Method for approximate construction of three-dimensional mass distribution function and gradient of an elipsoidal planet based on external gravitational field parameters
Received: February 03, 2018
Revised: November 07, 2018
Accepted: December 28, 2018
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University

Purpose. To investigate the technique for constructing a three-dimensional distribution function for the masses of the interior of the Earth and its derivatives, coordinated with the parameters of the planet's gravitational field to fourth order inclusive. By using the mass distribution function constructed, to make an interpretation of the features of the internal structure of an ellipsoidal planet. Methodology. Based on the created initial approximation of the function, which includes a reference density model, further refinements are built. Using Stokes constants up to the second order inclusive, we give the following approximation, which we subsequently take as zero. In this case, the use of Stokes constants up to the fourth order inclusive leads to the solution of systems of equations. It is established that the addition of one identity leads to uniqueness of the solution. One system with Stokes constants  is an exception. It is necessary to note that the computation process is controllable, since the power moments of the density derivatives are reduced to quantities that take into account the value of the density on the surface of the ellipsoid. Results. In contrast to the second-order model describing gross global inhomogeneities, the obtained distribution function gives a detailed picture of the location of the density anomalies (the deviation of the three-dimensional function from the averaged over the sphere is "isodense.") Analysis of maps at different depths 2891 km (core-mantle), 5150 km (internal-external core) allows us to draw preliminary conclusions about the global mass redistribution due to the rotating component of the force of gravity over the entire radius, as well as due to the horizontal components of the density gradient. On the contrary, the minimum of such a deviation is observed in the polar parts of the Earth, which also has its explanation: the magnitude of the rotational force decreases when approaching the pole. The mass distribution function is constructed using the proposed method to describe in more detail the picture of the mass distribution. Of particular interest are sketch maps of the components of the density anomaly function gradient, namely the component which coincides with the axis  - for the upper part of the shell which is negative, and for the lower part it is positive. This means that the gradient vector is directed toward the centre of mass. The nature of the values for other two components is different both in sign and in magnitude and depends on the placement point. The cumulative consideration and consideration of all the quantities makes possible a more complete interpretation of the processes inside the Earth. Originality.  In contrast to the traditional approach, the changes for the density derivatives of one variable (depth), obtained from the Adams-Williams equation, in this paper made an attempt to obtain derivatives with Cartesian coordinates. Used in the described method, the parameters of the gravitational field up to the fourth order inclusively increases the order of approximation of the mass distribution function of three variables from two to six, and its derivatives up to five. In this case, unlike the traditional method, the defining here is the construction of the derivatives, from which the mass distribution function and the use of geophysical information accumulated in the referential PREM model are reproduced. Practical significance. The resulting mass distribution function of the Earth can be used as the next approximation when using Stokes constants of higher orders in the presented algorithm. Its application makes it possible to interpret global anomalies of the gravitational field and to study geodynamic processes deeply inside the Earth.

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