viscoelastic planet with a core

We investigate the motion of a satellite in the gravitational field of a massive deformable planet. Planet is modeled as body that consists of a solid core and a viscoelastic shell of a Kelvin–Voigt material. The satellite is modeled as a point mass. The system of integro-differential equations for a motion of a mechanical system is got out from the variational principle of the d'Alembert–Lagrange according to the linear theory of elasticity. Approximate equations of motion in vector are constructed with asymptotic method of motions separation. This system of equations describes the dynamics of the «planet-satellite» with regard to the perturbations caused by elasticity and dissipation. To describe the evolution of the orbital parameters of a satellite, averaged differential equations were derived.
Phase trajectories were constructed for particular cases, their stationary solutions were found and investigated on stability. In the case of the existence of two stationary orbits stationary solution that corresponding to the motion along the orbit of larger radius is asymptotically stable, and the orbit of smaller radius is unstable. Some of the planets in the solar system and their satellites are considered as examples. This problem is a model for the study of the tidal theory of planetary motion.