Astronomy&Astrophysicsmanuscriptno.ms6613 c ESO2008 (cid:13) February5,2008 On the temporal variability classes found in long gamma-ray bursts with known redshift L.Borgonovo1,F.Frontera2,C.Guidorzi2,3,4,5,E.Montanari2,L.Vetere6,7,andP.Soffitta7 1 StockholmObservatory,SE-10691Stockholm,Sweden 7 2 DipartimentodiFisica,Universita`diFerrara,44100Ferrara,Italy 0 3 AstrophysicsResearchInstitute,LiverpoolJohnMooresUniversity,TwelveQuaysHouse,BirkenheadCH411LD 0 4 DipartimentodiFisica,Universita`diMilano-Bicocca,Italy 2 5 INAF,OsservatorioAstronomicodiBrera,viaBianchi46,23807Merate(LC),Italy n 6 DipartamentodiFisica,Universita`LaSapienza,PiazzaleA.Moro2,I-00185Roma,Italy a 7 INAF,IASF-SezionediRoma,viadelFossodelCavaliere,I-00133Roma,Italy J 1 Received;accepted 3 ABSTRACT 1 v 0 Context. BasedontheanalysisofasmallsampleofBATSEandKonusgamma-raybursts(GRBs)withknowredshiftithasbeenreportedthat 2 thewidthoftheautocorrelationfunction(ACF)showsaremarkablebimodaldistributionintherest-frameofthesource.However,theorigin 9 ofthesetwowell-separatedACFclassesremainsunexplained. 1 Aims. WestudythepropertiesoftheburstsbelongingtoeachACFclassandlookforsignificantdifferencesbetweenthem. 0 Methods. WecomplementpreviousACFanalysisstudyingthecorrespondingpowerdensityspectra(PDS).WiththeadditionofBeppo-SAX 7 dataandtakenadvantageofitsbroad-bandcapability,wenotonlyincreasetheburstsamplebutweextendtheanalysistoX-rayenergies. 0 Results. The rest-frame PDS analysis at γ-ray energies shows that the two ACF classes are not simply characterised by a different low / h frequency cut-off, but they have a distinct variability as a whole in the studied frequency range. Both classes exhibit average PDS with p power-lawbehaviour athighfrequencies(f′ 0.1Hz)butsignificantlydifferentslopes,withindexvaluesclosetothoseofBrownian( 2) ≥ − - andKolmogorov( 5/3)spectraforthenarrow andbroad classesrespectively.ThelatterspectrumpresentsanadditionalPDScomponent, o − a low-frequency noise excess with a sharp cut-off. At X-ray energies we find the power-law index unchanged for the broad class, but a r t significantly steeper slope in the narrow case ( 3). We interpret this as an indication that the broad class bursts have weaker spectral s ∼ − evolutionthanthenarrowones,assuggestedalsobyouranalysisoftheACFenergydependence.ThelowandhighfrequencyPDScomponents a : maythenarisefromtworadiatingregionsinvolvingdifferentemissionmechanisms.WecompareourGRBsampleconditionedbyafterglow v detectionswithacomplete,fluxlimitedBATSEsample,findingasignificantbiasagainstnarrowACFbursts. i X r a Keywords.gammarays:bursts–gammarays:observations–methods:dataanalysis–distancescale 1. Introduction tion time T > 2 s. Second, burst light curves (LCs) show a 90 remarkablemorphologicaldiversityandtheyappeartohavea Determiningtherelevanttimescalesforanyastronomicalphe- compositestructure.Whileasignificantfractionofbrightlong nomenonisessentialtounderstanditsunderlyingphysicalpro- bursts( 15%)exhibitsasinglesmoothpulsestructure,inmost cesses.However,inspiteofextensiveresearch,temporalstud- ∼ cases they appear to be the result of a complex, seemingly ies on the prompt emission phase of long gamma-ray bursts random distribution of several pulses. Burst pulses are com- (GRBs)arenotyetabletodescribeandexplaintheirbasictem- monlydescribedashavingfast-riseexponential-decay(FRED) poralproperties.Themainchallengesencounteredinthetem- shape,althoughthedecayisnotstrictlyexponential.Therefore, poral analysis of GRBs are related to intrinsic characteristics the second timescale that seems relevant for the description of the emitted signal. Firstly, bursts are non-repetitive short- of a burst is a “typical” pulse duration. However, analysis of term events. Consequently, the total duration of the emission thepulseparametershasshownbroadlog-normaldistributions in a given observational energy window is the first timescale notonlyamongdifferentbursts,butalso withina single burst used to characterised them. Through out this paper we will (see,e.g.,Norrisetal.1996).Considerationofothertimescales only consider the class of long bursts, i.e., those with dura- mightberelevant,e.g.,forGRBswithprecursors(Koshutetal. 1995) or when long quiescent periods occur (Nakar&Piran Sendoffprintrequeststo:L.Borgonovo,e-mail:[email protected] 2 L.Borgonovoetal.:VariabilityclassesinGRBs 2002),althoughthesetemporalfeaturesappearonlyinasmall thatwhencorrectedforcosmicdilationeffectstheACFsexhibit fractionofbursts. aclearbimodaldistribution.Usingasameasurethehalf-width Due to these characteristics, much of the GRB temporal at half-maximum, there is a highly significant gap between a analysishasbeendonedirectlyontheLCs,i.e.,modellingthe narrowandabroadwidthclass,theseparationinstandarddevi- pulsesandstudyingtheirshapeanddistribution.Standardlin- ationsbeing>7σ.Theestimatedlocalorintrinsicvalues(i.e., earanalysistoolsmustbeusedwithsomecaution,sincemost thosecalculatedattherest-frameofthesource)fortheaverage of the inferences based on them would in principle require widthswere 1.6s and 7.5s, andthe relativedispersionswere “long”stationarysignals,i.e.,themostsuitableburstsarethose 32%and4%forthenarrowandbroadclasses,respectively.It wherethedurationismuchlongerthanthetypicalpulsewidth. isremarkablethelowdispersionfoundinthelastsubset,which Thisis thecase ofthetemporalanalysisbasedonpowerden- comprised 1/3ofthetotalsample. ∼ sity spectra (PDS).IndividualPDSof GRB haveverydiverse This article builds on the ACF analysis done in B04. In shapes, and they do not seem to have common features, al- Sect. 2 we present our data samplesand in Sect. 3 we briefly thoughthelongestburstsshowspectrawithaconsistentpower- describethemethodsusedinthesubsequenttemporalanalysis. lawbehaviour.Onewaytoovercometheselimitationsistoes- In Sect. 4 we strengthenpreviousfindingsbased on the ACF, timate an average PDS from a sample GRBs. This approach expandingtheprevioussampleofGRBwithknownzbythein- will only produce physically meaningful results if each burst clusionofproprietarydatafromtheBeppoSAXmission,andwe canbeconsideredarealisationofthesamestochasticprocess, complementthetemporalanalysisestimatingtheintrinsicPDS i.e.,therearenosubclassesinthesample. foreachofthesubsetsidentifiedusingtheACF.Furthermore, Under this assumption Beloborodov,Stern,&Svensson usingthebroadbandcapabilityofBeppoSAXcombinedinstru- (1998, 2000) calculated with a large sample of bright GRBs, ments,in Sect. 5 we areable to extendthe studyto theX-ray anaveragePDSshowingaclearpower-lawbehaviourextended energies.In Sect. 6, we investigatewhetherthe typicalvalues over two frequency decades (approximately within the 0.01– ofseveralphysicalparameterscommonlyusedtocharacterised 1Hzfrequencyrange),andmoreremarkablywithanexponent GRBsdiffersignificantlybetweenthetwotemporalclasses.We valueapproximatelyequaltothatoftheKolmogorovspectrum lookintotheproblemoftheenergydependenceoftheACFand foundinfluidturbulence(Kolmogorov1941).Thesignificance discusspossiblebiasesinoursampleofburstswithknownzin ofathusobtainedaveragePDSdependsontwoimportantad- Sect.7.InSect.8wediscussourmainresults. ditionalfactors.First,lightcurveshavetobenormalisedtobal- ancetheweightbetweenburstsofdifferentbrightness.Itisnot 2. Data clear at this point which norm should be used to produce the most meaningfulaverage. Beloborodovetal. (1998) favoured This work is mainly based on the analysis of light curves the use of the peak flux normalisation, however they tested from GRBs with known redshift.Given the scarce numberof several other norms with qualitatively similar results. Thus, casesavailableforstudy,wecombineddata(inthegammaen- a good normalisation should give better convergence but the ergy band) from three instruments to improve our statistics. norm should not affect the final result for a sufficiently large Increasingasampleinthiswaypresentsanobvioustrade-off, sample. The second problem is the shift in frequencies pro- since we use counttime series and the differencebetween in- ducedby cosmic dilation effects. Having no redshiftsz deter- strumentresponsesintroduceanadditionaldispersionthatmay minedfortheirGRBsample,Beloborodovetal.(2000)didnot counteract the benefits. For this reason, we initially analysed correctfortheseeffects.However,theyarguedthatiftheunder- thedataoftheburststhatwereobservedbymorethanonein- lyingPDSshapeforeveryGRBisafeaturelesssinglepower- strument, evaluating whether the differences were acceptable lawwithaconstantexponent,thefrequencyshiftswillnotaf- forourpurposes. fectthe obtainedaveragePDS. If thisis the case, considering The comparisons were made taking BATSE as the refer- thatthetwodecadesrangeofthepower-lawismuchlargerthan ence instrument. Its data comprise half of our GRB sample thestandarddeviationoftheredshiftdistribution(σ 2based (inthe gammaenergyband),showingthebestsignal-to-noise z ∼ onthefewknownredshifts),indeedtheshiftsshouldjustsmear ratio (S/N) with relatively low directional dependence thanks thecut-offfrequencies. to the large collecting area of its eight Large Area Detectors Thesame statisticalapproachwasusedbyFenimoreetal. (LADs) placed on each corner of the Compton Gamma-Ray (1995, hereafterF95)intheirstudyoftheaverageautocorre- Observatory (CGRO; Fishmanetal. 1989), giving full sky lation function (ACF) of a sample of bright GRBs. The ACF coverage. It flew during the period 1991–2000 collecting the givesa measure of the correlation between differentpointsin largest GRB catalog up to date. The CGRO Science Support the light curve that are separated by a given time lag. Since Center (GROSSC) providesthe so-called concatenated64 ms it is the Fouriertransformof the PDS, it containsin principle burst data, which is a concatenation of the three standard thesameinformationthatcanbevisualisedinadifferentway. BATSE data types DISCLA, PREB, and DISCSC. All three Therefore,the same caveatsregardingthe averagePDS apply data types have four energy channels (approximately 25–55, to the average ACF. It was only after the discoveryof the af- 55–110, 110–320, and > 320 keV). The DISCLA data is a terglow emission (Costaetal. 1997) and the determination of continuous stream of 1.024 s and the PREB data covers the their redshift for a significant number of bursts that we were 2.048spriortothetriggertimeat64msresolution,bothtypes abletoaddresssomeofthoseissues.Borgonovo(2004, here- obtained from the 8 LADs. They have been scaled to overlap afterB04)showedforasampleof16brightGRBwithknownz the DISCSC 64 ms burst data, that was gathered by the trig- L.Borgonovoetal.:VariabilityclassesinGRBs 3 Table1.Sampleof22GRBswithknownredshift.ThecolumnsgivethenameoftheGRB,thesourceinstrumentfortheγ-ray data, the X-ray instrument when available, the measured redshift z, the corresponding reference, the ACF half-width at half- maximumintheγ-rayenergybandw ,thehalf-widthcorrectedfortimedilationw ,thecorrespondinghalf-widthsw andw γ ′γ X ′X fromtheX-raylightcurveswhenavailable,thewidthratiobetweenthetwoenergybandsw /w ,theindexξassumingawidth X γ energydependencew (E) E ξ,andtheACFwidthclass. γ − ∝ GRB Instrument(γ) Inst.(X) z Ref.b w w w w w /w ξc Classd γ ′γ X ′X X γ 2ch (55–320keV)a (2–28keV) (s) (s) (s) (s) 970228 GRBM WFC 0.695 (1) 1.3 0.1 0.77 0.06 3.5 0.2 2.1 0.1 2.7 n ± ± ± ± 970508 BATSE/GRBM WFC 0.835 (2) 2.7 0.1 1.47 0.05 10.3 0.4 5.6 0.2 3.8 0.06 0.05 n ± ± ± ± ± 970828 BATSE 0.9578 (3) 15.33 0.06 7.83 0.03 0.09 0.03 b ± ± ± 971214 BATSE/GRBM/Konus WFC 3.418 (4) 8.02 0.08 1.81 0.02 11.8 0.8 2.7 0.2 1.5 0.10 0.04 n ± ± ± ± ± 980326 GRBM WFC 1.2 (5) 1.34 0.1 0.61 0.04 2.35 0.4 1.05 0.2 1.7 n ± ± ± ± 980329 BATSE/GRBM/Konus WFC 3 1 (6) 5.96 0.02 1.5 0.5 8.8 0.4 2.2 0.7 1.5 0.05 0.01 n ± ± ± ± ± ± 980425 BATSE/GRBM WFC 0.0085 (7) 7.62 0.08 7.56 0.08 14.8 0.9 14.7 0.9 1.9 0.24 0.06 b ± ± ± ± ± 980703 BATSE 0.966 (8) 14.15 0.1 7.19 0.05 0.11 0.01 b ± ± ± 990123 BATSE/GRBM/Konus WFC 1.600 (9) 19.81 0.03 7.62 0.01 35.6 0.2 13.70 0.08 1.8 0.30 0.05 b ± ± ± ± ± 990506 BATSE/GRBM/Konus 1.3066 (10) 3.83 0.02 1.66 0.01 0.20 0.05 n ± ± ± 990510 BATSE/GRBM/Konus 1.619 (11) 2.54 0.03 0.97 0.01 0.20 0.01 n ± ± ± 990705 GRBM WFC 0.86 (12) 14.3 0.2 7.7 0.1 22.3 0.4 12.0 0.2 1.6 b ± ± ± ± 990712 GRBM WFC 0.433 (13) 4.1 0.2 2.85 0.1 4.8 0.2 3.35 0.1 1.2 n ± ± ± ± 991208 Konus 0.7055 (14) 3.67 0.04 2.15 0.02 n ± ± 991216 BATSE/GRBM/Konus 1.02 (15) 3.80 0.02 1.88 0.01 0.18 0.02 n ± ± ± 000131 BATSE 4.500 (16) 5.77 0.08 1.05 0.01 0.21 0.06 n ± ± ± 000210 GRBM/Konus WFC 0.846 (17) 2.4 0.2 1.3 0.1 5.35 0.3 2.9 0.2 2.2 n ± ± ± ± 000214 GRBM WFC 0.47 (18) 2.5 0.4 1.7 0.3 6.8 0.3 4.65 0.2 2.7 n ± ± ± ± 010222 GRBM/Konus WFC 1.477 (19) 3.68 0.07 1.48 0.03 42.2 0.4 17.0 0.2 11.5 n ± ± ± ± 010921 GRBM 0.451 (20) 9.8 0.3 6.75 0.2 b ± ± 011121 GRBM/Konus WFC 0.362 (21) 10.0 0.3 7.35 0.2 18.9 0.2 13.9 0.15 1.9 b ± ± ± ± 030329 Konus 0.1685 (22) 2.6 0.1 2.19 0.08 n ± ± a TheenergyrangeofBATSEdatawhichwastakenasreferenceinstrument.ForKonusandGRBMdatatheactualenergyrangesare50–200 keVand40–700keVrespectively. b (1) Bloometal. (2001); (2) Metzgeretal. (1997); (3) Djorgovskietal. (2001); (4) Kulkarnietal. (1998); (5) Bloometal. (1999); (6) Lamb,Castander,&Reichart (1999); (7) Tinneyetal. (1998); (8) Djorgovskietal. (1998); (9) Kulkarnietal. (1999); (10) Bloometal. (2003); (11) Beuermannetal. (1999); (12) Amatietal. (2000); (13) Vreeswijketal. (2001); (14) Dodonovetal. (1999); (15) Vreeswijketal. (1999); (16) Andersenetal. (2000); (17) Piroetal. (2002); (18) Antonellietal. (2000); (19) Jhaetal. (2001); (20) Priceetal.(2002);(21)Garnavichetal.(2003);(22)Greineretal.(2003). c IndexξhasbeenestimatedforanenergywindowwidthoftwoBATSEchannels. d (n)and(b)indicatenarrowandbroadwidthACFclass,respectively. geredLADs(usuallythefourclosertothelineofsight).This However,inB04acomparativeACFanalysisoftheburstsob- combineddataformatwasusedwhenavailable,sincethecon- servedbybothinstrumentsshowedagoodagreementforvery catenatedpre-burstdataallowsabetterestimationoftheback- brightbursts,andtheselectioncriteriaforKonuscaseswereset ground.In the case of GRB 970828the DISCSC data are in- requiringpeakcountrateslargerthan3000countss 1 andthe − complete,andweusedinsteadthe16-channelMERdatatype, availability of post-burstdata, resulting in the 5 KonusGRBs binned up into 4 DISCSC-like energy channels. All BATSE includedinB04burstsample. burstswithknownzwereconsideredforstudy,excludingtwo The BeppoSAX mission, that operated between the years cases were the data are incomplete or were no recorded (i.e., 1996–2002,had broad energy band capabilities thanks to the GRB980326andGRB980613),resultinginatotalof11cases. combined operation of several instruments. The Gamma Ray WealsoincludethesetofburstsselectedinB04thatwere Burst Monitor (GRBM; see Fronteraetal. 1997) covered the observed by Konus, which is a GRB detector on board the 40–700 keV energy range, roughly matching the range of Wind mission (Aptekaretal. 1995). Light curves of its bursts Konus and BATSE (i.e., 55–320 keV using 2 + 3 channels). are publicly available at 64 ms resolution in the 50–200 keV Note that since the ACF and the PDS are quadratic functions energy band. The collecting area of this experiment is about of the numberof counts, and generallythere are more counts 20timessmallerthantheoneofBATSE andconsequently,in at lower energies, the agreement for these temporal analysis most cases, the signal is too weak for our temporal studies. functionswilldependmainlyonhavingasimilarlower-enden- 4 L.Borgonovoetal.:VariabilityclassesinGRBs X-ray bands when available, the estimated redshift z, and the correspondingreference. TofurtherstudytheACFwidthenergydependencewese- lectedfromtheBATSEcurrentcatalogalllongbursts(i.e.,du- rationtimeT >2s)withapeakfluxmeasuredonthe1.024s 90 timescale F 4 photons cm 2s 1 in the 50–300 keV band 1s − − ≥ thathaveavailableconcatenated64msdata.Thisresultedina sample of188brightburstsforwhich,in mostcases, thered- shiftisunknown. 3. Methods For the autocorrelation function analysis we follow the same Fig.1.Examplesof GRB lightcurvesfromthe oursample of method presented in B04, that was based on earlier works burstswithknownredshifts,wherethe time t iscalculatedat of Linketal. (1993) and F95. Here, we will summarise the ′ thesourcerest-frame.Thelightcurvesontheleft(right)panels method,andwerefertoB04forfurtherdetails.Followingthe have broad (narrow) ACF widths. There are no obviousmor- same notation, from a uniformly sampled count history with phologicaldifferencesbetweenthetwoclasses,bothpresenting ∆T timeresolutionandNtimebins,letmibethetotalobserved casesofsimpleandcomplexstructures. counts at bin i. Also let bi be the corresponding background level and c = m b the net counts. The discrete ACF as a i i i − functionofthetimelagτ=k∆T is emrgsyLlCims,itb.uWtewmeabdiennuesdeothfethmeustpanidnatordah6ig2h.5remsoslutitmioenr7e.s8o1l2u5- A(τ=k∆T) XN−1cici+k−miδ0k, k=0,...,N 1, (1) ≡ A − tion to improve the S/N ratio. This has a negligible effect on i=0 0 the measurement of the time scales that concern us here and whereδistheKroneckerfunction.Heretheperiodicboundary it reduces some noise artifacts (i.e., the noise becomes more conditions(c =c )areassumed.Thenormalisationconstant i i+N Poissonian).Inaddition,itapproximatelymatchesthestandard A isdefinedas 0 64msBATSE temporalresolutionforbettercomparison.The LCs were dead-time corrected and backgroundsubtracted. In N 1 − Borgonovoetal.(2005)a comparativeACFanalysiswaspre- A0 ≡X(c2i −mi), (2) sented,including17GRBSdetectedbytheGRBMwithknown i=0 redshift from which 8 were also observed by BATSE. It was suchthatA(0)=1fork =0.Thetermm inEq.2subtractsthe i concludedtherethatalthoughthemeasureddispersionsofthe contributionoftheuncorrelatednoiseassumingthatitfollows localACFwidthswerelargerthanintheBATSEcase( 15% thePoissonstatistics. ∼ at half-maximum), the average sample values for each ACF For practical reasons, the actual calculation of Eq. 1 was width class were equal to within uncertainties. Therefore, at done using a Fast Fourier Transform(FFT) routine.Denoting leastinthecontextofthepresenttemporalvariabilityanalysis, by C the Fourier transformed of the background subtracted f wecanconsiderthatthesedatahavemoreintrinsicdispersion, light curve, then the definition of the power density spectrum buttheintroducederrorsaremainlystochastic. (PDS) can be written as P C 2. The noise contribution f f ≡ | | The two Wide Field Cameras (WFCs) also on board is subtractedfromthePDS assumingPoisson statistics. Since BeppoSAX covered the 2–26 keV energy range (Jageretal. thePDSandtheACFareFourierpairs(Wiener-Khinchinthe- 1997). During their operation time they detected 53 GRBs in orem), the latter is obtained by inverse transformingthe first. conjunctionwiththeGRBMallowingthefirstbroadbandstud- Zeropaddingofthetimeserieswasusedtoavoidtheartifacts iesofGRBs(see,e.g.,Amatietal.2002).Furthermore,acon- producedby the periodicboundarycondition.The normalisa- siderable fraction of the WFC bursts ( 36%) were also de- tionusedfortheACF,thatgivestoitscentralmaximumunity ∼ tectedbyBATSE.TheLCswereextractedwithatimeresolu- value,isequivalent(asidefromanoisecorrectionterm)tothe tionof62.5msanddiscriminatedinthreeenergychannels(i.e., scalingoftheLCbythesquarerootoftotalpower √P ,where tot 2–5,5–10,and10–26keV).Theenergyintervalswerechosen P c2.ThisnormalisationisanaturalchoicefortheACF tot ≡Pi i in order to have a similar amount of counts in each channel analysis and it makes the ACF of each burst independent (to for a typical GRB. However, except for the brightest GRBs, first order) of its brightness. Note however that for our PDS thesignalineachchannelistooweakforthepurposesofour analysiswefounditmoresuitabletoscaletheLCsbytheirre- temporalanalysisandtheLCshadtobeintegratedintoasin- spectivenetcountfluences c (orequivalentlydividing F ≡Pi i gle energychannel.Here we will focuson the analysisof the the PDS of the original LCs by 2). Since the a zero order 0 F 13GRBsforwhichwehaveredshiftestimations.Thesebursts Fouriercoefficientisequaltothefluence ,ournormalisation F constituteasubsetofoursampleofGRBswithknownredshift makesthePDSconvergetounitytowardsthelowfrequencies, in the γ energy band. Table 1 lists all these bursts indicating thereforeinFourierspaceitcanbeinterpretedalsoasnormalis- initsfirstcolumnstheirname,thesourceinstrumentsinγand ingbytheamountofpoweratthelowestfrequency.InSect.4.2 L.Borgonovoetal.:VariabilityclassesinGRBs 5 wewillfurtherdiscussthischoiceoverotherpossiblenormal- isations. The backgroundestimations were done by fitting with up toasecondorderpolynomialthepre-andpost-burstdata,that werejudgedbyvisualinspectiontobeinactive.Thisiscritical forweakburstsandparticularlyaggravatedin theKonuscase sincethepubliclyavailableLCshavealmostnopre-burstdata. This is the main reason why dim bursts were excludedin our sampleselectioninSect.2. In B04 it was found empirically, considering the analysis of the average ACFs of the narrow and broad width sets and theirdispersion,thatathalf-maximumtheseparationbetween thetwosetsismostsignificant.Therefore,theACFhalf-width athalf-maximumwgivenby A(τ=w) 0.5(herewedeviate ≡ fromtheB04 notationforsimplicity)was chosenas the mea- surethatbestcharacteriseseachclass.Sinceithasbeenshown that the average ACF decreases approximately following a stretchexponential(F95),theACFwidthwwascalculatedfit- ting the logarithm of the ACF in the range 0.4 A(τ) 0.6 ≤ ≤ withaseconddegreepolynomial. The uncertainties due to stochastic fluctuation were esti- matedusingaMonteCarlomethod.ForanygivenLCalarge number of realisations is generated assuming that the fluctu- ations on the gross number of counts in each bin m follows i a Poisson distribution. Then, the LCs are background sub- tractedandtheACFwidthswarecalculatedfollowingthesame method as for the original data. However, when subtracting the Poisson noise contribution in the ACF (Eq. 2) a factor of 2 must be introducedin the correspondingnoise term to take Fig.2. a) Autocorrelationfunctions(ACFs) of 22 GRBs with into accountthat the procedureto generate the synthetic LCs known redshifts calculated in the observer’s frame. Although doublesthenoisevariance,sincetheyareinasensesecondor- easily identifiedinthe rest-frame,the narrow(graylines) and derrealisations.Ingeneral,ifoneconsidersaPoissonprocess thebroad(solidlines)classesoverlapgivingtheimpressionof withexpectedvalueµanditerativelyassumesitscountsasex- aunimodaldistribution.b)LocalACFswherethecosmictime pectedcountsofanotherPoissonprocess,afterniterationsthe dilation effect has been corrected, being τ′ = τ/(1+ z). The expectedvalueis stillµ, butthe varianceisnµ(withn = 2in newlyaddedGRBMdata(dashedlines)reinforcethebimodal thiscase).Finally,thestandarddeviationoftheobtainedwidths patternpreviouslyfoundbyB04thatusedonlytheBATSEand isusedasestimatoroftheuncertainty.Nevertheless,themain Konusdata(herebothshownwithsolidlines). source of uncertaintyin dim bursts is the backgroundestima- tion,thatintroducesasystematicerrorinthedeterminationof theACFwidth.Therefore,thereporteduncertaintiesshouldbe pattern. In Fig. 2a the observed ACFs A(τ) are shown first regarded as lower limits. Another likely source of systematic for comparison, while Fig. 2b shows the rest-frame or local errorsisthenonuniformresponseofthedetectors,sincewedo ACFs A(τ),whereτ = τ/(1+z)isthetimelagcorrectedfor notdeconvolvethecountdata. ′ ′ cosmic dilation. We obtained mean values for the rest-frame ACF half-widthat half-maximumw¯ (n) = (1.56 0.15)s and 4. Analysisoftheγ-rayLCs w¯ (b) = (7.42 0.14) s, and also sa′mγ ple standa±rd deviations ′γ ± SomeLCexamplesfromoursampleofGRBswithknownred- σ(n) = 0.6 s and σ(b) = 0.5 s for the narrow and broad sub- γ γ shiftareshowninFig.1.Nomorphologicaldifferencesareev- sets respectively.Themeanvaluesareequaltothosereported identbetweenthenarrowandbroadclassesbysimpleinspec- in B04withinuncertainties.Thegapbetweenthe twosubsets tion.However,standardlinearanalysistoolsrevealclearvari- (definedas the meandifference)representsa 4.9σseparation, abilitydifferencesbetweenthetwo,aswereportinthissection. smaller than the 7σseparationreportedin B04 mainlydueto the increased dispersion in the broad class. However, thanks toalargersample,wehaveslightlyincreasedthesignificance 4.1.ACFs ofthisseparation.Theprobabilitypofarandomoccurrenceof With the inclusion of GRBM bursts we were able to expand suchgapwasestimatedusingMonteCarlomethods.Assuming thesamplepresentedinB04.TheaddedACFs(showninFig.2 that there are no characteristic timescales and an underlying indashedlines)alsofollowsabimodaldistributionwhencor- uniformprobabilitydistribution(themostfavourablecase)we rectedfortimedilationeffects,reinforcingthepreviouslyfound estimated p<4 10 7. − × 6 L.Borgonovoetal.:VariabilityclassesinGRBs sarybecauseevenifalltheLCshadthesametimeduration,the redshift correction would make the Fourier frequencies differ foreachburst(i.e.,sincetheobservedfrequency f transforms totherest-frameas f = f(1+z)).ThelowerpanelsofFigs.3 ′ and4showthecorrespondingcentralvaluesP˜ ,wherewehave f chosen the use of the medianoverthe mean as a more robust estimatoroftheexpectedor“underlying”spectrum.Although qualitativelysimilarresultsareobtainedinbothcases,theme- dian shows smaller fluctuations and it is less sensitive to the randomexclusionofafewburstsfromeachclass.Thefigures showalsothequartiledeviation(dashedlines)asameasureof thedispersionthatwaschosentobeconsistentwiththeuseof themedian(i.e.,the25%andthe75%quartilesaroundtheme- dianvalue).ComparingthetwomedianPDSitisevidentthat they haveremarkablydifferentshapes. Thenarrowclass PDS asexpectedshowsmorepowerathighfrequenciesanditiswell describedbyasinglepower-lawmodelP˜ 1/fα uptoalow f ∝ frequencycut-offat 0.1Hz. In Fig. 3b we show a fit to the ≈ medianPDSinthefrequencyrange0.1 f 10Hzforwhich ′ wefoundabest-fit-parameterα(n) = 1.9≤7 0≤.04(i.e.,γband- γ ± narrowclass).Theobtainedexponentisconsistentwiththatof Brownianorrednoise(i.e.,withα=2),whichsuggeststheal- ternativeuseofasingleLorentzianmodeltodescribetheentire PDSasinshotnoisemodels(Belli1992).However,asshown alsoinFig.3b,thereseemstobeaslightsystematicdeviation aroundthecut-offregion.ThebroadclassP˜ ontheotherhand f appearstohavetwodistinctcomponents,i.e.,abroadlowfre- quencycomponentwithasharpbreakandapower-lawcompo- Fig.3.Powerdensityspectra(PDS) ofthesubsetof15bursts nent(Fig.4b).Ifweassumethatthetwocomponentsareinde- with narrow width ACFs. The frequency scale has been cor- pendentthenthesinglepower-lawshouldhaveanindependent rected for cosmic time dilation effects and the noise levelhas cut-offatlowfrequencieswhichcannotbedeterminedbutonly been subtracted assuming Poisson statistics. The upper panel constrainedbyourdata.Thelowfrequencycomponentiswell showsallindividualPDSwherethefrequencydatahavebeen fitted by a stretched exponential P (f) = exp[( f/f )η]. The equallybinnedinthelogarithmicscale.Thelowerpanelshows 1 1 − best-fit-parametersfor this model will depend to some extent theestimatedsamplemedian(solidline)andthequartilesabout on the cut-off frequency parameter of the power-law compo- themedian(dashedlines)toindicatethedispersion.Thedecay nentandviceversa,howeverthiswillaffectmainlytheestima- phaseis wellmodelledbya power-law(grayline) withindex tionoftheexponentη.Takingintoaccounttheseuncertainties 1.97 0.04. Also shown, a Lorentzian function (dotted line) ± we found a characteristic frequency f = (0.025 0.003)Hz providesafairlygoodfitoverthewholefrequencyrange. 1 ± and an index η = 1.3 0.2. We considered other empirical ± models,concludingthatfunctionswith power-lawasymptotic Hereafteroursampleofburstswithknownredshiftsintheγ behaviour (e.g., a Lorentzian) do not decay fast enough to fit bandwillbedividedforanalysisintotwosubsetsof15narrow thedata.Forthehighfrequencycomponentwefoundabest-fit and7broadACFwidthburstsrespectively.InTable1weshow power-law index α(b) = 1.6 0.2 (i.e., γ band - broad class), γ ± thenewlyaddedACFswidthstogetherwiththosepresentedin consistent with the Kolmogorovspectral value 5/3. However, B04forcompleteness. notefromFig.4athatfordifferentburststhelattercomponent appearstovarymuchmorethanthefirst. As we discussed in Sect. 1, the obtained mean PDS (as 4.2.PDS well as the median) may depend on the chosen normalisation WecalculatethePDSforoursampleofGRBswithknownred- sinceeachburstwillbeweightedinadifferentway.IfallLCs shift, correcting the LCs for cosmic time dilation effects. We in the sample are realisations of the same stochastic process groupburstsfollowingtheclassesestablishedin 4.1basedon then the norm should simply improve the convergenceto the § theACFwidth.TheupperpanelsofFigs.3and4showforthe mean value but should give identical results for a sufficiently narrowandbroadclassesallindividualPDSoverlaidforcom- large sample. Aside from the fluence norm, we tested several parison. Given the chosen normalisation, all PDS must con- ways to scale the LCs, e.g., the peak flux used previously by vergetounityatlowfrequencies(seeSect.3).Thefrequencies Beloborodovetal. (1998, 2000), the Miyamotoetal. (1992) havebeenequallybinnedonthelogarithmicscale.Theproce- normalisation (which expresses the PDS in fractional root- durenotonlysmoothsoutthePDSstochasticvariationsbuten- mean-square),andtherootofthetotalpower √P usedforthe tot ableustoestimateanaveragePDSforeachclass.Thisisneces- ACF.WefoundinallcasesqualitativelythesamemedianPDS L.Borgonovoetal.:VariabilityclassesinGRBs 7 Fig.5. Median power density spectra of our whole sample of 22 GRBs with known redshift calculated in the observer’s frame. The spectrum shows a power-law behaviour approxi- matelywithinthe0.01–1Hzfrequencyrangeandabestfit(dot- tedline)givesanindex1.73 0.07,consistentwiththeresultsof ± Beloborodovetal.(1998)derivedwithoutredshiftcorrections. Notethatthequartiledispersion(dashedlines)isconsiderably largerthanthosefoundinFigs.3and4. intotheX-rayenergyrange.Theseburstsrepresentasubsetof thepreviouslyusedsampleintheγ-rayband(Sect.4).Wefind also at these energies significant differences between the two Fig.4. Power density spectra (PDS) of the subset of 7 bursts ACFclassesalthoughtheresultsinevitablehavelargerassoci- showing broad width autocorrelationfunctions(ACFs). As in ateduncertainties. Fig.3theindividualPDSandthesamplemedianareshownfor comparison.The medianPDS is significantly differentto that ofthenarrowclassandtwocomponentsappeartobepresent. 5.1.ACFs The low frequencycomponenthas a sharp cut-offand is well Although the ACF widths are substantially broader at these modelledbyastretchedexponential(grayline).Thehighfre- lowerenergies(2–26keV),thetwoACFclassesarestilleasily quencycomponentiswelldescribedbyapower-lawwithindex distinguishableasshowninFig.6.However,theGRB010222 1.6 0.2consistentwitha5/3Kolmogorovspectralindex. ± classifiedashavinganarrowACFintheγband(graylines)has broadedtosuchextentthatitfallsintothebroadwidthrange, for each class butsignificantly larger spreads. Obviously,this although its ACF decays more slowly than any of the broad doesnotprovethateachclasssampleis“uniform”sinceitisa widthcases.AsshowninTable1wherewelistalltheobserved necessarybutnotsufficientcondition.Inthisrespect,itisworth and the local ACF widths, this is the only outlier in the sam- mentioningthatwhencombinedintoasinglesampleandwith- ple. In column 10 we list as a measure of the broadeningthe outcorrectingforredshiftseffectsweobtainedamedianPDS ACFwidthratiofortheγandX-raybands.Whiletheaverage (showninFig.5)thatiswellfittedinthe0.01–1Hzfrequency widthratiois wX/wγ 2,thebroadeningintheGRB010222 h i≃ rangebyasinglepower-lawwithindex1.73 0.07,consistent case(wX/wγ 11.5)ismuchlargerthananyotherburstinthe withinuncertaintieswiththeKolmogorovspe±ctrumandinfull sample and it≃is most likely caused by a systematic error, as agreementwithBeloborodovetal.(1998).Therefore,oursam- wewilldiscusslaterinSect.7.Forthisreasonthiscasewillbe pleofGRBswithknownzisnotdifferent,atleastinthisregard, keptseparatefromthesamplefortherestofourWFCdatatem- tothegeneralsampleofbrightBATSEGRBsusedinprevious poralanalysis.Consequently,theobtainedaveragelocalwidth works. Note also that the overalldispersionaboutthe median ACFare w¯′(Xn) = (3.1±0.5)sandw¯′(Xb) = (13.6±0.6)s, with is significantly largerthan in Figs. 3 and 4 where, despite the sample standard deviations(relative dispersions) σ(n) = 1.9 s X smallersamplesizes,redshiftcorrectionsandACFclasseshave (60%)andσ(b) =1.6s(12%)forthenarrowandbroadsubsets X beentakenintoaccount. respectively. 5. AnalysisoftheX-rayLCs 5.2.PDS InthissectionweusetheWFCsampleof13GRBswithknown OnceagainusingoursampleofWFCburstswithknownz,we redshifts(seeTable1)toextendtheprevioustemporalanalysis estimate the localmedian PDS for each ACF width class fol- 8 L.Borgonovoetal.:VariabilityclassesinGRBs lowingthesamemethodsusedinSect.4.2.Weconsidersepa- ratelythespecialcaseGRB010222duetotheuniquebroaden- ingofitsACF.Figure7showsourresultsforthenarrowclass. As expectedfrom the ACF analysis the break appearsnow at approximatelyhalf the correspondingfrequencyat γ energies (Fig. 3), but most noticeably the behaviour towards the high frequenciesnowshowsamuchfasterdecay.Abest-fitforfre- quencies f &0.08HzusingaP 1/fαmodelgivesanexpo- ′ f ∝ nentα(n) =3.0 0.2,asignificantlysteeperpower-lawthanin X ± theγband.ThePDSofGRB010222(grayline)isalsoshown in Fig. 7 for comparison. Although with much less power at lowfrequenciesthantheotherburstsinthisclass, athighfre- quencies (f 0.2 Hz) it follows a very similar asymptotic ′ ≥ behaviour. Fig.6. Autocorrelation functions (ACFs) calculated at the Our estimation of the median PDS P˜ for the broad class source rest-framefor a sample of13 GRBs detectedby WFC f is shown in Fig. 8. In spite of the large uncertainties associ- inthe2–28keVenergyband.Inspiteofthebroadeningofthe ated with such a small sample, the two componentsfound in ACFatlowerenergies,thebimodaldistributionfoundintheγ- Fig. 4 are recognisable. The low frequencycomponentis not raybandisclearlyseen,althoughwithlargerrelativedispersion as prominent and broad as before, and consequently the data for each class. However, one burst (GRB 010222)previously are not enough to properly constrain the fit parameters of a classified as belonging to the narrow ACF width class (gray stretchedexponentialas it was previouslydone.However,we lines) appears wider than the broad ACF width class (solid estimatedthatthehighfrequencycomponentfollowsapower- lines). This behaviour is most likely not intrinsic but due to lawwithindexα(b) = 1.7 0.2consistentwithinuncertainties asystematicerror(seediscussioninSect.7). X ± with the Kolmogorov index found in Sect. 4.2 for the γ-ray LCs. When compared with the broad class P˜ in Fig. 8, the f GRB 010222 spectrum matches neither the P˜ general shape f noranyofitscomponents.Power-lawfitsusingtheindicesand thefrequencyrangesshowninFigs.7and8fortherespective high-frequencycomponentsresultedinsquaredresidualtotals per degrees of freedom χ2/ν = 16.2/11 and χ2/ν = 179./17 forthenarrowandbroadPDSrespectively.Basedonthesefits we conclude that the first model is acceptable while the sec- ond model can be rejected at a very high confidence level. Uncertainties were estimated using synthetic shot noise and calculating standard deviations on a sample of PDS follow- ingthesameproceduresandfrequencybinningsappliedtothe burstdata. Fig.7. Median power density spectra (PDS) from the subset 6. ComparingACFclasses of WFC bursts with known redshifts belonging to the narrow In order to understand the origin of the two classes found widthACFsclass,excludingtheoutliercaseGRB010222that based on our temporal analysis we looked for any additional isshownalongside(grayline).Thedecayathighfrequencies GRBcharacteristicorphysicalparameterthatmightdifferbe- (f > 0.1 Hz) approximatelyfollowsa power-law,but with a ′ tween them. As point out in Sect. 4, the visual inspection significantly steeper slope than the previouslyfoundin the γ- of the LCs reveals no trivial morphological differences be- rayenergyband(3.0 0.2).GRB010222showssimilarasymp- ± tweentheclasses(seeFig.1).Bothclassespresentcaseswith toticbehaviourathighfrequencies. simple and complex structure, showing from a few smooth pulses to many heavily overlapped sharp pulses. To cover a broad range of physical parameters, we made use of the we compared the distributions within the narrow and broad database of GRB redshifts and other burst parameters com- ACF classes using the standard Kolmogorov-Smirnov (K-S) piledinFriedman&Bloom(2005)andreferencestherein,that test(see,e.g.,Pressetal.1992),whichisparticularlysensitive contains much additional information about the 22 bursts in to median deviations, but we found no significant difference Table1,althoughnotallparameterestimationsareavailablefor forthecumulativedistributionsofanyofthoseparameters.No everyburst.Inparticular,weconsideredthefollowingparame- significantRcorrelationcoefficientswithw werefoundforE ′ p′k ters(primedquantitiesarecalculatedatthesourcerest-frame): and E (logarithmstakenin allcases),althoughmarginalre- iso the peak energy E , the isotropic equivalent gamma-ray en- sultswereobtainedforthetemporalparametersT (R = 0.44 p′k 9′0 ergyE ,thedurationtimeT ,thetimeoftheobservedbreak withsignificance p<0.04)andT (R=0.46withsignificance iso 9′0 b′ intheafterglowlightcurvesT ,andtheredshiftz.Inallcases p < 0.05 for only 18 bursts with determined values). Note b′ L.Borgonovoetal.:VariabilityclassesinGRBs 9 ditional redshift corrections to the accounted cosmic dilation werestudiedusingthetransformation w =w/(1+z)1+a (3) ′ fortheobservedwidths,wherearepresentsasmallerthanunity correction.Therelativedispersionofeachwidthclasswascal- culatedasafunctionoftheparameteralookingforminimum values(seeFig.6in B04)since,undertheassumptionthatthe w intrinsic distribution does not depend on z (i.e., evolution ′ effects are neglected), redshift dependencies should increase the observed dispersion. We obtained minimal dispersions at a(n) = 0.35 0.2anda(b) = 0.04 0.04forthenarrowand min − ± min ± broadclassesrespectively,equalwithinuncertaintiestoprevi- ousresults.Theuncertaintieswereestimatednumericallyusing Fig.8. Median power density spectra (PDS) from the subset thebootstrapmethod(see,e.g.,Pressetal.1992).Ifthedisper- of4WFC burstswithknownredshiftsbelongingtothe broad sioniscalculatedoverthewholesample(i.e.,withoutseparat- widthACFsclass.Despitethesmallsamplesize,thetwobroad ingthetwowidthclasses)nowell-definedminimumisfound, PDScomponentsidentifiedatγ-rayenergiesarerecognisable. with an approximately constant value within the free param- Theindexofthepower-lawcomponent(dottedline)isconsis- eter range a < 1, which further supports the separation into | | tentwith the Kolmogorovvalue as previouslyfoundin Fig. 4 twoclasses.However,whilethenarrowclassminimumagrees givenabest-fit-parameter1.7 0.2. withinerrorswiththeexpectedeffectfromtheenergyshift,the ± broadclass dispersion doesnotimprovemuchwith anyaddi- tionalredshiftcorrection.Actually,inbothcasesaperfectcan- cellation(i.e.,a = 0)cannotberuledout,althoughitwould min thatacorrelationbetweenwandT90hasbeenwellestablished appear coincidental that all other possible redshift dependen- (R=0.58),andalthoughintheobserver-frameispartiallydue cies wouldexactlycompensatethe energyshift.Anotherpos- to the cosmic temporal dilations, it should be found also in sibility is that for the broad class the ACF dependence with the rest-frame (Borgonovo&Bjo¨rnsson 2006). However, our energyisweaker.Atfacevaluethelowdispersionalreadysug- presentsampleistoosmalltodetermineitatahighconfidence gests this andin thiscase a nearly nulla correctionwould be level. More surprising is the finding of an indication of some muchmorelikely.Inwhatfollowswewillshowevidencethat correlation between a prompt (w′) and an afterglow (Tb′) pa- supportsthelatteralternative. rameter,butthiswouldneedbetterstatisticstobeconfirmed. Using 4 energychanneldata from a sample of 188 bright Other subdivisions of the long/soft GRB class have been BATSEburstsweanalysetheACFwidthdependencewithen- suggested in the literature but we found them to be unrelated ergyfor each individualcase using a single power-lawmodel to with the one under consideration. Evaluation of the distri- w(E) E ξ (thechannelsare approximatelyequallyspaced − ∝ butions of the observed T and E for the broad ACF class on a logarithmicscale). We foundthatin mostcases the rela- 90 pk excludes an association between these bursts and the “inter- tionw(E)inindividualburstsisnotwellfittedbyapower-law mediate” duration GRB group reported by Horva´th (1998) (lessthan33%oftheburstssatisfiedforthesumofthereduced and Mukherjeeetal. (1998). Likewise, we found no corre- residuals χ2 < 2), but for most cases it can be considered a r spondence between the ACF bimodality and the bimodality goodapproximationwithintheBATSEenergyrangewithtyp- of the afterglow optical luminosity distribution reported by icaltotalrelativeerrors< 10%.Theobtaineddistributionofξ Nardinietal.(2006)andLiang&Zhang(2006). indices is shown in Fig. 9. Since we foundfor ourfits poorer figures-of-merit(i.e.,χ2)forlargerξ values,thelargestvalues r arerelatedtothecasesthatstronglydeviatefromthepower-law modelforw(E)andarethereforeilldetermined.Forthissame 7. ACFenergydependence sample,apower-lawfitoftheenergydependenceofthewidth InouranalysiswehaveonlycorrectedtheLCsforcosmictime ofthemeanACFgivesanindex 0.40 0.03inagreementwith − ± dilation. However, the data are collected over a finite energy F95.However,noteinFig.9thatforsuchalong-taileddistri- band and therefore additional corrections might be needed to bution most individual bursts typically have a weaker energy account for the shift in energy, since in principle we should dependence and therefore the median ξ˜ = 0.29 is a more 1ch compareLCs emitted in the same energyband.AsLC pulses representativevalueofindividualindices. aresharperathigherenergies(Norrisetal.1996),correspond- In addition, since we are particularly interested in quan- ingly the ACF is narrower. For a sample of 45 bright bursts, tifying the effect of the energy dependence in our ACF anal- F95 found that the width of the mean ACF depends on the ysis where the observation energy window has the width of energy E as w(E) E 0.43. Since for large redshifts the in- two BATSE channels, we studied the ACF width energy de- − ∝ strumentwillseephotonsemittedathighermeanenergies,the pendencew(E) combiningthe 4-channeldata into three over- anti-correlationoftheACFwidthwiththeenergyshouldpar- lapped 2-channelwidth energy bands. The distribution of the tially counteract the time dilation effect. In B04 possible ad- thus obtained indices ξ is shown in Fig. 9 (dashed lines). 2ch 10 L.Borgonovoetal.:VariabilityclassesinGRBs halfoftheBATSEwdistributionwithoutredshiftsseemstrun- cated. Using the K-S test that compares cumulative distribu- tionswecalculateachanceprobability p < 0.02thattheyare drawnfromthesamedistribution.Weconcludethatthesample withknownzissignificantlybiasedtowardslargeACFwidths, mainlyaffectingtherepresentationofthenarrowclass.Taking intoaccountthistruncationandthefactthatforabursttobe- longtothebroadclassitsobservedACFwidthmustbew&7s independently of the redshift, we estimate that the unbiased BATSE fraction of broad cases should lie approximately be- tween0.08. f .0.22tobeconsistentwiththeobservations. b Having characterised the energy dependence of the timescale w with the index ξ, we investigated possible corre- lations with other GRB parameters, but bearing in mind that Fig.9. Distribution of the index ξ in the power-law relation the largest ξ values are most likely meaningless. We tested a w(E) E ξusedtomodeltheenergydependenceoftheACF broadrangeoftemporalandspectralparameters(e.g.,duration − width,∝obtainedusing4-channelconcatenateddata(solidline) time T90, emission time, ACF width w, time lag between en- forasampleof188brightBATSEbursts.Thedistributionhas ergychannels,νFν peakenergyEpk,lowandhighspectralin- anestimatedmedianξ˜ =0.29anditshowsanapproximately dicesαandβ)followingdefinitionsandmethodsdescribedin 1ch exponentialdecaytowardslargerξvalues.Forthepurposesof Borgonovo&Bjo¨rnsson (2006) for a multivariate correlation ourACFanalysisthatusesthetwocentralBATSEchannels,we analysis.Inparticular,weusedameasureoftheoverallspectral modelledalsothew(E)relationusinga2-channelenergywin- evolutionofaburstintroducedthere,wheretheLCisdividedin dowforeachdatapoint.Asexpected,theACFwidthshowsa twofluence-halvesandthepeakenergiesofthecorresponding weakerenergydependenceandconsequentlylowertypicalval- integratedspectraareestimatedandcompared.Inthatwayara- uesfortheξindex.Thethusobtaineddistribution(dashedline) tioofpeakenergiesisdefinedas Epk Epk(1)/Epk(2),sothat R ≡ isnarrowerandwithasignificantlylowermedianξ˜2ch =0.21. REpk > 1 implies an overallhard to soft evolution.Although we found no significant global correlations with the index ξ, thereareindicationsofadifferentbehaviouratlowξvaluesin The median ξ˜ = 0.21 is significantly smaller, as expected twoofthestudiedparameters,i.e.,wand E . 2ch pk R sincelesscorrectionwillbeneededthewidertheenergywin- In Fig. 10 we show scatter plots of the ACF width w and dow. If we consider the medianvalue of the ACF broadening the ratio of peak energies E versus the index ξ. The ξ ob- pk R w /w forthesampleinTable1(excludingGRB010222)we served range was divided into four bins with equal number X γ deriveamedianindexξ˜ 0.20,ingoodagreementwiththe2- of data points, and the correspondinggeometricmeans (more ≈ channelenergywindowdistributioninFig.9,eithertakingthe appropriateforlog-normaldistributions)anduncertaintiesare loweredgeofeachenergywindow(followingF95)ortheirge- shown.Duetothelargespreadthevariablesseemuncorrelated ometricmeans.InTable1welistourestimationsoftheindex based ontheir correlationcoefficientsR, neverthelessthe bin- ξ for the 11 BATSE GRBs with knownz. Both ACF width ning reveals a trend where lower ξ values correspond on av- 2ch classesshowtypicalvaluesthatseemconsistentwiththegen- erage to broaderACFwidths w and lowervaluesof the E pk R eral distribution in Fig. 9, but unfortunatelythe small sample ratio.Inaddition,aweakbutsignificantanti-correlationexists sizeonlyallowedustodistinguishverysubstantialdifferences. between w and E with coefficient R = 0.25 taken their pk R − Since we do not have redshifts for most of the BATSE logarithms. For ξ . 0.3 there is noticeably less dispersion in GRBs, the estimated distributionof indexξ was derivedmix- log E (i.e.,lessrelativedispersionin E )andaclearpos- pk pk R R ing both ACF width classes; neverthelessit is probablyfairly itive correlationthatcompriseshalf of the sample (with coef- representative of the narrow class that dominates the sample. ficient R = 0.45). Comparing a linear fit to those data to the In principle, we do not know how representative that general nocorrelationnullhypothesisgivesfortheF-test(thestandard distributionisofthebroadclass.Foroursampleof11BATSE test for a ratio of χ2 values) a level of rejection p < 0.001. GRBs with known z the fraction of broad cases is f = 4/11 However,since we do not have an objective criterionto trun- b (f = 7/22 for our whole sample). However, since the after- catetheindexξdistribution,therealsignificanceofthisfound b glow localisations needed for the redshift determinations al- correlationishardtoevaluate;thereforeweapproachtheprob- ways involved another instruments with different trigger re- lemusingMonteCarlomethods.Wedrawrandomsamplesfor sponses, that fraction is most likely not representative of the both parameters using the bootstrap method so that they are generalBATSEcatalog.AnanalysisoftheACFwidthsofthe uncorrelated.Thensortingthedatabasedontheirξvalues,we 188 brightest long BATSE bursts sample shows that the dis- look for the ξ-truncate that would give the maximum chance tribution is well described by a log-normal distribution with correlationwith E .Assumingconservativelythatthecorre- pk R a mean 10logw = 3.0 s and a standard deviation represent- lationhastocompriseatleast1/3ofthesampleand R > 0.3 h i | | ing a factor of 3.1, with no significant hint of an underlying then the trial is judged as successful. In this way we found a bimodality(see also Borgonovo&Bjo¨rnsson 2006). Since all probabilityof p < 0.008offindingasimilarorstrongercorre- 11 BATSE GRB with known redshift have w > 2.5 s almost lationbychance.AlthoughtheenergydependenceoftheACF