On the temporal variability classes found in long gamma-ray bursts with known redshift

0.43 MB·English

by  L. Borgonovo

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