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Preliminary analysis of the Free Core Nutation from VLBI data
Z. Malkin1 and D. Terentev1
1 2

,2

Institute of Applied Astronomy (RAS), St. Petersburg, Russia Sobolev Astronomical Institute, St. Petersburg State University, Russia

Summary: S everal VLBI EOP series were investigated with goal of determination of parameters of the Free Core Nutation (FCN). Both the amplitude and period of the FCN were studied. Our preliminary analysis reveals a variability of both the amplitude (known also from other investigations) and period of the FCN nutation term. The FCN amplitude varies in the range about 0.1­0.3 mas, and the FCN period -- in the range about 415­490 solar days.

1

Intro duction
In this paper we investigate variability of the FCN parameters. Whereas variations of the FCN amplitude was already investigated (see e.g. Herring et al., 2002; Shirai and Fukushima, 2001), variations of the FCN period is not been studied yet. Modern theory of nutation predicts the steady FCN period of 431.2 sidereal days (Dehant and Defraigne, 1997). The FCN period also have been estimated from VLBI observations, and found to be about 430­431 sidereal days or about 429­430 solar days (see, e.g. Table 4 in Shirai01 and Fukushima, 2001). In this paper we analyze four VLBI nutation series available in the IVS data base, sufficiently long and dense to obtain reliable estimates. We consider the differences between observed values of nutation angles and IAU2000A model (which is equivalent to MHB2000 model without FCN contribution). For our purpose, we interpret the unpredicted part of observed nutation series in the FCN frequency band as the FCN contribution.

2

Data used in analysis
The series used in our analysis are BKG00003, GSF2002C, IAAO0201, USN2002B. We analyzed both raw (i.e. given on original epochs) and smoothed (equally spaced by 0.05 year) series. For smoothed series we also computed weighted mean one. The parameter of smoothing was chosen in such a way to suppress oscillations with periods less then 1 month. Common time span for all series is 1984.0­2002.8. Figure 1 shows smoothed series used in our analysis.

3
3.1

Analysis and results
Sp ectral analysis For estimation of the power spectral density from both raw (unequally spaced) and smoothed (equally spaced) nutation series we used the FerrazMello's method (Ferraz-Mello, 1981) which allows us to process both types of data. For supplement testing, we also compute the power spectral density using the Burg's method (Marple, 1987). Figures 2--4 show the normalized results of spectral estimation, and Table 1 presents the estimates of the FCN period.

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Z. Malkin and D. Terentev: Preliminary analysis of the Free Core Nutation from VLBI data.

dPsi(FCN), indivdual series, mas 1.2 0 -1.2 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 dEps(FCN), indivdual series, mas 0.5 0 -0.5 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 dPsi(FCN), mean series, mas 1.2 0 -1.2 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 dEps(FCN), mean series, mas 0.5 0 -0.5 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004
Figure 1: FCN contribution in the individual and mean series. The average estimated value of the FCN period is of about 434 solar days (about 435 sidereal days). This value is substantially greater than one found in Shirai and Fukushima (2001) (431.0±0.6 sidereal days). However, when we used for spectral analysis only nutation series cut at the epoch 2000.2 which corresponds to the data span used in Shirai and Fukushima (2001), we obtain the FCN period of about 432 sidereal days which is close to found in Shirai and Fukushima (2001) (see the last line in each section of Table 1). 3.2 Wavelet analysis At the next step we applied wavelet analysis to all the nutation series. For this analysis we used program WWZ, developed by the American Association of Variable Star Observers and available at (1 ). Theoretical background of this method can be found in Foster (1996). Results of wavelet analysis are presented in Figures 5­9. Figures 5­7 show the skeletons (a period at which the wavelet is maximum). In all these figures the results for the first and the last 2-year periods are not shown since they are affected by the edge effect. All the periods in the figures are given in solar days, and all amplitudes computed from the smoothed values are to be multiplied by 1.01 to recover the smoothing effect.
1

http://www.aavso.org/cdata/wwz.shtml

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BKG, dPsi, raw 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 GSF, dPsi, raw 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 IAA, dPsi, raw 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 USN, dPsi, raw 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 1 0.5 1 0.5 1 0.5 1 0.5

BKG, dPsi, raw

0 300 325 350 375 400 425 450 475 500 525 550 GSF, dEps, raw

0 300 325 350 375 400 425 450 475 500 525 550 IAA, dEps, raw

0 300 325 350 375 400 425 450 475 500 525 550 USN, dEps, raw

0 300 325 350 375 400 425 450 475 500 525 550

Figure 2: Spectra of raw data, Ferraz-Mello's method, solar days.

BKG, dPsi, smoothed 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 GSF, dPsi, smoothed 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 IAA, dPsi, smoothed 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 USN, dPsi, smoothed 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 Mean, dPsi, smoothed 1 0.5 0 300 325 350 375 400 425 450 475 500 525 550 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5

BKG, dEps, smoothed

0 300 325 350 375 400 425 450 475 500 525 550 GSF, dEps, smoothed

0 300 325 350 375 400 425 450 475 500 525 550 IAA, dEps, smoothed

0 300 325 350 375 400 425 450 475 500 525 550 USN, dEps, smoothed

0 300 325 350 375 400 425 450 475 500 525 550 Mean, dEps, smoothed

0 300 325 350 375 400 425 450 475 500 525 550

Figure 3: Spectra of smoothed data, Ferraz-Mello's method, solar days.

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Z. Malkin and D. Terentev: Preliminary analysis of the Free Core Nutation from VLBI data.

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Z. Malkin and D. Terentev: Preliminary analysis of the Free Core Nutation from VLBI data.

BKG, dPsi, smoothed 1 0.5 0 380 390 400 410 420 430 440 450 460 470 480 GSF, dPsi, smoothed 1 0.5 0 380 390 400 410 420 430 440 450 460 470 480 IAA, dPsi, smoothed 1 0.5 0 380 390 400 410 420 430 440 450 460 470 480 USN, dPsi, smoothed 1 0.5 0 380 390 400 410 420 430 440 450 460 470 480 Mean, dPsi, smoothed 1 0.5 0 380 390 400 410 420 430 440 450 460 470 480 1 0.5 1 0.5 1 0.5 1 0.5 1 0.5

BKG, dEps, smoothed

0 380 390 400 410 420 430 440 450 460 470 480 GSF, dEps, smoothed

0 380 390 400 410 420 430 440 450 460 470 480 IAA, dEps, smoothed

0 380 390 400 410 420 430 440 450 460 470 480 USN, dEps, smoothed

0 380 390 400 410 420 430 440 450 460 470 480 Mean, dEps, smoothed

0 380 390 400 410 420 430 440 450 460 470 480

Figure 4: Spectra of smoothed data, Burg's method, solar days.

Table 1: Periods of the FCN contribution, solar days. Series Method BKG dPsi Raw Smoothed Smoothed Smoothed Ferraz-Mello Ferraz-Mello Burg Burg (­2000.2) 435.0 434.2 430.6 433.4 dEps Raw Smoothed Smoothed Smoothed Ferraz-Mello Ferraz-Mello Burg Burg (­2000.2) 435.4 435.4 438.4 431.7 432.2 433.1 438.6 428.5 435.0 432.9 436.2 429.5 432.9 433.5 438.7 430.1 -- 433.5 438.8 429.9 432.2 432.7 431.0 430.0 434.4 430.3 434.3 428.5 433.7 433.7 433.1 433.4 -- 432.5 431.9 431.9 GSF IAA USN Mean

Mean of dPsi and dEps Raw Smoothed Smoothed Smoothed Ferraz-Mello Ferraz-Mello Burg Burg (­2000.2) 435.2 434.8 434.5 432.6 432.2 432.9 434.8 429.2 434.7 431.6 435.2 429.0 433.3 433.6 435.9 431.8 -- 433.0 435.4 430.9

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Figure 9 presents the final results of the present investigation. It should be mentioned that based on comparison of FCN amplitudes found here and previous investigations (Malkin, 2002), we consider the results obtained before 1990 as not very reliable. Of course, an important question arising from the obtained result is whether the variations of the period found from our analysis is an actual geophysical signal or an artifact caused by inadequate computational procedures. One can see that large increasing of the FCN period after 1998 corresponds to relatively low amplitude of the FCN oscillation. We have performed some tests to estimate how result of wavelet analysis depend on variable amplitude of input signal. Our conclusion is that found variations of the FCN period cannot be explained by computational errors. Besides, results of spectral analysis made for different subset of data also corroborate our conclusion.

4

Discussion and conclusions
The results of our investigations allow us to make some preliminary conclusions. The FCN period most likely varies with time. Probably, change in the period is physically connected with change in amplitude. On the other hand, one can see that the variations of the FCN period show clear periodicity with a period about 5 years, whereas variations of the FCN amplitude does not show such an effect. Another reason of the observed behavior of the FCN period maybe a jump(s) in the FCN phase. Analogous effect was found also at the Chandler frequency (Vondrak, 1988), for which dependence of the period on amplitude, and the phase jump occurred during the period of the lowest amplitude were also found. It is interesting, that the Chandler wobble period also decreased in 1986­1988, and increased in 1989­1996 (see HЁ fner, 2003, Schuh op et al., 2001). Unfortunately, Polar Motion series studied in those papers are much shorter than one analyzed here to perform a reliable comparison. Variations of FCN amplitudes show several possible epochs of the excitation of the FCN, most of them are close to ones detected in Shirai and Fukushima (2001). Some tests we performed allow us to make a conclusion that investigated nutation series really contain such a signal with variable amplitude and period, however it's not clear if this corresponds to a known geophysical process(es). As stated above, we interpret the differences between observed nutation and the IAU2000A model as FCN contribution, which may be too strong assumption. We plan to continue our analysis to detect and investigate more carefully the complex observed signal and possible geophysical interpretation.

Acknowledgments
Authors are grateful to V. Vityazev for useful discussion of methods and results of spectral and wavelet analysis.

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Z. Malkin and D. Terentev: Preliminary analysis of the Free Core Nutation from VLBI data.

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Working Meeting on Europ ean VLBI for Geo desy and Astrometry

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Z. Malkin and D. Terentev: Preliminary analysis of the Free Core Nutation from VLBI data.

BKG, dPsi, raw 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 GSF, dPsi, raw 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 IAA, dPsi, raw 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 USN, dPsi, raw 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 510 470 430 510 470 430 510 470 430 510 470 430

BKG, dEps, raw

390 1986 1988 1990 1992 1994 1996 1998 2000 2002 GSF, dEps, raw

390 1986 1988 1990 1992 1994 1996 1998 2000 2002 IAA, dEps, raw

390 1986 1988 1990 1992 1994 1996 1998 2000 2002 USN, dEps, raw

390 1986 1988 1990 1992 1994 1996 1998 2000 2002

Figure 5: Variations of the FCN period with time, raw data, solar days.

BKG, dPsi, smoothed 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 GSF, dPsi, smoothed 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 IAA, dPsi, smoothed 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 USN, dPsi, smoothed 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 Mean, dPsi, smoothed 510 470 430 390 1986 1988 1990 1992 1994 1996 1998 2000 2002 510 470 430 510 470 430 510 470 430 510 470 430 510 470 430

BKG, dEps, smoothed

390 1986 1988 1990 1992 1994 1996 1998 2000 2002 GSF, dEps, smoothed

390 1986 1988 1990 1992 1994 1996 1998 2000 2002 IAA, dEps, smoothed

390 1986 1988 1990 1992 1994 1996 1998 2000 2002 USN, dEps, smoothed

390 1986 1988 1990 1992 1994 1996 1998 2000 2002 Mean, dEps, smoothed

390 1986 1988 1990 1992 1994 1996 1998 2000 2002

Figure 6: Variations of the FCN period with time, smoothed data, solar days.

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BKG, dPsi*sin(Eps), raw 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 GSF, dPsi*sin(Eps), raw 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 IAA, dPsi*sin(Eps), raw 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 USN, dPsi*sin(Eps), raw 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 0.4 0.2 0.4 0.2 0.4 0.2 0.4 0.2

BKG, dEps, raw

0 1986 1988 1990 1992 1994 1996 1998 2000 2002 GSF, dEps, raw

0 1986 1988 1990 1992 1994 1996 1998 2000 2002 IAA, dEps, raw

0 1986 1988 1990 1992 1994 1996 1998 2000 2002 USN, dEps, raw

0 1986 1988 1990 1992 1994 1996 1998 2000 2002

Figure 7: Variations of the FCN amplitude with time, raw data.
BKG, dPsi*sin(Eps), smoothed 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 GSF, dPsi*sin(Eps), smoothed 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 IAA, dPsi*sin(Eps), smoothed 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 USN, dPsi*sin(Eps), smoothed 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 Mean, dPsi*sin(Eps), smoothed 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 Mean, dEps, smoothed 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 USN, dEps, smoothed 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 IAA, dEps, smoothed 0.4 0.2 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 GSF, dEps, smoothed BKG, dEps, smoothed

Figure 8: Variations of the FCN amplitude with time, smoothed data.

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Z. Malkin and D. Terentev: Preliminary analysis of the Free Core Nutation from VLBI data.

Working Meeting on European VLBI for Geodesy and Astrometry


Working Meeting on Europ ean VLBI for Geo desy and Astrometry

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Z. Malkin and D. Terentev: Preliminary analysis of the Free Core Nutation from VLBI data.

Period, solar days 500 480 460 440 420 400 1986 1988 1990 1992 1994 1996 1998 2000 2002

Amplitude, mas 0.4

0.3

0.2

0.1

0 1986 1988 1990 1992 1994 1996 1998 2000 2002
Figure 9: Variations of the FCN period and amplitude with time; sin() (solid line), (dashed line), and mean of and (bold line).

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References
Herring, T. A., Mathews, P. M., Buffet, B. A.: Modelling of NutationPrecession: Very long baseline interferometry results. J. Geophys. Res., 2002, JB000165. Shirai, T., T. Fukushima: Did Huge Earthquake Excite Free Core Nutation. J. Geodetic Soc. Japan, 2001, 47, No 1, 198­203. Dehant, V., P. Defraigne: New Transfer Functions for Nutation of a Nonrigid Earth. J. Geophys. Res., 1997, 102, 27659­27687. Shirai, T., T. Fukushima: Construction of a New Forced Nutation Theory of the Nonrigid Earth. Astron. J., 2001, 121, 3270­3283. Ferraz-Mello, S.: Estimation of periods from unequally spaced observations. Astron. J., 1981, 86, No 4, 619­624. Marple, S. L., Jr.: Digital Spectral Analysis with Applications. PrenticeHall, Inc., Englewood Cliffs, N. J., 1987. Foster, G.: Wavelets for period analysis of unevenly sampled time series. Astron. J., 1996, 112, No 4, 1709­1729. Malkin, Z.: A Comparison of the VLBI Nutation Series with IAU2000 Model. In.: IVS 2002 General Meeting Proceedings, eds. N. R. Vandenberg, K. D. Baver, NASA/CP-2002-210002, 2002, 335­339. Vondrak J.: Is Chandler frequency constant? In: A. K. Babcock, G. A. Willis (eds.), The Earth's Rotation and Reference Frames for Geodesy and Geodynamics, Proc. IAA Symp. 128, 1988, 359­ 364. HЁ ner, J.: Low-frequency variations, Chandler and annual wobbles opf of polar motion as observed over one century. Scientific Technical Report STR03/01, GeoForschungsZentrum Potsdam, Germany. Schuh H., S. Nagel, T. Seitz. Linear drift and periodic variations observed in long time series in polar motion. J. Geod., 2001, 74, No 10, 701­710.

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