Celestial Mechanics and Dynamical Astronomy

, Volume 93, Issue 1–4, pp 113–146 | Cite as

Analytical Solutions of Love Numbers for a Hydrostatic Ellipsoidal Incompressible Homogeneous Earth

  • Marianne Greff-Lefftz
  • Laurent Métivier
  • Hilaire Legros


Tidal forces acting on the Earth cause deformations and mass redistribution inside the planet involving surface motions and variation in the gravity field, which may be observed in geodetic experiments. Because for space geodesy it is now necessary to achieve the mm level in tidal displacements, we take into account the hydrostatic flattening of the Earth in the computation of the elasto-gravitational deformations. Analytical solutions are derived for the semi-diurnal tides on a slightly elliptical homogeneous incompressible elastic model. That simple analytical Earth’s model is not a realistic representation of any real planet, but it is useful to understand the physics of the problem and also to check numerical procedures. We rediscover and discuss the Love’s solutions and obtain new analytical solutions for the tangential displacement. We extend these analytical results to some geodetic responses of the Earth to tidal forces such as the perturbation of the surface gravity field, the tilt and the deviation of the vertical with reference to the Earth’s axis.


body tides elasto-gravitational deformations Love numbers 


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  1. Alterman, Z., Jarosch, H., Pekeris, C. H. 1959‘Oscillation of the Earth’Proc. R. Soc. London.A2528095ADSGoogle Scholar
  2. Buffett, B. A., Mathews, P. M., Herring, T. A., Shapiro, I.I. 1993‘Forced nutations of the Earth: contributions from the effects of ellipticity and rotation on the elastic deformations’J. Geophys. Res.982165921676ADSGoogle Scholar
  3. Dahlen, F. A. 1968‘The normal modes of a rotating, elliptical earth’Geophys. J. Roy. Astron. Soc.16329367MATHGoogle Scholar
  4. Dahlen, F. A. 1976‘Reply’J. Geophys. Res.8149514956ADSGoogle Scholar
  5. Dahlen, F. A., Tromp, J. 1998Theoretical Global SeismologyPrinceton University PressNew JerseyGoogle Scholar
  6. Dehant, V. 1987‘Integration of the gravitational motion equations for an elliptical uniformly rotating earth with an inelastic mantle’Phys. Earth Planet. Inter.49242258ADSGoogle Scholar
  7. Dehant, V. 1991‘Review of the Earth tidal models and contribution of Earth tides in Geodynamics’J. Geophys. Res.962023520240ADSCrossRefGoogle Scholar
  8. Dehant, V.: 1995, ‘Report of the Working Group on theoretical tidal model’, Proc. 12th Int. Symp. on Earth tides, Science Press, Beijing China, 17–18.Google Scholar
  9. Dehant, V., Defraigne, P., Wahr, J. 1999‘Tides for a convective Earth’J. Geophys. Res.10410351058CrossRefADSGoogle Scholar
  10. Dziewonski, A. M., Anderson, D. L. 1981‘Preliminary Reference Earth Model PREM’Phys. Earth Planet. Int.25297356ADSGoogle Scholar
  11. Heiskanen, W., Moritz, H. 1967Physical GeodesyW. H. Freeman and CompanySan Fransisco, LondonGoogle Scholar
  12. Love, A. E. H. 1911Some Problems of GeodynamicsDoverNew YorkMATHGoogle Scholar
  13. Mathews, P. M., Buffett, B. A., Shapiro, I. I. 1995‘Love numbers for a rotating spheroidal Earth: new definitions and numerical values’Geophys. Res. Let.22579582CrossRefADSGoogle Scholar
  14. Melchior, P. 1966The Earth tidesPergamon PressOxford-London-Edinburgh-New York-Paris-FrankfurtGoogle Scholar
  15. Métivier, L., Greff-Lefftz, M., Diament, M. 2005‘A new approach to compute accurate gravity variations for a realistic Earth model with lateral variations’Geophys. J. Int.62570574Google Scholar
  16. Smith, M. 1974‘The scalar equations of infinitesimal elastic-gravitational motion for a rotating, slightly elliptical Earth’Geophys. J. Int. Astr. Soc.37491526MATHADSGoogle Scholar
  17. Wahr, J. M. 1981‘Body tides on an elliptical, rotating, elastic and oceanless Earth’Geophys. J. R. Astr. Soc.64677701ADSMATHGoogle Scholar
  18. Wahr, J. M., Bergen, Z. 1986‘The effects of mantle anelasticity on nutations, Earth tides, and tidal variations in rotation rate’Geophys. J. R. Astron. Soc.646336681981.ADSGoogle Scholar
  19. Wang, R. 1994‘Effect of rotation and ellipticity on Earth tides’Geophys. J. Int.117562565ADSGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Marianne Greff-Lefftz
    • 1
  • Laurent Métivier
    • 1
  • Hilaire Legros
    • 2
  1. 1.Institut de Physique du Globe de ParisParis 05France
  2. 2.E.O.S.T.StrasbourgFrance

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