Correlations between Lag, Duration, Peak Luminosity, Hardness, and Asymmetry in Long GRB Pulses

0.17 MB·English

by  Jon Hakkila

Correlations between Lag, Duration, Peak Luminosity,
Hardness, and Asymmetry in Long GRB Pulses
Jon Hakkila and Renata S. Cumbee†
∗
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0 DepartmentofPhysicsandAstronomy,CollegeofCharleston,Charleston,SC
∗
0 †DepartmentofPhysicsandAstronomy,FrancisMarionUniversity,Florence,SC
2

n Abstract. Continued study of the BATSE catalog verifies previously-identified correlations between pulse lag and pulse
a durationandcorresponding anti-correlationsbetweenbothpropertiesandpulsepeakfluxforalargesampleofLongGRB
J pulses; thestudyalsofindscorrelationsbetweenpulsepeaklags,pulseasymmetry, andpulsehardness. Thesecorrelations

0 apparentlycanbeusedtodelineateLongGRBsfromShortones.Correlatedpulsepropertiesrepresentconstraintsthatcan
2 beusedtoguidetheoreticalmodeling,whereasbulkpromptemissionpropertiesappeartobeconstructedbycombiningand
smearingoutpulsecharacteristicsinwaysthatpotentiallylosevaluableinformation.

]
E Keywords: gamma-raybursts,statisticalandcorrelativestudiesofgamma-rayburstproperties
PACS: 98.70.Rz,98.62.Ve
H
.
h
INTRODUCTION
p
-
o
Importantcorrelativecharacteristicshavebeenattributedovertheyearstopulsesingamma-rayburst(GRB)prompt
r
t emission. Examples of these characteristics include (1) temporal asymmetry characterized by longer decay than
s
rise rates, (2) hard-to-softspectral evolution, and (3) broadening at lower energies (e.g. [1, 2, 3, 4]). Despite these
a
[ suggestive correlations, GRB prompt emission studies have focused on the measurement of bulk properties, with

pulsesoftenregardedasnothingmorethanminorvariationstotheoverallsignal.
1
Recently, Hakkila et al. [5, 6] demonstrated that bulk GRB characteristics are composite properties formed
v
1 from characteristics belonging to distinct yet unresolved pulses. Each pulse in a small sample of GRBs with
7 known redshift was found to have its own lag, and the lag associated with the bulk gamma-ray emission (us-
1 ing the cross correlation function [7]) was shown to be a heterogeneous composite of the individual pulse lags
3 favoring the high intensity, short duration pulses. Since bulk GRB lags are correlated with GRB peak luminosi-
.
1 ties [8], this implies that pulse properties must be highly correlated to produce measurable correlations in bursts
0 composed of often many pulses. Pulses belonging to the small sample of BATSE GRBs with known redshifts
9 used in this study have unique, individual pulse lags that correlate with the pulse durations and that anti-correlate
0
with pulse peak luminosity. In order to verify that this is not a statistical discrepancy unique to this data set, we
:
v have set out to expand our GRB pulse database by sequentially analyzing bursts in the Current BATSE Catalog
i (http://www.batse.msfc.nasa.gov/batse/grb/catalog/current/).Theresultsallowustodeter-
X
minehowpervasivecorrelatedpulsepropertiesarewhilesamplingthe BATSECatalogdataina relativelyunbiased
r
a way.
PROCEDURE
Weuseasemi-automatedprocedureandcodetoidentifyandfitpulsesinmulti-channelobservationsofGRBprompt
emission using 64 ms data [5, 6]. The approach starts with a search for potential pulses in summed four channel
data.CandidatetimeintervalspotentiallycontainingpulsesareidentifiedusingtheBayesianBlocksmethodology[9];
a pulse model is optimized within each interval to see if it contains a statistically-significant pulse. The data from
intervalscontainingstatistically-insignificantmodelpulsesare mergedintothe surroundingintervals,andthe fitting
processbeginsagain.Iterationeventuallyproducesanoptimalsetofmodelpulsesforthesummedfour-channeldata.
Thefour-parameterpulsemodelused[10]assumesthefollowingfunctionalformforeachpulse:
I(t)=Al exp[−t1/(t−ts)−(t−ts)/t2], (1)
wheret is time since trigger, A is the pulse amplitude,t is the pulse start time, t and t are characteristicsof the
s 1 2
pulse rise and pulse decay,and l =exp[2(t /t )1/2]). A two-parameterbackgroundmodelis simultaneouslyfitted
1 2
alongwithanypulses.Theresulting4-channelpulsecharacteristicsareusedasstartingpointsfromwhichindividual
energychannelpulsefitsareobtained.Theaforementionedprocessisrepeateduntilconvergentsolutionsareobtained
forpulsesineachenergychannel.
Avarietyofmodel-dependentpulsepropertiesareextractedfromtheaforementionedpulseparametersinthefitting
process;theseincludepulsedurations,pulsepeakfluxes(256mstimescale),pulsepeaklags,pulseasymmetries,and
pulse spectral hardnesses. The pulse duration w is obtained from the summed four-channel data and is defined as
w=[9+12pt1/t2]1/2;thisistheintervalbetweentimeswhenthepulseamplitudeisAe−3.Thepulsepeakflux p256
is definedon the 256ms timescale in termsof the summed four-channeldata. Pulse peak lags l are the differences
betweenthepulsepeaktimesindifferentenergychannels(pulsepeaktimesare givenbytpeak =ts+√t1t2).Pulse
peaklagscanbeobtainedforanypulsebetweentwoenergychannels,althoughwedefinethestandardpulsepeaklag
l asthatmeasuredbetweenenergiesof100to300keV(BATSEchannel3)and25to50keV(BATSEchannel1).
31
Thepulse asymmetryk is definedas k =w/(3+2pt1/t2). HardnessratiosHR areconstructedby dividingpulse
fluences in two differentenergychannels;we use a hardnessratio HR defined by HR =S /S , where S is the
31 31 3 1 3
channel3fluenceandS isthechannel1fluence.
1
This analysis approach has been successfully applied to both BATSE and Swift bursts. However, difficulty in
normalizingBATdatatoBATSEdatahasthusfarpreventedusfrommergingmeasurementsfromthetwodatasets.
The procedure is fairly successful at pulse extraction, even though the pulse signal-to-noise ratio is often low in
more than one of the BATSE energychannels(most often in channel4). Pulse propertiesare cleanly extractedin a
relativelyunambiguousmannerfor GRBs containingnon-overlappingor isolated pulses.Typically,these tendto be
lowluminosity,longdurationbursts[10,11].However,theprocessisalsosuccessfulatidentifyingandfittingmany
pulsesincomplexGRBscontainingoverlappingpulses.Theapproachislesssuccessfulatfittinglowintensitypulses,
pulses that stronglyoverlap,and very short pulses, which can be indistinguishable from Poisson noise. Ambiguous
pulseidentificationsoftenresultinpoorc 2 goodness-of-fitmeasures,infitsthatappeartomergepulsesseparableto
theeye,and/orinpulsesthathavedisparatepropertiesorarenotobservedincontiguousenergychannels.Weexclude
fromouranalysisoverlappingandlowfluencepulsesthatareovertlyambiguous,butrecognizeexistenceofthelatter
bytentativelyidentifyingthemaslowfluenceevents(LFEs).
We also classify the bursts that we study. Although the definition of the Short and Long class of GRBs has
becomemorecomplexduringtheSwifterathanitwas,weuseaclassificationschemebasedondurationandspectral
hardnessoriginallyobtainedfromBATSEburstsusingmachinelearningalgorithms[11].Thisscheme[12]classifies
GRBs as Shortif theysatisfy the inequality(T90 <1.954)OR (1.954 T90<4.672AND HR >3.01),where
321
≤
HR =S /(S +S )[13],whereS isthechannel2fluence.IfGRBsdonotsatisfytheaforementionedinequality,
321 3 2 1 2
thentheyareclassifiedasLong.
When pulses have been fitted and the GRBs to which they belong have been classified, we are able to seek out
possiblecorrelationsamongthepropertiesofpulsesinLongandShortBATSEGRBs.
ANALYSIS
Ourcurrentdatabaseconsistsof307pulsesin106Longand46ShortGRBs. We havestartedsimultaneouslyatthe
endsoftheCurrentBATSECatalogandareworkingourwaythroughtherestoftheCatalogsequentially.Theresults
obtained with this larger database are consistent with those obtained previously for smaller datasets [5, 6]: pulse
properties,ratherthanbulkpropertiesofthepromptemission,underlyGRBmeasurements.
Figure 1 demonstrates an example of pulse fits for BATSE Trigger 0214. This is a Long GRB with three fitted
pulses.PulsepropertiesareextractedforthisandmanyotherBATSEburst,andcompiledinapulsedatabase.
Correlations
Pulse properties are compared and correlated for both the Long and Short GRB classes. The most rigorous
correlationfoundistheonebetweenpulsedurationandpulsepeaklag:theprobabilitythattheseareuncorrelatedin
LongGRBpulsesis2.3 10 19.Suchastrongcorrelationsuggeststhatthepulsedurationandthepulsepeaklagare
−
×
twodifferentrepresentationsofthesamephenomenon;thesealsoappearrelatedtohard-to-softpulseevolution.Both
FIGURE1. ExampleofpulsefitsforaLongBATSEGRB(Trigger0214):(a)channel1,and(b)channel3.Thepulseshavethe
followinglags:pulse1lag=0.679s,pulse2lag=0.368s,andpulse3lag=0.352s.
characteristics have been previously found to anti-correlate with pulse peak luminosity [5] and are thus luminosity
indicators. We note that it is much easier to accurately measure pulse durations than pulse peak lags, since pulse
durationsarelongerandsincethesmoothly-varyingpulsefittingfunctionallowsthepulsedurationtobeaccurately
determined.
Althoughthepulsedurationvs.pulsepeaklagrelationisgenerallyverytight,therearepulseswithspecificpulse
propertiesforwhichthisrelationmaynotcorrelate.PulsepeaklagsofShortGRBpulses,forexample,donotstrongly
correlatewith pulse durations;this couldbe becauseit is difficultto measurepulse peaklags forShortGRB pulses
using the 64 ms timescale. The lags of Short GRB pulses are so short that roughly half of them are negative; this
is consistent with a distribution centered around zero seconds and is likely due primarily to measurement error.
Additionally,a small number of Long GRB pulses are found to have short lags but long durations;these appear to
be long pulses with sharply-defined pulse peaks. These pulses also appear to be inconsistent with the Pulse Scale
Conjecture of Nemiroff [14] which assumes that all pulses have temporal structures that are similar when properly
scaled. Hakkila et al. [5] have previously suggested that these pulses might be external shock signatures, but it is
possiblethattheserepresentinaccurately-measuredprorpetiesofoverlappingpulses.
WenotethatmanyLongGRBscontainshortdurationpulses;thesetypicallyhavepulsepeaklagsnearzeroseconds.
ThisindicatesthatashortdurationpulsedoesnotnecessarilyidentifyaGRBbelongingtotheShortclass.Likewise,
thereare manyLongGRB pulseswith pulse lagsthatare demonstrablynegative.Figure3 (BATSE trigger1807)is
oneofthese:thenegative-lagpulsedoesnotappeartobeasinglepulsewithhomogeneousproperties,butinsteadhas
twopeaksindicativeofoverlappingpulses.Giventherelativerarityofnegative-lagpulses,wesuggestthatmanyof
theseare merelyunresolvedoverlappingpulses,althoughsomecouldbe pulsesofshortdurationwhoselagscannot
beaccuratelymeasured.
Figure4demonstratesstronganti-correlationsbetweenpulsedurationandpulsepeakfluxforbothLongandShort
GRB pulses. Both Long and Short GRB pulses exhibit anti-correlations, but these relations have different slopes:
usinga modelwherelogp =Alogw+B, we findthatA = 0.27(withcorrelationcoefficientR=0.36)and
256 Long
−
A = 0.67(R=0.64).Differentcorrelativerelationscouldindicatedifferentpulsemechanismsforthetwoburst
Short
−
classes. Figure 4 also demonstratesan anti-correlationbetweenspectral hardnessand pulse durationfor LongGRB
pulses. Long, low-luminosity pulses are softer than short, luminous pulses, supporting the argument that intrinsic
propertiesdominateovercosmologicaleffects.
Figure5ademonstratesacorrelationbetweenLongGRBpulsedurationandpulseasymmetry:shorterpulsestendto
bemoresymmetricthanlongerones.TheasymmetriesofShortGRBpulsesarenotplotted,sincetheshortdurations
of these pulses (often only a few temporal bins wide) tend to make them be observed as being very symmetric or
very asymmetric. Since duration is a luminosity indicator for Long GRB pulses, both spectral hardness and pulse
asymmetryarealsoluminosityindicators.
ThecorrelativepropertiesofGRBpulsesaresummarizedinTable1(LongGRBpulses)andTable2(ShortGRB
pulses). Although Short GRB pulse properties are difficult to accurately measure with 64 ms resolution, there is
FIGURE2. (a)Pulsedurationvs.pulsepeaklagforLong(square)andShort(*)GRBsinlogarithmicunits,showingthestrong
correlationbetweentheseparameters.(b)Asubsetofthesedatatypesinlinearunits,indicatingthatsomepulseshavenegativelags.
Manyoftheselagmeasurementsareconsistentwithpulselagsnearl31=0,butsomehaveabnormally-negativevalues.ShortGRB
pulsesaretypicallyfoundinregionS(shortlags,shortdurations),whilenormalLongGRBpulsesarefoundinregionL(following
thecorrelationshowninFigure2(a)).RegionH containsLongGRBpulsesthathaveextremelynegativelagsandwhichmightbe
marredbyhidden pulses,andRegionE containsLongGRBpulsesthathavecharacteristicspotentiallyassociatedwithexternal
shocks(shortpulselagscoupledwithlongpulsedurations).
FIGURE3. Exampleofanegative-lagpulse(BATSETrigger1807):(a)channel1,and(b)channel3.Thesecondfitpulse(with
ameasuredlagof-4.2s)appearstobecomposedoftwopulses,withthesecondonebeingspectrallyharderthanthefirst.Merged
pulsessuchasthiswouldexplainmanyofthepulseshavingnegativelags.
evidencetosuggestthatShortGRBpulsepropertiesarenotascorrelativeasLongGRBpulseproperties.Sucharesult
could provide a mechanism for more accurate GRB classification. Similarly, pulse property correlations (or a lack
thereof)couldwellprovideconstraintsnecessaryforconstructingmoreaccuratemodelsofGRBpromptemissionfor
LongandShortGRBs.
Despitetheinsightsgainedfromthesepulserelationships,thereisstillambiguityinthedefinitionofShortandLong
GRBsasobtainedfrombulkemissionandpulseproperties.Take,forexample,BATSEtrigger0809(Figure5b):this
burstishardwithaT durationoflessthantwoseconds,andwouldnormallybeclassifiedasaShortGRB.However,
90
thepulsepropertiesofpulseduration,pulsepeakflux,andpulsepeaklagcorrelatewithoneanotherinawaythatis
consistentwithitbeingaLongGRB.Thus,wheremanyobserversarenowmakingclaimstoexpandtheShortGRB
definitiontoreclassifymanyLongburstsasShort,wehavefoundanambiguousShortGRBthatmightactuallybea
LongGRBsolelybecauseitislackinginter-pulsedurations.
FIGURE4. (a)256mspeakfluxvs.pulsedurationforLong(square)andShort(*)GRBs.Best-fitlinesaregivenforLong(solid
line)andShort(dashedline)relationships.(b)Spectralhardnessvs.pulsedurationforLong(square)andShort(*)GRBs.
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FIGURE5. (a)Pulsedurationvs.asymmetryforLong(square)GRBs,alongwithabest-fitline.(b)Summedfour-channelpulse
fitstoBATSEtrigger0809,aShortGRBhavingpotentialcharacteristicsoftheLongGRBclass(pulse1lag=0.679s,pulse2lag
=0.368s,andpulse3lag=0.352s).
TABLE1. LongGRBpulsecorrelations.Spearmanrankordercor-
relationprobabilitiesthatthepulsecharacteristicsinquestionareran-
dom.Anti-correlationsareindicatedbya†,andasignificantcorrela-
tionoranti-correlationisindicatedinboldface.
w P HR k
256 31
l31 2.3 10−19 †4.8 10−3 †2.8 10−3 1.1 10−1
w ×— †8.8×10 7 †4.2×10 7 1.1×10 7
− − −
P256 — ×— 1.7 ×10−1 †5.3× 10−3
HR31 — — ×— †3.3×10−3
×
TABLE2. ShortGRBpulsecorrelations.
w P HR k
256 31
l31 2.1 10−1 †6.6 10−2 †2.2 10−2 4.4 10−1
w ×— †7.0×10 7 †1.3×10 1 †8.2× 10 2
− − −
P256 — ×— 7.3 ×10−2 7.9 ×10−1
HR31 — — ×— 2.3×10−2
×
CONCLUSIONS
A large GRB pulse sample verifies that each GRB is characterized by its own lag. The lag of a Long GRB pulse
is directly related to its duration, and both of these properties anti-correlate with the pulse peak luminosity. Via
correlations with pulse peak lag and pulse peak duration, it also appears that pulse spectral hardness and pulse
asymmetryareluminosityindicators.TheserelationshipsholdformostLongGRBpulses,butperhapsexcludelong
pulseswithshort,intensepulsepeaks(externalshocks?).
ShortGRBpulsesdonotreflectthesamecorrelativebehaviorsasthosefoundforLongburstpulses,althoughthis
resultmightpartlyreflectthelimitedtemporalresolutionusedinthisstudy.Perhapsthemostinterestingbehaviorof
Shortburstpulsesisananti-correlationbetweenpulsedurationandpulsepeakfluxwhichmightindicatearelationship
betweenpulsedurationandpulsepeakluminosity.
ThereisaneedforbettermeasuringandunderstandingGRBpulseproperties.Inprinciple,thebulkcharacteristics
of the promptemission can be derivedfrom knowledgeof pulse propertiesand the pulse decompositionof a burst.
Theconverse,however,isnottrue:wecannotinferthebasiccharacteristicsofthepulses,norcanweunderstandthe
relationshipbetweenbulkand(morefundamental)pulseproperties,withoutexplicitstudyoftheconstituentpulses.
ACKNOWLEDGMENTS
We thankTom Loredo,Rob Preece, Tim Giblin, Chris Fragile, and Jay Norrisfor valuablediscussions. R. Cumbee
acknowledgesfinancialsupportfromtheSouthCarolinaSpaceGrantprogramandfacultysupportfromDr.J.Myers
ofFrancisMarionUniversity.WealsogratefullythankUSRAandtheconferencepostercompetitionjudges.
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