A mathematical model of the gravitational potential of the planet taking into account tidal deformations

Abstract

Objectives. This paper investigates the gravitational potential of a viscoelastic planet moving in the gravitational field of a massive attracting center (star), a satellite and one or more other planets moving in Keplerian elliptical orbits relative to the attracting center. Celestial bodies other than a viscoelastic planet are modeled by material points. Within the framework of the linear model of the theory of viscoelasticity, the problem of finding the vector of elastic displacement has been resolved. Traditionally, a solid body model is used to determine the Earth’s gravitational field, while tidal deformations are taken into account in the form of small corrections to the coefficients of the geopotential model. In this work, the viscoelastic ball model is used to take into account tidal effects. The relevance of the research topic is associated with high-precision forecasting of the movement of artificial satellites of the Earth, high-precision measurement of the Earth’s gravitational field.Methods. In this study the asymptotic and analytical methods developed by V.G. Vilke are used for mechanical systems containing viscoelastic elements of high rigidity, as well as methods of classical mechanics, mathematical analysis. The graphs were plotted using the Octave mathematical package.Results. After resolving the quasi-static problem of elasticity theory by calculating triple integrals over a spherical area, a formula for the gravitational potential of a deformable planet was obtained. In addition, the gravitational potential of the Earth was also calculated taking into account solid-state tidal effects from the Moon, Sun, and Venus at an external point. Graphs were constructed to show the dependence of the Earth’s gravitational potential on time.Conclusions. The theoretical and numerical results established herein show that the main contribution to the gravitational potential of the Earth is made by the Moon and the Sun. The influence of other planets in the solar system is small. The value of the gravitational potential at the outer point of the Earth, taking into account tidal effects, depends both on the position of the point in the moving coordinate system and on the relative position of celestial bodies.