submittedtoTHEASTROPHYSICALJOURNALLETTERS,05/04/99 ANALYSISOFTEMPORALFEATURESOFGAMMA-RAYBURSTSINTHEINTERNALSHOCKMODEL M.SPADA1,2,A.PANAITESCU1&P.ME´SZA´ROS1 DepartmentofAstronomy&Astrophysics,PennsylvaniaStateUniversity,UniversityPark,PA16802 ABSTRACT InarecentpaperwehavecalculatedthepowerdensityspectrumofGamma-RayBurstsarisingfrommultiple shocksinarelativisticwind. Thewindopticalthicknessisoneofthefactorstowhichthepowerspectrumismost 0 sensitive, thereforewe have furtherdevelopedour modelby taking into accountthe photondown-scatteringon 0 0 thecoldelectronsinthewind. Foranalmostopticallythickwindweidentifyacombinationofejectionfeatures 2 and windparametersthatyield burstswith an averagepowerspectrumin agreementwith the observations,and with an efficiency of converting the wind kinetic energy in 50–300 keV emission of order 1%. For the same n set of model features the interval time between peaks and pulse fluences have distributions consistent with the a J log-normaldistributionobservedinrealbursts. 4 Subjectheadings:gamma-rays:bursts-methods:numerical-radiationmechanisms:non-thermal 1 2 1. INTRODUCTION physicalpropertiesoftheejecta. v 7 TheGamma-RayBurst(GRB)light-curvesarecomplexand 9 irregular,withoutanysystematictemporalfeatures(Fishman& 2. OUTLINEOFTHEMODEL 0 Meegan1995)andanunderstandingoftheoriginofthetempo- 8 ralbehaviorofGRBsremainsanopenissue. Statisticalstudies We simulate GRB light-curvesby addingpulsesradiatedin 0 a series of internal shocks that occur in a transient, unstable arenecessaryinordertoidentifythephysicalpropertiesofthe 9 relativisticwind. Aswe showedinPSM99the observedburst emission mechanism existent in all or a group of GRBs. Re- 9 variability time-scale depends mostly on the wind dynamics, cently Beloborodov et al. 1998, hereafter BSS98, have used / h the Fourier analysis of a sample of long GRB light-curves to its optical thickness and its radiative efficiency in the BATSE p study the statistical properties of their power density spectra window. Here we modelthe wind dynamicsandthe emission - processesas in PSM99, but we includea more accurate treat- (PDS).ThePDSfeaturestogetherwithothertemporalproper- o mentofthephotondown-scatteringonthecoldelectronsinthe r tiesoftheobservedGRBs,suchasthedistributionsofthetime t interval between peaks and of the pulse fluence (McBreen et wind. We calculate the effectof the photondiffusionthrough s thecollidingshellsand thewind onthe pulsedurationandon a al. 1994, Li & Fenimore 1996), can be used to constrain the : physicalcharacteristicsoftheGRBsource. theenergyoftheemergentphoton,ratherthanjustattenuating v thepulsefluenceaccordingtotheopticalthicknessofthewind i Intheframeworkoftheinternalshockmodel,therapidvari- X ability and complexity of the GRB light-curves is due to the throughwhichitpropagates.Howeverthecontributionofthese photons to the duration of the received pulses may be impor- r emission from multiple shocks in a relativistic wind (Rees & a tant for bursts that are not very optically thin, and the photon Me´sza´ros1994,Kobayashietal. 1997,Daigne&Mochkovitch down-scatteringshouldbetakenintoaccountformorereliable 1998).Theejectaarereleasedbythesourceduringatimecom- calculationsofGRBlight-curves. parable to the observed burst duration. The instability of the AsdescribedinPSM99,thewindisdiscretizedasasequence windleadstoshockswhichconvertafractionofthebulkkinetic energyininternalenergyatadistanceR 1012 1014cmfrom ofN =tw/tv shells,wheretw isthedurationtimeofthewind ∼ − ejectionfromthecentralsourceandt < 1photonsaredown-scatteredbythecold B B s c We assume that between two consecutivecollisionsthe thick- electronsbeforetheyescapetheemittingshell,leadingtoade- ness of the shell increases proportionally to the fractional increase in the photon energy and an increase of the pulse du- crease of its radius d∆/∆ dR/R. The shell internal en- ration. For down-scatterings occurring in the Thomson limit ergyincreasesineachcollisi∝onbythefractionofU′ thatisnot (ε′ m c2/γ )theenergyoftheemergentphotoncanbeap- rloasdsieaste.d, and decreases during the expansion due tso adiabatic pagroex≪nimumatbeeedrboyfeεs′dcsat=terεin′(g1s−suεff′e/rmedecb2y)τac2,pwhohteorne.τFc2oirsmthoereaveenr-- Theshock-acceleratedelectronsradiateandtheemittedpho- ergeticphotons,weevaluateε′ numerically,becausethecross ds tons can be up-scattered on the hot electrons (γe >> 1) or section depends on the photon energy and changes after each down-scattered by the cold ones (γe 1). Far from the photon-electroninteraction. For the set of parametersconsid- ≈ Klein-Nishinaregimetheopticaldepthtoup-scatteringisτic = eredinthispaper,theThompsonlimitisusuallyagoodapprox- σThζen′emin(ct′γ,∆′), where n′e is the co-moving electrons imationtotreatthedown-scatteringofthesynchrotronphotons densityandt′ =t′ /(1+y)istheradiativetimescale,witht′ during all the wind expansion. For the smaller collision radii γ sy sy thesynchrotroncoolingtimeandytheComptonizationparam- the inverseCompton emission peaksat large comovingframe eter(forτic < 1,y = γm2τic). Theopticalthicknessτc forthe energiesandthe generalcase hasto beconsidered. Figure1f cold electrons within the emitting shell is evaluated by taking shows the evolution of the synchrotron and inverse Compton into accountthe cold electronswithin the hot fluid, those that observerframe peak energiesfor a thick wind. At R 1012 were accelerated but have cooled radiatively while the shock cm,τ 103andthe 10keVsynchrotronemissionis≈down- c crossed the shell, and those within the yet un-shockedpart of scattere≈dbyanorder∼ofmagnitude,while 100MeVinverse theshell. Comptonradiationisdown-scatteredto 1∼0keV. Afractionmin(1,τ )ofthesynchrotronphotonsisinverse Weapproximatetheincreaseinthepu≈lsedurationduetothe ic Comptonscatterednic = max(1,τi2c)times,unlesstheKlein- diffusionthroughopticallythickshellsbythetimeδTd ittakes Nishinaregimeisreached.Theenergyoftheup-scatteredpho- toaphotontodiffusethroughthem,whichweaddtoδT tode- 0 tonandtheratiooftheComptontosynchrotronpowercanbe terminetheobservedpulsedurationδT.IntheThompsonlimit castintheforms: δT 5τ ∆/(24c); in the general case the diffusion time is d c hνic =min(cid:20)γmmec2,(cid:18)34γm2(cid:19)nichνsy(cid:21) , (1) goipvteicn≈albythδicTkdne=ss(cid:8)foPrnit=hse1[iτ-t(hε′is)c]−at1te(cid:9)ri∆ng/(a2ncd),nwshiserteheτ(nxuim)biserthoef down-scatteringsonthecoldelectrons,evaluatedrequiringthe P m c2 4 nic photonrandomwalkequaltotheshellwidth. Figure1eshows Pic =min(cid:26)γm hνe ,(cid:20)3γm2 min(1,τζ)(cid:21) (cid:27) , (2) theevolutionofthepulsedurationduringthewindexpansion: sy sy for smaller collision radii δT > δT and the pulse duration d 0 which take into account the upper limits imposed by the isdeterminedbyδT whichdecreaseswithR. Forlargerradii d Klein-Nishina effect. Figure 1c shows the evolution of the δT <δT ,thusδT δT andincreaseswithR. d 0 0 synchrotron and inverse Compton peak energies during the Foragivenpulse,≈weaddtothepulsedurationthediffusion windexpansion: theenergyislowerforlargercollisionsradii, time it takes the photon to propagatethrough all the shells of due to the increased shell volume and the less relativistic opticalthicknessaboveunity.AsshowninFigure1d,thewind shocks, which lead to lower magnetic fields and electron ran- optical thickness is 1–2 orders of magnitude smaller than the domLorentzfactors. 3 optical thickness of the emitting shell (τc). Nevertheless the 3. EFFECTOFTHEPHYSICALPARAMETERSONTHEGRBPOWER photondiffusionthroughtheopticallythickshellsinthe wind DENSITYSPECTRUM can contributeup to 30% to the pulse durationbecause of the Inthissectionweanalyzetheeffectofthemodelparameters broadeningoftheshellwidthduringthewindexpansion. ontwodistributionsthatcharacterizetheGRBtemporalstruc- The30–500keVpulseenergyisafractionofthekineticen- ture: the PDS and the distribution of the interval δ between p ergyofthecollidingshells,equaltotheproductofthedynami- peaks. Therelevantmodelparametersdescribethewindejec- cal(ǫd),theradiative(ǫr),andthewindowefficiency(ǫw). tion (tv, tw, Γmin, Γmax and Lw) and the energyrelease (ǫe, 1)Thedynamicalefficiencyisthefractionofthekineticen- ζ and ǫ ). In order to diminish the large PDS fluctuations, e B ergy that is converted to internal, and is given by the energy inthissectionweuse powerspectrathatareaveragedover10 andmomentumconservationinthecollisionofaforwardshell peak-normalized bursts. The light-curve peaks are identified (Mf,Γf)caughtupbyabackshell(Mb,Γb >Γf): withthepeakfindingalgorithm(PFA)describedbyLi&Feni- more(1996).ForeachtimebinT withaphotonfluxC higher MΓ p p ǫ =1 (3) thanthoseoftheneighboringtimebins,wesearchforthetimes d − ΓbMb+ΓfMf T1 < Tp andT2 > Tp whenthephotonfluxesC1 andC2 sat- isfyC C > N C . A peakis identifiedatT there whereM =Mb+Mf isthetotalmassand arenopti−meb1i,2nsbetwveaernpT1pandT2 withphotonfluxesphigher than C . The light-curve valleys are identified as the minima Γ M +Γ M 1/2 p Γ= b b f f (4) betweentwoconsecutivepeaks. (cid:20)Mb/Γb+Mf/Γf(cid:21) The energyrelease parametersǫe, ζe, and ǫB determinethe 50–300keV radiative efficiency of the pulses. In an optically isthefinalLorentzfactorofthemergedshell.Theǫ decreases d thin wind, the parameter that affects mostly the window effi- withΓ /Γ andismaximizedbyM = M , sotheinnercol- b f b f ciencyistheelectroninjectionfractionζ (foranopticallythick lisions, forwhichthe differencein the shells Lorentzfactor is e wind the photons are down-scattered before they escape the larger,arethemostdynamicallyefficient,withǫ >0.1. Dur- d shellsandthewindowefficiencydependsalso onτ ). Figures ing the wind expansion the collisions diminish th∼e initial dif- c 2aand2bshowthePDSandtheδ distributionsforathinwind ferenceintheLorentzfactorsandthedynamicalefficiencyde- withL=1052ergs−1,Γ =30p,andΓ =800,andfortwo creasesto1%orless. Asshowinthenextsection,amodulation differentζ (1and 10−3).mIn bothcasesMsynchrotronemission in theejectionLorentzfactorisnecessaryto dynamicallyeffi- e is the dominantradiative process the inverse Compton contri- cientcollisionsatlargerradii. butiontothetotalemissionbeing10%forζ =1andlessthen 2) The radiative efficiencyis the fraction of the internalen- 0.1%forζ = 10−3. Forζ = 1thesynchreotronemissionlies ergyconvertedinradiation,andisgivenby: e e mainlybelowtheBATSEwindow(Figure2c),thewindowef- ǫ =ǫ t−γ1 (5) fiiscfieonrmcyeddebcyrepauselssefsrowmithshaodrtuerrattoiolnonogfeδrTpulse1s,0t−h2esli,gahntd-cwurivthe r et−γ1+t−ad1 an average difference in the photon arrival≈time ∆T 0.2 s. ≈ Because∆T δT,thedistributionofintervalsbetweenpeaks where tad R/c is the adiabatic time-scale. The radiative isdetermined≫bythepulsesarrivaltimesandpeaksat0.1–0.2 ∼ efficiency decreases during the wind expansion and it’s upper s. Ifζ = 10−3 thesynchrotronemissionisabovetheBATSE e limit is the fraction ǫe of internal energy stored in electrons. windowforδT <0.3sandthewindowefficiencyismaximized Formagneticfieldsnottoofarfromequipartition,theradiative for δT 0.2 – 0.4 s. The light curve is formed by longer timescaleisdeterminedbythesynchrotronlosses. pulses,th≈elowerfrequencypowerinthePDSincreasesandthe 3)Thewindowefficiencyisthefractionoftheradiatedenergy intervaltimebetweenpeaksshiftstolongertime-scale. thatarrivesatobserverinthe50–300keVband.Thecalculation The 50–300 keV efficiency of the synchrotron and the in- of ǫw is based on the approximationof the synchrotron spec- verse Compton emissions is determined by the strength mag- trumbythreepower-laws,withbreaksatthecoolingfrequency neticfieldB. Whileforǫ >0.1theemissionisdominatedby B νc andthepeakfrequencyνsy (atwhichtheγm-electronsradi- synchrotronradiation,forvaluesofthemagneticfieldwellbe- ate). If the νc < νsy then the shape of the spectrum is given lowequipartition(ǫB < 0.01)theburstemissionisdominated by: by the inverse Compton. Because the shape of the PDS does ν1/3 ν <νc notchange,weconcludethatthePDSisnotsensitivetoǫB and F  ν−1/2 ν <ν <ν , (6) therelativecontributionofsynchrotronandinverseComptonin ν c sy ∝ ν−p/2 ν <ν thelight-curve. sy Theejectionparametersdeterminethedynamicsofthewind wherepistheindexoftheassumedpower-lawelectrondistri- andtheevolutionofthepulsedynamicalefficiencyǫ . Thelat- bution.Ifν <ν then d sy c terreflectstheevolutionofthedifferencesbetweentheLorentz factors of a pair of colliding shells. The first collisions re- ν1/3 ν <ν sy movetheinitialrandomdifferences,andthemergedshellshave F  ν−(p−1)/2 ν <ν <ν . (7) ν ∝ ν−p/2 νsy<ν c Lorentz factors near the ejection average value Γ = (Γm + c Γ )/2. IfthewindisuniformΓisthesameforalltheshells, M  resultinginasteadydecreaseofǫ duringthewindexpansion. The inverse Compton spectrum has the same shape but is d IftherangeofshellejectionLorentzfactorsisvariableontime shiftedtohigherenergybythefactorimpliedbyequation(1). scale of the order of t (a ”modulated” wind), Γ reflects the For optically thick emitting shells we approximate the burst w spectrum as given in equations (6) and (7), using the down- scatteredcoolingandpeakfrequencies. 4 initialmodulationinΓ andlargeradiicollisionsthataredy- int donotaffectmuchtheevolutionofuniformwinds.How- M w namicallyefficientarestillpossible. ever,for a modulatedwind, t also determinesthe numberof w Figure3a showstheeffecton thePDS ofsquare-sinemod- periodsintheLorentzfactor(ifthedurationofaperiodisinde- ulations of the upper limit Γ with periods P = t and pendentoft ),influencingthustheclumpingofshells. M w w P =t /4.Thei-thshellejectionLorentzfactorisgivenby The burst redshift determines the co-moving energy range w which is redshifted into the observing range, leading to a ωi change in the total pulse efficiency, and altering the observed Γ =Γ +a sin2 (Γ Γ ), (8) i m i (cid:18)N(cid:19) M − m pulse duration. Obviously, by increasing the burst redshift, powerisshiftedfromhighertolowerfrequencies. where a is a random number between 0 and 1, and ω = i 2πt /P. The modulation shifts the power from high to low w frequencies, and the magnitude of this shift depends on the 4. COMPARISONWITHTHEOBSERVATIONS modulation period. If P = t the effect of the modulation w for interaction radii less than 1014 cm (corresponding to An analysis of the PDS of real bursts was presented by δT 1s)isnegligibleandthe≈windevolvesasintheuniform BSS98. They calculated the Fourier transform of 214 long case≈: the ǫd decreases from 5% to 0.2% when δT increases (T90 > 20s)andbrightburst,andhavefoundthattheaverage from 0.01 s to 1 s (Figure 3c). For R > 1014 cm the modu- PDSisapower-law(P f−5/3,f isfrequency)overalmost f lation becomesrelevant: the wind is for∼medof groupsof few twoordersofmagnitude∝infrequency,between0.02Hzand2 massive shells with different Lorentz factors. The dynamical Hz, where a break is observed, indicating a paucity of pulses efficiency remains constant for subsequentcollisions between with duration less than 0.5 s. The distribution of intervals ≈ massiveshells,whichyieldlongpulses(δT = 0.3–10s)that betweenpeakshasbeenstudiedbyMcBreenetal. (1994)and carryasubstantialfractionofthetotalburstfluence. byLi&Fenimore(1996),whoshowedthatthedistributionsof Figure 3d shows that the dependence on δT of the syn- thepulsefluenceS andofthetimeintervalδ betweenpeaks p p chrotron efficiency ǫ of the FS pulses has a similar behav- areconsistentwithalog-normaldistribution. sy ior as that of ǫ , because the internal energy density in the Aswasshownintheprevioussection,ifthewindisoptically d shockedplasmadependsonǫ . Foran higherinternalenergy, thin and the ejection features are random, the pulse duration d the minimumelectron Lorentzfactorγ increases, leadingto increaseswith the collision radiusand the emission efficiency m a higherenergyemissionanda shorterradiativecoolingtime- decreases during the wind expansion. The short inner colli- scale. Therefore the synchrotron efficiency remains constant sionsyieldmostofthe50–300keVburstemissionandthein- on the same range of δT where is constant the dynamical ef- ternalshockmodelpredictsaflatPDSwithequalpoweratlow ficiency,contributingtoa shiftofpowertolow frequenciesin andhighfrequency. Thus,inordertoexplaintheobservedbe- thePDS. havior,weneedaconfigurationoftheparameterswhichshifts The optical thickness of the wind depends mostly on the powerfromtheshorttothelongtime-scalesinthelight-curves. rangeofshellLorentzfactors(Γ Γ )andonthewindlumi- Moreover,theδ distributionisnotlog-normal:Figures2b,3b, m M p − nosity(L ). Figure4ashowsPDSsfortworangesofLorentz and4bshowthatinGRBsarisingfromopticallythin,uniform w factors, 30–1000 and 10–150. In the former case the wind is windstherearetoomanyshortintervalsbetweenpeaksrespect essentiallyopticallythin,andthephotondiffusiondoesnotaf- toaGaussianlogδ distribution. p fect the pulses duration δT, that increases with R (Figure 4c) PSM99 have identified three possible ways to explain the from0.01sto1s,between1013and1015cm.Inthelattercase deficitofpulseswithδT <1s: the wind is optically thick, in 80% of the collisions τ > 1, (1) a reduction in the electron injection fraction. This in- c and the pulse durationis givenby the diffusiontime: δT decreases the photon energy, reducing the window efficiency of d creasesfrom 5sto 0.6sbetweenR = 3 1012 cmand theshortpulses(causingthehighenergybreak)andincreasing R=1013cm≈,whereδ≈T isdeterminedmainlyb×ytheshellcur- that of the longer ones. However the behavior of the PDS at vatureandthusincreaseswithR. Fortheopticallythickwind lowerfrequencyremainsflat(seeFigure2a). thelongpulsesaregeneratedatsmallerRwheretheefficiency (2) a modulation of the shell ejection Lorentz factor. This has the maximum value, and the pulse energy increases with allows different configurations for the collisions series and a δT (Figure4d).ThePDShasmorepoweratlowfrequencyand higherdynamicalefficienciesforlongerpulses(seeFigure3a). thetimeintervalsbetweenpeaksarelongerthanintheoptically (3) an increase of the optical thickness of the wind. In this thincase(Figure4b).The50–300keVefficiencyisofthesame casethedown-scatteringsufferedbythephotonsastheypropa- orderforthetwocases: 4 10−3and5 10−3 forarangeof gatethroughthewindincreasesthepulsedurationforthesmall × × Lorentzfactorsof30–1000and10–150,respectively. radii collisions, which yield the shorter duration pulses (see Anincreaseinthewindluminosityhasasimilareffectonthe Figure4a). PDSshapeasadecreaseinΓ andΓ . Inthelattercase the In Figure 5a we show a simulated light-curvefor a square- m M windbecomesthickerbecausetheshellsaremoremassive. sine modulated wind (with P = t ) The burst 50–300 keV w Thevariabilitytimescalet affectsthedynamicalevolution efficiency is 1%, and the 90% of the RS and 80% of the FS v oftheshellsinthefollowingway. Ifthetimeintervalsbetween propagateinopticallythickshells. IfN = 0.1(thefreepa- var successiveejectionsdelaysdecreasesthenthecollisionsoccur rameterofthePFA),wefind22pulsesinthelight-curveshown atsmallerradii,wherethewindismoreopticallythick.Thedif- in Figure 5a. In order to have more peaks we simulate four ferencesbetweentheLorentzfactorsdiminishfaster(thereare light-curveswiththesameinjectionfeaturesandwindparame- more shells for smaller t ), reducingthe dynamicalefficiency tersandwecalculatetheintervalbetweenpeaksδ (Figure5b) v p for short pulses. For the modulated wind this effect is more andpeakfluenceS (Figure5c)distributions.Thedistributions p relevantthanintherandomcase. Thedurationt ofthewind are similar to a log-normal one, and the choice of N does w var ejection determinesmainly the numberof shells, and changes notaffectstronglytheirshape. 5 In order to compare the PDS of the simulated bursts with than the dynamical time-scales of plausible GRB progenitors the observed one, we consider an ensemble of cosmologi- (Me´sza´ros et al 1999), and we do not consider this a viable cal GRBs. Some authors (Totani, 1997, Wijers, et al. 1998, possibility. ThechoicesofL /L andβ are consistentwith M m Krumholz et al. 1998, Hogg & Fruchter 1999, Mao & Mo the values foundby Reichart & Me´sza´ros (1997), Mao & Mo 1999 ) have used a GRB co-moving rate density proportional (1999), and Krumholz et al. (1998) from fits to the observed tothestarformationrate. Others(Reichart&Me´sza´ros1997) intensitydistribution. Thevalueschosenforε andε arenot e B have employed a power-law GRB density evolution with red- toofarfromthosedeterminedbyWijers&Galama(1999)from shift, whichwasfoundby(Bagotetal. 1998)tobeconsistent the emission features of the afterglows of GRB 970508 and for z < 2 with their results from population-synthesis com- 971214. The above value of electron index p is close to the putatio∼ns of binary neutron stars merger rates. Finally, other valuesimpliedbytheobservedslopesofthe afterglowoptical researchers (Krumholz et al. 1998, Hogg & Fruchter 1999), decays. have considered a constant GRB rate density. In this work, Figure 6b shows a burst-averaged PDS whose features are we use the power-law with redshift GRB density evolution similar to thatfoundby BSS98 in realbursts. The wind ejec- n (z) (1+z)D, mainly as a convenientparameterization. tionismodulatedbyasquaresine(eq.[8])witharandompe- c ∝ Ann (z)proportionaltothestarformationratewouldleadto riod between t /4 and t . About 40% of the 300 simulated c w w differentsets of modelparameters(see below), but the differ- bursts have peak photon fluxes brighter than 1γcm−2s−1 . encesareminor,becausethetwofunctionsdiffersubstantially Taking into account that the average redshift for these bursts in shape only for z > 1, where there is a strong decrease of is z = 0.90 the average burst duration T 1.5(1 + z)t b w ≈ theco-movingvolumeperunitredshiftandasmallerchanceof is close to the value T = 80 s of the bursts used by BSS98 b obtainingaburstthathasa50–300keVpeakphotonfluxbelow (the factor 1.5 was determinednumericallyand representsthe 1γcm−2s−1 (burstsdimmerthanthislimitarenotincludedin ratiobetweentheburstdurationatthesourceredshiftandt ). w thecalculationoftheaveragePDSandintensitydistribution). As can be seen in Figure 6b, P f−5/3 between 0.04 Hz f Giventheratedensityevolution,theGRBredshiftischosen and2 Hz andfalls offsteeperatfre∝quencieslargerthan2 Hz. fromaprobabilitydistribution The model parameters that led to the PDS of Figure 6b yield bursts whose integralintensity distribution is shown in Figure dP nc(z)dV , (9) 6a, which consistent with the distributionfoundby Pendleton dz ∝ 1+z dz etal.(1996): excludingtheburstsdimmerthan1γcm−2s−1 , themodelhasχ2 =9.5for9degreesoffreedom. wheredV/dz isthecosmologicalco-movingvolumeperunit redshift 5. CONCLUSION dV c 3 [q z (1 q )(√2q z+1 1)]2 We have calculated power density spectra of GRBs arising 0 0 0 =4π − − − . dz (cid:18)H (cid:19) q4(2q z+1)1/2(1+z)6 frominternalshocksinanunsteadyrelativisticwind.Bystudy- 0 0 0 (10) inghowthefeaturesofthesespectradependonthemodelpa- Weassumeq =0.5andH =75kms−1Mpc−1. rameters(Figures2,3,and4),wehaveidentifiedasetparame- 0 0 Theinferredisotropic50–300keVluminositiesoftheGRBs ters(Figure6)thatleadstoburstswhoseaveragePDSexhibits thathavemeasuredredshiftsspanmorethanoneorderofmag- anf−5/3 behavior(wheref isfrequency)for0.04Hz < f < nitude, therefore the standard candle approximation is not a 2Hz,asfoundbyBSS98inrealGRBs. Moreover,theintegral good approximation. We use an un-evolving power-law dis- intensitydistributionofthe simulatedburstsisconsistentwith tributionforthewindluminosity: thatobservedby Pendletonetal. (1996),and the distributions of the time intervals between peaks and of the pulse fluences Φ(L) L−β , L L L , (11) areconsistentwiththelog-normaldistributionsidentifiedbyLi m M ∝ ≤ ≤ &Fenimore(1996)inrealbursts. and zero otherwise. Note that this not the same as assum- The characteristics of the modeled bursts with the above ing that GRBs have a power-law distribution of their 50– mentionedfeaturesare: (1)asub-unityelectroninjectionfrac- 300 keV luminosities, as it is usually done (e.g. Reichart & tion, required to increase the radiative efficiency of the larger Me´sza´ros(1997),Krumholzetal. 1998,Mao&Mo1999),as collision radii, (2) a modulated Lorentz factor of the ejected the relationship between the wind and the 50–300 keV lumi- shells,necessarytoincreasethedynamicalwindefficiencydur- nositiesis setbythewindowefficiency(atthesource)and,in ing the wind expansion and, (3) a shells optical thickness to the case of winds that are optically thick, by the wind optical scattering on cold electrons above unity, required to increase thickness,bothofwhicharedependentonthewindluminosity. thedurationofthepulsesastheypropagatethroughthecollid- Infindingmodelparametersthatyieldburstsconsistentwith ingshellsandthewind. the observations, we held constant t = 25 ms, L /L = In the internal shock model, the most efficient collisions, v M m 100, β = 2, ε = 0.25, ε = 0.1, and p = 2.5 . The cho- with a dynamical efficiency of 10-20% and a radiative effi- e B sen t is short enough to ensure that the observed 2 Hz PDS ciency of 10 30%, happen in the first part of the wind ex- v breakfrequencyisbelow[(1+z)t ]−1 (burstswithz > 3are pansion wher−e the wind optically thickness is higher and the v rarelybrighterthan1γcm−2s−1),correspondingtothepulses angularspreadtime,theshellshock-crossingtimeandtheelec- that are partly suppressed by the choice of t . the PDS fre- tronscoolingtimeareshorter( 0.5s). Inordertoreproduce v quencies affected by the assumed t = 25 ms are > 10 Hz. theobservedbreakat2Hzinth≪ePDS,wehavepreviously(see v This may suggest a possible explanation for the P∼DS break PSM99)attenuatedthefluenceoftheseshortpulsesaccording observed by BSS98: the lack of pulses shorter than 1 s is toan highwindopticallythickness, with a resultinglow burst duetotheexistenceofa minimumwindvariabilitytim∼e-scale efficiency (10−4 for an uniform wind and 10−3 for a modu- of the same order. However, such t ’s would be much larger lated one). The study of the photondiffusion, presentedhere, v 6 allowedustofindmodelparametersthatyieldan1%efficiency simulated average PDS shows the break at 2 Hz with a burst of convertingthe wind kinetic energy into 50–300 keV emis- efficiencyclosetothemaximalvalue(few%)admittedbythe sion. Foranopticallythickwind,thepulsedurationofthefirst, model(seealsoKumar1999). efficient collisions at small radii is determinated by the time thephotonstaketoescapetheshells,thatdependsonlyonthe This research is supported by NASA NAG5-2857, NSF colliding shells width and optically thickness τc. If τc 1 PAY94-07194 and the CNR. We are grateful to Martin Rees, thediffusiontimefortheefficientcollisionsis>0.5san≫dthe SteinSigurdssonandMarcoSalvatiforstimulatingcomments. ∼ REFERENCES Bagot,P.,PortegiesZwart,S.F.,&Yungelson,L.R.1998,A&A,332,L57 Me´sza´ros,P.,Rees,M.J.,Wijers,R.1999,NewAstronomy,4,303 Beloborodov,A.M.,Stern,B.E.,&Svensson,R.1998(BSS98),ApJ,508,L25 McBreen,B.etal1994,MNRAS,271,662 Daigne,F.&Mochkovitch,R.,1998,MNRAS,296,275 Norris,J.P.etal.1996,ApJ,459,393 Fishman,G.J.&Meegan,C.A.1995,ARAA,33,415 Panaitescu,A.&Me´sza´ros,P.1999,ApJ,(astro-ph/9810258) Ghisellini,G.&Celotti,A.1999,ApJ,511,L93 Panaitescu,A.,Spada,M.,&Me´sza´ros,P.1999(PSM99),ApJL,522,L10 Hogg,D.W.&Fruchter,A.S.1999,ApJ,520,54 Pendleton,G.N.etal.1996,ApJ,464,606 Kobayashi,S.,Piran,T.&Sari,R.1997,ApJ,490,92 Rees,M.J.&Me´sza´ros,P.1994,ApJ,430,L93 Krumholz,M.,Thorsett,S.E.,&Harrison,F.A.1998,ApJ,506,L81 Reichart,D.E.&Me´sza´ros,P.1997,ApJ,483,597 Kumar,P.1999,ApJL,523,L113 Totani,T.1997,ApJ,486,L71 Li,H.&Fenimore,E.E.1996,ApJ,469,L115 Wijers,R.etal.1998,MNRAS,294,L7 Mao,S.&Mo,H.J.1999,A&A,339,L1 Wijers,R.&Galama,T.J.1999,ApJ,523,177 7 0 4. 1 νsyνic 15 νhsyνhic 3.5 hh 1 0 m] 3.m] 1 c c R [ R [ 14lg 12.5lg 0 2. 1 c f 5 13 1. 7 5 3 1 1 8 7 6 5 4 3 2 1 01 − ]Ve[ νh gl ]Ve[ νh gl 0 4. 1 b TTd 5 15 δδ 3. 1 0 m] 3.m] 1 c c R [ R [ 14lg 12.5lg θT∆TγT0 2.0 δδδδ 1 e 5 13 1. 2 0 2 4 0 1 21 − − − − ]s[ Tδ gl ]s[ Tδ gl 0 4. 1 τcτw 15 τcτw 3.5 1 0 m] 3.m] 1 c c R [ R [ 14lg 12.5lg 0 2. 1 a d 5 13 1. 0 1 2 3 4 3 2 1 0 1 21 − − − − − − τ gl τ gl Fig. 1.— Upperpanels: opticallythinwindwithLw = 1053ergs−1 and30 < Γ < 1000. PanelashowsthedependenceonthecollisionradiusRofthe opticalthicknessτcforscatteringonthecoldelectronsinsidetheemittingshell,andthatoftherestofthewind(τw). PanelbillustratestheR-dependenceofthe pulsedurationandofthetermsthatcontributetoit;whilegraphcshowstheR-dependenceofthesynchrotronandinverseComptonpeakenergies. Lowerpanels: opticallythickwindLw = 1053ergs−1 and5 < Γ < 300. GraphdshowstheR-dependenceofτcandτw,grapheshowstheevolutionofthepulseduration δT = δTd+(δTθ2 +δT∆2 +δTγ2)1/2 andthecontribution ofthediffusiontimethroughthewindδTd (i.e.excludingtheemittingshell). Graphf showsthe down-scatteredsynchrotronandinverseComptonenergypeaksversusthecollisionradius. Thedashedlinesinpanelf showtheevolutionofthesynchrotronand inverseComptonpeakbeforethedown-scattering,andthedottedlinesinpanelcandf showtheBATSEwindow.Parameters:tv =0.02s,tw =20s,ǫe=0.25, ǫB =0.1,ζe=1,burstredshiftz=1.Onlyasmallfractionofthetotalnumberofpulsesisshown;thedensityofthepointsillustratestheradiusdistribution.The curvesshownarelog-logspacefitsforthemostefficientpulses.Theactualvaluesarescatteredaroundthefit. 8 d b 1 0 1. 1 =e ζ 3 0s] s] − T [ [ 0 δ δp 1 g =e o ζ l 1 − 1 2 0. − 3 2 1 0 0 5 0 5 0 5 0 ))pδgl(d tN(/Nd 0. 0. 1. 1. 2. 2. 3. − − − − − − wε gol 0 0. 1 1 c a 3 0 − 0 1 =e s] 1.0f [Hz] ζ δg T [ o l 1 −30 ζ=e 1 11 − == e e ζζ 1 2 0. − 0 0 1 8 6 4 2 0 2 0. 1. 0. − 1 ysνh gol f P f 3/5 Fig. 2.— Effectoftheinjectionfractionζe ontheburstPDSs(panela),distributionofintervalbetweenpeaks(panelb)forNt ∼ 130peaks,energyofthe synchrotronspectrumpeak(panelc),andwindowefficiency(paneld).Parameters:Lw=1052ergs−1,Γm=30,ΓM =800,otherparametersareasforFigure 1.Panelscanddshowlog-logspacepolynomialfits,illustratingthusonlythetrends.TheRSemissionisrepresentedwithcirclesandtriangles,whiletheFSoneis shownwithcrossesandstars. 9 0 1. b 5 1 0. 0 0. s] δ [s]p −0.5δg T [ l 0 1. − 5 1. − d 0 0 2. 2 ))pδ(gltN(/1Nd 0 −0.5 −1.0 −1.5 −2.0 −2.5− )SF( ysε gl 0 0. 1 0 1. 4 4 d w/w /wd w anP=tP=t P=tanP=t 0.5 r r 0 1. 0 0. ] s] Hz T [ f [ 0.5δg −l 0 1. 1 − 0. 5 1. − c a 0 2. 0.0 1.0 0.1 0.0 −1.7 −2.2 −2.7 −3.2 −3.7 −4.2− 1 f P f dε gl 3/5 Fig. 3.— Effectofamodulationofthewindejection. PanelsaandbshowacomparisonbetweenthePDSsandtheδpdistributionsforanuniformwind(solid line),andasquare-sinemodulatedejectionwithperiodtw(dottedline)andtw/4(dashedline). Parameters: Lw =1052ergs−1,Γm =30,ΓM =1000,other parametersareasforFigure2.Lowerpanels:thedependenceofthedynamicalefficiency(panelc),andFSsynchrotronefficiency(paneld)versusthepulseduration. 10 2 b 1 1 0 00 s] ] 05 [ s 11 [ << 0T p ΓΓ δ δ << g 00 l 31 1 − 0 d 2 − 3 2 ))pδ(gltN(/Nd1 0 −2 −3 slpε gl −4 −5 0 0. 16 1 a 5 1 0 1. 4 1] m ] z c H [ [ R 000 f g 05 3l 11 1 << 1 ΓΓ 0. << 00 31 2 1 c 0.0 11 00.0 10.0 1.0 0.1 0.0 0.0 1 ]0s[ tδ gl −1 −2 1 f P f 3/5 Fig. 4.— Effect ofthe optical thickness. Panels aand bshow acomparison between the PDSsandthe distributions ofδp foranoptically thin windwith 30 < Γ< 1000(Nt = 70peaks),andanopticallythickonewith10< Γ < 150(Nt = 40peaks). Lw = 1053ergs−1,otherparametersareasforFigures 2and3. ThedependenceofδT onthecollisionradiusandofthepulse50–300keVfluenceonδT areshowninpanelscanddrespectively,fortheopticallythick (triangle)andthin(circle)winds.Thepulsefluenceisthefractionofthetotalfluenceinthelightcurvecarriedbyeachpulse.

Analysis of Temporal Features of Gamma Ray Bursts in the Internal Shock Model

Similar Analysis of Temporal Features of Gamma Ray Bursts in the Internal Shock Model