The duration distribution of Swift Gamma-Ray Bursts
aa r X i v : . [ a s t r o - ph . H E ] A p r The duration distribution of Swift Gamma-Ray Bursts
Author • I. Horv´ath • B. G. T´oth Abstract
Decades ago two classes of gamma-raybursts were identified and delineated as having dura-tions shorter and longer than about 2 s. Subsequentlyindications also supported the existence of a third class.Using maximum likelihood estimation we analyze theduration distribution of 888 Swift BAT bursts observedbefore October 2015. Fitting three log-normal func-tions to the duration distribution of the bursts pro-vides a better fit than two log-normal distributions,with 99.9999% significance. Similarly to earlier results,we found that a fourth component is not needed. Therelative frequencies of the distribution of the groups are8% for short, 35% for intermediate and 57% for longbursts which correspond to our previous results. Weanalyse the redshift distribution for the 269 GRBs ofthe 888 GRBs with known redshift. We find no evi-dence for the previously suggested difference betweenthe long and intermediate GRBs’ redshift distribution.The observed redshift distribution of the 20 short GRBsdiffers with high significance from the distributions ofthe other groups.
Keywords
Gamma-rays: theory – Gamma rays: ob-servations – Gamma-ray burst: general – Methods:data analysis – Methods: statistical – Cosmology: ob-servations
Decades ago (Mazets et al. 1981) and (Norris et al.1984) suggested that there is a separation in the du-ration distribution of gamma-ray bursts (GRBs). To-day it is widely accepted that the physics of the short
AuthorI. Horv´athB. G. T´oth National University of Public Service, Budapest, Hungary and long GRBs are different, and these two kinds ofGRBs are different phenomena (Norris et al. 2001;Bal´azs et al. 2003; Fox et al. 2005; Zhang et al.2009; L¨u et al. 2010).In the Third BATSE Catalog (Meegan et al. 1996)— using uni- and multi-variate analyses — (Horv´ath1998) and (Mukherjee et al. 1998) found a third typeof GRBs. Later several papers (Hakkila et al. 2000;Balastegui et al. 2001; Rajaniemi & M¨ah¨onen 2002;Horv´ath 2002; Hakkila et al. 2003; Borgonovo 2004;Horv´ath et al. 2006; Chattopadhyay et al. 2007) con-firmed the existence of this third (”intermediate”in duration) group in the same database. In theSwift data the intermediate class has also been found(Horv´ath et al. 2008; Huja et al. 2009). There arealso more recent works (Horv´ath 2009; Balastegui et al.2011; de Ugarte Postigo, A. et al. 2011; L¨u et al. 2014)in this field.In the Swift database, the measured redshift distri-bution for the two groups are also different, for shortburst the median is 0.4 (O’Shaughnessy et al. 2008)and for the long ones it is 2.4 (Bagoly et al. 2006).The paper is organized as follows. Section 2 brieflysummarizes the method and fits, Section 3 contains thecalculations of the fits and their results, Section 4 con-tains the comparison of the redshift distributions of thedifferent classes and Section 5 summarizes the conclu-sions of this paper.
On the Swift web page there are 997 GRBs; 957 ofthese bursts were catalogued prior to October 2015.Multi-variate analyses have demonstrated that flux and http://swift.gsfc.nasa.gov/archive/grb table fluence values are also needed for a robust classification.There were 30 GRBs without fluence information and19 GRBs without peak flux information. Another 20GRBs are excluded from this analysis because their flu-ence uncertainties are larger than 50%. Figure 1 showsthe log T (time to accumulate the central 90% of theburst fluence) distribution of the remaining 888 GRBs.The maximum likelihood (ML) method assumes thatthe probability density function of an x observable vari-able is given in the form of g ( x, p , ..., p r ) where p , ..., p r are parameters of unknown value. Having N observa-tions on x , one can define the likelihood function in thefollowing form: l = N Y i =1 g ( x i , p , ..., p r ) , (1)or in logarithmic form (the logarithmic form is moreconvenient for calculations): L = log l = N X i =1 log ( g ( x i , p , ..., p r )) . (2)The ML procedure maximizes L according to p , ..., p r .Since the logarithmic function is monotonic, the loga-rithm reaches its maximum at the same p , ..., p r pa-rameter set where l does. The confidence region of theestimated parameters is given by the following formula,where L max is the maximum value of the likelihoodfunction and L is the likelihood function at the truevalue of the parameters (Kendall & Stuart 1976):2( L max − L ) ≈ χ r , . (3) Similar to our previous work (Horv´ath 2002; Horv´ath et al.2008), we fit the T distribution using ML with a Table 1
The best parameters for the two log-normal fit ofthe GRB duration distribution. center ( logT ) σ ( logT ) wshort .
386 0 .
484 188 . long .
700 0 .
480 699 . k log-normal components, each ofthem having two unknown parameters to be fitted with N = 888 measured points. The choice to use log-normalfunctions to fit the duration distribution is based on theresults of (Bal´azs et al. 2003; Shahmoradi & Nemiroff2015; Zitouni et al. 2015). Our goal is to find the mini-mum value of k suitable to fit the observed distribution.Assuming a weighted superposition of k log-normal dis-tributions, one has to maximize the following likelihoodfunction: L k = N X i =1 log k X m =1 w m f m ( x i , log T m , σ m ) ! (4)where w m is a weight and f m is a log-normal functionwith log T m mean and σ m standard deviation havingthe form of f m = σ m √ π exp (cid:16) − ( x − log T m ) σ m (cid:17) (5)and due to a normalization condition k X m =1 w m = N . (6)We used a simple C++ code to find the maximum of L k . Assuming only one log-normal component, the fitgives L max = 4978 .
88 but in the case of k = 2 onegets L max = 5082 .
246 with the parameters given inTable 1. The solution is displayed in Fig. 2.Based on Eq. (3) we can infer whether the addi-tion of a further log-normal component is necessary tosignificantly improve the fit. We make the null hypoth-esis that we have already reached the the true value N log T90 Fig. 1
The duration distribution of Swift BAT bursts. of k . Adding a new component, i.e. moving from k to k + 1, the ML solution of L kmax has changed to L ( k +1) max , but L remained the same. In the meantimewe increased the number of parameters with 3 ( w k +1 , logT k +1 and σ ( k +1) ). Applying Eq. (3) on both L kmax and L ( k +1) max we get after subtraction2( L ( k +1) max − L kmax ) ≈ χ . (7)For k = 1, L max is greater than L max by more than100, which gives for χ an extremely low probability. Itmeans that the fit with two log-normal distributions isreally a better approximation for the duration distribu-tion of GRBs than the fit with one.Thirdly, a three-log-normal fit is made combin-ing three f k functions with eight parameters (threemeans, three standard deviations and two independentweights). For the best fitted parameters see Table 2.The highest value of the logarithm of the likelihood( L max ) is 5098.361. For two log-normal functions themaximum is 5082.246. The maximum has thus im-proved by 16. Twice this value is 32, which gives usa probability of 0.00006% for the difference between L max and L max being only by chance. Therefore,there is a high probability that a third log-normal distri-bution is needed. In other words, the three-log-normalfit (see Fig. 3) is better and there is a 0.0000006 proba-bility that it was caused only by statistical fluctuation.In one of our previous papers (Horv´ath et al. 2008),we published a similar analysis on 222 GRBs of the FirstBAT Catalog. One should compare these results withthe results published in that paper. The centers of thedistributions change by only a very small amount. Inthe current analysis, the center of the distribution ofthe short bursts is at -0.508 (0.311 s) which was previ-ously at -0.473 (0.336 s). For the intermediate ones, the N log T90 Fig. 2
Fit with two log-normal component for the durationdistribution of BAT bursts. N logT90 Fig. 3
Fit with three log-normal components for the du-ration distribution of BAT bursts.
Table 2
The best parameters for the three log-normal fitof the GRB duration distribution. center ( logT ) σ ( logT ) wshort − .
508 0 .
439 68 . long .
897 0 .
367 506 . intermediate .
076 0 .
448 312 . center is at 1.076 which was at 1.107 and for the longbursts at 1.897, which was at 1.903 in our previous anal-ysis. The relative frequencies of the distribution of thegroups now are 8% for short, 35% for intermediate and57% for long bursts (see Table 2). Using a sample fourtimes smaller in 2008 (Horv´ath et al. 2008), the rela-tive frequencies of the distribution of the groups were7% for short, 35% for intermediate and 58% for longbursts. Therefore, both analyses give us very similarresults. This does not mean that the three-Gaussian isthe best approximation for the duration distribution ofthe GRBs, but strongly suggests that the smaller andthe larger sample duration distributions are almost thesame. Neither the nature of the GRBs nor the Swiftdetectors changed during the years.One should also calculate the likelihood for four log-normal functions. The best logarithm of the ML is5098.990. It is larger with 0.63 than it was for threelog-normal functions. This gives us a low significance(26%), therefore the fourth component is not needed.In Table 3 we summarize the improvement of the like-lihood and the corresponding significances. Among the 888 GRBs there are 269 GRBs which haveredshift information. Since the duration distribution ofthe three groups overlap one cannot be sure to whichgroup does a specific burst belong. However, 2.5 s(0.4 in logarithmic scale) and 31.6 s (1.5 in logarithmicscale) seems to be an approximate border between theshort, intermediate and long GBRs. Using these bor-ders, there are 20 short, 79 intermediate and 170 longGRBs. We cut the biggest population, the long burstgroup, into two parts: shorter than 100 s (91 GRBs)and longer than 100 s (79 GRBs). Figure 4 showsthe cumulative distribution of the short, intermediate,long1 and long2 bursts. One can use the Kolmogorov-Smirnov (KS) test to compare the distributions. Theredshift distribution of the intermediate, the long1 and
Table 3
The improvement of the likelihood and the sig-nificances. i L imax L imax − L ( i − max signif icance . .
361 16 . . .
99 0 .
63 0 .
26 the long2 groups do not differ from each other. Theprobabilities can be seen in Table 4. However, the ob-served redshift distribution of the short bursts differsfrom the other three distributions with high significance(more than 99.9 %).This latter result is well-known in the literature. Forthe intermediate bursts, there was a suggestion with avery low significance that the redshift distribution ofthese bursts are different from the redshift distributionof the long GRBs. Based on the results discussed above,we are not able to confirm this suggestion. The redshiftdistributions of the long and the intermediate GRBsseem to be very similar.
We presented that fitting the duration distribution of888 Swift BAT GRBs with three log-normal functionsis better than the fit using only two. Though this maybe the result of statistical fluctuations and maybe thereare only two types of GRBs, the probability that thethird component is not needed is only 0.00006%. Onecan compare the parameters of the burst groups withprevious results. In (Horv´ath et al. 2008) the relativefrequencies were 7% for short, 35% for intermediate and58% for long bursts. Now our results show 8% for short,35% for intermediate and 57% for long ones. The centerof the groups are also nearly the same.We have shown with very high significance that theredshift distribution of the short bursts is different fromthe redshift distributions of the other (longer) GRBs.However, the redshift distribution of the intermediateGRBs seems to be similar to the redshift distributionof the long GRBs.
Acknowledgements
This research was supportedby OTKA grant NN111016.
Table 4
The KS probabilities assuming the same redshiftdistribution for the classes. p ( KS ) intermediate long long short .
000 0 .
000 0 . intermediate .
90 0 . long . P e r c en t il e z Fig. 4
The cumulative redshift distribution of the short(pink), intermediate (blue), long1 (green) and long2 (red)bursts.
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