Search for gamma-ray burst classes with the RHESSI satellite

J. Řípa; A. Mészáros; C. Wigger; D. Huja; R. Hudec; W. Hajdas

doi:10.1051/0004-6361/200810913

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Volume 498 / No 2 (May I 2009)

A&A, 498 2 (2009) 399-406

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Issue		A&A Volume 498, Number 2, May I 2009


Page(s)		399 - 406
Section		Extragalactic astronomy
DOI		https://doi.org/10.1051/0004-6361/200810913
Published online		19 March 2009

Search for gamma-ray burst classes with the RHESSI satellite

J. Rípa¹ - A. Mészáros¹ - C. Wigger^2,4 - D. Huja¹ - R. Hudec³ - W. Hajdas²

1 - Charles University, Faculty of Mathematics and Physics, Astronomical Institute, V Holesovickách 2, 180 00 Prague 8, Czech Republic
2 - Paul Scherrer Institute, 5232 Villigen, Switzerland
3 - Astronomical Institute, Academy of Sciences of the Czech Republic, 251 65 Ondrejov, Czech Republic
4 - Kantonsschule Wohlen, 5610 Wohlen, Switzerland

Received 4 September 2008 / Accepted 2 February 2009

Abstract
Aims. We study a sample of 427 gamma-ray bursts (GRBs), measured by the RHESSI satellite, statistically with respect to duration and hardness ratio.
Methods. Standard statistical tests were used, such as $\chi ^2$ , F-test, and the maximum likelihood ratio test, to compare the number of GRB groups in the RHESSI database with that of the BATSE database.
Results. Previous studies based on the BATSE Catalog claim the existence of an intermediate GRB group, besides the long and short groups. Using only the GRB duration T₉₀ as information and $\chi ^2$ or F-test, we do not find any statistically significant intermediate group in the RHESSI data. However, maximum likelihood ratio test reveals a significant intermediate group. Using the 2-dimensional hardness/T₉₀ plane, the maximum likelihood analysis also reveals a significant intermediate group. In contrast to the BATSE database, the intermediate group in the RHESSI data set is harder than the long one.
Conclusions. The existence of an intermediate group follows not only from the BATSE data set, but also from the RHESSI one.

Key words: gamma rays: bursts

1 Introduction

In the years 1991 - 2000, approximately 3000 gamma-ray bursts (GRBs) were detected by the BATSE instrument onboard the Compton Gamma-Ray Observatory (Meegan et al. 2001). After the end of this mission (June 2000), the frequency of GRB detections decreased (to ${\sim} 100{-}200$ GRBs annually) due to different observational methods employed by the operating satellites. Any observational GRB database, compiled since the year 2000 and later, can therefore be of great significance, such as the GRB observations completed by the RHESSI satellite (Holman et al. 2008) $\sim$ 70/year.

Originally it was found (Kouveliotou et al. 1993) that two GRB classes exist: a short one of duration ${\la}2$ s and a long one of duration ${\ga} 2$ s. This was confirmed by GRB data from the Konus-Wind instrument (Aptekar et al. 1998). However, some articles point to the existence of three classes of GRBs in the BATSE database with respect to their durations (Horváth 1998, 2002). By applying a maximum likelihood ratio test to durations, the work of Horváth et al. (2008) indicates that there is a statistically significant intermediate group in the Swift data set. Horváth et al. (2004) and Horváth et al. (2006) also claimed that, when using a 2-dimensional plane of hardness ratio versus duration, three classes of GRBs can be found in the BATSE data set. Mukherjee et al. (1998) pointed to the existence of three GRB classes in multiparameter space. In another multidimensional analysis of the BATSE catalog by Chattopadhyay et al. (2007), it was argued that at least three clusters of GRBs are found. Some articles also say that the third class (of intermediate duration), observed by BATSE, is a bias caused by an instrumental effect (Hakkila et al. 2000). In Hakkila et al. (2004), there is a review and discussion of GRB classification, based on statistical clustering and data-mining techniques, placing the intermediate group as a separate source population in doubt.

The purpose of this paper is to investigate the number of GRB groups in another data set, namely in the GRB data set provided by the RHESSI satellite. Although the main goal of the RHESSI satellite is the study of solar physics, it also contains a useful set of GRB observations covering the period 2002 - 2004. Hence, its study can be maximally useful. Trivially, any comparison of different catalogs using different instruments is useful for an independent confirmation of previous results.

In the first step, we analyse the 1-dimensional duration distribution of GRBs observed by RHESSI, and in the second step, we use the two-dimensional plane of hardness ratio versus duration. To determine the number of GRB groups, standard statistical tests described in Trumpler & Weaver (1953), Press et al. (1992), and Zey et al. (2006) are used.

The paper is organized as follows. In Sect. 2, the RHESSI satellite and the analysed data set are described. In Sect. 3, we present the duration distribution for the RHESSI GRBs and its analysis. In Sect. 4, the 2-dimensional hardness ratio versus duration distribution and the maximum likelihood fit of these data are presented. In Sects. 5 and 6, we present our discussion and conclusions. At the end of our paper, we list the RHESSI data sample.

2 The RHESSI data sample

The Ramaty High Energy Solar Spectroscopic Imager (RHESSI) is a NASA Small Explorer satellite designed to study hard X-rays and gamma-rays from solar flares (Lin et al. 2002). It consists mainly of an imaging tube and a spectrometer. The spectrometer consists of nine germanium detectors (7.1 cm in diameter with a height of 8.5 cm) (Smith et al. 2002). They are only lightly shielded, ensuring that RHESSI is also useful for detecting non-solar photons from any direction (Smith et al. 2003). The energy range for GRB detection extends from about 30 keV to 17 MeV. For a wide range of energies and GRB incoming directions, the effective area is around 150 cm² (Wigger et al. 2006b). With a field of view of about half of the sky, RHESSI observes about one or two GRBs per week. Photon hits are stored event-by-event in onboard memory with a time sampling of $\Delta t$ = 1 $\mu$ s resolution. The energy resolution for lines is excellent at $\Delta E$ = 3 keV for 1000 keV.

We used the RHESSI GRB Catalog (Wigger et al. 2008) and the Cosmic Burst List (Hurley 2008) to detect 487 GRBs in the RHESSI data between 2002 February 14 and 2008 April 25. We should describe the strategy by which RHESSI GRBs were found. There is no automatic GRB search routine. Only if there is a message from any other instrument of the IPN (Hurley 2007), the RHESSI data are searched for a GRB signal. Therefore, in our data set there are only GRBs, which are also observed by other instruments. The biggest overlap is with Konus-W. About 85% of all RHESSI GRBs are also observed by Konus-W (Wigger et al. 2006a).

For a deeper analysis, we chose a subset of 427 GRBs with data of a signal/noise ratio higher than 6. We used the SolarSoftWare (Freeland et al. 2008) program developed in the Interactive Data Language (RSI IDL) programming application as well as our own IDL routines to derive count light curves (with a time resolution higher than 10% of the burst's duration for the vast majority of our entire data set) and count fluences from the rear detectors' segments (except number R2) of the RHESSI spectrometer (Smith et al. 2002) in the energy band from 25 keV to 1.5 MeV. This data set (and the time resolutions of derived light curves) are listed in Table 7.

3 Duration distribution

First, we study the one-dimensional duration distribution. We use T₉₀ as the GRB duration, i.e., the time interval during which the cumulative counts increase from 5% to 95% above background (Meegan et al. 2001). The T₉₀ uncertainty consists of two components. We make an assumption that one is given by the count fluence uncertainty during T₉₀ ( $\delta t_{\rm s}$ ), which is given by Poissonian noise, and the second one is the time resolution of derived light curves ( $\delta t_{\rm res}$ ). The total T₉₀ uncertainty $\delta t$ was calculated to be $\delta t = \sqrt{ \delta t_{\rm s}^2 + \delta t_{\rm res}^2 } .$

The histogram of the times T₉₀ is a distribution with two maxima at approximately 0.2 s and 20 s (Fig. 1). The histogram consists of 19 equally wide bins on a logarithmic scale (of base 10) starting at 0.09 s and ending at 273.4 s.

We follow the method completed by Horváth (1998) and fitted one, two (Fig. 1), and three (Fig. 2) log-normal functions and used the $\chi ^2$ test to evaluate these fits. The minimal number of GRBs per bin is 4 (last bin), and hence the use of the $\chi ^2$ test is possible.

In the case of the fit with one log-normal function, we obtained $\chi^2 \simeq 157$ for 17 degrees of freedom (d.o.f.). Therefore, this hypothesis is rejected on a smaller than 0.01% significance level.

The fit with two log-normal functions is shown in Fig. 1 and the fit with three log-normal functions in Fig. 2. The parameters of the fits, the values of $\chi ^2$ , the degrees of freedom, and the goodness-of-fits are listed in Table 1.

The assumption of two groups being represented by two log-normal fits is acceptable, the fit with three log-normal functions even more. The question is whether the improvement in $\chi ^2$ is statistically significant. To answer this question, we used the F-test, as described by Band et al. (1997), in their Appendix A. The F-test gives a probability of 6.9% of the improvement in $\chi ^2$ being accidental. This value is remarkably low, but not low enough to reject the hypothesis that two log-normal functions are still enough to describe the observed duration distribution.

$\begin{figure} \par\includegraphics[width=90mm,clip]{0913fig1.eps} \end{figure}$	Figure 1: Duration distribution of the 427 RHESSI bursts with the best $\chi ^2$ fit of two log-normal functions. Number of bins is 19, $\rm d.o.f. = 14$ and $\chi ^2 \simeq 19.1$ which implies the goodness-of-fit ${\simeq } 16\%$ . The bar errors are standard deviations of the number of GRBs per bin for ten different simulated duration distributions as described in the text.
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$\begin{figure} \par\includegraphics[width=90mm,clip]{0913fig2.eps} \end{figure}$	Figure 2: Duration distribution of the 427 RHESSI bursts with the best $\chi ^2$ fit of three log-normal functions. Number of bins is 19, $\rm d.o.f. = 11$ and $\chi ^2 \simeq 10.3$ which implies the goodness-of-fit ${\simeq } 50\%$ . The bar errors are the same as described in Fig. 1.
Open with DEXTER

To determine how the T₉₀ uncertainties affect our result, we randomly selected one half of the bursts and shifted their durations by the full amount of their uncertainties to lower values and the second half to higher values. We then compiled a histogram and recalculated the best-fit model parameters, $\chi ^2$ , and F-test. The results for ten such calculations are listed in Table 2. This method also gives us information about how the fitted parameters vary, and thus informs us of their uncertainties. From Table 2, we see that, on average, the improvement in $\chi ^2$ is insignificant. Therefore, we cannot proclaim acceptance of three groups by using this statistical method.

Since the number of GRBs is low for many bins, we also used the maximum likelihood method (see Horváth 2002, and the references therein) to fit two and three log-normal functions to the RHESSI data set. The parameters are listed in Table 3.

Since the difference of the logarithms of the likelihoods $\Delta \ln L = 9.2$ should be half of the $\chi ^2$ distribution for 3 degrees of freedom (Horváth 2002), we infer that the introduction of a third group is statistically significant on the 0.036% level (of being accidental).

To derive an image of how the T₉₀ uncertainties affect our result, we proceed as in the $\chi ^2$ fitting and generate ten different data sets randomly changed in durations by the full amount of their uncertainties. The results are presented in Table 4. From this table, it can be seen that all ten simulations give probabilities, of introducing a third group being accidental, that are much lower than 5%. Thus, the hypothesis of introducing a third group is highly acceptable.

Table 1: Parameters of the best $\chi ^2$ fits of two and three log-normal functions on the RHESSI GRB T₉₀ distribution. $\mu$ are the means, $\sigma$ are the standard deviations and w are the weights of the distribution. Given uncertainties are standard deviations of the parameters obtained by ten different fittings of randomly changed histogram of durations by their uncertainties.

Table 2: The minimal $\chi ^2$ , corresponding goodness-of-fits and F-tests for fitted two and three log-normal functions on the RHESSI GRB T₉₀ distribution for ten different changes of durations by their uncertainties.

Table 3: Parameters of the best fit with two and three log-normal functions done by the maximum likelihood method on the RHESSI data. $\mu$ are the means, $\sigma$ are the standard deviations, w are the weights of the distribution and L₂, L₃ are the likelihoods. Given uncertainties are standard deviations of the parameters obtained by ten different fittings of data sets, in which the durations were randomly changed by their uncertainties.

Table 4: The maximal likelihoods and corresponding probabilities that introducing of the third group is accidental for maximum likelihood fittings (one-dimensional) with two and three log-normal functions of ten different changes of durations by their uncertainties.

4 Hardness ratio versus duration

Two-dimensional scatter plots of RHESSI GRBs are shown in Figs. 3 and 4. One axis is the duration T₉₀, used in the previous section, the other axis is a hardness ratio. The hardness ratio is defined to be the ratio of two fluences F in two different energy bands integrated over the time interval T₉₀. For the RHESSI data set, we used the energy bands (25 - 120) keV and (120 - 1500) keV, i.e., H=F_120-1500/F_25-120.

Using the maximum likelihood method (see Horváth et al. 2004, 2006 and the references therein), we fit two and three bivariate log-normal functions to search for clusters. In Fig. 3., we show the best-fit solution of two bivariate log-normal functions (11 independent parameters, since the two weights must add up to 100%).

The parameters are listed in Table 5. One result is that the short GRBs are on average harder than long GRBs. After a closer look at the GRB distribution within the short class, one can see that the points within the 1 $\sigma$ ellipse are not evenly distributed. They cluster towards the shortest durations (Fig. 3).

The fitting of the sum of three groups (17 independent parameters) is shown in Fig. 4. The best-fit model parameters are listed in Table 5. The former short group is clearly separated into two parts. As far as one can tell by sight, the data points scatter evenly within (and around) the 1 $\sigma$ ellipses.

Since the difference in the logarithms of the likelihoods $\Delta \ln L = 10.9$ should equal one half of the $\chi ^2$ distribution for 6 degrees of freedom (Horváth et al. 2006), we find that the introduction of a third group is statistically significant at the 0.13% level (of being accidental).

To derive an image of how GRB durations and hardness ratio uncertainties effect our result, we proceed similarly as in the $\chi ^2$ fitting and generated ten different data sets randomly changed in duration and hardness ratio by the full amount of their uncertainties. The results are presented in Table 6. From this table, it is seen that almost all of the simulations infer probabilities, of introducing the third group being accidental, that are much lower than 5%. Thus, the hypothesis of introducing the third group is highly acceptable.

$\begin{figure} \par\includegraphics[width=9cm,clip]{0913fig3.eps} \end{figure}$	Figure 3: Hardness ratio vs. T₉₀ of the RHESSI GRBs with the best fit of two bivariate log-normal functions.
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$\begin{figure} \includegraphics[width=9cm,clip]{0913fig4.eps} \end{figure}$	Figure 4: Hardness ratio vs. T₉₀ of the RHESSI GRBs with the best fit of three bivariate log-normal functions.
Open with DEXTER

Table 5: Parameters of the best fit with two and three bivariate log-normal functions done by the maximum likelihood method on the RHESSI data. $\mu _{{x}}$ are the means on the x-axis ( $x = \log T_{90}$ ), $\mu _{{y}}$ are the means on the y-axis ( $y = \log H$ ), $\sigma _{{x}}$ are the dispersions on the x-axis, $\sigma _{{y}}$ are the dispersions on the y-axis, r are the correlation coefficients, w are the weights of the distribution and $L_{{\rm 2}}, L_{{\rm 3}}$ are the likelihoods. Given uncertainties are standard deviations of the parameters obtained by ten different fittings of data sets, where the durations and hardness ratios were randomly changed by their uncertainties.

Table 6: The maximal likelihoods and corresponding probabilities that introducing of the third group is accidental for maximum likelihood fittings (two-dimensional) with two and three bivariate log-normal functions of ten different changes of durations and hardness ratios by their uncertainties.

Table 7: The RHESSI GRB data set including I. GRB names which correspond to dates (the letters after GRB names are internal and do not have to be in accordance with e.g. GCN GRB names), II. GRB peak time, III. T₉₀ duration, IV. time resolution $\delta t_{\rm res}$ (described above) and V. hardness ratios.

5 Discussion

The analysis of the one-dimensional duration distribution, by $\chi ^2$ fitting, has revealed the class of so-called long GRBs (about 83% of all RHESSI GRBs) with typical durations from 5 to 70 s, the most probable duration being $T_{90} \approx 19$ s. Another class are short GRBs (about 9% of all RHESSI GRBs) with typical durations from 0.1 to 0.4 s, the most probable duration being $T_{90} \approx 0.21$ s. By fitting 3 log-normal functions, we have identified a third class (about 8% of all RHESSI GRBs) with typical durations from 0.8 to 3 s, the most probable duration being $T_{90} \approx 1.5$ s. The existence of the intermediate class from the RHESSI T₉₀ distribution is not confirmed to a sufficiently high significance using only the $\chi ^2$ fit. However, the maximum likelihood ratio test on the same data reveals that the introduction of a third class is statistically significant. The $\chi ^2$ method might not be as sensitive and hence robust as the likelihood method, because of the low number of bursts in our data-sample (Horváth et al. 2008, $2{\rm nd}$ section, $1{\rm st}$ paragraph).

The hardness ratio versus duration plot for the RHESSI sample demonstrates further the existence of a third class. The typical durations are similar to those obtained with the one-dimensional analysis, the percentages being slightly different ( ${\approx}86$ % long, ${\approx}9$ % short, ${\approx}5$ % intermediate).

Three classes of GRBs have also been identified for the BATSE GRBs (Horváth et al. 2006) and the Swift GRBs (Horváth et al. 2008). For BATSE, ${\approx}65$ % of all GRBs are long, ${\approx}24$ % short, and ${\approx}$ 11% intermediate (Horváth et al. 2006, Table 2 of that article). The typical durations found for BATSE are roughly a factor of 2 longer than for RHESSI, but consistent for all three classes. As is known from BATSE, also in the RHESSI data set, the short GRBs are on average harder than the long GRBs. The most remarkable difference is the hardness of the intermediate class. In the BATSE data, the intermediate class has the lowest hardness ratio, which is anticorrelated with the duration (Horváth et al. 2006), whereas we find for the RHESSI data that its hardness is comparable with that of the short group and correlated with the duration, although this correlation is inconclusive because of its large error. The hardness of the intermediate class found for the RHESSI data is surprising since the intermediate class in the BATSE data was found to be the softest. This discrepancy might by explained by the different definitions of the hardnesses. The hardness H for the RHESSI data is defined as H=F_120-1500/F_25-120, whereas for the BATSE data H=F_100-320/F_50-100, where the numbers denote energy in keV (the BATSE fluences at higher energies than 320 keV are noisy (Bagoly et al. 1998)). This means that hardnesses do not measure the identical behaviour of bursts. The situation differs even more significantly if we compare hardnesses in the Swift and RHESSI databases, because the Swifts' hardnesses are defined as H=F_100-150/F_50-100 and H=F_50-100/F_25-50 (Horváth et al. 2008; Sakamoto et al. 2008).

The shorter durations of the RHESSI GRBs compared to the BATSE GRBs can be understood in the following way. For RHESSI, which is practically unshielded, the background is high (minimum around 1000 counts per second in the (25 - 1500) keV band) and varies by up to a factor 3. Additionally, RHESSI's sensitivity declines rapidly below ${\approx}50$ keV. Weak GRBs (in the sense of counts per second) and soft GRBs are not so well observed by RHESSI. Since GRBs tend to be softer and weaker at later times, they should soon fall bellow RHESSI's detection limit, resulting in a shorter duration being inferred.

For Swift, ${\approx}58$ % of all GRBs are long, ${\approx}7$ % short and ${\approx}35$ % intermediate (Horváth et al. 2008). The percentage of each group depends obviously on the used instrument.

6 Conclusion

The RHESSI data confirm that GRBs can be separated into short and long classes, and that the short GRBs are on average harder than the long ones. A two-dimensional analysis of the hardness/duration plane as well as a maximum likelihood fit of the duration distribution also indicate a third class with intermediate duration and similar hardness as the short class.

Acknowledgements

This study was supported by the GAUK grant No. 46307, by the OTKA grants No. T48870 and K 77795, by the Grant Agency of the Czech Republic grant No. 205/08/H005, by the Research Program MSM0021620860 of the Ministry of Education of the Czech Republic, by the INTEGRAL PECS Project 98023 and by the grant GA CR 205/08/1207. We appreciate help of K. Hurley with the RHESSI GRB list, valuable discussion with L.G. Balázs and useful remarks of O. Wigger. Thanks are due to the anonymous referee for the worthwhile notes.

References

Aptekar, R. L., Butterworth, P. S., Cline, T. L., et al. 1998, AIPC, 428, 10 [NASA ADS] (In the text)
Band, D. L., Ford, L. A., Matteson, J. L., et al. 1997, ApJ, 485, 747 [NASA ADS] [CrossRef], Appendix A (In the text)
Bagoly, Z., Mészáros, A., Horváth, I., Balázs, L. G., & Mészáros, P. 1998, ApJ, 498, 342 [NASA ADS] [CrossRef] (In the text)
Chattopadhyay, T., Misra, R., Chattopadhyay, A. K., & Naskar, M. 2007, ApJ, 667, 1017 [NASA ADS] [CrossRef] (In the text)
Freeland, S. L., et al. 2008, http://www.lmsal.com/solarsoft (In the text)
Hakkila, J., Haglin, D. J., Pendleton, G. N., et al. 2000, ApJ, 538, 165 [NASA ADS] [CrossRef] (In the text)
Hakkila, J., Giblin, T. W., Roiger, R. J., et al. 2004, Balt. Astron., 13, 211 [NASA ADS] (In the text)
Holman, G. D. 2008, http://hesperia.gsfc.nasa.gov/hessi (In the text)
Horváth, I. 1998, ApJ, 508, 757 [NASA ADS] [CrossRef]
Horváth, I. 2002, A&A, 392, 791 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Horváth, I., Mészáros, A., Balázs, L. G., & Bagoly, Z. 2004, Balt. Astron., 13, 217 [NASA ADS]
Horváth, I., Balázs, L. G., Bagoly, Z., Ryde, F., & Mészáros, A. 2006, A&A, 447, 23 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Horváth, I., Balázs, L. G., Bagoly, Z., & Veres, P. 2008, A&A, 489, L1 [NASA ADS] [CrossRef] [EDP Sciences] (In the text)
Hurley, K. 2007, http://www.ssl.berkeley.edu/ipn3/index.html (In the text)
Hurley, K. 2008, http://www.ssl.berkeley.edu/ipn3/masterli.html (In the text)
Kouveliotou, C., Meegan, C. A., Fishman, G. J., et al. 1993, ApJ, 413, 101 [NASA ADS] [CrossRef] (In the text)
Lin, R. P., Dennis, B. R., Hurford, G. J., et al. 2002, Sol. Phys., 210, 3 [NASA ADS] [CrossRef] (In the text)
Meegan, C. A., et al. 2001, Current BATSE Gamma-Ray Burst Catalog, http://gammaray.msfc.nasa.gov/batse/grb/catalog (In the text)
Mukherjee, S., Feigelson, E. D., Jogesh, B. G. et al. 1998, ApJ, 508, 314 [NASA ADS] [CrossRef] (In the text)
Press, W. H., Feigelson, E. D., Jogesh, B. G., et al. 1992, Numerical Recipes in C (Cambridge University Press) (In the text)
RSI IDL, http://rsinc.com/idl/ (In the text)
Sakamoto, T., Barthelmy, S. D., Barbier, L., et al. 2008, ApJS, 175, 179 [NASA ADS] [CrossRef] (In the text)
Smith, D. M., Lin, R. P., Turin, P., et al. 2002, Sol. Phys., 210, 33 [NASA ADS] [CrossRef] (In the text)
Smith, D. M., Lin, R. P., Hurley, K. C., et al. 2003, Proc. SPIE, 4851, 1163 [NASA ADS] (In the text)
Trumpler, R. J., & Weaver, H. F. 1953, Statistical Astronomy (Berkeley: University of California Press) (In the text)
Wigger, C., et al. 2006, http://grb.web.psi.ch/publications/talk_venice.pdf (In the text)
Wigger, C., Hajdas, W., Zehnder, A., et al. 2006, Il Nuovo Cimento B, 121, 1117 [NASA ADS] (In the text)
Wigger, C., et al. 2008, http://grb.web.psi.ch (In the text)
Zey, C, et al. NIST/SEMATECH, e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/ (In the text)

All Tables

Table 1: Parameters of the best $\chi ^2$ fits of two and three log-normal functions on the RHESSI GRB T₉₀ distribution. $\mu$ are the means, $\sigma$ are the standard deviations and w are the weights of the distribution. Given uncertainties are standard deviations of the parameters obtained by ten different fittings of randomly changed histogram of durations by their uncertainties.

Table 2: The minimal $\chi ^2$ , corresponding goodness-of-fits and F-tests for fitted two and three log-normal functions on the RHESSI GRB T₉₀ distribution for ten different changes of durations by their uncertainties.

Table 3: Parameters of the best fit with two and three log-normal functions done by the maximum likelihood method on the RHESSI data. $\mu$ are the means, $\sigma$ are the standard deviations, w are the weights of the distribution and L₂, L₃ are the likelihoods. Given uncertainties are standard deviations of the parameters obtained by ten different fittings of data sets, in which the durations were randomly changed by their uncertainties.

Table 4: The maximal likelihoods and corresponding probabilities that introducing of the third group is accidental for maximum likelihood fittings (one-dimensional) with two and three log-normal functions of ten different changes of durations by their uncertainties.

Table 5: Parameters of the best fit with two and three bivariate log-normal functions done by the maximum likelihood method on the RHESSI data. $\mu _{{x}}$ are the means on the x-axis ( $x = \log T_{90}$ ), $\mu _{{y}}$ are the means on the y-axis ( $y = \log H$ ), $\sigma _{{x}}$ are the dispersions on the x-axis, $\sigma _{{y}}$ are the dispersions on the y-axis, r are the correlation coefficients, w are the weights of the distribution and $L_{{\rm 2}}, L_{{\rm 3}}$ are the likelihoods. Given uncertainties are standard deviations of the parameters obtained by ten different fittings of data sets, where the durations and hardness ratios were randomly changed by their uncertainties.

Table 6: The maximal likelihoods and corresponding probabilities that introducing of the third group is accidental for maximum likelihood fittings (two-dimensional) with two and three bivariate log-normal functions of ten different changes of durations and hardness ratios by their uncertainties.

Table 7: The RHESSI GRB data set including I. GRB names which correspond to dates (the letters after GRB names are internal and do not have to be in accordance with e.g. GCN GRB names), II. GRB peak time, III. T₉₀ duration, IV. time resolution $\delta t_{\rm res}$ (described above) and V. hardness ratios.

All Figures

$\begin{figure} \par\includegraphics[width=90mm,clip]{0913fig1.eps} \end{figure}$	Figure 1: Duration distribution of the 427 RHESSI bursts with the best $\chi ^2$ fit of two log-normal functions. Number of bins is 19, $\rm d.o.f. = 14$ and $\chi ^2 \simeq 19.1$ which implies the goodness-of-fit ${\simeq } 16\%$ . The bar errors are standard deviations of the number of GRBs per bin for ten different simulated duration distributions as described in the text.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=90mm,clip]{0913fig2.eps} \end{figure}$	Figure 2: Duration distribution of the 427 RHESSI bursts with the best $\chi ^2$ fit of three log-normal functions. Number of bins is 19, $\rm d.o.f. = 11$ and $\chi ^2 \simeq 10.3$ which implies the goodness-of-fit ${\simeq } 50\%$ . The bar errors are the same as described in Fig. 1.
Open with DEXTER
In the text

$\begin{figure} \par\includegraphics[width=9cm,clip]{0913fig3.eps} \end{figure}$	Figure 3: Hardness ratio vs. T₉₀ of the RHESSI GRBs with the best fit of two bivariate log-normal functions.
Open with DEXTER
In the text

$\begin{figure} \includegraphics[width=9cm,clip]{0913fig4.eps} \end{figure}$	Figure 4: Hardness ratio vs. T₉₀ of the RHESSI GRBs with the best fit of three bivariate log-normal functions.
Open with DEXTER
In the text

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