A distinct peak-flux distribution of the third class of gamma-ray bursts: A possible signature of X-ray flashes?
aa r X i v : . [ a s t r o - ph . H E ] O c t Draft version September 12, 2018
Preprint typeset using L A TEX style emulateapj v. 11/10/09
A DISTINCT PEAK-FLUX DISTRIBUTION OF THE THIRD CLASS OFGAMMA-RAY BURSTS: A POSSIBLE SIGNATURE OF X-RAY FLASHES?
P. Veres
Department of Physics of Complex Systems, E¨otv¨os University, H-1117 Budapest, P´azm´any P. s. 1/A, HungaryDepartment of Physics, Bolyai Military University, H-1581 Budapest, POB 15, Hungary
Z. Bagoly
Dept. of Physics of Complex Systems, E¨otv¨os University, H-1117 Budapest, P´azm´any P. s. 1/A, Hungary
I. Horv´ath
Department of Physics, Bolyai Military University, H-1581 Budapest, POB 15, Hungary
A. M´esz´aros
Charles University, Faculty of Mathematics and Physics, Astronomical Institute, V Holeˇsoviˇck´ach 2, 180 00 Prague 8, Czech Republic
L. G. Bal´azs
Konkoly Observatory, H-1505 Budapest, POB 67, Hungary
Draft version September 12, 2018
ABSTRACTGamma-ray bursts are the most luminous events in the Universe. Going beyond the short-longclassification scheme we work in the context of three burst populations with the third group of inter-mediate duration and softest spectrum. We are looking for physical properties which discriminate theintermediate duration bursts from the other two classes. We use maximum likelihood fits to establishgroup memberships in the duration-hardness plane. To confirm these results we also use k-means andhierarchical clustering. We use Monte-Carlo simulations to test the significance of the existence of theintermediate group and we find it with 99 .
8% probability. The intermediate duration population hasa significantly lower peak-flux (with 99 .
94% significance). Also, long bursts with measured redshifthave higher peak-fluxes (with 98 .
6% significance) than long bursts without measured redshifts. Asthe third group is the softest, we argue that we have related them with X-ray flashes among thegamma-ray bursts. We give a new, probabilistic definition for this class of events.
Subject headings:
Gamma rays: bursts, observations – Methods: data analysis, observational, statis-tical, maximum likelihood INTRODUCTION
Gamma-ray bursts (GRBs) are the most powerful ex-plosions known in the Universe (for a review see M´esz´aros(2006)). To discern the physical properties of GRBs asa whole, we need to understand the number of phys-ically different underlying classes of the phenomenon(Zhang et al. 2009; L¨u et al. 2010).Before the launch of BATSE (Fishman et al.1994), there were hints of two distinct populations(Mazets et al. 1981; Norris et al. 1984). The bimodalitywas established using BATSE observations of theduration by Kouveliotou et al. (1993). The subsequentclasses were dubbed short and long type GRBs refer-ring to their durations. More sophisticated statisticalmethods based on more data using one classificationparameter (Horv´ath 1998) and more than one observ-able property, showed three populations in the BATSEdata (Mukherjee et al. 1998). These were furtherconfirmed by subsequent analyses (Hakkila et al. 2003;Horv´ath et al. 2006; Chattopadhyay et al. 2007): thethird population is intermediate in duration (Horv´ath [email protected] SAMPLE
The First Swift BAT Catalog (Sakamoto et al. 2008b)was augmented with bursts up to August 7, 2009 withmeasured T and hardness ratio. After excluding theoutliers and bursts without measured parameters, thesample consisting of the Sakamoto et al. (2008b) sampleand our extension has a total of 408 GRBs (219 from theCatalog and 189 newer bursts).Data reduction was carried out using HEAsoft version6.3.3 and calibration database version 20070924. Forlight curves and spectra we ran the batgrbproduct pipeline. To obtain the spectral parameters we fittedthe spectra integrated for the duration of the burst witha power law model and a power law model with an ex-ponential cutoff. As in Sakamoto et al. (2008b) we havechosen the cutoff power law model if the χ of the fitimproved by more than 6.The most widely used duration measure is T , whichis defined as the period between the 5% and 95% of theincoming counts. To find the fluences ( S E min ,E max ) weintegrated the model spectrum in the usual Swift energybands with 15 − − − −
150 keV as their bound-aries. We define the hardness ratio ( H ij , where i and j mark the two energy intervals) as the ratio of the flu-ences in different channels for a given burst. For example H = S − S − , where S − is the fluence of the burstfor the entire duration measured in the 50 −
100 keVrange. Different hardness ratios are possible to defineand we have used them to check our results.Bursts have a wealth of measured parameters and itis possible to use many variations of them. The choiceof T has some draw-backs (Qin et al. 2010). It is notsensitive to quiescent episodes between the active phaseof bursts (e.g., bursts with precursors). Also it cannotdifferentiate between bursts with an initial hard peak anda soft extended emission from bursts with constant longemission. In turn this latter type of burst with a hardinitial spike and an extended soft emission can bias H as well (Gehrels et al. 2006). Nevertheless, keeping inmind these draw-backs of T , this quantity is still oneof the most important measures of GRBs, and hence itsuse is straightforward. This question is discussed also inSection 3.5. http://heasarc.nasa.gov/lheasoft/ CLASSIFICATION
The choice of variables
There are many indications that the phenomenonwhich we observe as gamma-ray bursts has more thanone underlying population. The goal is to identify classeswhich are physically different. We choose the durationand the hardness ratio of bursts as the principal mea-sure. This choice has been made by other studies as well(Dezalay et al. 1996; Horv´ath et al. 2006).The choice of variables for the clustering deserves somejustification. Satellites generally observe many proper-ties and subsequent observations add to the volume ofthe parameters belonging to a burst. Bagoly et al. (1998)showed that two principal components are enough todescribe the data in the BATSE Catalog satisfactorily.Horv´ath et al. (2006) followed these arguments and usedthe H hardness and the T duration to classify thebursts. By using T and H we include a basic tempo-ral and spectral characteristic of the bursts.The reality of any classification is hard to assess. Agood way to make sure that the classification is robustand has some physical significance is to check the groups’stability with respect to various classification methods.We carry out three types of classifications: model-basedmultivariate classification, k-means clustering and hier-archical clustering.In the mathematical literature there is a wealth ofclassification schemes. We use the algorithms imple-mented in the R software . The clustering methods canbe divided to parametric and non-parametric schemes.Parametric schemes postulate that the data follows apre-defined model (in our case a superposition of multi-variate Gaussian distributions) and give a membershipprobability for each gamma-ray burst belonging to agiven group. Thus each burst will have assigned k num-ber of membership probabilities, where k is the numberof multivariate components (groups). This is called afuzzy clustering (Yang 1993). The non-parametric tests(k-means and hierarchical clustering), on the contrary,assign definitive memberships to each burst. However,here one needs to define the distance or similarity mea-sure between the cases. Model based clustering
As discussed in Horv´ath et al. (2006), we can as-sume that the observed distribution of bursts on theduration-hardness plane is a superposition of two ormore groups. The conditional probability density( p (log T , log H | l )), together with the probabilityof a burst being from a given group ( p l ) using the law offull probabilities: p (log T , log H ) = k X l =1 p (log T , log H | l ) p l , (1)where k is the number of groups.Studies show that for example the distribution of thelogarithm of the duration can be adequately describedby a superposition of three Gaussians (Horv´ath 1998). http://cran.r-project.org (R Development Core Team 2008) In this section we thus use the model based on bivariateGaussian distributions. We suppose that the joint dis-tribution of the parameters can be described as a super-position of Gaussians. Previously Horv´ath et al. (2006) carried out a similar analysis on the duration-hardnessplane of the BATSE Catalog where data was fitted withbivariate Gaussians.One bivariate Gaussian will have the following jointdistribution function: p (log T , log H | l ) = 12 πσ log T σ log H √ − r exp − − r ) (log T − log T C ) σ T ++ (log H − log H C ) σ H + 2 r (log T − log T C )(log H − log H C ) σ log T σ log H !! , (2)where log T C and log H C are the ellipse centercoordinates, σ log T and σ log H are the two standarddeviations of the distribution and r is the correlationcoefficient.Here we find the model parameters using the maximumlikelihood method. The procedure is called Expectation-Maximization (EM). This consists of appointing a mem-bership probability to each burst using an initial value ofthe parameters (E step). Then we calculate the param-eters of the model using these memberships (M step).Using this new model we re-associate each burst to thegroups and calculate the model parameters. We repeatthese steps until the solution converge. It is proved thatthis procedure converges to the maximum likelihood so-lution of the parameters (Dempster et al. 1977). Number of groups
It is important to decide on the true number of com-ponents to fit (the number of classes). In the model-based framework we have a better grip on this problemcompared to the non-parametric methods. For our cal-culations we use the Mclust package (Fraley & Raftery2000) of R.In the most general case the best model is found bymaximizing the likelihood. It is possible to penalize amodel for more degrees of freedom. A widely used versionof this method is called the Bayesian Information Crite-rion (BIC) introduced by Schwarz (1978) (for astronom-ical applications see e.g. Liddle (2007)). The function tobe maximized to get the best fitting model parametershas an additional term besides the log-likelihood: BIC = 2 L max − m ln N, (3)where L max (=ln l max ) is the logarithm of the maximumlikelihood of the model, m is the number of free parame-ters, and N is the size of the sample. This method takesinto account the complexity of our model by penalizingfor additional free parameters.We use the BIC to find the most probable model (in-cluding the number of components) and the parametersof this model. In a two-dimensional fit the number of freeparameters of a single bivariate Gaussian component is 6(two coordinates for the mean, two values of the standarddeviations in different directions, a correlation coefficientand a weight). For k bivariate Gaussians the number of free parameters is 6 k −
1, since the sum of the weights is1. In the most general model all 6 parameters of eachcomponent can be varied. Some of the parameters mayhave interrelations between the components (e.g., allcomponents have the same weight or shape, there is nocorrelation between the variables ( r = 0), etc.). In thisway we construct models with less degrees of freedom.The possible interrelations between the parameters ofthe Gaussians are taken into account by trying differ-ent models with different types of constraints (for thelist of models see the Mclust manual ).We have applied this classification scheme on our sam-ple. We found that the model with three compo-nents gives the best fit for the data in the BIC sense,where the shape of the bivariate Gaussians is the same( σ log T ,i = σ log T ,j and σ log H ,i = σ log H ,j for i, j = { short, long, intermediate } ) for each group, onlytheir weights are different with no correlation. This iscalled the EEI model in Mclust. The description of themodel follows from its name: equal volume (E), equalshape (E) and the axes are parallel with the coordinateaxes (I). In other words this is the model with optimalinformation content describing the data (see Fig. 1).We find that the best model has a value of BIC = − .
14. This model has three bivariate components.In the general case the maximum number of free param-eters would thus be m = 17. Taking into account theconstraints of this model, the degrees of freedom herewill be m = 10 (three coordinate pairs for the center ofthe distributions, two standard deviations common forall three components and two weights).The clustering method based on this model shows thata model containing three bivariate components is themost preferred. Models with two components have thebest BIC ∼ −
276 and for models with four componentsthe best
BIC ∼ − Fig. 1.—
Bayesian information criterion values for different mod-els (different lines show different models) in function of the numberof bivariate Gaussian components: the higher the value of the BIC,the more probable the model. The best model is marked in blackand the highest value is reached for k = 3. Also, some of the othermodels have their peak at k = 3. The inset shows the magnifiedpeak region. TABLE 1Bivariate model parameters for the best-fitted (EEI)model. The standard deviations in the direction of thecoordinate axes and the correlation coefficients areconstrained by the model.
Groups p l lg T C lg H C σ lg T σ lg H r N l short 0 . − .
331 0 .
247 0 .
509 0 .
090 0 31interm. 0 .
12 1 . − .
116 0 .
509 0 .
090 0 46long 0 .
80 1 .
699 0 .
114 0 .
509 0 .
090 0 331
The best-fit model has 10 free parameters and has threebivariate Gaussian components. The parameters of themodel as well as the number of the bursts in the groupsare in Table 1. The shortest and hardest group will bedesignated short, the longest and of moderate hardnesswill be called long and the intermediate duration groupwith the softest spectrum will be called intermediate. For the relation of the intermediate class to other studies onthis topic see the discussion.To calculate the significance of three populations wecarried out a Monte-Carlo simulation. We tested the hy-pothesis that the presence of the third population is onlya statistical fluctuation. We generated 10000 randomcatalogs, with the best k = 2 model. We found that withthe classification method only 0 .
2% of the cases yielded athree component model while 99 .
8% of cases produced atwo component fit. This means that the probability thatthe third group is only a statistical fluctuation is 0 . .
1% of the cases compared to the three populationmodel.There is another three component model which has avery similar BIC value (the difference is only ∼
1, seethe inset in Fig. 1). This model is called VEI and it hasvariable volume, meaning the product of the standarddeviations is the same (V), equal shape (E) and the axesare parallel with the coordinate axes (I). It has 12 degreesof freedom (three coordinate pairs for the centers of thedistributions, three pairs of standard deviations with therestriction that their product is the same (four degrees offreedom) and two weights). Information Criterion givesonly a weak hint as to which model is preferred. TheVEI model gives a visibly different group structure. Ifwe compare it to the three component EEI model theratio of differently classified bursts is 26 . I l (log T , log H ) = P l × P (log T , log H | l ) X l ∈{ short , interm ., long } P l × P (log T , log H | l ) . (4)Here P (log T , log H | l ) is the conditional probabil-ity density of a burst, assuming it comes from class l . P l is the probability of the l class. The indicator functionassigns a probability for a burst that it belongs to a givengroup. In this framework there is no definite answer to thequestion: ”To which class does a specific burst belong?” ,rather there is a probability of a burst belonging to agiven group as given by the indicator function. If the con-tribution of a component is dominant, the membership for the detailed classification results with the EEI model see:http://itl7.elte.hu/ ∼ veresp/swt90h32gr408.txt determination is straightforward. If the two (or three)highest membership probabilities for a burst are approx-imately equal the uncertainty in the classification is high.To check for contamination from other groups we carryout our analysis on a sub-sample where only the morecertain memberships are taken into consideration.The model has three components with equal standarddeviations in both directions and with no correlation ( r =0). The components are as follows (see also Table 1):1. The first component is the known short class ofGRBs (shortest duration and hardest spectra).The average duration is 0 .
47 s and the averagehardness ratio is 1 .
77. It has 31 members, and the -2 -1 0 1 2 3log T -0.4-0.20.00.20.40.6 l og H XRF
Fig. 2.—
GRB populations on the duration-hardness plane. Dif-ferent symbols mark different groups. One and two sigma ellipsesare superimposed on the figure to illustrate the model componentsfound as described in the text. Filled symbols mark bursts withmeasured redshifts. The dashed line indicates the definition of X-ray flashes (XRFs) given by Sakamoto et al. (2008a). weight of this model component is 0 . long class, also identified in many previousstudies. It has an average duration of 50 . .
30. It has 331 membersand the weight of the model component is 0 . intermediate in dura-tion. It has overlapping regions with previous def-initions of the intermediate class (Horv´ath et al.2010). The average duration is 13 . .
77. It has 46members and the weight of the model componentis 0 . r = 0. Non-parametric clustering
A major draw-back of the model based clustering isthat it assumes the distribution of the underlying pop-ulations is of a given functional form (bivariate Gaus-sian in our case). Non-parametric clustering does notassume any a priori model. We need to define a met-ric to measure distances on the duration-hardness plane.Here we scale the variables with their standard devia-tions because the clustering algorithms are sensitive tothe distance scale of the variables. If one of the vari-ables has a standard deviation, for example, one orderof magnitude larger than the other, the method will usethat variable with greater weight. Non-parametric clus-tering gives definite membership values for each burstwithout providing any information on the uncertaintiesof the clustering. Here we perform k-means and hier-archical clustering to substantiate our findings with themodel based method.
K-means clustering
We apply k-means clustering to the dataset (for an ap-plication of this method see e.g., Chattopadhyay et al.
Number of Clusters W i t h i n g r oup s s u m o f s qua r e s Fig. 3.—
The evolution of the sum of squares while increasingthe number of groups in k-means clustering. An ”elbow” is clearlyvisible at k = 3. The inset shows the group structure for k = 3groups on the duration-hardness plane. TABLE 2Group structure properties using k-means clustering forthree populations.
Group N (%) Center( T [s]) Center( H )short 48 (11 .
8) 0 .
96 1 . .
7) 20 . . .
5) 65 . . k = 3. The number of bursts in each group for k = 3as well as the center of the groups is shown in Table 2.This result strongly supports the group structure foundwith the model-based method. Hierarchical clustering
Another method of classifying bursts is the hierarchicalclustering algorithm (Murtagh & Heck 1987). We startfrom a state, where there are N groups (each burst is aseparate group) and step by step we merge two groupsusing some pre-defined criterion. In N − Fig. 4.—
The within group sum of squares in function of the num-ber of groups in the case of the hierarchical classification. Again,an ”elbow” is visible at k = 3. The inset shows bursts classifiedwith hierarchical clustering. The structure of the groups is similarto the model-based classification. We need to make a choice for the distance measurebetween two points. We choose the simple Euclideandistance. This choice is motivated by the small corre-lation between the two variables (correlation coefficient r = − . k = 3groups. This resembles the group structure found withmodel based clustering. As a justification for k = 3groups, we plot the within group sum of squares in func-tion of the number of groups as in the case of the k-meansclustering and also see an ”elbow” feature at k = 3. Wethus conclude that three groups describe the sample sat-isfactorily. Robustness of the clustering
Using both model-based and non-parametric methodswe have experimented by using T instead of T , byusing different hardness ratios (e.g., H = S − S − , H = S − S − + S − etc.). The classification remainedessentially the same.The most ”stable” group is the shortest and hardestpopulation. The elements of this group are clearly dif-ferent from the other two and the membership remainsthe same as we use other variants of hardness or dura-tion. The rest of the bursts is divided between the longand the intermediate classes. At the border between theclasses bursts have a high class-uncertainty. This resultsin a slight change of the membership of these bursts. Inother words, the separation of the long and the interme-diate population is fuzzy.A classification is well founded if different methods givesimilar results. To compare the similarities and differ-ences between the hierarchical and the model based clas-sification (EEI model) we construct a so called contin- TABLE 3Contingency table for the hierarchical (HC) and themodel based clustering.
Model basedShort Intermediate Long TotalShort 28 0 0 28HC Intermediate 0 39 8 47Long 3 7 323 333Total 31 46 331 408
TABLE 4Contingency table for the k-means (KM) and the modelbased clustering.
Model basedShort Intermediate Long TotalShort 31 0 17 48k-means Intermediate 0 46 59 105Long 0 0 255 255Total 31 46 331 408 gency table. This shows the number of bursts classifiedin the same- and different groups by the two methods.Table 3. shows that hierarchical and model based classi-fication schemes are consistent. The off-diagonal or miss-classified elements ratio is only 4 . . PEAK-FLUX DISTRIBUTION
Peak-flux is measured by Swift in the one-second inter-val about the highest peak in the lightcurve. Counts aresummed from this interval in the 15 −
150 keV range in 58energy channels and deconvolved with the instrument’sspectral response matrix via a forward-folding method.From the spectrum one can obtain the peak-flux by in-tegrating the best spectral model in the 15 −
150 keVinterval. The peak-flux is measured in units of ergs cm − s − .It is important to analyze if the intermediate popula-tion is in any ways different from the other two. In theprevious section we have analyzed with different meth-ods the number of classes of bursts and determined theindividual burst’s group membership. After classify-ing the bursts on the duration-hardness plane we com-pared the peak-flux distribution of the three classes usingKolmogorov-Smirnov test. In the following we use theclasses obtained by the EEI model-based classification.We found that the intermediate group has a differentpeak-flux distribution with high significance (6 × − ;see Table 5) when compared to the long population. Inother words bursts which belong to this group tend tohave a lower peak-flux than both the long and the shortpopulation. It is worth mentioning that the other twonon-parametric classification methods led to similar con-clusions.We thus found a difference in the peak-flux of the in- -9 -8 -7 -6 -5log (Peak Flux [15-150 keV])0.00.20.40.60.81.0 shortintermediatelong Fig. 5.—
Peak-flux cumulative distribution for the differentgroups. The distribution of the short and long population is notsignificantly different. The distribution of peak-fluxes of the in-termediate class (dotted curve) is significantly different from theshort (0 . . TABLE 5Comparison of the three subgroups’ peak-flux in the − keV range. This shows that the peak-fluxdistribution of the intermediate population differssignificantly from the long and the short population. Groups KS distance Error prob.short-long 0 .
221 0 . .
478 0 . .
448 6 × − termediate group of bursts. It is also possible to includethe peak-flux in the classification scheme. Including thepeak-flux in the classification, the main difference will bethat the long duration group will be split in two, whilethe short and intermediate groups will have the essen-tially the same members. POPULATIONS WITH- AND WITHOUT MEASUREDREDSHIFT
We have included the available redshift measurementsfor the bursts. The distribution of each class was in-spected for potential differences between the groups. Wefound that 23 % of bursts classified as short have mea-sured redshift (7 out of 31). The ratio is slightly higher(30% (14 out of 46) and, 36% (119 out of 331)) for theintermediate and long population respectively (for burstswith redshift see Fig. 2).We have analyzed the distribution of the peak-flux ofbursts in different groups comparing the bursts with andwithout measured redshift. We found that the peak-fluxdistribution of the long class with redshift measurementis significantly different from the population without it.Bursts with redshift tend to have higher peak-flux thanbursts without redshift (see Fig. 5). In other words,bursts with higher peak-flux have a better chance of hav-ing a redshift measured. There is no significant differencebetween the other populations (see Table 6).Next, we compare the distribution of the redshifts forthe three groups with each other. It is well-known,that short and long bursts differ significantly in theirredshifts (Bagoly et al. 2006) and Swift bursts have alarger mean redshift than previous spacecrafts’ sample
TABLE 6Comparison of the three subgroups’ peak-flux in the − keV range. Here we compare peak-fluxes ofdifferent populations with and without redshifts. Wefind there is a significant difference in the peak-fluxdistribution of long bursts with measured redshift andthe bursts without. Sample KS dist. Error prob.Short z and non-z sample 0 .
208 0 . .
250 0 . .
183 0 . -8.5 -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0log (Peak Flux [15-150 keV])0.00.20.40.60.81.0 with redshiftwithout redshift Fig. 6.—
Cumulative distribution of the long population with-and without measured redshifts. Long bursts with redshift have aclearly higher peak-flux distribution. (Jakobsson et al. 2006). We also find that the distribu-tion of the short bursts is markedly different when com-pared to the long or the intermediate class (the errorprobability is 0 .
002 and 0 . ∼ . .
79 to 0 . DISCUSSION
Physical interpretation
Our analysis on the Swift GRBs supports the earlierresults that there are three distinct groups of bursts.Again, besides the long and the short population, theintermediate duration class appears to be the softest.In this study, however, the structure of the intermedi- C u m u l a t i v e f r a c t i on shortinterm(P>0.8)itermlong Fig. 7.—
Cumulative redshift distribution of the three groups.As previously known, short bursts are on average closer than thelong GRBs. The intermediate population distribution hints lower z values than the long class, but the two distributions are still com-patible with the hypothesis of being drawn from the same parentdistribution. If we truncate the probability of the intermediateclass at 80%, we find the difference is more apparent, but still notsignificant. ate class is not exactly the same as in previous studies,mostly due to the different mathematical approach. Dueto the differences, it is possible to give a different phys-ical interpretation of the intermediate group, comparedto previous studies, for example linking them to X-rayflashes.A physical relationship of the intermediate class withthe short population is unlikely. This is suggested byalmost non-existent contamination of short bursts withintermediate in the cross-tabulated values with both hier-archical and k-means clustering versus model based clus-tering.The different model based classification algorithms re-veal a significant overlap between the distribution of theintermediate and the long class in the duration-hardnessplane. One possibility is that the intermediate class is adistinct class by its physical nature. This may indicatethere is a third type of progenitor.Also it is possible that the intermediate bursts do notform a different class by themselves, but are related tothe long population through some physically meaningfulparameter or parameters. This could be the observ-ing angle to the jet axis (Zhang 2007), a less energetic central engine possibly related through the angular mo-mentum of the central black hole and the accretion rate(Krolik & Hawley 2010), a baryon-loaded jet with lowerLorentz factor (Dermer et al. 1999) or a combination ofthese. This way the intermediate population representsa continuation of the long population.The intermediate bursts’ peak-flux are systematicallylower than the long ones, while their redshift range iseither lower or similar. We thus conclude that the inter-mediate class is intrinsically dimmer. If the intermediatepopulation is part of the long population, the lower peak-flux requires a physical explanation. The observationalproperties show that intermediate bursts are the softestamong the three groups, meaning that their emission isconcentrated to low-energy bands. Relation to X-ray flashes
As the intermediate population is the softest, it isworth searching for a link with the similar and softerphenomenon compared to classical gamma-ray bursts,the X-ray flashes (XRFs) (for a review, see Hullinger(2006)). Sakamoto et al. (2008a) gives a working def-inition for X-ray flashes (XRF) and X-ray rich GRBs(XRR) for Swift using the fluence ratio. The S fluenceratio is the reciprocal of the hardness ( H = ( S ) − ).Current understanding of XRFs indicate that they arerelated to long bursts and they form a continuous distri-bution in the peak energy ( E peak ) of the νF ν spectrum(Sakamoto et al. 2008a).X-ray Flashes were first defined using BeppoSAX(Heise et al. 2001). The criteria for an X-ray Flash wasto trigger the Wide Field Camera (sensitive between2 −
30 keV) instrument but not in the GRBM (sensi-tive between 40 −
700 keV). 9 out of 10 XRFs detectedby BeppoSAX were found in the BATSE data as untrig-gered events (Kippen et al. 2003) with their bulk prop-erties similar to GRBs.The clustering methods identifies on the duration-hardness plane the location of the bursts in the interme-diate class. According to the fuzzy classification modelwe do not get a definite membership for a given burst,rather a probability that a burst belongs to a group. Toidentify the intermediate population (and tentatively theX-ray flashes), we use the indicator function: I Interm . (log T , log H ) = P interm . × P (log T , log H | ” Interm. ”) X l ∈{ short , interm ., long } P l × P (log T , log H | l ) (5)The values of the parameters in this equation shouldbe taken from Table 1. This yields the probability that aburst belongs to the third group given its two measuredparameters. The joint distribution function of the fittedmodel can be seen in gray on Fig. 8 and the probabilitycontours of the third population are drawn in black withprobability level contours shown. We have also plottedthe borders in hardness for the working definitions ofXRRs and XRFs in Fig. 8 (dotted and dashed horizontal lines respectively).Sakamoto et al. (2008a) define XRFs as events withfluence ratio S > .
32. This translates to a hardnessratio H < .
76. This definition aims to transformthe limit of XRFs and X-ray rich GRBs found with Bep-poSAX and HeteII. The limit is found using a pseudo-burst with spectral parameters: α = − , β = − . E peak = 100 keV for a Band spectrum (Band et al. 1993).Based on this definition we identify 24 bursts from our Fig. 8.—
Contour plot of the duration-hardness distributionbased on the EEI model with three components in light-gray.Points show individual bursts. The broken lines in black show theprobability contours of a given region belonging to the intermedi-ate population. Also bursts classified as XRFs and XRR GRBsare marked on the plot with horizontal lines. One can observe aremarkable coincidence between the XRFs and the third group asshown by the indicator function.
TABLE 7X-ray flashes as defined by Sakamoto et al. (2008a) andthe probability that they belong to the intermediategroup.
Name T [ s ] H .
50 0 .
48 1 . .
70 0 .
73 0 . .
90 0 .
72 0 . .
70 0 .
62 0 . .
60 0 .
59 0 . .
00 0 .
75 0 . .
10 0 .
68 0 . .
90 0 .
67 0 . .
50 0 .
71 0 . .
60 0 .
70 0 . .
00 0 .
68 0 . .
50 0 .
57 1 . .
50 0 .
75 0 . .
00 0 .
55 1 . .
00 0 .
66 0 . .
78 0 .
53 1 . .
00 0 .
76 0 . .
40 0 .
60 1 . .
00 0 .
66 0 . .
84 0 .
46 1 . .
00 0 .
63 0 . .
00 0 .
69 0 . .
00 0 .
70 0 . .
44 0 .
72 0 .
408 burst sample. Table 7. contains data of bursts alongwith the probabilities that they belong to the third pop-ulation. The average of these probabilities (i.e. the XRFbelongs to the intermediate group) is 95%. This highvalue allows us to conclude that all XRFs belong to theintermediate group defined by the EEI model with highprobability.Based on Fig. 8. we propose that the members of the -0.4 -0.2 0.0 0.2 0.4 0.6log H N Fig. 9.—
Hardness distribution of the bursts in the sample. Thefilled portion marks the intermediate population. The vertical lineshows the limit defined to identify X-ray flashes by Sakamoto et al.(2008a), it identifies 24 XRFs. An additional 22 XRFs are proposedby the model fit probabilities. third component are probably the X-ray flashes. There-fore, using the model based classification method we cangive probabilistic definition for the X-ray flashes basedon the duration-hardness distribution. This definitiondefines 22 additional bursts that belong to the interme-diate population and hence to the XRFs.All the X-ray flashes are in the region where the thirdcomponent has the highest probability, but not all thirdcomponent bursts can be unambiguously classified as X-ray flashes according to the Sakamoto et al. (2008a) cri-terion. In other words the third component in the EEImodel contains all the X-ray flashes and some additional,very soft bursts.To give further support to our point, we make a his-togram with the hardness ratios of the bursts (see Figure9). The vertical line represents the working definitionof XRFs and the filled part of the histogram representsputative X-ray flashes identified as the third, soft classin this study. The limiting contours are not horizontal,as the centers of the long and intermediate classes havedifferent T values. Furthermore, there are some shortXRFs T ≈
1s which are harder than the working defi-nition limit.The mechanism behind the X-ray flashes is still notclear. There are various scenarios that could producethese phenomena (e.g. dirty fireballs, inefficient inter-nal shocks, structured jets with off-axis viewing angle,etc., for a review of the models see (Zhang 2007)). Amore precise experimental definition of XRFs can resultin more stringent constraints on the models.
Relation to a recent study on groups of GRBs usingSwift data
A recent work by Horv´ath et al. (2010) also confirmedthe existence of the third class in the Swift database.Horv´ath et al. (2010) used a maximum likelihood fit withEM algorithm. In their model they applied no restric-tions for the parameters of the ellipses. This can be re-lated to the VVV model in the MClust package withmaximum number of degrees of freedom. In our case theVVV model is disfavored because of the lower BIC valuecaused by the higher number of free parameters. How-ever, one model (VEI) with only marginally lower BIC0
TABLE 8Contingency table comparing the common bursts fromHorv´ath et al. (2010) and the EEI solution used in thisstudy. There are X-ray flashes in this sample all ofwhich are classified as intermediate by both methods, i.e.they are included in the intermediate-intermediate fieldwith elements. Horv´ath et al. (2010) classificationShort Intermediate Long TotalShort 24 0 0 24Model based Interm. 0 31 1 32(EEI) Long 0 55 213 268Total 24 86 214 324 value than the best model has a similar structure as theone found by Horv´ath et al. (2010). We have constructedcontingency tables comparing the common bursts in theHorv´ath et al. (2010) with the model based results. (seeTables 8 and 9 for the comparison with the EEI and VEImodels)The common sample with the Horv´ath et al. (2010)study consists of 324 GRBs. According to the contin-gency table (Table 8) there are 31 bursts which are clas-sified as intermediate in both studies. The main differ-ence between the classifications can be seen in the the to-tal number of intermediate bursts: Horv´ath et al. (2010)classify 86 bursts as intermediates, and this study findsonly 32 (according to the EEI model).The other important question is the number of X-ray flashes in the two kinds of classifications. Usingthe Sakamoto et al. (2008a) definition there are 19 X-ray flashes in the common sample. These belong with-out exception in the intermediate class with the highestprobability according to both
Horv´ath et al. (2010) andthis study. In other words the 31 bursts classified as in-termediate by both Horv´ath et al. (2010) and this study,contains all the X-ray flashes. Horv´ath et al. (2010) clas-sifies 55 bursts in the intermediate class, whereas here weclassify them as long. Based on this we can state thatthe model presented here is more efficient to identify theX-ray flashes with high probability. The ratio of X-rayflashes in the intermediate class is 59% in the EEI modeland 22% in the Horv´ath et al. (2010) study.The VEI model with a marginally lower BIC value hasonly 32 off-diagonal elements in the contingency table(see Table 9) when compared to the Horv´ath et al. (2010)study. The structure revealed by this model is more sim-ilar to the one in Horv´ath et al. (2010). In this model allthe 19 X-ray flashes are also classified as intermediate.In this case the the number of intermediate bursts is 118which means there are many bursts classified as interme-diate which are not X-ray flashes. The ratio of XRFs tointermediate class members is 16%. The reason for finding different group structure for theintermediate population lies in the fact that this group issignificantly overlaid with the long population and it isvery much sensitive to the mathematical approach used. CONCLUSION
The results of this paper can be summarized as follows:- We have established with multiple methods - inconcordance with previous studies - that SwiftGRB data can be best modelled using three popula-
TABLE 9Contingency table comparing the common bursts fromHorv´ath et al. (2010) and the VEI model.
Horv´ath et al. (2010) classificationShort Intermediate Long TotalShort 22 0 0 22Model based Interm. 2 86 30 118(VEI) Long 0 0 184 184Total 24 86 214 324 tions. Both the model independent and the modelbased methods showed three groups with high sig-nificance.- We found that the third population of GRBs, inter-mediate in duration and with the softest spectrum,has a peak-flux distribution that significantly dif-fers from the other two classes. This group has thelowest average peak-flux.- Furthermore, the redshifts of the intermediate pop-ulation do not differ significantly from that of thelong class, although their average redshift is lower.Considering this and their lower average peak-fluxit indicates that the intermediate GRBs are inher-ently dimmer than the longer ones.- We have also found evidence that the intermediatepopulation is closely related to X-ray flashes: allthe previously identified Swift X-ray flashes belongto the third, soft population. Therefore, we relatethe intermediate class to the X-ray flashes. Thus,we give a new, probabilistic definition for this phe-nomenon.This work was supported by OTKA grant K077795, byOTKA/NKTH A08-77719 and A08-77815 grants (Z.B.),by the GA ˇCR grant No. P209/10/0734 (A.M.), by theResearch Program MSM0021620860 of the Ministry ofEducation of the Czech Republic (A.M.) and by a BolyaiScholarship (I.H.). We thank Peter M´esz´aros, G´aborTusn´ady, L´ıdia Rejt˝o, Jakub ˇR´ıpa and the anonymousreferee for valuable comments on the paper.
REFERENCESBagoly, Z., M´esz´aros, A., Horv´ath, I., Bal´azs, L. G., & M´esz´aros,P. 1998, ApJ, 498, 342Bagoly, Z., et al. 2006, A&A, 453, 797Band, D., et al. 1993, ApJ, 413, 281Chattopadhyay, T., Misra, R., Chattopadhyay, A. K., & Naskar,M. 2007, ApJ, 667, 1017 Dempster, A., Laird, N., & Rubin, D. 1977, Royal statisticalSociety B, 39, 1Dermer, C. D., Chiang, J., & B¨ottcher, M. 1999, ApJ, 513, 656Dezalay, J. P., Lestrade, J. P., Barat, C., Talon, R., Sunyaev, R.,Terekhov, O., & Kuznetsov, A. 1996, ApJ, 471, L27Fishman, G. J., et al. 1994, ApJS, 92, 2291