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Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory of Russian Academy of Science St.Petersburg Space Research Centre of Polish Academy of Sciences Warszawa 2004

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The aim of the investigation: Construction of a new high-precision series for the rigid Earth rotation, dynamically consistent with DE404/LE404 ephemeris and based on the SMART97 developments. A L G O R I T H M: 1.Numerical solutions of the rigid Earth rotation are constructed. Discrepancies of the comparison between our numerical solutions and the SMART97 ones are obtained in Euler angles. 2.Investigation of the discrepancies was carried out by the least squares (LSQ) and by the spectral analysis (SA) methods. The secular and periodic terms were determined from the discrepancies. 3.New precession and nutation series for the rigid Earth, dynamically consistent with DE404/LE404 ephemeris, were constructed.

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SA method calculate periodical terms Initial conditions from SMART97 Numerical integration of the differential equations Discrepancies: Numerical Solutions minus SMART97 LSQ method calculate secular terms 6-th degree Polinomial of time Precession terms of SMART97 Compute new precession parameters New precession and nutation series Construct a new nutation series Remove the secular trend from discreapancies

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Fig.1. Difference between our numerical solution and SMART97 a) in the longitude. Kinematical case Dynamical case Secular terms of… Secular terms of… smart97 ( as)- d ( as) smart97( as)- d ( as) 7.00 6.89 50384564881.3693 T- 206.50 T50403763708.8052 T - 206.90 T - 107194853.5817 T 2 - 3451.30 T 2 - 107245239.9143 T 2 - 3180.80 T 2 - 1143646.1500 T 3 1125.00 T 3 - 1144400.2282 T 3 1048.00 T 3 1328317.7356 T 4 - 788.00 T 4 1329512.8261 T 4 - 306.00 T 4 - 9396.2895 T 5 - 57.50 T 5 - 9404.3004 T 5 - 65.50 T 5 - 3415.00 T 6 - 3421.00 T 6 The calculations on Parsytec computer with a quadruple precision.

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Fig.1. Difference between our numerical solution and SMART97 b) in the proper rotation. Kinematical case Dynamical case Secular terms of… Secular terms of… smart97 ( as)d ( as) smart97 ( as)d ( as) 1009658226149.36916.58 1009658226149.36916.53 474660027824506304.0000 T99598.30 T474660027824506304.0000 T97991.40 T - 98437693.3264 T 2 - 7182.30 T 2 98382922.2808 T 2 - 6934.40 T 2 - 1217008.3291 T 3 1066.80 T 3 -1216206.2888 T 3 1004.00 T 3 1409526.4062 T 4 - 750.00 T 4 1408224.6897 T 4 - 226.00 T 4 - 9175.8967 T 5 - 30.30 T 5 - 9168.0461 T 5 - 37.80 T 5 - 3676.00 T 6 - 3682.00 T 6

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Fig.1. Difference between our numerical solution and SMART97 c) in the inclination. Kinematical case Dynamical case Secular terms of… Secular terms of… smart97( as)d ( as) smart97( as)d ( as) 84381409000.0000 1.4284381409000.0000 1.39 - 265011.2586 T - 96.61 T - 265001.7085 T- 96.73 T 5127634.2488 T 2 - 353.10 T 2 5129588.3567 T 2 - 595.90 T 2 - 7727159.4229 T 3 771.50 T 3 - 7731881.2221 T 3 - 945.10 T 3 - 4916.7335 T 4 - 84.50 T 4 - 4930.2027 T 4 - 76.50 T 4 33292.5474 T 5 - 86.00 T 5 33330.6301 T 5 - 70.00 T 5 - 247.50 T 6 - 247.80 T 6

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Fig.2. Difference between our numerical solution and SMART97 after formal removal of secular trends. Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. Kinematical case Dynamical case 12 3

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 Kinematical case Dynamical case B A

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A Kinematical case Dynamical case II I

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A-I Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A-II Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A-II Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A-II (zoom) Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B (zoom) Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B (zoom) Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B (zoom2) Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 2 Kinematical case Dynamical case

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Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 3 Kinematical case Dynamical case

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Fig.4. Difference between our numerical solution and SMART97 after formal removal the secular trends and 9000 periodical harmonics. Kinematical case Dynamical case

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Fig.5. Repeated Numerical Solution minus New Series. Kinematical case Dynamical case

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Fig.6. Numerical solution minus New Series after formal removal secular trends in the proper rotation angle. Kinematical case Dynamical case

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Fig.6. Numerical solution minus New Series after formal removal secular trends in the proper rotation angle. (zoom) Kinematical case Dynamical case

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Fig.7. Numerical solution minus New Series after formal removal secular trends in the proper rotation angle. Kinematical case Dynamical case The calculations on PC with a double precision.

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Fig.8. Sub diurnal and diurnal spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. Kinematical case Dynamical case

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Fig.9. Numerical solution minus New Series including sub diurnal and diurnal periodical terms after formal removal secular trends in the proper rotation angle. Kinematical case Dynamical case

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CONCLUSION Spectral analysis of discrepancies of the numerical solutions and SMART97 solutions of the rigid Earth rotation was carried out for the kinematical and dynamical cases over the time interval of 2000 years. Construction of a new series of the rigid Earth rotation, dynamically consistent with DE404/LE404 ephemeris, were performed for dynamical and kinematical cases. The power spectra of the residuals for the dynamical and kinematical cases are similar. The secular trend in proper rotation found in the difference between the numerical solutions and new series is considerably smaller than that found in the difference between the numerical solutions and SMART97.

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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 Russian Academy of Science and the Space Research Centre of Polish Academy of Science, under a financial support of the Cooperation between Polish and Russian Academies of Sciences, Theme No 25 and of the Russian Foundation for Fundamental Research, Grant No 02-02-17611.

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The massive-parallel computer system Parsytec CCe20 Parsytec CCe20 is a supercomputer of massive-parallel architecture with separated memory. It is intended for fulfilment of high-performance parallel calculations. Hardware: 20 computing nodes with processors PowerPC 604e (300MHz); 2 nodes of input-output; The main memory: o 32 Mb on computing nodes; o 64 Mb on nodes of an input / conclusion; disk space 27 Gb; tape controller DAT; CD-ROM device; network interface Ethernet (10/100 Mbs); communication interface HighSpeed Link (HS- Link) Center for supercomputing applications http://www.csa.ru/ Massive-parallel supercomputers Parsytec is designed by Parsytec GmbH, Germany, using Cognitive Computer technology. The system approach is based on using of PC technology and RISC processors PowerPC which are ones of the most powerful processor platforms available today and are clearly outstanding in price / performance. There are 5 Parsytec computers in CSA now.

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Discrepancies after removal the secular trend LSQ method compute amplitude of power spectrum of discrepancies LSQ method determine amplitudes and phases of the largest rest harmonic if |Am| > | | Until the end of specta Set of nutation terms of SMART97 Nutation terms of SMART97 Construct a new nutation series Remove this harmonic from discrepancy and Spectra YesNo Compute a new nutation term SA method for cleaning the discrepancies calculated periodical terms

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Quadruple precision corresponding to 32- decimal representation of real numbers. Double precision corresponding to 16- decimal representation of real numbers.

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Fig.5 Repeated Numerical Solution minus New Series. Kinematical case Dynamical case

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