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Consistent post-Newtonian theory for nutation/precession in a realistic Earth model

Due to currently achievable VLBI accuracies, present nutation/precession theories have to model the relevant angles with a precision at a level of 1 μas or better. This implies that the various sub-systems of the Earth like mantle, outer and inner core, atmosphere, ocean, continental hydrology and their complex interactions have to be taken into account. At this level of accuracy also effects from relativity can no longer be neglected. A consistent theory of nutation/precession has to be therefore formulated within the post-Newtonian framework of Einstein’s theory of gravity.

The goal of the project is such a consistent post-Newtonian theory for nutation/precession in the framework of a realistic Earth model. To achieve this goal the project is subdivided into two stages. In the first three years the post-Newtonian model of rigidly rotating multipoles will be worked out for a rigid Earth. In the second stage this model will be extended towards an non-rigid basic three layered Earth with mantle, fluid outer core and solid inner core.

The work on the rigid Earth was successfully finished in the first phase of project P3, where dynamical equations for Euler-angles in the GCRS have been derived and numerically integrated (see Klioner et al., 2007). The underlying model considers post-Newtonian multipole moments and principle moments of inertia, that are assumed to rotate rigidly in the GCRS. The relevant post-Newtonian torques are calculated numerically using cartesian symmetric and trace-free tensors. We have demonstrated that our numerical code can reproduce SMART97 in the Newtonian limit within the full accuracy of the latter. In the post-Newtonian limit we showed that the currently used corrections for the largest relativistic effect (geodetic precession/nutation) were applied incorrectly up to now. Using consistent dynamical equations our post-Newtonian theory represents therefore the best theory for the rigid Earth at the moment. For further details see Klioner et al. (2010).

Currently our work of the first phase of funding is extended towards a realistic Earth model, where relevant subsystems like atmosphere, ocean and continental hydrology will be taken into account. Here the Newtonian approximation will be used where it can be justified and a post-Newtonian treatment is superfluous (e.g., for the treatment of atmosphere, ocean, hydrosphere, core-mantle coupling etc.).

Involved institutions:

Scientists / Staff:

  • Prof. Dr. Michael Soffel, Institut für Planetare Geodäsie IPG, Technical University Dresden
    Tel.: +49 351 463 34200, E-Mail: soffel (at)
  • Prof. Dr. Sergei Klioner, Institut für Planetare Geodäsie IPG, Technical University Dresden
    Tel.: +49 351 463 32821, E-Mail: klioner (at)
  • Enrico Gerlach, Institut für Planetare Geodäsie IPG, Technical University Dresden
    Tel.: +49 351 463 32050, E-Mail: enrico.gerlach (at)


  • Klioner, S.A., Gerlach, E., Soffel, M. (2010): Relativistic aspects of rotational motion of celestial bodies. In: Relativity in Fundamental Astronomy, Proc. of the IAU Symposium 261, S. Klioner, K. Seidelmann, M. Soffel (eds.) Cambridge University Press, Cambridge, 112-123.
  • Soffel, M., Klioner, S, Müller, J., Biskupek, L.: Gravito-Magnetism and LLR. Physical Review D, submitted 2008.
  • Klioner, S., Soffel, M., Le Poncin-Lafitte, C. (2007): Towards a relativistic theory of precession and nutation. In: N.Capitaine (ed.), Proc. of Les Journees
    2007, Systemes de reference spatio-temporels, 139-142.
  • Soffel, M., Klioner, S., (2007): Relativistic aspects of Earth’s rotation, Highlights of Astronomy 14, 469.
  • Müller, J., M. Soffel, & S. Klioner (2007): Geodesy and Relativity. Journal of Geodesy 81, 13 p., doi 10.1007/s00190-007-0168-7.

Oral presentations:

  • M. Soffel: Eine neue post-Newtonsche Theorie eines starren Erdmodells. Statusseminar DFG Forschergruppe FOR 584 “Erdrotation und globale dynamische Prozesse”, Wettzell, 12. – 14.03.2008.
  • S.Klioner: Post-Newtonian theory of Earth rotation, Seminaire ‘Temps et Espace’, Paris, 3 December 2007.
  • S.Klioner, M.Soffel, C.LePoncin Lafitte: Earth rotation and relativity: first results of dynamical modelling, Sternberg State Astronomical Institute, Moscow State University, Moscow; 25 October 2007.
  • S.Klioner, M.Soffel, C.LePoncin Lafitte: Towards a relativistic theory of precession and nutation, Les Journees 2007, 18 September 2007.
  • S.Klioner, M.Soffel, C.LePoncin Lafitte: Theory of Earth Rotation: post-Newtonian theory for rigidly rotating multipoles; Meeting of the DFG research unit (FOR 584), Lohrmann-observatory Dresden, 30 May 2007.

Historical publications relevant to the project:

  • Müller, J., Kutterer, H., Soffel, M. (2005): Earth Rotation and global dynamic processes – joint research activities in Germany. In Fundamental Astronomy: New concepts and models for high accuracy observations. Proceedings of the Journees Systemes de reference spatio-temporels (ed. by N.Capitaine), Paris 2004, P. 121-125.
  • Soffel, M., Klioner, S. (2004): Relativity in the problems of Earth rotation and astronomical reference systems: status and prospects. In: N.Capitaine (ed.), Proc. Of Les Journees 2004, 191-195.
  • Klioner, S., Soffel, M., Xu, Ch., Wu, X. (2001): Earth rotation in the framework of general relativity: rigid multipole moments. In: N.Capitaine (ed.), Proc. Of Les Journees 2001, 232-238; also available from arXiv as astro-ph/0303376.
  • Soffel, M., Klioner, S. (1998): Nutation, Transfer Functions and Relativity: Relativistic Nutation of a Non-Rigid Earth. In: N.Capitaine (ed.), Proc. Of Les Journees 1998, 79-85.
  • Klioner, S., Soffel, M. (1998): Rotational Motion of Celestial Bodies in the Relativistic Framework. In: Ferraz-Mello, J.Henrard (eds.), Proc. Of the IAU Colloquium 172, Kluwer, Dordrecht, 435-436.
  • Klioner, S., Soffel, M. (1997): Relativistic Considerations for Precession and Nutation. In: J.Anderson (ed.), Highlights of Astronomy, 11A, 173-176.
  • Klioner, S. (1996): Angular velocity of extended bodies in general relativity. In: S.Ferraz-Mello, B.Morando, J.Arlot (eds.), Dynamics, ephemerides and astrometry in the solar system, Kluwer, Dordrecht, 309-320.
  • Klioner, S. (1995): Rotation of Deformable Bodies in General Relativity, Proc. Of the 20th General Assembly of the European Geophysical Society, Annales Geophysicae, Suppl. to Volume 13, C34.

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Modification date: 20 Dec 2010

© 2007 - Bundesamt für Kartographie und Geodäsie. Alle Rechte vorbehalten.
Gefördert durch Deutsche Forschungsgemeinschaft (FOR 584).