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Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation

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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Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation

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  1. 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

  2. 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: • 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. • 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. • New precession and nutation series for the rigid Earth, dynamically consistent with DE404/LE404 ephemeris, were constructed.

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

  4. The calculations on Parsytec computer with a quadruple precision. Fig.1. Difference between our numerical solution andSMART97 a) in the longitude. Kinematical caseDynamical case

  5. Fig.1. Difference between our numerical solution andSMART97 b) in the proper rotation. Kinematical caseDynamical case

  6. Fig.1. Difference between our numerical solution andSMART97 c) in the inclination. Kinematical caseDynamical case

  7. Fig.2. Difference between our numerical solution andSMART97 after formal removal of secular trends. Kinematical caseDynamical case

  8. Fig.3. Spectra of discrepancies between our numerical solution andSMART97 for proper rotation angle. Kinematical caseDynamical case

  9. Fig.3. Spectra of discrepancies between our numerical solution andSMART97 for proper rotation angle. Kinematical caseDynamical case 3 1 2

  10. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 Kinematical caseDynamical case

  11. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 Kinematical caseDynamical case B A

  12. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 A Kinematical caseDynamical case

  13. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 A Kinematical caseDynamical case II I

  14. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 A-I Kinematical caseDynamical case

  15. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 A-II Kinematical caseDynamical case

  16. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 A-II Kinematical caseDynamical case

  17. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 A-II (zoom) Kinematical caseDynamical case

  18. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 B Kinematical caseDynamical case

  19. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 B Kinematical caseDynamical case

  20. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 B (zoom) Kinematical caseDynamical case

  21. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 B (zoom) Kinematical caseDynamical case

  22. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 1 B (zoom2) Kinematical caseDynamical case

  23. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 2 Kinematical caseDynamical case

  24. Fig.3. Spectra of discrepancies between our numerical solutionandSMART97 for proper rotation angle. DETAIL 3 Kinematical caseDynamical case

  25. Fig.4. Difference between our numerical solution andSMART97 after formal removal the secular trends and 9000 periodical harmonics. Kinematical caseDynamical case

  26. Fig.5. Repeated Numerical SolutionminusNew Series. Kinematical caseDynamical case

  27. Fig.6. Numerical solutionminus New Series after formal removal secular trends in the proper rotation angle. Kinematical caseDynamical case

  28. Fig.6. Numerical solutionminus New Series after formal removal secular trends in the proper rotation angle. (zoom) Kinematical caseDynamical case

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

  30. Fig.8. Sub diurnal and diurnalspectra of discrepancies between our numerical solution andSMART97 for proper rotation angle. Kinematical caseDynamical case

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

  32. CONCLUSION • Spectral analysis of discrepancies of the numerical solutions andSMART97 solutions of the rigid Earth rotationwas 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 properrotation 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.

  33. 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.

  34. http://www.csa.ru/ The massive-parallel computer system Parsytec CCe20 Center for supercomputing applications • Parsytec CCe20 is a supercomputer of massive-parallel architecture withseparated memory. It is intended for fulfilment of high-performanceparallel 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) 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 powerfulprocessor platforms available today and are clearly outstanding in price / performance. There are 5 Parsytec computers in CSA now.

  35. SA method for cleaning the discrepanciescalculated periodical terms Discrepancies after removal the secular trend Set of nutation terms of SMART97 LSQ methodcompute amplitude of power spectrum of discrepancies LSQ methoddetermine amplitudes and phases of the largest rest harmonic Nutation terms of SMART97 if |Am| > || Compute a new nutation term Construct a new nutation series No Yes Until the end of specta Remove this harmonic from discrepancy and Spectra

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

  37. Fig.5 Repeated Numerical SolutionminusNew Series. Kinematical caseDynamical case

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