Multifrequency model of Earth tidal deformations within framework of three-body problem

Abstract

The Earth tidal deformation is studied. Earth is considered as a visco-elastic axially symmetrical body with a solid core. For the description of the visco-elastic Earth part the linear theory of visco-elasticity is used. It is assumed that the Earth - Moon center of mass moves on the constant elliptic orbit around the Sun, that is considered as attractive center. From the d’Alembert - Lagrange principle the system of motion equations is obtained. Then the displacement vector is expanded in series of Earth’s oscillation modes. The system of ordinary differential equations for modal variables is obtained. This system describes process of oscillations, but in real situation we can consider these deformations as quasi-stationary ones. Hence, we can discard inertial members in equations. We can expand right parts of equations in series by orbit’s eccentricities of the Earth - Moon center of mass and of the Earth (Moon). Then we can look, that the right parts of the equations are the series with coefficients depending of angles combinations, hence modal variables can be represented also as the similar series. From these conditions we can find approximately of Sun - Moon’s tidal frequencies. We find well known frequencies, corresponding periods as year, half year, month, half month, day, half day, frequencies combinations, near above mentioned and large quantity of other combinations.