Spectral lag of gamma-ray burst caused by the intrinsic spectral evolution and the curvature effect

0.46 MB·English

by  Z. Y. Peng

AstronomischeNachrichten,24January2011
Spectral lag of gamma-ray burst caused by the intrinsic spectral evolu-
tion and the curvature effect
Z.Y.Peng1,Y.Yin2,X.W.Bi3,Y.Y.Bao4,andL.Ma1,⋆
1 DepartmentofPhysics,YunnanNormalUniversity,Kunming,Yunnan650092,China
2 DepartmentofPhysics,LiupanshuiNormalCollege,Liupanshui553004,China
3 DepartmentofPhysics,HongheUniversity,Mengzi661100,China
1
4 DepartmentofPhysics,YuxiNormalCollege,Yuxi653100,China
1
0
2 Thedatesofreceiptandacceptanceshouldbeinsertedlater

n
a Keywords gamma-raybursts:gamma-raytheory:relativity
J
Assuminganintrinsic‘Band’shapespectrumandanintrinsicenergy-independentemissionprofilewehaveinvestigated
1
theconnectionbetweentheevolutionoftherest-framespectralparametersandthespectrallagsmeasuredingamma-ray
2
burst(GRB)pulsesbyusingapulsemodel.Wefirstfocusourattentionontheevolutionofthepeakenergy,E0,p,and
] neglecttheeffectofthecurvatureeffect.ItisfoundthattheevolutionofE0,palonecanproducetheobservedlags.When
E E0,p variesfromhardtosoftonlythepositivelagscanbeobserved.ThenegativelagswouldoccurinthecaseofE0,p
H varyingfromsofttohard.WhentheevolutionofE0,pandthelow-energyspectralindexα0varyingfromsofttohardthen
tosoftwecanfindtheaforesaidtwosortsoflags.Wethenexaminethecombinedcaseofthespectralevolutionandthe
.
h curvatureeffectoffireballandfindtheobservedspectrallagswouldincrease.Asampleincluding15singlepulseswhose
p spectralevolutionfollowshardtosofthasbeeninvestigated.Allthelagsofthesepulsesarepositive,whichisingood
- agreementwithourtheoreticalpredictions.Ouranalysisshowsthatonlytheintrinsicspectralevolutioncanproducethe
o spectrallagsandtheobservedlagsshouldbecontributedbytheintrinsicspectralevolutionandthecurvatureeffect.Butit
r
isstillunclearwhatcausethespectralevolution.
t
s
a
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[

1
v 1 Introduction ter,whichwillcausethesoftradiationtobeupscatteredby
2 inverse Comptonization. The photons get harder the more
6 The phenomenonof the observed spectral lag of Gamma- scatterings they suffer and thus they are more delayed. A
0
rayburst(GRB)isverycommon.SinceChengetal.(1995) definitiveexplanationoftheoriginofthemhasnotyetbeen
4
. first analyzed the spectral lags of GRBs based on BATSE given.Theoriginofthespectrallagsarestillunclear.
1
Large Area Detector channel 1 (25-50 keV) and channel Kocevski&Liang(2003)pointedoutthattheobserved
0
3 (100-300keV) light curves,several authorshave carried lagisthedirectresultofspectralevolution.Theymeasured
1
1 out more analysis work on the GRB lags (e.g., Norris et the rate of Ep decay (Φ0) for a sample of clean single-
: al.,2000;Wu&Fenimore2000;Chenetal.2005;Yietal. peakedburstswithmeasuredlagandusedthisdatatopro-
v
2006;Pengetal.2007). vide an empirical relation that expresses the GRB lag as
i
X Severalattemptstoexplaintheoriginofthespectrallag a function of the burst’s spectral evolution rate. While the
r havebeenputforward.Thedominantbehaviorobservedin studiesofGRBspectrahaveshownthatGRBspectralevo-
a
GRBs, positive lags, is that the soft radiation lagging the lutionisacommonphenomenon(e.g.,,Kargatisetal.1994;
hard,canbeproducedbyseveralreasons.Anintrinsiccool- crideretal.1997;Band1997;Kanekoetal.2006;Butler&
ing of the radiating electrons cause the radiation to domi- Kocevski2007).
nate at lowerenergies.A Comptonreflectionof a medium Recently,Shenetal.(2005)tentativelystudiedthecon-
atasufficientdistancefromtheinitialhardsourcewillalso tribution of curvature effect of fireball to the lag, and the
cause a positive lag. While the photons emit at relativis- resultinglagsareveryclosedtotheobservedone.Luetal.
tic speeds fromdifferentlatitudesof a convexsurface(the (2006)alsostudiedthespectrallagsmoredetailedbasedon
curvatureeffect)willcausetheradiationtobedelayedand Dopplereffectoffireball(orinsomepapers,thecurvature
softened,thusproducinga observedpositive lag (e.g.,Qin effect) (see, Qin 2002; Qin et al. 2004; Peng et al. 2006
2002;Ryde & Petrosian 2002;Qin et al. 2004;Shen et al. for more detailed description) and confirmed that the cur-
2005).Thenegativelagisdefinedasthehardradiationlag- vature effect can produce the observedspectral lags. They
gingbehindthesoft.However,fewattemptsaremadetoex- performedmoreprecisecalculationwithbothformulaepre-
plaintheoriginofthenegativelag.Thepossiblescenariois sented by Shen et al. (2005) and Qin et al. (2004). Some
ahotmedium,sayaleptoncloudsurroundingacooleremit- conclusionsobtainedbyShenetal.(2005)wereascertained,
and more complete conclusions on spectral lags resulting
⋆ Correspondingauthor:e-mail:[email protected] fromcurvatureeffectwereobtained.Luetal.(2006)argued
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2 Z.Y.Pengetal.:Spectrallagofgamma-rayburstcausedbytheintrinsicspectralevolutionandthecurvatureeffect
that,aslongasthewholefireballsurfaceisconcerned,both which could be assigned to any values of t , and R is
θ c
formulaswereidenticalinthecaseofultra-relativisticmo- the radius of the fireball measured at t = t . Variable
θ c
tions.OthercasesdidnotstudybyShenetal.(2005)were τ is a dimensionless relative observation time defined by
investigatedbyLuetal.(2006)andsomenewconclusions τ [c(t t ) D + R ]/R , where D is the distance
c c c
≡ − −
weredrawn.Theseseemshowthatthecurvatureeffectcan ofthefireballtotheobserver,andtistheobservationtime
indeedproducethespectrallags.Inaddition,severalstudies measuredbythedistantobserver.
show thatthe curvatureeffectalso playsan importantrole Informula(1),I(τ )representsthedevelopmentofthe
θ
in the early X-ray afterglow of GRBs where the so-called intensitymagnitudeofradiationintheobserverframe,called
softeningphenomenonisobserved(Qin2008a,2008b;Qin asa localpulse funection,andg0,ν(ν0,θ)describesthe rest-
2009). frameradiationmechanisms.
Aspointedoutabovethatthefundamentaloriginofthe ForthesakeofsimplicitywefirstadoptGaussianpulse
observed lag is the evolution of the GRB spectra and the as rest-frame local pulse. As for the rest-frame radiation
curvatureeffectcanproducetheobservedlags.Moreoverit spectrum we employ the most frequently used Band func-
ismentionedclearlythatwhilstthecurvatureeffectcanac- tion (Band et al. 1993) since it is rather successfully em-
count for the main part of the hardnessratio curvesof the ployed to fit the spectra of the GRBs. The two forms are
GRBs concerned and an intrinsic spectral evolution is re- rewrittenasformulas(2)and(3).
quiredtofullyexplainthe observeddata(Qinetal. 2006).
τ τ
Thesemotivateourinvestigationsbelow.Howtheinfluences I(τ )=I exp[ ( θ− θ,0)2] (τ τ ), (2)
θ 0 θ,min θ
− σ ≤
ofintrinsicspectralevolutiononthespectrallagsarewhen
thecurvatureeffectplaysanimportantroleordoesnot?Can e
thenegativelagsbeproducedinthecourseofintrinsicspec- (ν0,θ)1+α0exp[ (2+α )ν0,θ],
tral evolution?We introducethe theoreticalformulaof the ν0,p (ν0,θ−< α0−β00)ν0,p
GisntuRgdBtihepdeusilnpseeScitenrcaStlieolcantgi3os.nwI2nh.eSVneancrtieiogoulnesc4ptotwshseeibcilnuevrcevasatstiuegsraetoeefftpfhereoctdlauagrces- g0,ν(ν0,θ)=(α20+−αβ00)α0−β0(eνx0ν,pθ0(,pβ0α−02−α+β0α0)0(),νν00,,θp)1+β0
ν0,p ≥ 2+α0
wmhueltnansepoeucstrlayl.Weveoleumtiopnloaynadscaumrvpaletutroeteefsftetchtetapkreedeifcfteioctnsiin- whereI ,σandτ areconstants,α andβ arethelo(w3)-
0 θ,min 0 0
Section 5. In the last section, we give the conclusionsand andhigh-energyindicesintherestframe,respectively,and
discussion. ν istherest-framepeakfrequency.
0,p
Light curves determined by formula (1) are dependent
2 Theoretical formula on the integral limits τθ,min and τθ,max, which are deter-
minedbytheconcernedareaofthefireballsurface,together
withtheemissionrangesoftheradiatedfrequencyandthe
Theobservedgamma-raypulsesarebelievedtobeproduced e e
localtime.Theintegrallimitsareonlydeterminedby
in a relativistically expanding and collimated fireball be-
cause of the large energies and the short time-scales in-
τ =max τ ,τ−1+cosθmax
volved. To account for the observed pulses and spectra of τθ,min =min{τθ,min,τ1−−1β+ccoossθθmmaixn }, (4)
bursts, the Doppler effect (or curvature effect in some pa- ( θ,max { θ,max (1−βcosθmin}
per)overthewholefireballsurfacewouldplayanimportant e
where τ and τ are the lower and upper limit of
role(e.g.,MeszarosandRees1998;Haileyetal.1999;Qin θ,mine θ,max
τ confiningI(τ ), andθ andθ areconfinedbythe
2002; Qin et al. 2004). The Doppler effect is the photons θ θ min max
concernedareaofthefireballsurface.Theradiationsareob-
emitted from the region on the line of sight and those off
the line of sightan angleof θ are Doppler-boostedby dif- servablewithientherangeof
ferentfactorsandtraveldifferentdistancestotheobserver.
(1 cosθ )+(1 βcosθ )τ τ
Therefore,we adopt the modelderivedby Qin (2002)and − min − min θ,min ≤ ≤ (5)
(1 cosθ )+(1 βcosθ )τ
max max θ,max
Qin et al. (2004) to explore the spectral lags. But our at- − −
tentionisconcentratedonthecaseofθ 0.Thecountrate Due to the constraintto the lower limit of τ , which is
≃ θ
withinenergychannel[ν1,ν2]isrewrittenasformula(1). τθ,min > 1/β (see, Qin et al. 2004), we assign τθ,0 =
−
10σ +τ so that the interval between τ and τ
C(τ)=[2πRc3 τeτeθθ,,mmianxI(τθ)(1+βτθ)2(1−τ +τθ)dτθ wouldbeθ,lmarigneenoughtomaketherisingphaθs,emoinfthelocθa,0l
× νν12 g0,ν(νν0,θ)Rdν]×[hecD2Γ3(1−β)2(1+ 1−ββτ)2]−(11) pounlesecacnloosbetatointhatoftheGaussianpulse.Fromformula(2)
R
wheretheτ isdimensionlessrelativelocaltimedefinedby
θ ∆τ =2√ln2σ, (6)
τ c(t t )/R , t is the emission time in the ob- θ,FWHM
θ θ c c θ
≡ −
server frame, called local time, of photons emitted from
whichleadsto
the concerned differential surface ds of the fireball (θ is
θ
the angle to the line of sight), t is the initial local time σ =∆τ /2√ln2, (7)
c θ,FWHM
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where∆τ istheFWHM (fullwidthathalfmax- assign the σ = 1.0 and take Γ = 100, 200,300, 400, 500,
θ,FWHM
imum)oftheGaussianpulse.Fromtherelationbetweenτ 600,700,800,900,1000,respectivelysincetheLorentzfac-
θ
andt ,onegets toroffireballaregenerallylargerthan100.Wefindthatall
θ
thespectrallagslag inthiscaseare0.Thissuggeststhat
13
∆t =(R /c)∆τ . (8)
θ,FWHM c θ,FWHM there has no spectrallags when the rest-framespectralpa-
In thefollowingsections,we assign(2πR3I )/(hcD2)=1, rametersareconstantandthecurvatureeffectdoesnotplay
c 0
R =3 1015cm,τ =0. a role on the spectral lag. Fig. 1 (panel (a)) gives a exam-
c θ,min
× pleplotforthiscase.Thelinesofthepeaktimesofthetwo
Becausepeaktimesofdifferentlightcurvesassociated
pulsesaresuperposedeachother.
with different frequency intervals [ν ,ν ] or different en-
1 2
ergybands[E ,E ]aredifferent,followingShenetal.(2005)
1 2
andLuetal.(2006),wedefinethespectrallagsasthedif- 3.2 Spectrallagsoftherest-framespectral
ferencesbetweenthe pulsepeaktimes in two differenten- parametersvaryingfromhardtosoft
ergychannels,alowerenergychannel[E ,E ]andahigher
1 2
energychannel[E ,E ].Inthefollowinganalysisweonly Many investigations of the spectra of GRB show that the
3 4
considertheenergychannelpair:BATSEchannels1(25–50 spectralsofteningisanuniversalphenomenon.Inaddition,
keV)and3(100–300keV)whichwedenotethelagaslag13 Koceviski and Liang (2003) pointed out as the Ep of νFν
sinceitwasinvestigatedbymostofauthors(e.g.,Norriset spectra decays throughthe four BATSE channelsand pro-
al.2000;Chenetal.2005). duce what we measure as lag. Schaefer (2004) found that
only the evolution of E can produce observed lag using
p
the general Liang-Kargatis relation (which describes how
3 Spectral lagscaused by the spectral
thepeakphotonenergyinthespectrumchangeswithtime).
evolutionalone
Therefore, we first investigate the case of E in the rest-
p
framevaryingfromhardtosoft.FollowingQinetal.(2005),
Since Kocevski & Liang (2003) pointed out that the ob-
weassumeasimpleevolutionofthepeakenergyE ,which
0,p
servedlag isthe resultofspectralevolution,letusfirstin-
followslogE =0.1 k(τ τ )/(τ τ )(keV)
0,p θ θ,1 θ,2 θ,1
vestigatethespectrallagsduetothespectralevolutionalone. − − −
forτ τ τ .Forτ < τ ,logE = 0.1(keV),
θ,1 θ θ,2 θ θ,1 0,p
Thatisweconsidertheradiationsemittedfromaverysmall ≤ ≤
whileforτ >τ ,logE =0.1 k(keV).
areawithθ =0andθ =10−5 1/Γwherethepho- θ θ,2 0,p −
min max × HerewealsotakeΓ=100,200,300,400,500,600,700,
tonsemittedtravelalmostsamedistancestotheobserver.In
800, 900, 1000, respectively and α = 1, β = 2.25,
this case we think that the curvature effect has no role on 0 − 0 −
σ = 1.0areassigned.ForeachΓweassumek =0.1,0.2,
theproducingofspectrallag.
0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3and1.4(they
Observations show that the spectral parameters of the
correspond to different rates of decreasing) to investigate
GRBs vary with time. In addition, the parameters of the
thespectrallag’sdependence.
Band function low- and high-energy indices α, β and the
ThelocalGaussianpulse(seeequation(2))isemployed
peak energy of GRB’s νF spectrum E in the observed
ν p to study this issue. We adopt τ = 8σ +τ , τ =
frame are mainly distributed within 1.5 — 0.5, 3.2 θ,1 θ,min θ,2
— 2.1 and 100 — 700 keV and th−eir typica−l value−s are 12σ+τθ,minandτθ,0 =10σ+τθ,min.
− Inthisway,wecancomputethespectrallags.Itisfound
α = 1 and β = 2.25, E = 250 keV, respectively
p
− − that all of the lags are positive. A example light curve is
(Preeceetal.1998,2000;Kanekoetal. 2006).Qin(2002)
showed in Fig. 1 (panel(b)), which illustrates the connec-
showedtheDopplereffectcannotaffecttheintrinsicspec-
tionbetweenthespectralevolutionandthespectrallag.This
tral shapes. When we take the Doppler effect of fireballs
indicatesthemaximumofthephotoncountratesatthehigher
intoaccount,theobservedpeakenergywouldberelatedto
energy arrives earlier than at the lower energy in the case
the peak energy of the rest-frame Band function spectrum
of hard-to-soft. Therefore, we can deduce that only hard-
(α = α, β = β) by E 1.67ΓE (see, Table 4 in
0 0 p 0,p
≃ to-softspectral evolutioncan producethe positive spectral
Qin 2002).In the following we first consider the common
lags.
evolutionarymodeloftherest-framespectralparametersto
investigate the spectral lags. Then the other possible cases
arealsotakenintoaccount. 3.3 Spectrallagsoftherest-frameradiationform
varyingfromsofttohardwithtime
3.1 Spectrallagsoftherest-framespectral
Band(1997)foundthatasmallfractionofGRBspectra(
parametersremainconstant ∼
10%)varyfromsofttohardwithtime.Kargatisetal.(1994)
We first explore if the spectral lags would occur when the alsofoundthiscase.Therefore,wealsoinvestigatethecase
rest-frame spectral parametersremain unchangedto check of the rest-frameradiationformvaryingfromsoftto hard.
theeffectofspectralevolutiononthespectrallag.Thetypi- Wealsoassumeasimplecasethattheevolutionofpeaken-
calvaluesofthelow-andhigh-energyindicesα = 1and ergyE follows:logE = 0.8+k(τ τ )/(τ
0 0,p 0,p θ θ,1 θ,2
− − − −
β = 2.25andE =250/1.67ΓkeVaretaken.Herewe τ ) (keV) for τ τ τ . For τ < τ , and
0 0,p θ,1 θ,1 θ θ,2 θ θ,1
− ≤ ≤
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4 Z.Y.Pengetal.:Spectrallagofgamma-rayburstcausedbytheintrinsicspectralevolutionandthecurvatureeffect
Fig.1 Example light curves of the Band spectral parameters E remain unchanged (a), the E varying from hard
0,p 0,p
tosoft(b)andsofttohard(c),wherethesolidanddashedcurvesrepresentthelightcurvesofBATSEchannels1and3,
respectively.
logE = 0.8 (keV), while for τ > τ , logE = tweenk1andk2comingtoatleast0.5.Thatisiftheincreas-
0,p θ θ,2 0,p
−
0.8+k (keV). The Γ, k, α and β are taken the same ingrateofsofttohardislargerthanthedecreasingrateof
0 0
−
values as the case of hard-to-soft. We also adopt τ = hardtosoftbyatleast0.5thenegativelagswouldoccur.As
θ,1
8σ+τ ,τ =12σ+τ andτ =10σ+τ . the increasing of Γ the difference needed would decrease.
θ,min θ,2 θ,min θ,0 θ,min
Accordingto ourcomputationthe soft-to-hardspectral The aboveassumption is reasonable for the tracking pulse
evolutioncanresultinthenegativespectrallagsalone.Fig. becauseitisconsistentwiththeresultofthehardnessevo-
1(panel(c))alsodemonstratestwoexamplelightcurvesin- lutionarycharacteristicsofthe trackingpulsesinvestigated
dicatingtheconnectionbetweenthespectralevolutionand byPengetal.(2009a)thatthedurationoftherisephaseare
thespectrallaginthecaseoftherest-frameradiationform muchshorterthanthatofthedecayphase.Fig.2illustrates
varyingfromsofttohardwithtime. fourexamplelightcurves.
3.5 Spectrallagsofasuddenlyshiningandgradually
3.4 Spectrallagsoftherest-frameradiationform
dimmingintrinsicemissionprofile
varyingfromsofttohardthentosoft
If the different intrinsic emission profiles are different in
Asseveralauthorspointedoutthattheevolutionofthelow-
producingthespectrallags.SimilartoQinetal.(2004)and
energyindexαandE exhibitthe“tracking”(theobserved
p
Shenetal.(2005)wealsoconsidertheone-sidedexponen-
spectral parameters follow the same pattern as the flux or
tialdecayandpower-lawdecayprofile,whicharelistedas
countratetimeprofile,i.e.theevolutionofspectralparam-
follows,respectively:
etersexhibitsoft-to-hard-to-soft)behaviors(e.g.,Fordetal.
1995; Crider et al. 1997; and Kaneko et al. 2006; Peng et
al.2009a,2009b).SoweinvestigatethecaseofE0,p vary- I(τ )=I exp[ (τθ−τθ,min)] (τ τ ), (9)
ing from soft to hard then to soft. In order to ensure that θ 0 θ,min θ
− σ ≤
the three parameters are in the range of observed values
we adopted the following evolutionary form: logE = e
0,p
0.8+k1(τ τ )/(τ τ )(keV)forτ τ τ τ
− θ− θ,1 θ,0− θ,1 θ,1 ≤ θ ≤ I(τ )=I (1 θ− θ,min )µ (τ τ ),
(τkθ,e0V, l)ogfoEr0τ,p = −τ0.8+τk1,−lokg2E(τθ −=τθ,0)0/.8(τ+θ,2k−1 τθ,k02) θ 0 − τθ,max−τθ,min θ,min ≤ θ(10)
θ,0 θ θ,2 0,p
≤ ≤ − − e
(keV)forτθ > τθ,2,whileforτθ < τθ,1,logE0,p = −0.8 We assume the τθ,min = 0 and µ = 2 in the follow-
(keV). where the k1 and k2 denote the increasing rate of inganalysis.Thehard-to-softandsoft-to-hardspectralevo-
soft-to-hard phase and the decreasing rate of hard-to-soft lution are investigated to check if the two sorts of local
phase, respectively.We take Γ = 100,200,300,400,500, pulsesalso resultin the same resultsas those ofthe Gaus-
600, 700, 800, 900, 1000, respectively and k1 = 0.3, 0.4, sian pulses. The same evolutional patterns adopted above
0.5,0.6,0.7,0.8,0.9,1.0,and1.1,k2 = 0.1,0.2,0.3,0.4, areemployedtoexplorethisissue.Wefindtheresultsarein
0.5, 0.6, 0.7, and 0.8, respectively. We also adopt τθ,1 = agreementwiththatofGaussianlocalpulses.Theexample
8σ+τθ,min,τθ,2 =12σ+τθ,minandτθ,0 =10σ+τθ,min. plotsinFigs.3and4alsoillustratetheconnectionbetween
We use this form to investigate the spectral lags and the spectral evolution and spectral lag for the exponential
find that the negativelag disappears. We explore E and decayandpower-lawdecay,respectively.Thissuggeststhe
0,p
α varying from soft to hard then to soft. The same form intrinsic emission profile has no effect on the connection
0
as E is also adopted for α . We find there are negative between the spectral evolution and the spectral lags. Like-
0,p 0
spectral lags in this case but it does notoccurin the small wise, the hard-to-softevolutionproducespositive lagsand
Γ =100. When the Γ is larger than 100, the negative lags thesoft-to-hardevolutionleadstonegativelagsifweadopt
willcomeintobeingonlyinthecaseofthedifferencesbe- otherrest-framelocalpulses.
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ANheaderwillbeprovidedbythepublisher 5
0.00010
0.000012
E0,psoft-hard-soft
0.000010 E0,psoft-hard-soft 0.00008 k1=0.7,k2=0.3
k1=0.9,k2=0.2
0.000008
) ) 0.00006
C( 0.000006 C(
0.00004
0.000004
0.00002
0.000002
0.000000 0.00000
0.000020 0.000025 0.000030 0.00020 0.00025 0.00030 0.00035
Fig.2 Example light curvesof the negative(left panel) and positive (rightpanel) spectral lags for the case of soft-to-
hard-to-soft,wherethesymbolsarethesameasthoseadoptedinFig.1.
1.60E-008
6.00E-008 1.40E-008
5.00E-008 0,phard-to-soft 1.20E-008 0,psoft-to-hard
k=0.5 k=0.5
1.00E-008
4.00E-008 )
) C( 8.00E-009
C( 3.00E-008
6.00E-009
2.00E-008
4.00E-009
1.00E-008 2.00E-009
0.00E+000 0.00E+000
0.00000 0.00002 0.00004 0.00006 0.00000 0.00002 0.00004 0.00006
Fig.3 ExamplelightcurveofE variesfromhardtosoft(leftpanel)andsofttohard(rightpanel)forthecaseofthe
0,p
exponentialdecaylocalpulse,wherethesymbolsarethesameasthoseadoptedinFig.1.
4 Spectral lagscaused by the spectral getherinFig.5(panel(a))byconsideringasetofparame-
evolutionandthe curvature effect tervalues.Notethatthehistogramplotsmaymeannothing
because when one consider another set of parameter val-
uestheygeta differentresult. Itis shownthat(i) theaver-
As previous section pointed out that the curvature effect
agevalueofspectrallagscausedbyspectralevolutionand
mustbeplayaroleontheproducingoftemporalandspec-
combinedeffectare0.1086sand0.1692s,respectivelyand
tralprofilewealsotaketheeffectintoaccounttostudythe
(ii)thecorrespondingmediansare0.007sand0.016s,re-
spectral lag. The evolutionary form, hard-to-soft and soft-
spectively. These show the lags caused by the combined
to-hard, are combined with curvature effect to explore the
effect increase indeed. Whether the curvature effect also
influenceofthecombinedeffectsonthespectrallags.That
make the lags increase in the case of soft-to-hard or not?
is we consider the radiation emitted from a big area with
We also check the spectral lags resulting from the curva-
θ = 0 and θ = π/2 where the photons emitted
min max
tureeffectandevolutionofE followingsoft-to-hard.The
travel different distances to the observer. The photons on 0,p
lagsproducedbyspectralevolutionaloneandthecombined
the line of sight arrive at us earlier than that of off sight-
effect are also compared. The mean values are -0.1200 s
line. In this way the lags due to curvatureeffect is always
and -0.0114 s for spectral evolution and combined effect,
positiveandthehard-to-softevolutionalsomakethelagis
respectively.Whereasthecorrespondingmedianvaluesare
positive.Thereforewethinkthespectrallagscausedbythe
-0.0102sand-0.0005s,respectively.Fig.5(panel(b))also
combinedeffectmustbeincrease.
shows that the lags indeed increase in the case of soft-to-
We first consider the case of E0,p varying from hard hard. In addition, we also find due to the curvature effect
to softcombinedwith the curvatureeffect.Inaddition,we
somenegativelagschangeintopositiveones.Inviewofthe
also assume the evolutionof E0,p followsthe same means analysis of the above two cases we can conclude that the
as theabovesection.Tocomparethelagsproducedbythe
curvatureeffectindeedmakethelagsincreasesignificantly.
spectralevolutionalonewith thatbythe spectralevolution
and curvature effect together we plot two sorts of lags to-
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6 Z.Y.Pengetal.:Spectrallagofgamma-rayburstcausedbytheintrinsicspectralevolutionandthecurvatureeffect
0.0000040
0.000008 0.0000035 0,psoft-to-hard
0.000007 0,phard-to-soft 0.0000030 k=0.5
k=0.5
0.000006
0.0000025
)
) 0.000005
(
C( 0.000004 C 0.0000020
0.0000015
0.000003
0.0000010
0.000002
0.000001 0.0000005
0.000000 0.0000000
0.0000 0.0002 0.0004 0.0006 0.0008 0.0000 0.0002 0.0004 0.0006 0.0008
Fig.4 ExamplelightcurveofE variesfromhardtosoft(leftpanel)andsofttohard(rightpanel)forthecaseofpower
0,p
lawdecaylocalpulses,wherethesymbolsarethesameasthoseadoptedinFig.1.
Fig.5 Thecomparisonsoflagdistributionbetweenthecaseofhard-to-soft(panela)andsoft-to-hard(panelb)evolution
alone(solidlines)andthecaseofspectralevolutionandcurvatureeffectsimultaneously(thedashedlines).
5 Observed spectral lagsofthe hard-to-soft apparentlyreproducestheobservedlagdistributioninclud-
pulses ingthe measurednegativelags.With this, a goodcase can
be made that negative lags do not exist. The existence of
multiple overlappingpulses could easily produce negative
The above analysis shows that the spectral lags resulting
lags,eventhoughthelagsforeachpulsearepositive.Allit
fromthehard-to-softspectralevolutionarepositiveandthe
takesistwonearlyoverlappingpulseswiththesecondone
negative lags may come from the soft-to-hard or soft-to-
being harder than the first. In all, we are not convinced in
hard-to-soft spectral evolution. We wonder if the observa-
thesignificantexistenceofnegativelags.
tionsareconsistentwiththepredictions.Thepulsesexhibit
hard-to-soft,soft-to-hardandsoft-to-hard-to-softspectralevo- Therefore,weselectasampleonlycontainingtheFRED
lution should be checked. However, the spectral parame- (fast rise and exponential decay) pulses provided by Ko-
ter evolution of observed pulses generally exhibit hard-to- cevskietal.(2003)tocheckthetheoreticalpredicationsince
soft and “tracking” (i.e. soft-to-hard-to-soft). Both behav- these bursts exhibit clean, single-peaked or well-separated
iors are defined by E and/or low-energyindex evolution, in multi-peaked events. We can obtain directly the spec-
p
which are found by several authors (Kaneko et al. 2006; tral parametersofbrightbursts fromKanekoet al. (2006).
Fordetal.1995;Crideretal.1997).WithordinaryPoisson TheweakerburstsarealsoanalyzedusingRMFITsoftware
noiseinGRBlightcurves,therewillbesignificantscatterin and fitted with Band model. Similar to Peng et al. (2009a,
themeasuredlags.Thepercentageoflagsincreasesubstan- 2009b,2010)andRyde&Svensson(2002)asignal-to-noise
tially towards small values, with many (the high luminos- ratio(S/N)oftheobservationsofatleast 30togethigher
≥
ityevents)havingnear-zerolags.Insuchacase,weexpect timeresolutionareadopted.Forthehard-to-softpulsesonly
theretobemany“measured”negativelags,evenifalllags thatcontainingatleast1datapointsofE beforethepeak
p
are positive-definite. That is, a convolution of a positive- time of pulse and 4 in all are included in the our analysis.
definitelagdistributionwiththeknownmeasurementerrors Finally, we select 15 hard-to-soft pulses as our sample to
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ANheaderwillbeprovidedbythepublisher 7
Fig.6 SpectralevolutionofE andtemporalbehaviorsofGRBpulsesofBATSEchannels3and1inoursampleforthe
p
caseofhard-to-soft,wheretheblacklinesrepresentthelightcurvesofchannel3.Thepositivelagsareevident.
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8 Z.Y.Pengetal.:Spectrallagofgamma-rayburstcausedbytheintrinsicspectralevolutionandthecurvatureeffect
testourpredictions.IllustratedinFig.6arethespectraland
temporalbehaviorsforoursample.
Wefirstemploythewidelyusedcross-correlationfunc- 0.25
tion(CCF)tocomputethespectrallagsbetweenthepulses
0.20
ofthesameburstseenintheBATSEchannels1and3.Sim-
ilar to Norrisetal. (2000)we adopta cubicpolynomialto 0.15
fitthepeakoftheresultingdiscreteCCFfunctionandtake ag13
thepeakofthecubicpolynomialformasthelag.Theerror L 0.10
intheaverageCCFlag,lag ,isfoundbysimulationswith
13 0.05
Gaussian noise added to each of the two channels. These
spectrallagsresultingfromtheevolutionofhard-to-softare 0.00
listedinTable1.Wefindnonegativelagsinthehard-to-soft
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
pulses.
CCF13
Recently Hakkila et al. (2008) point out that the CCF
and pulse peak lags occasionally disagree. In order to fur- Fig.7 Peaklag(s)versusCCFlag(s).Agoodlinearcor-
ther test it we also find pulse peak lags which are defined respondenceexists,asexpected.
as thedifferencesbetweenthe pulsepeakinBATSE chan-
nels1and3.Inordertofindthepeaktimeofpulsewemust
andfindthatthespectrallagsresultingfromthesedifferent
select a pulse modelto fit these pulses. Following Peng et
localpulsesareconsistentwiththoseoftheGaussianpulse.
al. (2006) we only select the function presented in equa-
Let us check whether there are any differences in the
tion (22) of Kocevski et al. (2003). We fit those selected
measurements between peak lags, Lag , adopted in the
background-subtractedpulsesusingtheKRLfunction.Our 13
aboveanalysisand the widelyused crosscorrelationfunc-
fittingsareconstrainedonthechannels1and3tofindtheir
tionlags,CCF (e.g.,Norrisetal.2000).SimilartoBand
differenceofpeaktimes.Thebestfittingparametersarede- 13
(1997) and Norris et al. (2000) we fitted a cubic polyno-
pended on which χ2 of the fitting model is smaller. The
ν mialtothepeakoftheresultingdiscreteCCFfunctionand
pulsepeaklagsalongwiththefittingχ2 inchannels1and3
ν find the differenceof the correspondingpeak time. In Fig.
arealsolistedinTable1.FromTable1wefindallthelags
7 the peak lag, Lag , is plotted as a function of the CCF
of hard-to-soft pulse are positive. We also check the CCF 13
lag,CCF .Alinearfitisgivenwhichshowsthatthereisa
lags and peak lags by smoothing the light curves with the 13
goodcorrespondencebetweenthetwomethods.Thebestfit
DB3 wavelet(Qin et al. 2004)and obtainthe same results
isLag =( 0.03 0.03)+(1.28 0.23) CCF .The
(see,Table1). 13 − ± ± × 13
valuesofinterceptandslopeareconsistentwiththeresults
of Ryde et al. (2005).Therefore,the peak lags we used in
6 Conclusions and discussions thisstudywouldnotaffectouranalysisresults.
Weselectasampleconsistingof15pulseswhosepeak
SpectralevolutionisanestablishedcharacteristicinGRBs. energyE followhard-to-softevolutiontotestourtheoreti-
p
Moreoverthe trend for their high-energyphotonsto arrive calpredictions.OuranalysisshowsthatalloftheCCFlags
before the lower-energy ones (“hard-to-soft” evolution) is and analytic lags of hard-to-softpulses are positive within
universal phenomenon, which leads to positive lags. Ko- the range of uncertainty.In order to decrease the effect of
cevski & Liang (2003) showed the robust connection be- noise on our results we smoothed the data with the DB3
tween the spectral lag measuredin GRBs and the hard-to- wavelet (the second-class decomposition) using the MAT-
softevolutionoftheburstspectrabyemployingasampleof LAB software (Qin et al. 2004). Then we re-calculate the
cleansingle-peakedburstswithmeasuredlag.Inthispaper, two types of lags (see, Table 1) and find the lags are also
we haveinvestigatedtheconnectionbetweentheevolution positive.Thisindicateshard-to-softspectralevolutionmay
of the rest-frame spectral parameters and the spectral lag indeed produce positive lags, which is in good agreement
using a theoretical model derived by Qin (2002) and Qin withourtheoreticalprediction.
et al. (2004). We first assume an intrinsic Band spectrum Currently we can not ascertain if the negative lag pre-
andaGaussianemissionprofiletostudytheissue.Ourthe- dicted by our adopted model are consistent with observa-
oretical analysis shows that the connectionexists not only tion because we are not convinced in the significant exis-
betweenthe hard-to-softevolutionof the burstspectra and tenceofnegativelagsinourobservation.Themainreasons
the spectrallag butbetween the soft-to-hardevolutionand mentionedintheprevioussectionarethenear-zerolagsof
the spectral lag as well as between the soft-to-hard-to-soft mostpulseaswellastheexistenceofmultipleoverlapping
andthespectrallag.Inaddition,thehard-to-softspectralpa- pulses. Even then we can make a theoretical predication
rametersevolutioncanproducethepositivelags,whilethe abouttheproducingofthenegativelags,i.e.thesoft-to-hard
negative lags may be resulting from the evolution of soft- andthesoft-to-hard-to-softspectralevolutionmaygiverise
to-hardaswellassoft-to-hard-to-soft.Wealsotakeaccount to the negative lags, which reserves the further investiga-
of the other local pulse formsstudied by Qin et al. (2004) tionsinthefutureobservations.
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ANheaderwillbeprovidedbythepublisher 9
Table1 AlistofvariousobservedlagsofGRBpulseswhosespectralevolutionfollowingfromhardtosoft
trigger st(s) et(s) lag13,p(s) dlag13,p(s) lag13,c(s) dlag13,c(s) χ2ν,1 χ2ν,3
1406 -2.0 15.0 1.864±0.169 1.857±0.169 1.152±0.019 0.487±0.509 1.05 1.07
2138 -2.0 15.0 0.988±0.302 1.001±0.316 1.024±0.057 0.604±0.178 1.10 0.99
2387 -2.0 20.0 2.188±0.204 2.169±0.207 1.856±0.019 1.574±0.177 1.12 1.15
2662 -2.0 15.0 1.331±0.285 1.355±0.272 0.768±0.159 0.712±0.139 1.01 1.17
2665 -2.0 15.0 1.552±0.196 1.555±0.196 1.507±0.503 1.417±0.148 0.89 0.91
3256 -2.0 15.0 2.120±0.428 2.828±0.390 1.532±0.806 1.325±0.147 1.04 1.07
3257 -1.0 15.0 1.667±0.517 1.606±0.509 1.283±0.075 1.134±0.094 0.97 0.92
3875 -1.0 5.0 0.262±0.029 0.257±0.031 0.162±0.026 0.169±0.011 1.40 1.18
4368 -2.0 10.0 0.346±0.047 0.346±0.048 0.265±0.027 0.265±0.013 1.03 0.82
5621 3.2 4.5 0.032±0.022 0.076±0.011 0.028±0.002 0.030±0.001 1.24 1.71
6397 -2.0 20.0 1.158±0.082 1.160±0.082 0.942±0.031 0.934±0.171 1.17 1.70
7548 2.0 12.0 0.814±0.075 0.816±0.075 0.596±0.232 0.628±0.063 0.88 1.10
7588 -2.0 15.0 2.230±0.120 2.220±0.120 1.125±0.292 1.427±0.390 0.87 0.93
7638 -2.0 15.0 1.124±0.114 1.420±0.134 0.968±0.012 0.931±0.162 1.15 1.04
7648 -2.0 15.0 2.490±0.243 2.501±0.248 1.574±1.202 1.528±0.386 1.01 1.03
Note:standetdenotesthestartandendtimesinceburststriggerofselectedpulses,respectively.lag13,pandlag13,crepresentspeak
timelagandCCFlag,whiledlag13,panddlag13,carethecorrespondinglagsbutthelightcurvehavingbeensmoothedwiththeDB3
wavelet,χ2ν,1andχ2ν,3isthefittingχ2ν ofchannels1and3,respectively.
When taking the spectral evolution and curvature ef- thoughttheinterpretationofpeakenergyE (andhenceits
p
fectintoaccountsimultaneouslywefindthelagsarelarger evolution)dependsontheradiationmechanismthatisused
than those associated with the case that we only consider toexplaintheGRBspectra.Moreovertheyalsoattemptedto
the spectral evolution (see, Fig. 5). The two factors must examineseveralmechanismsthatcouldproducethe decay
playtherolessimultaneouslyontheproducingofthespec- oftheGRBspectraanddiscusshowthisevolutioncouldbe
trallagssincethecurvatureeffectmustbeplayaimportant connectedtotheburstsluminosityforeachmodel.Whereas
role on the producing of spectral and temporal profile of for the soft-to-hard-to-softspectral evolution few attempts
the GRB. In other words, the observed lags should suffer have been made to explain it. Peng et al. (2009a) argued
from this two factors simultaneously. Therefore we argue thatthephenomenonmaybecausedbybothkinematicand
thatthenumberoftheobservednegativelagshouldbemuch dynamicprocess.Howevertheydidnotprovidefurtherev-
smallerthanpositivelags.Recently,Chenetal.(2005)used idencestodistinguishwhichoneisdominant.Theissueof
the data acquiredin the time-to-spill(TTS) mode for long spectralevolutionofhard-to-softandsoft-to-hard-to-softis
GRBs by BATSE and found that positive lags is the norm still unclear,whichdeservescarefulstudiesandfurtherin-
in this selected sample set of long GRBs. While relatively vestigations.
fewinnumber,somepulsesofseverallongGRBsdoshow
negativelags, which is also consistentwith our theoretical Acknowledgements. ThisworkwassupportedbytheScienceFund
oftheEducationDepartmentofYunnanProvince(08Y0129,09Y0414),
predication.
theNaturalScienceFundofYunnanProvince(2009ZC060M),the
Boththetheoryandobservationsuggesttheconnection
keyprojectofappliedbasicstudyofYunnanProvince(2008CC011),
betweenthespectralevolutionandthespectrallagsindeed NationalNaturalScienceFoundationofChina(No.10778726).
exists.Therefore,wemustmakeclearthemechanismofin-
trinsic spectral evolution in order to reveal the underlying
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