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 Observatoryof Russian Academy of Science

St.PetersburgSpace Research Centre of Polish Academy of Sciences

Warszawa

2004

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.

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

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.

Fig.1. Difference between our numerical solution and SMART97 b) in the proper rotation.



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

Fig.1. Difference between our numerical solution and SMART97 c) in the inclination.



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

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


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


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


1 2

3

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




B

A

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


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


II

I

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


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


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


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


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


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


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


Fig.5. Repeated Numerical Solution minus New Series.


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


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


Fig.7. Numerical solution minus New Series after formal removal secular trends in the proper rotation angle.


The calculations on PC with a double precision.

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


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

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.

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.

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.

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

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

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

Fig.5 Repeated Numerical Solution minus New Series.


application of the spectral analysis for the mathematical modelling of the rigid earth rotation...

Documents

g o r i t h

smart97 ones

smart97 b

smart97 numerical integration

smart97 developments

smart97 lsq method

periodic terms

numerical solutions