Construction of the 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 2007

The last years a lot of attempts to derive a high-precision theory of the non-rigid Earth rotation was carried out. For these purposes used the different transfer functions, which usually applied to the nutation in longitude and in obliquity of the rigid Earth rotation with respect to the ecliptic of date. Aim: Construction of a new high-precision non-rigid Earth rotation series (SN9000), dynamically adequate to the DE404/LE404 ephemeris over 2000 years, which is expressed as a function of the three Euler angles with respect to the fixed ecliptic plane and equinox J2000.0.

S T E P S: The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing.

S T E P - 1: Numerical integration of the differential equations Discrepancies: numerical solution minus SMART97 Initial conditions from SMART97

Problem expressed by the Rodrigues – Hamilton parameters: S T E P - 1: Numerical integration of the differential equations Discrepancies: numerical solution minus SMART97 Initial conditions from SMART97 LAGRANGE DIFFERENTIAL EQUATIONS OF THE SECOND KIND: VECTOR OF THE GEODETIC ROTATION: Problem expressed by the Rodrigues – Hamilton parameters:

Fig.1 Numerical solution for Rigid Earth rotation minus solution SMART97 in the longitude of the ascending node of the Earth equator. Kinematical (relativistic) case Dynamical (Newtonian) case Secular terms of… Secular terms of… smart97 (as) d (as) smart97(as) 7.00 6.89 50384564881.3693 T - 206.50 T 50403763708.8052 T - 206.90 T - 107194853.5817 T2 - 3451.30 T2 - 107245239.9143 T2 - 3180.80 T2 - 1143646.1500 T3 1125.00 T3 - 1144400.2282 T3 1048.00 T3 1328317.7356 T4 - 788.00 T4 1329512.8261 T4 - 306.00 T4 - 9396.2895 T5 - 57.50 T5 - 9404.3004 T5 - 65.50 T5 - 3415.00 T6 - 3421.00 T6

Fig.2 Numerical solution for Rigid Earth rotation minus solution SMART97 in the angle of the proper rotation of the Earth. Kinematical (relativistic) case Dynamical (Newtonian) case Secular terms of… Secular terms of… smart97 (as) d(as) d (as) 1009658226149.3691 6.58 6.53 474660027824506304.0000 T 99598.30 T 97991.40 T - 98437693.3264 T2 - 7182.30 T2 98382922.2808 T2 - 6934.40 T2 - 1217008.3291 T3 1066.80 T3 -1216206.2888 T3 1004.00 T3 1409526.4062 T4 - 750.00 T4 1408224.6897 T4 - 226.00 T4 - 9175.8967 T5 - 30.30 T5 - 9168.0461 T5 - 37.80 T5 - 3676.00 T6 - 3682.00 T6

Fig.3 Numerical solution for Rigid Earth rotation minus solution SMART97 in the inclination angle. Kinematical (relativistic) case Dynamical (Newtonian) case Secular terms of… Secular terms of…  smart97(as) d(as) 84381409000.0000 1.42 1.39 - 265011.2586 T - 96.61 T - 265001.7085 T - 96.73 T 5127634.2488 T2 - 353.10 T2 5129588.3567 T2 - 595.90 T2 - 7727159.4229 T3 771.50 T3 - 7731881.2221 T3 - 945.10 T3 - 4916.7335 T4 - 84.50 T4 - 4930.2027 T4 - 76.50 T4 33292.5474 T5 - 86.00 T5 33330.6301 T5 - 70.00 T5 - 247.50 T6 - 247.80 T6

S T E P S: The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing.

S T E P S: The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing. Investigation of the discrepancies is carried out by the least squares (LSQ) and by the spectral analysis (SA) algorithms (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2005”, 2005). The high-precision rigid Earth rotation series S9000 is determined (V.V.Pashkevich and G.I.Eroshkin, 2005 ).

S T E P - 1: Numerical integration of the differential equations Discrepancies: numerical solution minus SMART97 Initial conditions from SMART97 S T E P - 2

S T E P S - 1 and 2: Numerical integration of the differential equations Discrepancies: numerical solution minus SMART97 Initial conditions from SMART97 Precession and GMST terms of SMART97 Calculation of the secular terms by the LSQ method Set of nutation terms of SMART97 Computation of the new precession and GMST parameters Removal of the secular trends from the discreapancies Calculation of the periodical terms by the SA method High-precision series S9000 Construction of the new nutation series

Fig.4. The numerical solution of the rigid Earth rotation minus S9000 after formal removal of the secular trends in the proper rotation angle. Kinematical case Dynamical case

Fig.5. Differences between S9000 and SMART97. Kinematical case Dynamical case

S T E P S: The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing. Investigation of the discrepancies is carried out by the least squares (LSQ) and by the spectral analysis (SA) algorithms (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2005”, 2005). The high-precision rigid Earth rotation series S9000 is determined (V.V.Pashkevich and G.I.Eroshkin, 2005 ).

S T E P S: The high-precision numerical solution of the rigid Earth rotation have been constructed (V.V.Pashkevich, G.I.Eroshkin and A.Brzezinski, 2004), (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2004”). The initial conditions have been calculated from SMART97 (P.Bretagnon, G.Francou, P.Rocher, J.L.Simon,1998). The discrepancies between the numerical solution and the semi-analytical solution SMART97 were obtained in Euler angles over 2000 years with one-day spacing. Investigation of the discrepancies is carried out by the least squares (LSQ) and by the spectral analysis (SA) algorithms (V.V.Pashkevich and G.I.Eroshkin, Proceedings of “Journees 2005”, 2005). The high-precision rigid Earth rotation series S9000 is determined (V.V.Pashkevich and G.I.Eroshkin, 2005 ). The new high-precision non-rigid Earth rotation series (SN9000), which expressed in the function of the three Euler angles, is constructed by using the method (P.Bretagnon, P.M.Mathews, J.-L.Simon: 1999) and the transfer function (Mathews, P. M., Herring, T. A., and Buffett B. A., 2002).

Expressions for Euler angles: S T E P - 3: Expressions for Euler angles:

Classical A l g o r i t h m

Classical A l g o r i t h m Expressions for the non-rigid Earth nutations in longitude and obliquity:

TRANSFER FUNCTIONS J.M.Wahr ,1981; V. Dehant and P. Defraigne, 1997: T. Shirai and T. Fukushima, 2001:

TRANSFER FUNCTIONS P.Bretagnon, P.M. Mathews, J.-L. Simon,1999: P.M. Mathews, T.A. Herring, B.A. Buffett, 2002:

P.M. Mathews, T.A. Herring, B.A. Buffett, 2002: TRANSFER FUNCTIONS P.M. Mathews, T.A. Herring, B.A. Buffett, 2002: Geophisical model includes : The effects Electromagnetic coupling The Ocean effects Mantle inelasticity effects Atmospheric effects Change in the global Earth dynamical flattening and in the core flattening

Algorithm of Bretagnon et al. 1999 1.The rigid Earth angular velocity vector:

Expressions for the Earth’s angular velocity vector: The coefficients of the developments of p, q in case of the rigid Earth:

The components of the Rigid Earth angular velocity vector:

The components of the coefficients of the developments of p, q in case of the rigid Earth:

The amplitudes of the prograde and retrograde components: Complex transformation of the componets of the angular velocity vector:

Algorithm of Bretagnon et al. 1999 1.The rigid Earth angular velocity vector: 2.The non-rigid Earth angular velocity vector is obtained by:

The components of the coefficients of the developments of p, q in case of the non-rigid Earth:

The prograde and retrograde components of the coefficients of the developments of p in case of the non-rigid Earth:

The prograde and retrograde components of the coefficients of the developments of q in case of the non-rigid Earth:

The sin coefficients of the developments for the non-rigid Earth angular velocity vector:

The cos coefficients of the developments for the non-rigid Earth angular velocity vector:

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

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

Iterative solution of the algorithm of Bretagnon et al. 1999 Details: Iterative solution of the algorithm of Bretagnon et al. 1999 1 iteration

Iterative solution of the algorithm of Bretagnon et al. 1999 Details: Iterative solution of the algorithm of Bretagnon et al. 1999 n iteration Iterations are repeated until the absolute value of the difference between iterations K-1 and K exceedes some DEFINITE values 

DD- Dehant and Defraigne, 1997; BMS- Bretagnon et al., 1999; Results: Table 1. Comparison of different solutions Argument (λ3 +D-F ) with a 18.6 year period Solution Ψ(sin) μas Ψ(cos) θ (sin) θ (cos) φ (sin) φ (cos) S9000+Wahr 1981 17199072 -431 35 -9202843 15777192 -395 S9000+DD 17202786 -9203936 15780599 -396 S9000+SF 17205158 -2701 -1269 -9204660 15782776 -2479 S9000+BMS 17206626 -3351 -1486 -9205148 15784122 -3075 S9000+MHB 17206664 -3386 -1505 -9205211 15784185 -3106 DD- Dehant and Defraigne, 1997; BMS- Bretagnon et al., 1999; SF-Shirai and Fukushima, 2001; MHB- Mathews et al., 2002.

DD- Dehant and Defraigne, 1997; BMS- Bretagnon et al., 1999; Results: Table 1. Comparison of different solutions Argument (λ3 +D-F ) with a 18.6 year period Solution Ψ(sin) μas Ψ(cos) θ (sin) θ (cos) φ (sin) φ (cos) SMART97+Wahr 1981 17199072 -431 35 -9202843 15777192 -395 SMART97+DD 17202786 -9203936 15780599 -396 SMART97+SF 17205158 -2701 -1269 -9204660 15782776 -2479 SMART97+BMS 17206626 -3351 -1486 -9205148 15784122 -3075 SMART97+MHB 17206664 -3386 -1505 -9205211 15784185 -3106 DD- Dehant and Defraigne, 1997; BMS- Bretagnon et al., 1999; SF-Shirai and Fukushima, 2001; MHB- Mathews et al., 2002.

Table 2. Comparison of different solutions For some arguments Argu-ment, Period (days) Ψ(sin) μas Ψ(cos) θ (sin) θ (cos) φ (sin) φ (cos) SMN SN9000 λ3 +D-F 6798.38 17206664.8552 -3386.9864 -1505.7856 -9205211.3617 15784185.0624 -3106.7674 2λ3 182.62 -1318596.8582 -1318596.6148 -748.6166 -748.0244 -451.5369 -451.1548 572973.2222 572973.2727 -1209767.5649 -1209767.3333 -658.9659 -658.4234 2λ3+2D 13.66 -227640.1800 -227640.1593 297.0399 297.0236 135.7237 135.7144 97843.9277 97843.9181 -208855.8620 -208855.8434 265.7426 265.7281 2λ3 +2D-2F 3399.19 207463.2276 207463.2277 -72.8532 -72.8533 -28.7639 -28.7598 -89756.6705 -89756.6616 190409.8464 190409.8336 -66.9362 -66.9358 λ3 365.24 -37360.1264 -123079.8222 16721.8818 623.4780 -34188.8930 -112726.8142 3λ3 121.75 11532.5978 11532.7794 50400.5095 50400.6781 21873.2512 21873.3218 -5006.6994 -5006.7717 10581.9069 10582.0752 46242.6261 46242.7801 SMN=SMART97+MHB; SN9000=S9000+MHB; MHB- Mathews et al., 2002.

Fig.6. The differences between SN9000 and SMN after removal of the secular terms. (non-rigid Earth rotation) Dynamical case μas Δφ Δθ Δψ YEARS

Fig.7. The differences between S9000 and SMART97 after removal of the secular terms. (rigid Earth rotation) Dynamical case μas Δφ Δθ Δψ YEARS

Fig.8. The differences between SN9000 and SMN after removal of the secular terms. (non-rigid Earth rotation) Kinematical case μas Δφ Δθ Δψ YEARS

Fig.9. The differences between S9000 and SMART97 after removal of the secular terms. (rigid Earth rotation) Kinematical case μas Δφ Δθ Δψ YEARS

Fig.10. The differences between S9000 and SN9000. Dynamical case μas Δφ Δθ Δψ YEARS

Fig.11. The differences between S9000 and SN9000. Kinematical case μas Δφ Δθ Δψ YEARS

Fig.12. The differences between SMART97 and SMN. Dynamical case μas Δφ Δθ Δψ YEARS

Fig.13. The differences between SMART97 and SMN. Kinematical case μas Δφ Δθ Δψ YEARS

Fig.14. The discrepancies between S9000 and SN9000 minus discrepancies between SMART97 and SMN. Dynamical case μas Δφ Δθ Δψ YEARS

Fig.15. The discrepancies between S9000 and SN9000 minus discrepancies between SMART97 and SMN. Kinematical case μas Δφ Δθ Δψ YEARS

Kinematical solution of the rigid Earth rotation= Dynamical solution of the rigid Earth rotation + Geodetics corrections Dynamical solution of the non-rigid Earth rotation= Dynamical solution of the rigid Earth rotation + TRANSFER FUNCTIONS Original Kinematical solution of the non-rigid Earth rotation= Dynamical solution of the rigid Earth rotation + TRANSFER FUNCTIONS + Geodetics corrections Kinematical solution of the non-rigid Earth rotation= Kinematical solution of the rigid Earth rotation + TRANSFER FUNCTIONS

CONCLUSION The exact expressions for the algorithm of Bretagnon et al. (1999) are obtained. The new semi-analytical solution of the non-rigid Earth rotation SMN (SMART97+MHB2002) is derived. The high-precision non-rigid Earth rotation series SN9000, which is expressed as a function of the three Euler angles and is dynamically adequate to the ephemerides DE404/LE404 over 2000 years, has been constructed.

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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 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.