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![Page 1: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/1.jpg)
Application of the spectral analysis for the mathematical modelling of
the rigid Earth rotation
V.V.Pashkevich
Central (Pulkovo) Astronomical Observatoryof Russian Academy of Science
St.PetersburgSpace Research Centre of Polish Academy of Sciences
Warszawa
2004
![Page 2: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/2.jpg)
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:
c) 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.
d) 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.
e) New precession and nutation series for the rigid Earth, dynamically consistent with DE404/LE404 ephemeris, were constructed.
![Page 3: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/3.jpg)
SA methodcalculate
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
Precessionterms of
SMART97
Compute new precession parameters
New precessionand nutation
series
Construct a newnutation series
Remove the secular trend
from discreapancies
![Page 4: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/4.jpg)
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 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
The calculations on Parsytec computer with a quadruple precision.
![Page 5: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/5.jpg)
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)
1009658226149.3691 6.58 1009658226149.3691 6.53
474660027824506304.0000 T 99598.30 T 474660027824506304.0000 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
![Page 6: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/6.jpg)
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 1.42 84381409000.0000 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
![Page 7: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/7.jpg)
Fig.2. Difference between our numerical solution and SMART97 after formal removal of secular trends.
Kinematical case Dynamical case
![Page 8: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/8.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle.
Kinematical case Dynamical case
![Page 9: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/9.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle.
Kinematical case Dynamical case
1 2
3
![Page 10: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/10.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1
Kinematical case Dynamical case
![Page 11: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/11.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1
Kinematical case Dynamical case
B
A
![Page 12: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/12.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A
Kinematical case Dynamical case
![Page 13: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/13.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A
Kinematical case Dynamical case
II
I
![Page 14: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/14.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A-I
Kinematical case Dynamical case
![Page 15: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/15.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A-II
Kinematical case Dynamical case
![Page 16: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/16.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A-II
Kinematical case Dynamical case
![Page 17: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/17.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 A-II (zoom)
Kinematical case Dynamical case
![Page 18: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/18.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B
Kinematical case Dynamical case
![Page 19: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/19.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B
Kinematical case Dynamical case
![Page 20: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/20.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B (zoom)
Kinematical case Dynamical case
![Page 21: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/21.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B (zoom)
Kinematical case Dynamical case
![Page 22: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/22.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 1 B (zoom2)
Kinematical case Dynamical case
![Page 23: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/23.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 2
Kinematical case Dynamical case
![Page 24: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/24.jpg)
Fig.3. Spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle. DETAIL 3
Kinematical case Dynamical case
![Page 25: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/25.jpg)
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
![Page 27: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/27.jpg)
Fig.6. Numerical solution minus New Series after formal removal secular trends in the proper rotation angle.
Kinematical case Dynamical case
![Page 28: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/28.jpg)
Fig.6. Numerical solution minus New Series after formal removal secular trends in the proper rotation angle. (zoom)
Kinematical case Dynamical case
![Page 29: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/29.jpg)
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.
![Page 30: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/30.jpg)
Fig.8. Sub diurnal and diurnal spectra of discrepancies between our numerical solution and SMART97 for proper rotation angle.
Kinematical case Dynamical case
![Page 31: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/31.jpg)
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
![Page 32: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/32.jpg)
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.
![Page 35: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/35.jpg)
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 endof specta
Set of nutation terms of SMART97
Nutation terms of SMART97
Construct a newnutation series
Remove this harmonic from discrepancy and Spectra
YesNo
Compute a new nutation term
SA method for cleaning the discrepancies calculated periodical terms
![Page 36: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/36.jpg)
Quadruple precision corresponding to 32- decimal representation of real numbers.
Double precision corresponding to 16- decimal representation of real numbers.
![Page 37: Application of the spectral analysis for the mathematical modelling of the rigid Earth rotation V.V.Pashkevich Central (Pulkovo) Astronomical Observatory.](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649cce5503460f949997cf/html5/thumbnails/37.jpg)
Fig.5 Repeated Numerical Solution minus New Series.
Kinematical case Dynamical case