The main equations for determining the harmonic expansion coefficients of the gravitational potential of the Earth according to the GOCE satellite

: pp. 149 - 154
Received: April 03, 2013
Accepted: October 23, 2013
Lopushansky A.M.

The last achievements of the physical geodesy is the project of European Space Agency – the GOCE satellite (Gravity of field and steady – state of Ocean Circulation Explorer) which uses the satellite gradientometry method. The gravitational field of the Earth is represented traditionally as a series of spherical harmonic functions, i.e. to model the gravitational field of the finite number of parameters, the so–called coefficients Cnm, Snm. In this paper we analyze the basic relationships for the determination of these coefficients.


  1. Casotto S, Fantino E (2009) Gravitational gradients by tensor analysis with application to spherical coordinates. Journal of Geodesy. Vol. 83, 621–634.
  2. Eötvös L. (1896) Studies in the field of gravity and magnetics. In: “Three fundamental papers of Loránd Eötvös”, Transl. from Hungarian, ELGI Budapest, 1998, p. 83–125.
  3. Fantino E, Casotto S (2009) Methods of harmonic synthesis for global geopotential models and their first–, second– and third–order gradients. Journal of Geodesy. Vol. 83, 595–619.
  4. GOCE Level 2 Product Data Handbook.
  5. Lundberg J.B., Schutz B.E. (1988) Recursion formulas for Legendre functions for use with nonsingular geopotential models. J. Guidance Control Dyn. 11(1), 31–38.
  6. Marchenko A.N. (1998) Parameterization of the Earth’s Gravity Field: Point and Line Singularities. Published by Lviv Astronomical and Geodetic Society. Lviv, Ukraine, 1998, 210 p. 8. Гофман–Велленгоф Б., Мориц Г. Физическая геодезия. – М.: МииГАиК, 2007.– С. 285–286.