The transformation of the second-degree harmonic coefficients and in the case of a finite commutative rotation was derived instead of the traditional Lambeck’s approach based on an infinitesimal rotation. The modified Lambeck’s formulae avoid uncertainty in the deviatoric part of inertia tensor and allow simple transformation of the 2nd-degree harmonic coefficients and zonal coefficients of an arbitrary degree (including their temporal changes) via orthogonal matrixes. These formulae together with exact solution of the eigenvalue-eigenvector problem are applied to determine static components and accuracy of the Earth’s tensor of inertia from the adjustment in the principal axes system of , from recent four gravity field models (EGM2008, GGM03S, ITG-GRACE03S, and EIGEN-GL04S1) and eight values HD of the dynamical ellipticity all reduced to the common MHB2000 precession constant at the epoch J2000. The second solution contains the same parameters based on these four sets of and only one HD from the MHB2000 model and corresponds better to the IERS Conventions 2003 and latest gravity field determinations. Two solutions for static components consist of the adjusted five 2nd-degree harmonic coefficients related to the IERS reference pole given by the conventional mean pole coordinates at the epoch 2000 (IERS Conventions 2003), the orientation of principal axes in this system, the principal moments (A, B, C) of inertia, and other associated parameters. The evolution with time of the above-mentioned static parameters was estimated in the principal axes system from the GRACE time series of , derived in five different centers of analysis over the time interval from 2002 to 2008. Special attention is given to the direct computation of temporally varying principal axes and moments of inertia based on , and the estimation of their mean values together with periodic constituents on given time-period. Stability of the positions of the equatorial inertia axes (, ) and the angle between two quadrupole axes located in the plane of the axes and of inertia is found. The estimated longitude of the principal axis as the parameter of the Earth’s triaxiality in the precession-nutation theory and precession rate of the precession constant are recommended for the Earth’s rotation theory. Additionally to some permanent constituents periodic components at seasonal and shorter time scale were evaluated.
- 1. Bretagnon P., Francou G., Rocher P., Simon J.L. (1998). SMART97: A new solution for the rotation of the rigid Earth. Astronomy & Astrophysics, 329, pp. 329-338.
2. Bourda G., Capitaine N. (2004). Precession, nutation, and space geodetic determination of the Earth's variable gravity field. Astronomy & Astrophysics, 428, pp. 691-702.
https://doi.org/10.1051/0004-6361:20041533
3. Bursa M. (1995). Primary and Derived Parameters of Common Relevance of Astronomy, Geodesy, and Geodynamics. Earth, Moon and Planets, 69, pp. 51-63.
https://doi.org/10.1007/BF00627769
4. Capitaine N., Wallace P.T., Chapront J. (2003). Expressions for IAU 2000 precession quan-tities. Astronomy & Astrophysics, 412, pp. 567-586.
https://doi.org/10.1051/0004-6361:20031539
5. Chen J.L., Wilson C.R., Tapley B.D. (2005). Inter-annual Variability of Low-degree Gravitational Change, 1980-2002. Journal of Geodesy, 78, pp. 535-543.
https://doi.org/10.1007/s00190-004-0417-y
6. Dehant V., Arias .F, Bizouard Ch., Bretagnon P., Brzezinski A., Buffett B., Capitaine N., Defraigne P., DeViron O., Feissel M., Flie-gel H., Forte A., Gambis D., Getino J., Gross R., Herring T., Kinoshita H., Klioner S., Mathews P.M., McCarthy D., Moisson X., Petrov S., Ponte R.M., Roosbeek F., Sal-stein .D, Schuh H., Seidelmann K., Soffel M., Souchay J., Vondrak J., Wahr J., Wallace P., Weber R., Williams J., Yatskiv Y., Zharov V., Zhu S.Y., (1999). Considerations concerning the non-rigid Earth nutation theory. Celestial Mechanics and Dynamical Astronomy, 72, pp. 245-309.
https://doi.org/10.1023/A:1008364926215
7. Fernández L.I. (2007). Analysis of Geophysical Variations of the C20 Coefficient of the Geopotential. In: Sideris M (ed.). Observing our Changing Earth, Proceedings of the 2007 IAG General Assembly, Perugia, Italy, July 2-13, 2007. International Association of Geodesy Symposia, Vol. 133, pp. 493-500, Springer, 2009.
https://doi.org/10.1007/978-3-540-85426-5_59
8. Förste C., Schmidt R., Stubenvoll R., Flechtner F., Meyer U., König R., Neumayer H., Bianca-le R., Lemoine J.-M., Bruinsma S., Loyer S., Barthelmes F., Esselborn S. (2008). The GeoForschungsZentrum Potsdam / Groupe de Recherche de Gèodésie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGEN-GL04C. Journal of Geodesy, Vol. 82, pp. 331-346.
https://doi.org/10.1007/s00190-007-0183-8
9. Gross R.S., Lavallée D.A., Blewitt G. and Clar-ke P.J. (2007). Consistency of Earth Rotation, Gravity, and Shape Measurements. In: Sideris M. (ed.) Observing our Changing Earth, Proceedings of the 2007 IAG General Assembly, Perugia, Italy, July 2-13, 2007. International Association of Geodesy Sym-posia, Vol. 133, pp. 463-471, Springer, 2009
https://doi.org/10.1007/978-3-540-85426-5_56
10. Groten E. (2000). Parameters of Common Relevan-ce of Astronomy, Geodesy, and Geodynamics. Journal of Geodesy, 74, pp. 134-140.
https://doi.org/10.1007/s00190-000-0134-0
11. Groten E. (2004). Fundamental parameters and current (2004) best estimates of the parameters of common relevance to astronomy, geodesy and geodynamics. Journal of Geodesy, Vol. 77, pp. 724-731.
https://doi.org/10.1007/s00190-003-0373-y
12. Hartmann T., Soffel M., Ron C. (1999). The geophysical approach towards the nutation of a rigid Earth, Astronomy & Astrophysics, Supplement Series, 134, pp. 271-286.
https://doi.org/10.1051/aas:1999139
13. Lambeck K. (1971). Determination of the Earth's pole of rotation from laser range observations to satellites. Bulletin Géodésique, 101, pp. 263-281.
https://doi.org/10.1007/BF02521878
14. Lemoine J.M., Bruinsma S., Loyer S., Biancale R., Marty J.-C., Perosanz F., Balmino G. (2007). Temporal gravity field models inferred from GRACE data. Advances in Space Research, Vol. 39, No. 10, pp. 1620-1629.
https://doi.org/10.1016/j.asr.2007.03.062
15. Krasinsky G. (2006). Numerical theory of rotation of the deformable Earth with the two-layer fluid core. Part 1: Mathematical model. Celes-tial Mechanics and Dynamical Astronomy, vol. 96, pp. 169-217.
https://doi.org/10.1007/s10569-006-9038-5
16. Madelund E. (1957). Die Mathematischen Hilf-smittel des Physikers, Berlin, Gottingen, Heidelberg, Springer-Verlag.
https://doi.org/10.1007/978-3-662-30168-5
17. Marchenko A.N. (1979). The gravitational qua-drupole of a planet. Soviet Astronomy Letters, vol. 5, pp. 106-107.
18. Marchenko A.N. (1998). Parameterization of the Earth's gravity field. Point and line singu-larities. Lviv Astronomical and Geodetic So-ciety. Lviv.
19. Marchenko A.N. (2003). A note on the eigenvalue - eigenvector problem. In: Festschrift dedi-cated to Helmut Moritz on the occasion of his 70th birthday. Ed.: N. Kühtreiber, Graz Uni-versity of Technology, pp. 143-154, Graz 2003.
20. Marchenko A.N., Schwintzer P. (2001). Principal axes and principal moments of inertia from recent satellite gravity field solutions. In: San-so F., Sideris M. (eds.). Vistas for Geodesy in the New Millennium, Proceedings of the IAG 2001 Scientific Assembly, 2-7 September, 2001, Budapest, Hungary, International Association of Geodesy Symposia, Vol. 125, pp. 138-143, Springer, 2002
https://doi.org/10.1007/978-3-662-04709-5_23
21. Marchenko A.N., Schwintzer P. (2003). Estimation of the Earth's tensor of inertia from recent global gravity field solutions. Journal of Geodesy, Vol. 76, p. 495-509, 2003.
https://doi.org/10.1007/s00190-002-0280-7
22. Marchenko A.N. (2007). Current estimation of the Earth's mechanical and geometrical para-meters. In: Sideris M. (ed.). Observing our Changing Earth, Proceedings of the 2007 IAG General Assembly, Perugia, Italy, July 2-13, 2007. International Association of Geodesy Symposia, Vol. 133, pp. 473-481, Springer, 2009.
https://doi.org/10.1007/978-3-540-85426-5_57
23. Mayer-Guerr T. (2007). ITG-Grace03s: The latest GRACE gravity field solution computed in Bonn, Paper presented at the Joint International GSTM and DFG SPP Symposium, 15-17 Oct 2007, Potsdam.
24. Mayer-Guerr T. (2008). Privat communication.
25. Mathews P.M. (2000). Improved models for precession and nutation. In: Proc. of IAU Colloquium 180 "Towards Models and Con-stants for Sub-Microarcsecond Astrometry", Naval Observatory, Washington, 27-30 March, pp. 212-222.
https://doi.org/10.1017/S0252921100000312
26. Mathews P.M., Herring T.A., Buffet B.A. (2002). Modeling of nutation-precession: New nutation series for nonrigid Earth, and insights into the Earth's interior, J. Geophys. Res., Vol. 107, No. B4, 10.1029/2001JB000390.
https://doi.org/10.1029/2001JB000390
27. McCarthy D., Petit G. (2004). IERS Conventions (2003), IERS Technical Note No.32, Verlag des Bundesamts fur Kartographie und Geodasie, Frankfurt am Main, 2004.
28. Rapp R.H. (1989): The treatment of permanent tidal effects in the analysis of satellite altimeter data for sea surface topography. Manuscripta geodaetica, 14, pp. 368-372
29. Melchior P. (1978). The tides of the planet Earth. Pergamon.
30. Pavlis N.K. (2008). Privat communication.
31. Pavlis N.K., Holmes S.A., Kenyon S.C., Factor J.K. (2008). An Earth Gravitational Model to Degree 2160: EGM2008. Geophysical Research Abstracts, Vol. 10, EGU2008-A-01891, 2008, EGU General Assembly 2008 (Paper presented at the EGU General Assembly 2008, Vienna, Austria, April, 2008).
https://doi.org/10.1190/1.3063757
32. Reigber Ch. (1981). Representation of orbital element variations and force function with respect to various reference systems. Bulletin Géodésique, 55, pp. 111-131.
https://doi.org/10.1007/BF02530834
33. Reigber Ch., Jochmann H., Wünsch J., Neuma-yer K.-H., Schwintzer P. (2003). First Insight into Temporal Gravity Variability from CHAMP. In: Reigber Ch., Lühr H., Schwintzer P. (eds.). First CHAMP Mission Results for Gravity, Magnetic and Atmospheric Studies. Springer-Verlag Berlin-Heidelberg, pp. 128-133, 2003.
https://doi.org/10.1007/978-3-540-38366-6_19
34. Rochester M.G., Smylie D.E. (1974). On changes in the trace of the Earth's inertia tensor. Journal of Geophysical Research, 79 (32), pp. 4948-4951
https://doi.org/10.1029/JB079i032p04948
35. Roosbeek F., Dehant V. (1998). RDAN97: An analytical development of rigid Earth nutation series using the torque approach. Celestial Mechanics and Dynamical Astronomy, 70, pp. 215-253
https://doi.org/10.1023/A:1008350710849
36. Souchay J., Kinoshita H. (1996). Corrections and new developments in rigid Earth nutation theory: I. Lunisolar influence including indirect planetary effects, Astronomy and Astrophysics, 312, pp. 1017-1030
37. Tapley B., Ries J., Bettadpur S., Chambers D., Cheng M., Condi F., Poole S. (2007). The GGM03 Mean Earth Gravity Model from GRACE. Eos Trans AGU 88(52), Fall Meet Suppl, Abstract G42A-03.
38. Williams J.G. (1994). Contributions to the Earth's obliquity rate, precession and nutation, The Astronomical Journal, 108, pp. 711-724.
https://doi.org/10.1086/117108
39. Yoder C.F., Williams J.G., Dickey J.O., Schutz B.E., Eanes R.J., Tapley B.D. (1983). Secular variation of Earth's gravitational harmonic J2 coefficient from Lageos and nontidal acceleration of Earth rotation. Nature, 303, pp. 757-762.
https://doi.org/10.1038/303757a0