1. Investigation of the short periodic terms of the rigid and non-rigid Earth rotation series V.V. Pashkevich Central (Pulkovo) Astronomical Observatory of the Russian Academy of Science St.Petersburg Space Research Centre of the Polish Academy of Sciences Warszawa 2008

2. Investigation of the short periodic terms (diurnal and sub- diurnal) in a long time new high-precision series of the rigid (S9000A) and non-rigid Earth rotation (SN9000A). Choice of the optimal spectral analysis scheme for this investigation. Method 1.To update the previous version of the high-precision numerical solution of the rigid Earth rotation (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”) by using the integration step of 1 day and the interpolation step of 0.1 day. All calculations have been carried out with the quadruple precision (Real 16) on PC Intel® Core™2 Duo E6600 (2.4GHz). We considered only the Kinematical (relativistic) case. Aims:

3. Method (continued) 2. The initial conditions have been calculated from the model SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were expressed in Euler angles over 2000 years with 0.1 day spacing. 3.Investigation of the discrepancies has been carried out by the least squares (LSQ) and by the spectral analysis (SA) algorithms (V.V.Pashkevich and G.I.Eroshkin, Proc. “Journees 2005”). 4.The high-precision rigid Earth rotation series S9000 (V.V.Pashkevich and G.I.Eroshkin, 2005 ) has been update by including the short periodic terms (diurnal and sub-diurnal). 5. The new high-precision non-rigid Earth rotation series (SN9000A) has been constructed. This series is expressed in the function of the three Euler angles and containing the short periodic terms. We applied the method of P.Bretagnon et al. (1999) and the transfer function of P. M. Mathews et al. (2002).

4. Part I: The rigid Earth rotation

5. Fig.1 Numerical solution for the rigid Earth rotation minus solution SMART97 in the longitude of the ascending node of the Earth equator. the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case Secular terms of… Secular terms of…  smart97 (  as)d  (  as)  smart97 (  as)d  (  as) 7.00 9.51 50384564881.3693 T- 206.50 T50384564881.3693 T - 211.16 T - 107194853.5817 T 2 - 3451.30 T 2 - 107194853.5817 T 2 - 3481.61 T 2 - 1143646.1500 T 3 1125.00 T 3 - 1143646.1500 T 3 1142.76 T 3 1328317.7356 T 4 - 788.00 T 4 1328317.7356 T 4 - 712.75 T 4 - 9396.2895 T 5 - 57.50 T 5 - 9396.2895 T 5 - 71.86 T 5 - 3415.00 T 6 - 3464.88 T 6

6. Fig.2 Numerical solution for the rigid Earth rotation minus solution SMART97 in the angle of the proper rotation of the Earth. Secular terms of… Secular terms of…  smart97 (  as)d  (  as)  smart97 (  as)d  (  as) 1009658226149.36916.58 1009658226149.36918.90 474660027824506304.0000 T99598.30 T474660027824506304.0000 T -236.95 T - 98437693.3264 T 2 - 7182.30 T 2 - 98437693.3264 T 2 -7235.36 T 2 - 1217008.3291 T 3 1066.80 T 3 - 1217008.3291 T 3 524.24 T 3 1409526.4062 T 4 - 750.00 T 4 1409526.4062 T 4 -684.84 T 4 - 9175.8967 T 5 - 30.30 T 5 - 9175.8967 T 5 -40.93 T 5 - 3676.00 T 6 -3717.78 T 6 the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case

7. Fig.3 Numerical solution for the rigid Earth rotation minus solution SMART97 in the inclination angle. Secular terms of… Secular terms of…  smart97(  as)d  (  as)  smart97(  as)d  (  as) 84381409000.0000 1.4284381409000.00001.59 - 265011.2586 T - 96.61 T - 265011.2586 T- 97.64 T 5127634.2488 T 2 - 353.10 T 2 5127634.2488 T 2 - 355.10 T 2 - 7727159.4229 T 3 771.50 T 3 - 7727159.4229 T 3 775.68 T 3 - 4916.7335 T 4 - 84.50 T 4 - 4916.7335 T 4 - 79.67 T 4 33292.5474 T 5 - 86.00 T 5 33292.5474 T 5 - 89.51 T 5 - 247.50 T 6 -250.65 T 6 the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case

8. Fig.4. Differences between the numerical solutions and SMART97 after the formal removal of secular trends. the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case

9. D after removal of the secular trend LSQ method computes the amplitudes of the spectrum of D for the all periodical terms LSQ method determines the amplitude and phase of the largest from the residual harmonics if |Am| > |  | till the end of the spectrum Set of nutation terms of SMART97 Removal of this harmonic from the D and from the Spectrum YesNo Spectral Analysis (SA) A L G O R I T H M – 1 for removal of the periodical terms from the Discrepancies (D) … Construction of the new nutation series New high- precision series S9000A1

10. D after removal of the secular trend LSQ method computes the amplitudes of the spectrum of D for the long periodical terms LSQ method determines the amplitude and phase of the largest from the residual harmonics if |Am| > |  | till the end of the spectrum Set of nutation terms of SMART97 Removal of this harmonic from the D and from the Spectrum YesNo Spectral Analysis (SA) A L G O R I T H M – 2 for removal of the periodical terms from the Discrepancies (D) … Construction of the new nutation series Construction of the new D-2 and apply the algorithm - 1 New high- precision series S9000A

11. Fig.5. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle. for only long periodical terms for all periodical terms ALGORITHM – 1 the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case

12. Fig.5. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle. DETAIL for only long periodical terms for all periodical terms ALGORITHM – 1 the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case

13. for ALGORITHM – 1 Fig.6. The numerical solution - 2 NS-2 (NS-2A1) of the rigid Earth rotation minus S9000 (S9000A1) after formal removal of the secular trends in the proper rotation angle.

14. Fig.7. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle. for only long periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 1 1-iteration The personal computer Intel® Core™2 Duo case

15. Fig.7. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle. DETAIL The personal computer Intel® Core™2 Duo case for only long periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 1 1-iteration

16. Fig.8. The numerical solution – 2 NS-2A2 (NS-2A1 ) of the rigid Earth rotation minus S9000A2 (S9000A1) after formal removal of the secular trends in the proper rotation angle. The personal computer Intel® Core™2 Duo case The end of 1-iteration of the ALGORITHM – 2 (for only long periodical terms) ALGORITHM – 1 ( for all periodical terms) The end of the

17. for all periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 1 2-iteration fig.7 Fig.9. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle. The personal computer Intel® Core™2 Duo case

18. Fig.9. Spectra of the discrepancies between numerical solutions and SMART97 for the proper rotation angle. DETAIL The personal computer Intel® Core™2 Duo case for all periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 1 2-iteration fig.7

19. Fig.10. The numerical solution – 3(2) NS-3(NS-2A1) of the rigid Earth rotation minus S9000A (S9000A1) after formal removal of the secular trends in the proper rotation angle. The personal computer Intel® Core™2 Duo case for all periodical terms ALGORITHM – 2 for all periodical terms ALGORITHM – 1 2-iteration fig.8

20. Part II: The non-rigid Earth rotation

21. TRANSFER FUNCTION P.M. Mathews, T.A. Herring, B.A. Buffett, 2002: Geophisical model includes the following effects: electromagnetic coupling, ocean tides, mantle inelasticity, atmospheric influence, change in the global Earth dynamical flattening and in the core flattening

22. Expressions for Euler angles:

23. Algorithm of Bretagnon et al. (1999) 1.The rigid Earth angular velocity vector: 3.The derivatives of Euler angles for the non-rigid Earth rotation: 2.The non-rigid Earth angular velocity vector is obtained by:

24. Cascade method: Common form of the periodic part of Euler angles:

25. 1-th iteration Details: Iterative solution of the algorithm of Bretagnon et al. (1999) (Pashkevich, 2008)

26. n-th iteration Iterations are repeated until the absolute value of the difference between iterations K-1 and K exceedes some a priori adopted values  Details: Iterative solution of the algorithm of Bretagnon et al. 1999 (Pashkevich, 2008)

27. Period (days), Argument Solution rigid Earth rotation Ψ(sin) μas Ψ(cos) μas Solution non-rigid Earth rotation Ψ(sin) μas Ψ(cos) μas 1.03505 λ 3 +D-  SMART97 S9000A -34.8213 4.2714 SMN SN9000A -33.6604 1.5930 1.03474 3λ 3 +3D-2F-  SMART97 S9000A -0.0133 -0.0806 0.0016 -0.0612 SMN SN9000A -0.0225 -0.0898 -0.0100 -0.0728 1.00000 λ 3 -  SMART97 S9000A -0.1996 0.0275 SMN SN9000A -0.1969 0.0005 0.99758 λ 3 +D-l-  SMART97 S9000A -19.8544 -19.8521 2.4906 2.4941 SMN SN9000A -19.8316 -19.8293 -1.0929 -1.0894 0.96215 λ 3 +D+  SMART97 S9000A 38.1283 4.6945 SMN SN9000A 36.3795 2.1620 0.527517 3λ 3 +3D-2  SMART97 S9000A -0.1783 0.2578 SMN SN9000A -0.3211 0.4389 1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB. Results: Table 1. Comparison of different solutions for selected durnal and semidurnal periodical terms in the longitude of the ascending node of the Earth equator.

28. Period (days), Argument Solution rigid Earth rotation Ψ(sin) μas Ψ(cos) μas Solution non-rigid Earth rotation Ψ(sin) μas Ψ(cos) μas 0.51753 2λ 3 +2D- 2  SMART97 S9000A 25.3564 14.5558 SMN SN9000A 26.3775 21.4672 0.507904 λ 3 +D- 2  SMART97 S9000A -0.2193 0.3213 SMN SN9000A -0.3258 0.3536 0.50000 2λ 3 -2  SMART97 S9000A 10.6324 6.0997 SMN SN9000A 11.2675 9.2705 0.49863 2  SMART97 S9000A 31.8523 -18.2836 SMN SN9000A 33.8043 -27.8520 0.49860 λ 3 +D-F- 2  SMART97 S9000A 4.3198 2.4797 SMN SN9000A 4.5837 3.7774 0.49795 λ 3 +2  SMART97 S9000A -0.1957 -0.2142 SMN SN9000A -0.2732 -0.2130 1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB. Results: Table 1. Comparison of different solutions for selected durnal and semidurnal periodical terms in the longitude of the ascending node of the Earth equator.

29. Table 2. Comparison of different solutions for selected durnal and semidurnal periodical terms in the inclination angle. Period (days), Argument Solution rigid Earth rotation θ (sin) μas θ (cos) μas Solution non-rigid Earth rotation θ (sin) μas θ (cos) μas 1.03505 λ 3 +D-  SMART97 S9000A -1.6400 -13.3708 SMN SN9000A -0.5515 -12.6886 1.03474 3λ 3 +3D-2F-  SMART97 S9000A -0.0010 0.0073 -0.0085 -0.0065 SMN SN9000A -0.0044 0.0039 -0.0006 0.0014 1.00000 λ 3 -  SMART97 S9000A -0.0109 -0.0790 SMN SN9000A -0.0001 -0.0777 0.99758 λ 3 +D-l-  SMART97 S9000A -0.9877 -0.9891 -7.8729 -7.8720 SMN SN9000A 0.4389 0.4375 -7.8522 -7.8513 0.96215 λ 3 +D+  SMART97 S9000A 1.8625 -15.1270 SMN SN9000A 0.8531 -14.4120 0.527517 3λ 3 +3D-2  SMART97 S9000A 0.0201 0.0131 SMN SN9000A 0.0915 0.0471 1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB. Results:

30. Table 2. Comparison of different solutions for selected durnal and semidurnal periodical terms in the inclination angle. Period (days), Argument Solution rigid Earth rotation θ (sin) μas θ (cos) μas Solution non-rigid Earth rotation θ (sin) μas θ (cos) μas 0.51753 2λ 3 +2D- 2  SMART97 S9000A -5.7900 10.0863 SMN SN9000A -8.5395 10.4933 0.507904 λ 3 +D- 2  SMART97 S9000A -0.1128 -0.0770 SMN SN9000A -0.1071 -0.1077 0.50000 2λ 3 -2  SMART97 S9000A -2.4263 4.2293 SMN SN9000A -3.6876 4.4822 0.49863 2  SMART97 S9000A -7.2728 -12.6702 SMN SN9000A -11.0792 -13.4478 0.49860 λ 3 +D-F- 2  SMART97 S9000A -0.9867 1.7190 SMN SN9000A -1.5033 1.8250 0.49795 λ 3 +2  SMART97 S9000A -0.0852 0.0778 SMN SN9000A -0.0847 0.1086 1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB. Results:

31. Period (days), Argument Solution rigid Earth rotation φ (sin) μas φ (cos) μas Solution non-rigid Earth rotation φ (sin) μas φ (cos) μas 1.03505 λ 3 +D-  SMART97 S9000A -32.2917 3.9612 SMN SN9000A -31.2266 1.5038 1.03474 3λ 3 +3D-2F-  SMART97 S9000A -0.0129 -0.0754 0.0016 -0.0564 SMN SN9000A -0.0213 -0.0838 -0.0090 -0.0670 1.00000 λ 3 -  SMART97 S9000A -0.1834 0.0253 SMN SN9000A -0.1809 0.0005 0.99758 λ 3 +D-l-  SMART97 S9000A -18.2334 -18.2314 2.2873 2.2908 SMN SN9000A -18.2125 -18.2105 -1.0005 -0.9974 0.96215 λ 3 +D+  SMART97 S9000A 35.0904 4.3205 SMN SN9000A 33.4859 1.9970 0.527517 3λ 3 +3D-2  SMART97 S9000A -0.1117 0.1630 SMN SN9000A -0.2427 0.3291 1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB. Results: Table 3. Comparison of different solutions for selected durnal and semidurnal periodical terms in the angle of the proper rotation of the Earth.

32. Period (days), Argument Solution rigid Earth rotation φ (sin) μas φ (cos) μas Solution non-rigid Earth rotation φ (sin) μas φ (cos) μas 0.51753 2λ 3 +2D- 2  SMART97 S9000A -0.1279 -0.0728 SMN SN9000A 0.8089 6.2684 0.507904 λ 3 +D- 2  SMART97 S9000A -0.2471 0.3617 SMN SN9000A -0.3448 0.3914 0.50000 2λ 3 -2  SMART97 S9000A -0.4033 -0.2352 SMN SN9000A 0.1794 2.6739 0.49863 2  SMART97 S9000A 31.9704 -18.3513 SMN SN9000A 33.7614 -27.1303 0.49860 λ 3 +D-F- 2  SMART97 S9000A 4.7817 2.7448 SMN SN9000A 5.0238 3.9354 0.49795 λ 3 +2  SMART97 S9000A -0.1964 -0.2148 SMN SN9000A -0.2675 -0.2137 1997; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB. Results: Table 3. Comparison of different solutions for selected durnal and semidurnal periodical terms in the angle of the proper rotation of the Earth.

33. Kinematical solution of the rigid Earth rotation= Dynamical solution of the rigid Earth rotation + Geodetics corrections Kinematical solution of the non-rigid Earth rotation= Kinematical solution of the rigid Earth rotation + TRANSFER FUNCTION

34. CONCLUSION The optimal spectral analysis scheme for this investigation (with respect to the accuracy) has been determined. It is algorithm-2, which used two iterations. The first iteration investigated the discrepancies for only the long periodical harmonics, while the second iteration investigated the discrepancies for all harmonics. The new version the high-precision rigid Earth rotation series S9000A has been constructed. It contains the short periodic terms (diurnal and sub-diurnal). The high-precision non-rigid Earth rotation series SN9000A (containing the short periodic terms) has been constructed. This series is expressed in the function of the three Euler angles with respect to the fixed ecliptic plane and equinox J2000.0 and is dynamically adequate to the ephemerides DE404/LE404 over 2000 years.

35. R E F E R E N C E S 1.P.Bretagnon and G.Francou. Planetary theories in rectangular and spherical variables // Astronomy and Astrophys., 202, 1988, pp. 309–-315. 2.P.Bretagnon, G.Francou, P.Rocher, J.L.Simon. SMART97: A new solution for the rotation of the rigid Earth // Astron. Astrophys., 1998, 329, pp. 329-338. 3.P.Bretagnon,P.M.Mathews,J.-L.Simon. Non Rigid Earth Rotation // in Proc. Journees 1999: Motion of Celestial Bodies, Astrometry and Astronomical Reference Frames Les Journees 1999 \& IX. Lohrmann - Kolloquium, (Dresden, 13-15 September 1999), p. 73-76. 4.V.A.Brumberg, P.Bretagnon. Kinematical Relativistic Corrections for Earth’s Rotation Parameters // in Proc. of IAU Colloquium 180, eds. K.Johnston, D. McCarthy, B. Luzum and G. Kaplan, U.S. Naval Observatory, 2000, pp. 293–302. 5.V.Dehant and P.Defraigne. New transfer functions for nutations of a non-rigid Earth // J. Geophys. Res., 1997,102, pp.27659-27688. 6.P. M. Mathews and P.Bretagnon. Polar Motions Equivalent to High Frequency Nutations for a Nonrigid Earth with Anelastic Mantle // Astron. Astrophys., 2003, 400, pp. 1113-1128. 7.Mathews, P. M., Herring, T. A., and Buffett B. A.. Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth's Interior // J. Geophys. Res., 2002, 107, B4, 10.1029/2001JB000390. 8.V.V.Pashkevich, G.I.Eroshkin and A. Brzezinski. Numerical analysis of the rigid Earth rotation with the quadruple precision // Artificial Satellites, Vol. 39, No. 4, Warszawa, 2004, pp. 291–304. 9.V. V. Pashkevich and G. I. Eroshkin. Spectral analysis of the numerical theory of the rigid Earth rotation // in Proc. of “Journees 2004”, Fundamental Astronomy: New concepts and models for high accuracy observations» (Observatoire de Paris, 20-22 September 2004.), pp. 82-87. 10.V.V.Pashkevich and G.I.Eroshkin. Choice of the optimal spectral analysis scheme for the investigation of the Earth rotation problem // in Proc. of “Journees 2005”, Earth dynamics and reference systems: five years after the adoption of the IAU 2000 Resolutions (Space Research Centre of Polish Academy of Sciences, Warsaw, Poland, 19-21 September 2005), pp. 105-109. 11.V.V.Pashkevich and G.I.Eroshkin. Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation // Artificial Satellites, Vol. 40, No. 4, Warszawa, 2005, pp. 251–260. 12.V.V. Pashkevich. Construction of the Non-Rigid Earth Rotation Series // Artificial Satellites, 2008 in press. 13.T.Shirai and T.Fukushima. Construction of a new forced nutation theory of the nonrigid Earth // The Astron. Journal, 121, 2001, pp.3270-3283. 14.J.M.Wahr. The forced nutationsof an elliptical, rotating, elastic and oceanless Earth // Geophys. J. R. Astron. Soc., 1981, 64, pp.705-727.

36. A C K N O W L E D G M E N T S The investigation was carried out at the Central (Pulkovo) Astronomical Observatory of the Russian Academy of Sciences and at the Space Research Centre of the Polish Academy of Sciences, under a financial support of the agreement cooperation between the Polish and Russian Academies of Sciences, Theme No 31.