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Spherical harmonic series for derivatives of all orders of the gravitational potential of a planet and their application in satellite geodesy and space navigation

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Series of spherical harmonics are constructed for derivatives of all orders of the gravitational potential of an arbitrary three-dimensional body, including the Earth, Moon and other planets. These series have a common structure, as simple as the potential itself. They differ from each other and from the series for the potential only by numerical coefficients of the spherical functions, by the degree of a numerical multiplier of the sum of double series, and by the limits of summation. The constructed series can be applied in solving many problems of celestial mechanics, satellite geodesy, and space navigation.

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References

  1. Kaula, W.M., Theory of Satellite Geodesy, Waltham, MA: Blaisdell, 1966. Translated under the title Sputnikovaya geodeziya: teoreticheskiye osnovy, Moscow: Mir, 1970.

    Google Scholar 

  2. Reed, G.B., Application of Kinematical Geodesy for Determining the Short Wave Length Components of the Gravity Field by Satellite Gradiometry, Ohio State University, Dept. of Geod. Sciences, Rep. no. 201. Columbus, Ohio, 1973.

  3. Belikov, M.V., Fully Numerical Experiment to Reveal Efficiency of Gradiometry, Preprint of S.-Petersburg Inst. of Theor. Astronomy, Ruus. Acad. Sci., St.-Petersburg, 1996, no. 52.

  4. Koop, R. and Stelpstra, D., On the Computation of the Gravitational Potential and Its First and Second Order Derivatives, Manuscripta Geodaetica, 1989, vol. 14, p. 373.

    Google Scholar 

  5. Balmino, G., Barriot, J-P., Koop, R., et al., Simulation of Gravity Gradients: A Comparison Study, Bulletin Geodesique, 1991, vol. 5, no. 4, pp. 218–229.

    Article  ADS  Google Scholar 

  6. Cunningham, L., On the Computation of the Spherical Harmonic Terms Needed during the Numerical Integration of the Orbital Motion of an Artificial Satellite, Celestial Mechanics, 1970, vol. 2, pp. 207–216.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Metris, G., Xu, J., and Wytrzyszczak, I., Derivatives of the Gravity Potential with Respect to Rectangular Coordinates, Celestial Mechanics and Dynamical Astronomy, 1999, vol. 71, pp. 137–151.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Fantino, E. and Casotto, S., Methods of Harmonic Synthesis for Global Geopotential Models and Their First-, Second- and Third-Order Gradients, Journal of Geodesy, 2009, vol. 83, no. 7, pp. 595–619.

    Article  ADS  Google Scholar 

  9. Montenbruck, O. and Gill, E., Satellite Orbits-Models, Methods, and Applications, Berlin: Springer, 2000.

    MATH  Google Scholar 

  10. Arsov, K. and Pail, R., Assessment of Two Methods for Gravity Field Recovery from GOCE GPS-SST Orbit Solutions, Advances in Geosciences, 2003, vol. 1, pp. 121–126.

    Article  ADS  Google Scholar 

  11. Pavlis, N.K., Holmes, S.A., Kenyon, S.C., and Factor, J.K., An Earth Gravitational Model to Degree 2160: EGM2008 (2008), General Assembly of the European Geosciences Union, Vienna, Austria, April 2008, pp. 13–18.

  12. Konopliv, A.S., LP-L-RSS-5-GRAVITY-V1.0, NASA Planetary Data System, 1999.

  13. Zuber, M., Mars Reconnaissance Orbiter Derived Gravity Data. MRO-M-RSS-5-SDP-V1.0, NASA Planetary Data System, 2008.

  14. Sjogren, W.L., MGN-V-RSS-5-GRAVITY-L2-V1.0, NASA Planetary Data System, 1997.

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Authors and Affiliations

  1. Central (Pulkovo) Astronomical Observatory of the Russian Academy of Sciences, Pulkovskoe Shosse 65/1, 196140, St.-Petersburg, Russia

    M. S. Petrovskaya & A. N. Vershkov

Authors
  1. M. S. Petrovskaya
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  2. A. N. Vershkov
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Correspondence to M. S. Petrovskaya.

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Original Russian Text © M.S. Petrovskaya, A.N. Vershkov, 2012, published in Kosmicheskie Issledovaniya, 2012, Vol. 50, No. 2, pp. 158–165.

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Petrovskaya, M.S., Vershkov, A.N. Spherical harmonic series for derivatives of all orders of the gravitational potential of a planet and their application in satellite geodesy and space navigation. Cosmic Res 50, 152–159 (2012). https://doi.org/10.1134/S001095251201008X

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  • DOI: https://doi.org/10.1134/S001095251201008X

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