Deformation and Gravity Potential Change Inside the Earth Caused by Tidal Force of Celestial Body

Abstract

Tidal Love numbers based on the Wang’s Earth model was obtained by solving simultaneous first-order partial differential equations （PDEs） with Runge-Kutta 4th order method considering the frequency dependency of the tidal force. The PDEs are basically according to Alterman et al. （1959） and partly according to Takeuchi & Saito （1972）, and the boundary conditions are improved in order to be applied for the case of small angular velocities, especially between solids and fluids layers. The initial values at the center of the solid core are obtained by converted from the three independent solutions U, V, P calculated based on the method of Pekeris & Jarosh （1958）. The obtained results are considered to be good even when compared with those by the conventional methods, and there is almost no difference between the effects of frequency difference in the range between 6 and 24 hours, where the dominant tidal constituents exist. The power of the Runge-Kutta 4th order method was clear, and good results were obtained even the number of steps in each layer was reduced to 14 to 30. In addition, the number of processes of the integration from the center to the surface of the earth was one enough. The three independent vectors, （0,1,0,0,0,0）, （0,0,0,1,0,0）, （0,0,0,0,0,1） used as the initial values for the Runge-Kutta method were proved to be good enough since almost the appropriate results were obtained in any cycle. It was confirmed that the determinant is almost zero at the periods 6.656, 17.617, and 19.886 hours, and the sign changed across the periods. These zero crossings seem to correspond some eigen periods of such as the fluid core, but since a stable solution was obtained at a distance from these periods, it was not necessary to divide the integration process to dynamic and static ones. Finally, comparison between the results obtained by our method and those by the method of Pekeris & Jarosh （1958） for a simple uniform earth model was made.