On the Solution of the Exterior Boundary Value Problem with the Aid of Series

The exterior gravitational field depending on the Earth’s non-sphericity is usually determined from the analysis of satellite data or by the solution of the exterior boundary value problem. In the latter case some integral equations are solved which correlate the exterior potential with the known vector of gravity and the shape of the Earth’s surface (molodensky problem). In order to carry out the integration the small parameter method is applied. As a result, all the quantities which involve the equations should be expanded in powers of a certain small parameter, among these being the heights of the Earth’s surface points as well as the inclination α of the Earth’s physical surface. Since the angle α can be significant, especially in mountains, and in fact does not depend on any small parameter then the solution of integral equations is possible only for the Earth’s surface which is smoothed enough.