Solar and Lunar Tides in Magma

Abstract

A planet consisting of an ideal incompressible fluid under the action of natural gravitation and attraction of celestial bodies is considered. The inhomogeneity and the presence of a solid core are modeled by a point mass concentrated at the center of the homogeneous planet. A formula is obtained that expresses the mass of the core in terms of the coefficient of flattening of the planet and the angular velocity of its rotation about its axis. The height of tides caused by the action of the Sun and satellites is determined. For the Earth, the numerical values have the same orders of magnitude as the height of tidal waves in the ocean. The initial assumption is valid if the energy of elastic deformation due to tidal forces is much smaller than the kinetic energy of the relative tidal motion of the planetary mass. This is shown to be typical of giant planets and, to a lesser degree, of the Earth. Exact Dirichlet, Riemann, and Ovsyannikov solutions and matrix calculus are used.