The motion of a satellite in the gravitational field of a viscoelastic planet

The motion of a satellite in the gravitational field of a massive deformable planet is investigated. The planet is modelled by a homogeneous isotropic viscoelastic body of spherical shape in the natural undeformed state, while the satellite is modelled by a point mass. A potential energy functional of elastic deformations is introduced in accordance with the classical theory of elasticity of small deformations, while the dissipative force functional corresponds to the Kelvin-Voigt model. A system of integro-differential equations of motion of the system is derived from the d’Alembert–Lagrange variational principle. Assuming that the stiffness of the viscoelastic planet is high, a small parameter is introduced, inversely proportional to Young's modulus, and an approximate system of ordinary differential equations is constructed in vector form by the method of separation of motions, describing the translational-rotational motion of the planet-satellite system, when perturbations due to elasticity and dissipation are taken into account. The system of equations obtained has a stationary solution, corresponding to the motion of the satellite in a circular orbit in a plane orthogonal to the constant vector, in which the number of stationary orbits cannot exceed two. It is shown that, in the case when two stationary orbits exist, the stationary solution, corresponding to the motion in an orbit of greater radius, is asymptotically stable, while in an orbit of smaller radius it is unstable. An evolutionary system of equations of motion of the satellite in Delaunay variables is obtained, which describes the change in the orbit parameters. Averaging was carried out over the “fast” angular variable – the mean anomaly.