Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.gao.spb.ru/english/personal/malkin/publ/zm13c.pdf Äàòà èçìåíåíèÿ: Fri Nov 8 18:00:59 2013 Äàòà èíäåêñèðîâàíèÿ: Thu Feb 27 23:20:39 2014 Êîäèðîâêà:

Advancing the Physics of Cosmic Distances Proceedings IAU Symposium No. 289, 2012 Richard de Grijs, ed.

c Intern atio nal A stro nom ical U nion 2013 doi:10.1017/S1743921312021825

Statistical analysis of the determinations of the Sun's Galacto centric distance
Zinovy Malkin1
1 2

,2

Pulkovo Observatory, St. Petersburg 196140, Russia St. Petersburg State University, St. Petersburg 198504, Russia email: malkin@gao.spb.ru

Abstract. Based on several tens of R0 measurements made during the past two decades, several studies have b een p erformed to derive the b est estimate of R0 . Some used just simple averaging to derive a result, whereas others provided comprehensive analyses of p ossible errors in published results. In either case, detailed statistical analyses of data used were not p erformed. However, a computation of the b est estimates of the Galactic rotation constants is not only an astronomical but also a metrological task. Here we p erform an analysis of 53 R0 measurements (published in the past 20 years) to assess the consistency of the data. Our analysis shows that they are internally consistent. It is also shown that any trend in the R0 estimates from the last 20 years is statistically negligible, which renders the presence of a bandwagon effect doubtful. On the other hand, the formal errors in the published R0 estimates improve significantly with time. Keywords. Galaxy: center, Galaxy: fundamental parameters

1. Intro duction
Accurate knowledge of the distance from the Sun to the center of the Galaxy, R0 , is important in many fields of astronomy and space science. In particular, the primary motivation for this study was a wish to improve the accuracy of modeling the Galactic aberration (Malkin 2011). Over the past decades, many tens of R0 determinations have been made, making use of different principles and observing methods, and thus characterized by different random and systematic errors. Therefore, deriving a best R0 estimate from these data is not only an astronomical but also a metrological task, similar to deriving the best estimates of the fundamental constants in physics. For the latter, several statistical methods have been developed to obtain best estimates--as well as their realistic uncertainties--from heterogeneous measurements. In this study, we applied those methods to the R0 data. Another goal of this study was to investigate a possible trend in the multi-year series of R0 estimates, as discussed by many authors. However, estimates of any such trend differ significantly among papers. Therefore, we tried to clarify this issue using the latest results. More details are provided by Malkin (2012).

2. Data used
For the present study, we used all R0 measurements published in the period 1992­ 2011, with the exception of a number of results that were revised in subsequent papers. We did not use the results of Glushkova et al. (1998; revised by Glushkova et al. 1999), Paczynski & Stanek (1998; revised by Stanek et al. 2000), Eisenhauer et al. (2003; revised by Eisenhauer et al. 2005, which was in turn revised by Gillessen et al. 2009). In total, 53 estimates (listed in Table 1) were used. 406


Statistical analysis of R0 determinations
Table 1. R0 estimates used in this study.
R
0

407

(k p c ) ±(1 ) 0 .8 1 .1 0 .6 0 .4 0. 3 1 .0 0 .5 0 .5 0 .5 1 .0 0. 98 0. 4 0. 7 0 .4 0 .5 0. 54 0 .1 5 0 .4 0. 29 0 .2 1 1 .0 0 .3 0 .3 0. 4 0 .7 0. 42 0. 6 0 .3 0 .1 5 0. 44 0 .6 0 .3 0. 36 0 .7 0 .3 0. 45 0. 35 0. 5 0 .3 5 0 .5 0. 40 0 .3 0 .6 0. 5 0 .7 5 0. 43 0 .3 3 0 .4 0 .6 1 .1 0. 26 0 .2 3 0 .1 6

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7. 8. 7. 7. 8. 7. 7. 8. 7. 8. 8. 7. 7. 8. 8. 7. 8. 7. 8. 8. 8. 7. 7. 8. 8. 8. 8. 8. 7. 8. 8. 7. 7. 8. 7. 7. 8. 8. 8. 8. 7. 7. 8. 7. 7. 8. 8. 7. 8. 8. 7. 8. 8.

9 1 6 6 09 5 0 8 1 3 21 95 55 1 5 66 1 1 51 2 6 4 9 67 2 24 05 3 7 01 7 2 52 1 4 94 07 16 33 7 58 2 4 75 9 24 28 7 1 3 80 3 29

Where both random (statistical) and systematic uncertainties were given, they were summed in quadrature. If two different values were given for the lower and upper boundaries of the confidence interval, the mean value of these boundaries was used as the uncertainty in the result (the lower and upper boundaries were close to each other in all cases, so that this approximation procedure does not significantly affect the final result). Where authors gave several estimates of R0 without a final preference, the unweighted average of these estimates was computed.


408
9.5 9.0 8.5 R0, kpc 8.0 7.5 7.0 6.5 1990

Z. Malkin
1.2 1.0 R0 uncertainty, kpc 1995 2000 Year 2005 2010 0.8 0.6 0.4 0.2 0.0 1990

1995

2000 Year

2005

2010

Figure 1. R0 estimates used in this study. The weighted (solid line) and unweighted (dashed line) trends are also shown.

Figure 2. Uncertainty in R0 estimates used in this study.

3. Results
Our analysis used two approaches. First, we investigated a possible drift in the R0 values (see Fig. 1), which may indicate the presence of a bandwagon effect. We obtained a linear slope in R0 as a function of time of +0.003 ± 0.010 kpc yr-1 for the weighted case and +0.009 ± 0.010 kpc yr-1 for the unweighted case. Thus, any trend in R0 estimates published in the last 20 years is statistically insignificant, which renders a significant bandwagon effect doubtful. This conclusion is confirmed by application of the Abbe criterion (Malkin 2013). The latter paper also contains a detailed discussion on this sub ject. On the other hand, the trend in the reported R0 uncertainties (see Fig. 2) is statistically significant, i.e., -0.0103 ± 0.0053 kpc yr-1 . This is particularly interesting because we expect that there are two tendencies in the changes in the errors in R0 measurements with time. First, the errors should decrease with progress in observational and analysis methods. On the other hand, one can expect that the published errors should increase because most authors of recent papers now pay more attention to the correct computation of the uncertainties in their result. Evidently, the former tendency prevails. To assess the internal consistency of the R0 measurements, several statistical methods were used. They can be divided into two groups. The first group consists of computation of the unweighted and classical weighted means, the author's modification with respect to computation of the mean uncertainty (Malkin 2001), and three other weighted-mean modifications, which allow to account for input data discrepancies, i.e., limitation of relative weights (Nichols 2004), normalized residuals (James et al. 1992), and the Mandel­ Paule method (Paule & Mandel 1982). If all variants of the weighted mean produce the same mean value and standard error, the input data are consistent. The second group of methods includes determination of the median with error estimation (Muller 1995) and ¨ bootstrap median methods. Detailed descriptions of these methods are given in Malkin (2012). The results of the computations are presented in Table 2.

4. Concluding remarks
Although the published R0 estimates were obtained based on different methods and samples of ob jects, they are consistent rather than discrepant. The results of the computation of a mean R0 value obtained using different statistical techniques converge at the level of approximately 0.1 kpc, which is much smaller than the uncertainty in the average value. Note, however, that this conclusion is not evident. Similar analysis of 0 measurements has shown that the latter are much less consistent.


Statistical analysis of R0 determinations

409

Table 2. Average estimates of R0 obtained with different statistical techniques (see text). Metho d Unweighted mean Classical weighted mean Mo dified weighted mean Limitation of relative weights Normalized residuals Mandel­Paule metho d Median Bo otstrap median 7 7 7 7 7 7 8 8 . . . . . . . . R 0 , kp c 979 967 967 967 967 958 090 060 ± ± ± ± ± ± ± ± 0 0 0 0 0 0 0 0 . . . . . . . . 061 048 073 048 048 058 062 072

As significant experience in deriving best estimates of the physical constants has shown, using various statistical methods to evaluate the best R0 estimate is very important to assess data consistency and obtain a realistic uncertainty in the final result. Therefore, careful astronomical consideration of the published measurements should be accompanied by a careful statistical analysis. It should be recognized that the computation of the new conventional IAU R0 value is not only an astronomical, but also a metrological task. Another result of this study is that any trend in the R0 estimates obtained during the last 20 years is statistically insignificant, which makes it unlikely that R0 results are significantly affected by a bandwagon effect. On the other hand, the formal errors in the published R0 estimates improve significantly with time. Note that the average value R0 = 8.0 ± 0.25 kpc computed in this study differs from the results of the latest direct measurements obtained from stellar orbits about Sgr A*, trigonometric parallaxes to Sgr B2, and over 60 trigonometric parallaxes of masers, which give R0 = 8.4 ± 0.2 kpc (M. J. Reid, priv. comm.), although these values are still statistically consistent. On the other hand, it seems important to properly combine all results obtained based on different methods, because this provides better systematic stability of the average result. Indeed, the systematic and random errors of all these results should be assessed and the corresponding correction should be applied when possible before averaging (although this is a very difficult task). References
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