Satellite motion in the gravitational field of a viscoelastic planet with a core

The motion of a "planet-satellite" system in a gravitational field of mutual attraction forces is investigated. The planet is modeled by a body consisting of a solid core and a viscoelastic shell made of Kelvin-Voigt material. The satellite is modeled by a material point. A system of integro-differential equations of motion of a mechanical system is derived from the variational d'Alembert-Lagrange principle within the linear model of the theory of elasticity. Using the asymptotic method of separation of motions, an approximate system of equations of motion is constructed in vector form. This system describes the dynamics of a system with allowance for disturbances caused by elasticity and dissipation. The solution of the quasistatic problem of elasticity theory for the deformable shell of a planet is obtained in the explicit form. An averaged system of differential equations describing the evolution of satellite's orbital parameters is derived. For partial cases phase trajectories are constructed, stationary solutions are found, and their stability is investigated. As examples, some planets of the Solar system and their satellites are considered. This problem is a model for studying the tidal theory of planetary motion.