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THE LEVEL SURFACE OF THE GENERALIZED PROBLEM OF TWO CENTERS

Prof. Dr. Valentin Gushchin, Assist. Prof. Dr. Adam Khasanov

Abstract

In this paper the properties of level surface of potential of gravity arising in the generalized problem of two centers are investigated. Interest in this research is related to the desire to use the opportunity of an analytical approximation of the EarthпїЅs potential through the potential arising when solving the generalized problem of two centers (intermediate EarthпїЅs potential) to modeling the Earth's surface. Further, for a such approximation the level surface of potential of gravity arising in the generalized problem of two centers is used. Based on this approximation, the analytical describing is proposed for the shape of the Earth's surface and the focus is on the study of the properties of this describing. A symmetric case is considered. In Lemmas 1, 2 and in Theorem 1, the smoothness of the selected representation of the investigated surface is proved. A rigorous justification of the necessary and sufficient conditions of the existence of a smooth level surface of gravity in the generalized problem of two centers is given in terms of the parameter space. In Lemmas 3, 4 and in Theorem 2 strict convexity of the level surface is proved on the subset of the parameter space. The relative position of the ellipsoid and the level surface is analytically studied for various parameters of the problem.

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DOI resolver: doi.org/10.5593/sgem2017/22/s09.085

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