The Use of Ellipsoidal Harmonics for the Representation of the Geopotential

  • H. G. Walter
Conference paper
Part of the COSPAR-IAU-IAG/IUGG-IUTAM book series (IUTAM)


The potential of the Earth is developed in ellipsoidal harmonics, and the mathematical tools required, which are the generation of Lame’s functions and the relationship between rectangular and ellipsoidal coordinates, are compiled. Brief reference is made to a procedure utilized for the determination of the gravity coefficients in the expansion ot the geopotential in ellipsoidal harmonics when precise satellite tracking data is available.

With the aim of carrying out a numerical integration of the Lagrangian equations of planetary motion, the functional dependence of the disturbing earth potential on the orbital elements for elliptic motion is given. In particular, formulae for the partial derivatives of the disturbing potential with respect to the orbital elements are derived, thus making possible the numerical calculation of these partial derivatives from orbital elements.


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  1. Anderle, R. J.: Computational methods employed with Doppler observations and derivation of geodetic results, in Iutam Symposium Trajectories of Artificial Celestial Bodies (pp. 178–193). Paris, Apr. 1965. Berlin/Heidelberg/New York: Springer 1966.Google Scholar
  2. Brouwer, D., Clémence, G. M.: Methods of celestial mechanics, New York: Academic Press 1961.Google Scholar
  3. Doubochine, G. N.: Sur le développement du potentiel de la terre par les fonctions de Lamé, in Iutam Symposium Trajectories of Artificial Celestial Bodies, Paris Apr. 1965, Berlin/Heidelberg/New York: Springer 1966.Google Scholar
  4. Hagihara, Y.: Recommendations on notation of the earth potential. Astron. J. 67 (1962) 108.CrossRefGoogle Scholar
  5. Heine, E.: Handbuch der Kugelfunctionen, Würzburg: Physica Verlag 1961.Google Scholar
  6. Hobson, E. W.: The Theory of Spherical and Ellipsoidal Harmonics, New York: Chelsea Publishing Company 1955.Google Scholar
  7. Hotine, M.: Downward Continuation of the Gravitational Potential (paper presented at the International Association of Geodesy, Lucerne, Switzerland, 1967).Google Scholar
  8. Iszak, I. G.: Tesseral Harmonics of the Geopotential and Corrections to Station Coordinates. J. Geophys. Res. 69(1964) 2621–2630.CrossRefGoogle Scholar
  9. Kaula, W. M.: Analysis of Gravitational and Geometric Aspects of Geodetic Utilization of Satellites. Geophys. J. 5 (1961) 104–133.zbMATHCrossRefGoogle Scholar
  10. King-Hele, D. G., Cook, G. E., Rees, J. M.: Determination of the Even Harmonics in the Earth’s Gravitational Potential. Geophys. J. 8 (1963) 119–145.zbMATHCrossRefGoogle Scholar
  11. King-Hele, D. G., Cook, G. E., Scott, D. W.: Evaluation of Odd Zonal Harmonics in the Geopotential, of Degree less than 33, from the Analysis of 22 Satellite Orbits. Royal Aircraft Establishment, Tech. Rep. No. 68202, 1968.Google Scholar
  12. Kozai, Y.: Improved Values for Coefficients of Zonal Spherical Harmonics in the Geopotential, in Geodetic Satellite Results during 1967, edited by C. A. Lund-quist. Smith. Astrophys. Obs., Special Report No. 264 (1967) 43–56.Google Scholar
  13. Lense, J.: Kugelfunktionen, p. 2, Leipzig: Akademische Verlagsgesellschaft 1955.Google Scholar
  14. Rutishauer, H.: Ausdehnung des Rombergschen Prinzips. Numerische Mathematik 5 (1963) 48–54.MathSciNetCrossRefGoogle Scholar
  15. Walter, H. G.: The Association of Gravity Coefficients in the Expansion of the Geopotential in Spherical and Ellipsoidal Harmonics (prov. title), publication in preparation.Google Scholar

Copyright information

© Springer Verlag, Berlin/Heidelberg 1970

Authors and Affiliations

  • H. G. Walter
    • 1
  1. 1.European Space Operations Centre DarmstadtGermany

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