Researching the influence of the mass distribution inhomogeneity of the ellipsoidal planet’s interior on its stokes constants
Received: March 27, 2019
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. Parameters of Earth’s gravitational field ( ) are determinated by its figure and internal filling (mass distribution) that have a different influence on their formation. Using a well-known representation of the planet masses distribution functions in the biorthogonal series form it is necessary to establish the Stokes constants  presentation through the planet potential expansion coefficients  and liner combinations of ellipsoid geometric parameters. Based on these formulas, it is the objective to investigate the possible influence of the inhomogeneity of the mass distribution function of the Earth’s interior and the representation of its shape with an ellipsoid of rotation onto the values of the Stokes constants and to explore the contribution of the radial distribution of the Earth’s mass density to these constants. Methodology. The presentation of the planet's interior density function as a sum of the Legendre polynomials of three variables and the approximation of its surface by an ellipsoid, as well as the representation of internal spherical functions in a rectangular coordinate system, makes it possible to integrate expressions for Stokes constant    and obtain the relation between these values of different orders and the linear combination of the planet potential expansion coefficients  and geometric parameters of ellipsoid .  Numerical data obtained from the derived relationships and the constructed graphs make it possible to analyze the influence of the inhomogeneity of the mass’s interior distribution of an ellipsoidal planet onto the value of the Stokes constants and determine the intervals of maximum impact. Results. The general relations between the expansion coefficients  of the distribution function and the integrals from spherical functions on an ellipsoidal surface that determine Stokes constants of a definite order are established. Herewith Stokes constants of n order are expressed in terms of values ,  of lower orders. The presented calculations give a procedure for the formation of Stokes constant values, which clearly implies the conclusion about the small effect of the planet’s ellipsoidal form on the magnitude and three-dimensionality of the Earth’s gravitational field as a result of the inhomogeneous of its interior masses distribution. Also known dependence of the values  on the geometric compression of the biaxial Earth ellipsoid of constant density is confirmed. Scientific novelty.  The formulas for the relation between Stokes constants of different orders and linear combinations of parameters   are determined.  The calculations and verification of the obtained relations for different sets of potential expansion coefficients  allow us to conclude that the three-dimensional gravity field of the Earth predominantly contributes to the Stokes constants, except , and the constructed graphs determine its maximum contribution to the mass distribution in depth. Practical significance. The obtained dependences allow us to check the approximation degree of the constructed density model of ellipsoidal planet by comparing Stokes constants which are calculated using model and are obtained from the observations. In addition, it is possible to optimally reconcile the geometric characteristics of the planet’s ellipsoid with its gravitational field.

1. Abrikosov, O. (1986). On computation of a derivatives of the Earth's gravitational potential for satellite geodesy and geodynamics. Kinematics and Physics of Celestial Bodies, 2(14), is.2, 51-58. (in Russian).
2. Antonov, B., Timoshkova, Ye, & Kholshevnikov, K. (1988). Introduction to the theory of Newtonian potential. Сhief editor of physical and mathematical literature, 272 p. (in Russian).
3. Bateman, H., & Erdelyi, A. (1953). Higher transcendental functions. MC Graw-hill Book Company, inc.
4. Chernyaga, P. G., & Fys, M. M. (2012). A new approach to the use of Stokes constants for the construction of functions and its derivatives of mass distribution of planets. Collection of scientific works of Western geodesic society UTGK "Modern achievements in geodetic science and production". II (24), 40-43. (in Ukrainian).
5. Cunningham, L. (1970). On the computation of the spherical harmonic terms needed during the numerical integration of the orbital motion of an artificial satellite. Celestial Mechanics and Dynamical Astronomy, 2, 207-216.
6. DeWitt, R. (1962). Derivatives of Expressions Describing the Gravitational Field of the Earth. U.S. Naval Weapons Laboratory, Defense Technical Information Center.
7. Dzewonski, A., & Anderson, D. (1981). Preliminary reference Earth model. Physics of the earth and planetary interiors, 25(4), 297-356. doi: 10.1016/0031-9201(81)90046-7.
8. Fys, M. (1982). On the calculation of the model Stokes constant of the Earth, corresponding to the representation of its density by the partial sum of the generalized Fourier series. Geodesy, cartography and aerial photography, 36, 103-107. (in Russian).
9. Fys, M., Zazuliak, P., & Zajats', O. (2004). On the question of determining spherical functions in a general planetary coordinate system Collection of scientific works of Western geodesic society UTGK "Modern achievements in geodetic science and production". I (7), 401-408. (in Ukrainian).
10. Hobson, Е. (1953). The theory of spherical and ellipsoidal. Foreign literature publishing house, 476 p. (in Russian).
11. Kholshevnikov, K., Milanov, D. & Shaidulin, V. (2017). Stokes constants of an oblate ellipsoid of revolution with equidensites homothetic to its surface. Vestnik SPbSU. Mathematics. Mechanics. Astronomy. 4 (62), issue 3, 516-524. doi: 10.21638/11701/spbu01.2017.313 (in Russian).
12. Kholshevnikov, K., & Shaidulin, V. (2015). Existence of a class of irregular bodies with a higher convergence rate of Laplace series for the gravitational potential. Celestial Mechanics and Dynamical Astronomy. 122(4), 391-403.
13. Meshcheriakov, G. (1991). Problems of potential theory and generalized Earth. M: Science, Сhief editor of physical and mathematical literature, 216 p. (in Russian).
14. Ostach, O., & Ageeva, I. (1982). Approximation of the Earth's external gravitational field by a model of gravitating point masses. Proceedings of the III Orel Conference "Studying the Earth as a planet using astronomy, geophysics and geodesy". Кyiv: Naukova dumka. (in Russian).
15. Pavlis N.K., Holmes S.A., Kenyon S.C. & et al. (2008). An Earth Gravitational Model to Degree 2160: EGM2008. EGU General Assembly. Geophysical Research Abstracts. Vol. 10, 2. (EGU2008- A-01891).
16. Shabat, B. (1976). Introduction to the complex analysis. - Мoscow: Nauka, 720 p. (in Russian).
17. Tarakanov, Yu, & Cherevko, Yu. (1979). Interpretation of the largest gravitational anomalies of the Earth. Izvestiya of the Academy of Sciences of the USSR. Physics of the Solid Earth, 4, 25-42. (in Russian).
18. Tarakanov, Yu. & Karagioz, O. (2012). Inverse problem of the planets' gravitational field as a physical problem. Geophisical journal. 34(1), 32-49. (in Russian).
19. Vinnik, L., Lukk, L., & Mirzokurbonov, M. (1978). Sources of the largest geoid undulations from seismic and gravity data. Reports of the USSR Academy of Sciences. 241(4), 789-793. (in Russian)