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

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:

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).

Part I: The rigid Earth rotation

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) T T T T T T T T T T T T T T T T T T T T T T 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) T T T T T T T T T T T T T T T T T T T T T T 6 the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case

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) T T T T T T T T T T T T T T T T T T T T T T 6 the computer system Parsytec CCe20 case the personal computer Intel® Core™2 Duo case

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

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

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

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

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

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.

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

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

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

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

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

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

Part II: The non-rigid Earth rotation

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

Expressions for Euler angles:

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:

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

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

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 (Pashkevich, 2008)

Period (days), Argument Solution rigid Earth rotation Ψ(sin) μas Ψ(cos) μas Solution non-rigid Earth rotation Ψ(sin) μas Ψ(cos) μas λ 3 +D-  SMART97 S9000A SMN SN9000A λ 3 +3D-2F-  SMART97 S9000A SMN SN9000A λ 3 -  SMART97 S9000A SMN SN9000A λ 3 +D-l-  SMART97 S9000A SMN SN9000A λ 3 +D+  SMART97 S9000A SMN SN9000A λ 3 +3D-2  SMART97 S9000A SMN SN9000A ; 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.

Period (days), Argument Solution rigid Earth rotation Ψ(sin) μas Ψ(cos) μas Solution non-rigid Earth rotation Ψ(sin) μas Ψ(cos) μas λ 3 +2D- 2  SMART97 S9000A SMN SN9000A λ 3 +D- 2  SMART97 S9000A SMN SN9000A λ 3 -2  SMART97 S9000A SMN SN9000A  SMART97 S9000A SMN SN9000A λ 3 +D-F- 2  SMART97 S9000A SMN SN9000A λ 3 +2  SMART97 S9000A SMN SN9000A ; 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.

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 λ 3 +D-  SMART97 S9000A SMN SN9000A λ 3 +3D-2F-  SMART97 S9000A SMN SN9000A λ 3 -  SMART97 S9000A SMN SN9000A λ 3 +D-l-  SMART97 S9000A SMN SN9000A λ 3 +D+  SMART97 S9000A SMN SN9000A λ 3 +3D-2  SMART97 S9000A SMN SN9000A ; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB. Results:

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 λ 3 +2D- 2  SMART97 S9000A SMN SN9000A λ 3 +D- 2  SMART97 S9000A SMN SN9000A λ 3 -2  SMART97 S9000A SMN SN9000A  SMART97 S9000A SMN SN9000A λ 3 +D-F- 2  SMART97 S9000A SMN SN9000A λ 3 +2  SMART97 S9000A SMN SN9000A ; BMS- Bretagnon et al., 1999; MHB- Mathews et al., 2002,SMN=SMART97+MHB; SN9000A=S9000A+MHB. Results:

Period (days), Argument Solution rigid Earth rotation φ (sin) μas φ (cos) μas Solution non-rigid Earth rotation φ (sin) μas φ (cos) μas λ 3 +D-  SMART97 S9000A SMN SN9000A λ 3 +3D-2F-  SMART97 S9000A SMN SN9000A λ 3 -  SMART97 S9000A SMN SN9000A λ 3 +D-l-  SMART97 S9000A SMN SN9000A λ 3 +D+  SMART97 S9000A SMN SN9000A λ 3 +3D-2  SMART97 S9000A SMN SN9000A ; 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.

Period (days), Argument Solution rigid Earth rotation φ (sin) μas φ (cos) μas Solution non-rigid Earth rotation φ (sin) μas φ (cos) μas λ 3 +2D- 2  SMART97 S9000A SMN SN9000A λ 3 +D- 2  SMART97 S9000A SMN SN9000A λ 3 -2  SMART97 S9000A SMN SN9000A  SMART97 S9000A SMN SN9000A λ 3 +D-F- 2  SMART97 S9000A SMN SN9000A λ 3 +2  SMART97 S9000A SMN SN9000A ; 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.

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

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 J and is dynamically adequate to the ephemerides DE404/LE404 over 2000 years.

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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.