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AN ANALYSIS OF SOURCE MOTIONS DERIVED FROM POSITION TIME SERIES
Z. MALKIN, E. POPOVA Central Astronomical Observatory at Pulkovo of RAS Pulkovskoe Ch., 65, St. Petersburg 196140, Russia e-mail: malkin@gao.spb.ru

ABSTRACT. In this paper, an attempt is made to extract a systematic part from the source apparent
motions obtained from the position time series provided by nine IVS Analysis Centers in the framework of the ICRF-2 pro ject. Our preliminary results show that the radio source velocities and the parameters of the systematic part of the velocity field differ substantially between the source position time series.

1. INTRODUCTION
Many radio sources observed during astrometric/geodetic VLBI sessions show progressive variations in its position derived from single session solutions. Several physical effects can cause systematic apparent movement of celestial ob jects. Hence investigation of the radio source apparent velocity field can help in investigations in various fields, such as fundamental physics, cosmology, etc. Several analysis strategies for computation of systematic part in the radio source velocities can be used: a) estimate source position and velocities from global solution, then fit spherical harmonics to the velocities (Gwinn et al. 1997); b) compute the coefficients of spherical harmonics as global parameters (MacMillan 2005; Titov 2008); c) compute velocities from position time series, then fit spherical harmonics to the velocities. In this paper, we will test the latter approach which, hopefully, can provide a possibility for supplement comparisons and accuracy assessment. For this work, we have used 26 source position time series computed at nine VLBI analysis centers in the framework of the ICRF-2 pro ject1 making use of six different software, which provides a good opportunity for comparisons. For more rigorous comparison we also selected the data at common epochs for 17 series. Only the time series having at least 5 sessions and 3-year time span were used. Time series statistics is shown in Table 1.

2. COMPARISON OF VELOCITIES AND SPHERICAL HARMONICS
The source velocities were computed as weighted linear drift of the submitted source positions with weights inversely proportional to the reported variances of source positions. Since some time series contain positions with unlikely small errors (down to 1 µas in the iaa series), which leads to problems with computing the velocity as the weighted trend, it was decided to use a minimal error value of 20 µas, i.e. all errors less then this value were replaced by 20 µas. No series except iaa were substantially affected by this procedure. At this stage we could compare both values of the source velocities and their errors obtained from different time series. Comparison of velocities showed that they can differ by several times. Median errors in velocities are shown in Table 1; they can serve as an index of the scatter of position time series. One can see that some time series are much more noisy than others. Then we compute the coefficients of two spherical harmonics H12 and H3 using the following formulas (Titov 2008): µ = -H
12

sin 2 ,

µ = -

H 4

12

cos 2 sin 2 +

H3 sin 2. 2

The results of computation are presented in Table 1. It can be noted that using more strict criteria, such as minimum 10 sessions and 10 years of observations gives statistically similar result, with smaller value of the formal error in the harmonics coefficients when more observations are used. In the last row
1

http://ivscc.gsfc.nasa.gov/ivsmisc/ICRF2/timeseries

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of the table the results are presented corresponding to the cumulative solution including all the velocity estimates in the input time series. Table 1: Median errors in velocities V cos , V , and spherical harmonics H12 , H3 . Unit: µas
Series aus000a aus001a aus002a aus003a bkg000c dgf000a dgf000b dgf000c dgf000d dgf000e dgf000f dgf000g gsf001a gsf002a iaa000b iaa000c mao000b opa000a opa000b opa001a opa002a sai000b usn000d usn001a All data Nsou 71 343 308 322 537 277 476 476 476 476 531 531 582 592 458 481 555 384 510 392 511 501 572 572 V cos 7 19 18 18 14 19 15 15 15 15 16 16 13 13 15 16 23 15 16 15 17 25 13 21 All V 11 28 26 29 18 25 18 21 19 19 23 19 17 17 22 22 31 19 23 18 19 37 18 29 sessions H12 -8.76±3.15 -4.53±1.08 -0.34±1.22 -3.80±1.12 -1.01±0.83 -3.84±1.44 -1.66±0.73 -0.27±0.90 -1.94±0.75 -2.08±0.73 -0.29±0.86 -2.09±0.71 -0.62±0.70 -0.39±0.65 1.23±1.37 1.15±1.34 0.05±1.05 0.11±1.10 -6.14±1.09 0.20±0.99 0.66±0.87 -1.96±1.18 -0.70±0.78 -5.94±1.22 -1.24±0.19 H 3 1.02±2.33 -2.13±0.87 -0.47±0.99 -2.47±0.90 0.85±0.74 4.88±1.49 1.12±0.70 1.23±0.76 1.14±0.72 1.08±0.71 1.27±0.71 1.18±0.67 -0.09±0.64 0.64±0.59 0.80±1.16 2.16±1.15 1.01±0.86 -0.46±0.95 0.46±0.88 -0.37±0.87 -0.53±0.77 1.79±0.95 0.18±0.70 1.53±1.02 0.62±0.17 Nsou -- -- -- -- 350 -- 350 350 350 350 350 350 350 350 350 350 350 -- 350 -- 350 350 350 350 Comm V cos V ­ ­ ­ ­ ­ ­ ­ ­ 17 21 ­ ­ 18 21 19 26 19 23 19 22 19 26 19 23 15 19 17 20 18 22 20 23 25 34 ­ ­ 19 28 ­ ­ 16 20 30 45 17 20 25 34 on sessions H12 -- -- -- -- -0.27±0.91 -- -1.06±0.74 0.88±0.93 -1.47±0.77 -1.71±0.77 0.85±0.97 -1.83±0.80 -0.77±0.86 0.50±0.83 3.49±1.21 2.75±1.24 0.14±1.26 -- -10.55±1.52 -- -0.21±0.94 0.27±1.37 -0.25±0.90 -6.57±1.67 -0.54±0.23 H 3 -- -- -- -- -1.27±0.83 -- -0.73±0.71 -1.08±0.80 -0.71±0.73 -0.57±0.73 -0.88±0.81 -0.43±0.75 -0.76±0.78 -1.46±0.76 1.70± 1.08 3.44± 1.17 -0.35±1.07 -- -1.01±1.21 -- 0.15± 0.85 -1.39±1.14 -1.44±0.81 -0.28±1.38 -0.51±0.21

3. CONCLUDING REMARKS
Although most results obtained in this paper are formally statistically reliable, they differ substantially between input time series, and also between various sets of data selected. This fact, along with results of velocity comparison, may indicate that source position time series should be used with care for analysis of the fine effects in the source motions. Further study is needed to investigate a possibility to use combined or cumulative solution as the most reliable estimate of spherical harmonics. In particular, careful selection of input series should be performed. For instance, in our cumulative solution dgf data are clearly overweighted due to 6 series used, often with very similar position estimates. On the other hand, it seems to be inappropriate to use only one series from one analysis center because some centers compute two and more series using quite different approaches, and this would be important to compare all of them, because there is no indisputable proof in favor of only one approach.

4. REFERENCES
Gwinn, C.R., Eubanks, T.M., Pyne, T., Birkinshaw M., Matsakis D.N., 1997, "Quasar proper motions and low-frequency gravitational waves", ApJ 485, pp. 87-91. Macmillan, D.S., 2005, "Quasar Apparent Proper Motion Observed by Geodetic VLBI Networks", In: Future Directions in High Resolution Astronomy: The 10th Anniversary of the VLBA, ASP Conference Proceedings, V. 340. Eds. J. Romney, M. Reid, San Francisco, 2005, pp. 477­481. Titov. O., 2008, "Proper motions of reference radio sources", In: Proc. Journ´es Syst`mes de R´f´rence e e ee Spatio-temporels 2007, Meudon, France, 17-19 Sep 2007, N. Capitaine (ed.), pp. 16­19.

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