A common stochastic process rules gamma-ray burst prompt emission and X-ray flares

0.26 MB·English

by  C. Guidorzi

DRAFTVERSIONJANUARY13,2015
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
ACOMMONSTOCHASTICPROCESSRULESGAMMA–RAYBURSTPROMPTEMISSIONANDX–RAYFLARES
C.GUIDORZI1,S.DICHIARA1,F.FRONTERA1,2,R.MARGUTTI3,A.BALDESCHI1,L.AMATI2
DraftversionJanuary13,2015
ABSTRACT
Promptγ–rayandearlyX–rayafterglowemissioningamma–raybursts(GRBs)arecharacterizedbyabursty
5 behaviorandareofteninterspersedwithlongquiescenttimes. ThereiscompellingevidencethatX–rayflares
1 arelinkedtopromptγ–rays.However,thephysicalmechanismthatleadstothecomplextemporaldistribution
0 of γ–raypulses andX–rayflares is notunderstood. Here we show thatthe waiting time distribution(WTD)
2 of pulses and flares exhibitsa power–lawtail extendingover4 decadeswith index∼2 and can be the man-
ifestation of a common time–dependent Poisson process. This result is robust and is obtained on different
n
catalogs. Surprisingly, GRBs with many (≥8) γ–ray pulses are very unlikely to be accompaniedby X–ray
a
J flares after the end of the prompt emission (3.1σ Gaussian confidence). These results are consistent with a
simpleinterpretation: anhyperaccretingdiskbreaksupintooneorafewgroupsoffragments,eachofwhich
2
is independently accreted with the same probability per unit time. Prompt γ–rays and late X–ray flares are
1
nothingbutdifferentfragmentsbeingaccretedatthebeginningandattheend,respectively,followingthevery

samestochasticprocessandlikelythesamemechanism.
]
E Subjectheadings:gamma-ray:bursts,waitingtimedistribution
H
.
h 1. INTRODUCTION natureofthestochasticprocess.
p In particular,it revealsthe degreeof memoryand correla-
Thefirstelectromagneticmessengerofagamma–rayburst
- tion and constrains the physical process responsible for the
o (GRB) is the so–called γ–raypromptemission, followedby
discontinuousandburstyreleaseofenergy.
r the early X–ray afterglow on a timescale from minutes to
t Processesshowingsimilar on–offintermittency,or, equiv-
s hours.Longduration(>2–3s)GRBsarenowadaysknownto
alently, bursty behavior or clusterization, can be found in
a beassociatedwiththecorecollapseofsomekindofmassive
[ starsridofhydrogenenvelopes(seeWoosley&Bloom2006; manyfields(Plattetal.1993). ThecorrespondingWTDsof-
tenshowpower–lawtailsatlongwaitingtimes(WTs),whose
Hjorth&Bloom2012forreviews). Promptγ–rays(withen-
1 indexdependsonthedegreeofclusterizationofthetimese-
ergies in the keV–MeV range) are observed within a given
v ries. Examplesencompasstheaftershocksequenceobserved
6 GRB as a sequence of pulses (typically a few up to several
in earthquakes, describedby Omori’s law (Utsu 1961), neu-
0 dozens). In addition, for a sizable fractionof GRBs the fol-
ronal firing activity, as well as a wide range of dynamical
7 lowing decaying X–ray emission, which marks the end of
systemsofhumanactivity,suchasmailandemailexchanges
2 the γ–rays, is characterized by the presence of X–ray flares
(Oliveira&Barabási2005;Eckmannetal.2004),phonecalls
0 whicharesometimesobservedaslateas105s(Burrowsetal.
(Karsaietal.2012andreferencestherein)allthewaytovio-
. 2005;Chincarinietal.2007;Falconeetal.2007;Curranetal.
1 lent conflicts (Picolietal. 2014). These processes are often
2008; Bernardinietal. 2011). Although mounting evidence
0 modeledandinterpretedinthecontextofself–organizedcrit-
5 exists that X–ray flares, like γ–ray pulses, result from the icality(SOC),whereanonlineardynamicalsystemreachesa
1 GRB inner engine activity rather than from external shocks stable critical point in which continuous energy input is re-
: (Lazzati&Perna2007;Chincarinietal.2010;Marguttietal.
v 2010),keyquestionsremainunanswered:whatradiationpro- leased intermittently through avalanches and in a scale–free
i way. SOC naturally predictspower–lawsin energyand WT
X cess(es)? What information on the inner engine can we ex-
distributions. See Aschwandenetal. (2014) for a recent re-
tract? Isthereacommonprocessrulinginnerengineactivity
r viewonthemanyareasdisplayingSOCbehavior.
a acrossseveraldecadesintime?
InastrophysicsWTDsarestudiedformanydifferentkinds
As a matter of fact, both emissions represent a tempo-
of sources, such as outbursting magnetars (Gögˇüs¸ etal.
ral point process, i.e. a time series characterized by the
1999; Gögˇüs¸etal. 2000; Gavriiletal. 2004), flare stars
discrete occurrence of impulsive events superposed on a
(Arzner&Güdel 2004), and particularly the activity of the
continuum. Intense bursting periods are often interspersed
Sun throughout its cycle. WTDs of solar X–ray flares ex-
with relatively long (several up to tens of seconds) inter-
hibitpower–lawtailswithindicesintherange2.0–2.4across
vals with very low activity, compatible with the detector
several decades (Boffettaetal. 1999; Wheatland 2000), de-
background, which are often referred to as quiescent times
pending on the class of flares and flux thresholds. Related
(QTs; Ramirez-Ruiz&Merloni 2001; Nakar&Piran 2002;
burstyemissionfromtheSunsuchascoronalmassejections
Quilliganetal. 2002; Drago&Pagliara 2007). The study of
(CMEs) are found to show very similar WTDs, whose in-
the waiting time distribution (WTD), i.e. of how time inter-
dex ranges from ∼1.9 to ∼3.0 in low to high–activity pe-
valsbetweenadjacentpeaksdistribute,providescluesonthe
riodsof the solarcycle (Wheatland2003). Likewise, WTDs
1DepartmentofPhysicsandEarthSciences,UniversityofFerrara,via of solar radio storms (Eastwoodetal. 2010), of solar ener-
Saragat1,I–44122,Ferrara,Italy getic particle and of solar electron events show very similar
2INAF–IASFBologna,viaGobetti101,I–40129,Bologna,Italy power–law indices (Lietal. 2014). Such power–law tailed
3Harvard-Smithsonian CenterforAstrophysics, 60GardenSt., Cam- WTDsareusuallyinterpretedasthe consequenceofa time–
bridge,MA02138,USA.
2 Guidorzietal.
varyingPoissonprocessproducedbySOCsystems,inwhich unlessstatedotherwise.
the energy input rate is intermittent and directly affects the
degreeofclusterizationofflares(Aschwanden&McTiernan 2. DATAANALYSIS
2010;Lietal.2014). Inthismodel,theburstyenergyrelease We searched all long–duration γ–ray light curves with
is the result of avalanches produced in active regions where MEPSA4 (Guidorzi2015, 2014), a peaksearchalgorithmde-
magnetic flux is twisted by the moving footpoints, leading signedand calibrated to this goal. The advantageof MEPSA
to a series of independentmagneticreconnectioneventsand compared with analogous algorithms such as the one by
consequentplasmaacceleration. AlternativelytoSOC,inter- Li&Fenimore(1996)(LF)istwofold:
pretationsinthecontextoffullydevelopedMHDturbulence
have also beenproposedto explain the bursty dynamicsand • ithasalowerfalsepositiverate.Thisisparticularlytrue
thepower–lawWTD:theintermittentcharacteristheresultof forthetimeintervalsinwhichthesignaldropstoback-
anonlineardynamicswhichmakestheconvectivemotionof ground between two adjacent activity periods: in the
thefluidandmagneticfieldswingbetweenlaminarandturbu- bestcasestheLFfalsepositiverateis3–5×10- 3bin- 1,
lentregimesrepeatedlyandchaotically(Boffettaetal. 1999; while the MEPSA one is 1–2×10- 5 bin- 1 (Guidorzi
Lepretietal.2004). 2015);
The WTD between adjacent peaks in GRB γ–ray prompt
emission profiles was found to be described by a lognor- • ithasahighertruepositiverate,especiallyatlowsignal
mal–whichimpliessomedegreeofmemory–(Li&Fenimore tonoiseratios(∼4–5).
1996) with an excess at long values due to QTs
(Nakar&Piran2002;Quilliganetal.2002;Drago&Pagliara 2.1. Sampleselection
2007). However,when thepeakdetectionefficiencyiscare-
2.1.1. Swift/BATdata
fully taken into account, it is found that the intrinsic WTD
at short values is also compatible with an exponential, that We started with the GRBs detected by Swift/BAT in burst
iswhatisexpectedforaconstantPoisson(thusmemoryless) modefromJanuary2005to September2014, collecting825
process (Baldeschi&Guidorzi 2015). On the other side of GRBs. We extracted the mask–weighted light curves in the
thedistributionlongQTscouldeithermarktheinnerengine 15–150 keV energy band with a uniform bin time of 64 ms
temporarily switching off, or result from modulation of the following the standard procedure recommendedby the BAT
relativisticwindofshells(Ramirez-Ruizetal.2001),orthey team5 and applied MEPSA. We then imposed a minimum
couldbe due to a differentphysicalmechanismfrom thatof thresholdof5σ significance,whichensuresa verylow false
shortWTs(e.g.,Dragoetal.2008). positive rate (<10- 5 bin- 1; Guidorzi2015) and selected the
In spite of the impressivedata that are routinelybeing ac- GRBs with at least two peaks. We then removed from our
quired in the Swift era, little progress has been reported on sample the short duration GRBs (both with and without ex-
WTDs in GRBs. Recently, energy and WT distributions tendedemission)bycrosscheckingwiththeclassificationpro-
for GRB X–ray flares have been shown to have power–law videdintheBATcatalog(Sakamotoetal.2011),astheywill
tails very similar to what is observed for solar X–ray flares. thesubjectoffutureinvestigation.Sincethiscatalogdoesnot
In particular, the WTD has a power–law index of 1.8±0.2 includeGRBsfrom2010,fortheseGRBsweusedtheT90val-
(Wang&Dai 2013). These results were interpreted as ev- uesaspublishedintheBAT refinedGCN circularsregularly
idence for SOC possibly driven by magnetic reconnection publishedbythe BAT team andseta conservativeminimum
episodes triggered in magnetized shells emitted by differen- thresholdof T90 >3 s. A couple of GRBs detected by BAT
tiallyrotatingmillisecondpulsarsor,alternatively,byanhy- exhibitedaverylongdurationwhichcouldnotbecovereden-
peraccretingdiskaroundablackhole(Pophametal.1999). tirely in burst mode: in one case we used the WIND/Konus
Yet, there are several crucial issues which can be tackled lightcurveforGRB091024(Virgilietal.2013),whileinthe
with WTDs: is there additional evidence for a link between case of GRB130925A we used the peak times as they have
promptγ–raysandlateX–rayflares? TowhatextentdoQTs beenobtainedbyEvansetal.(2014)fromthecorresponding
differ from short WTs? Is it possible to provide a common Konuslightcurve.Finally,weendedupwithasampleof418
description of short WTs, QTs, X–ray flares? What about long GRBs with at least two significant (>5σ) peaks each,
rest–frameproperties? IsthereevidenceformemoryinGRB totaling2000peaksand1582WTs. Hereafter,werefertothis
engines? Whatcan beinferredonGRB enginesthroughthe sampleastheBATset.
WTDstudy?
In this paper we address these issues through the analy- 2.1.2. CGRO/BATSEdata
sis ofthe WTD ofGRB promptpeaksforthreeindependent We took the concatenated64–msburst data distributed by
data sets: Swift/BAT, CGRO/BATSE, and Fermi/GBM. For theBATSEteam6. Foreachcurveweinterpolatedtheback-
theSwiftGRBswhichhavealsobeenpromptlyobservedwith groundbyfitting with polynomialsofupto fourthdegreeas
XRT, we present a joint analysis of γ–ray peaks and X–ray suggestedbytheBATSEteam(e.g.,Guidorzi2005). Likein
flares merged together. Section 2 describes the data sample theBATcase,weappliedMEPSAtoaninitialsampleof2024
selection and how we modeled the WTDs. Here we delib- light curves in the full passband. We applied the same se-
eratelydid notconsiderthe energydistributionofpeaksand lectionon the peaksignificance and selected the long GRBs
flares, because even thoughour peak search algorithmiden- by requiring T >2 s, where T was taken from the GRB
90 90
tified moderately overlapped pulses, estimating their energy catalog7 (Paciesasetal. 1999). We ended up with a sample
would require specific assumptions on their temporal struc-
ture. We therefore decided to postpone it for future investi- 4http://www.fe.infn.it/u/guidorzi/new_guidorzi_files/code.html
gation. The results, their implications and interpretation are 5http://swift.gsfc.nasa.gov/analysis/threads/bat_threads.html
reportedin Sections 3 and 4, respectively. Hereafter, uncer- 6ftp://cossc.gsfc.nasa.gov/compton/data/batse/ascii_data/64ms/
tainties on best fit parameters are given at 90% confidence, 7http://gammaray.msfc.nasa.gov/batse/grb/catalog/current/tables/duration_table.txt
WaitingtimesinGRBs 3
of 1089 long GRBs with at least two 5–σ significant peaks. for the previous sets, we selected those GRBs with at least
Overall we collected 7649 peaks and 6560 WTs. Hereafter, two(eitherγ orX–ray)peaks,soastohaveatleastoneWT.
thiswillbereferredtoastheBATSEsample. Weendedupwithasampleof1098(954γ–and144X–ray)
We also applied the same selection procedure to the light peaksin244GRBs (01/2005–12/2012). We hereafterrefer
curves corresponding to the sum of the two softest energy tothisjointsetasBAT-Xsample.
channels(1and2)andtothesumofthetwohardestchannels Finally,weselectedthesubsamplewithknownredshift,so
(3and4),respectivelywithinthe25–110keVand>110keV astoderivetheWTDinthesourcerest–frame.Thiswasdone
bands.Wecollected1065and922GRBs,with5156and4912 bysimplycorrectingforcosmicdilationandthusdividingthe
WTs, respectively. These two groups will be hereafter re- observedWTsbythecorresponding(1+z). Unlikethewidth
ferredtoasBATSE12andBATSE34sets,respectively. ofagivenpulse,whichisaffectednotonlybycosmicdilation
but also by the energy passband shift, for their nature WTs
2.1.3. Fermi/GBMdata are affected by the latter to a much lesser extent. We found
359WTs in 94GRBs with knownredshift. Thesubsetwith
We selected 586 long GRBs detected with Fermi/GBM
knownredshiftwillbereferredtoasBAT-Xz.
(Meeganetal.2009)fromJuly2008toDecember2013. We
As an independent check, we in parallel considered
extractedthe light curvesof the two brightestGBM unitsin
the X–ray flare catalogs of Chincarinietal. (2010) and
theenergyband8–1000keVwith64msresolutionandsub-
Bernardinietal. (2011), which respectively include 113
tracted the backgroundthrough interpolation with a polyno-
mial of up to third degree. We selected the long GRBs by early–time(t<103s)X–rayflaresfromApril2005toMarch
imposingT >2s,whereT wastakenfromtheofficialcat- 2008 and 36 late–time (t >103 s) flares from April 2005 to
90 90
alog.8 We restricted to time intervals whose median values December 2009. However, due to lower statistics, we here-
range from - 30 to 300 s with reference to the trigger time. afterfocusontheBAT-Xsample.
This corresponds to the time interval covered by the time
taggedevent(TTE)datatypeintriggermode(Paciesasetal. 2.2. Waitingtimedistributionmodeling
2012; Gruberetal. 2014). Before - 30 s and after 600 s the InphysicsaPoissonprocessisusuallyassumedtobechar-
timeresolutionisthatofCTIMEdata,0.256s. Inmostcases acterizedbyaconstantexpectedrate. TheWTD ofthispro-
wedidnotconsiderintervalst>300s,becauseinterpolation- cessisexponentialwithe–foldingτ =1/λ,
estimatedbackgroundoftenbecomescriticalandtherequired
effortforaproperestimateisbeyondourscope(Gruberetal. P(∆t) = 1e- ∆t/τ = λe- λ∆t , (1)
2011; Fitzpatricketal. 2012). We did not consider GRBs τ
showingprolongedactivitybeyondthistimeinterval.Wethen
where λ is the constantmean rate and τ is the mean WT. A
applied the same selection criteria as for the previous sets.
time–varying Poisson process is characterized by a variable
Finally, by visual inspection we removed phosphorescence
meanrateλ(t):theprocessislocallyPoisson,buttheexpected
spikesdueto high–energyparticles(Meeganetal. 2009), by
rate changes with time as described by λ(t). According to
comparing the same profiles in different GBM units. We
thisdefinition,theresultingprocessisthecombinationoftwo
endedupwithafinalsampleof2383peaksoutof544GRBs
differentprocessesatplayandis oftenreferredto as a “Cox
withatleasttwosignificantpeaks. ThetotalnumberofWTs
process”(e.g.,Cox&Isham1980):
is1839.
(a): atagiventimeteventsaregeneratedaccordingtoaPois-
2.1.4. Swift/XRTX–rayflares
sonprocesswithrateλ=λ(t)and,assuch,arestatisti-
We considered the catalog of 498 X–ray flare candidates callyindependent;
detected with Swift/XRT obtained by Swenson&Roming
(2014). Thiswas extractedfrom680XRT lightcurvesfrom (b): the expected rate λ is itself a function of time, which
January 2005 to December 2012 with a method based on can vary either randomly or deterministically as time
the identificationof breakpointsin the residuals of the fitted passes.
piecewisepower–lawlightcurves: thesepointsmarksudden
changesin the mean value due to unfitted features. The op- To derive the corresponding WTD, one may approximate
timal set of breakpoints was then found by minimizing the λ(t) as a piecewise constant function in a number of ad-
residualsumofsquaresagainstpiecewiseconstantfunctions. jacent time intervals ti (i = 1,...,n) and treat it as a se-
To counter the effect of overfitting with unnecessary break- quenceof severalPoisson processeswith rate λi. Following
points, they madeuse of the Bayesian InformationCriterion Aschwanden&McTiernan(2010)andreferencestherein,the
(see Swensonetal. 2013; Swenson&Roming 2014 for fur- resultingWTDis
therdetails). Inthiscatalogeachcandidateisassignedacon-
fidencevalue.Weconservativelyselectedthesubsamplewith P(∆t) ≈ φ(λi)λie- λi∆t , (2)
a minimum confidence of 90%, ending up with 205 X–ray Xi
flarecandidates.
where
We separately merged each X–ray flare catalog with the λ t
BAT one by joining, for each GRB, the sequence of γ–ray φ(λi) = i i , (3)
λ t
peaktimesandflarepeaktimesintoauniquesequenceoftem- j j j
P
poralpeaks.Indoingthis,everypeakwhichwasseeninboth
is proportionalto the expected numberof WTs in intervalt
i
instruments was tagged as a γ–ray peak and not considered
whereλ=λ. Inthecontinuouslimit,Eq.(2)becomes
i
anymoreasanX–rayflare. Analogouslytotherequirements
Tλ(t)2e- λ(t)∆tdt
8http://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html P(∆t)= 0 , (4)
R T
λ(t)dt
0
R
4 Guidorzietal.
whereT isthetotalduration.Whenλ(t)iseitherunknownor 14 stat PP
hardtotreat,itispossibletodefine f(λ)suchthat f(λ)dλ= non-stat PP
dt/T, that is the fraction of time during which the expected 12
ratelieswithintherange[λ,λ+dλ]. Equation(4)becomes,
+∞f(λ)λ2e- λ∆tdλ -1s s] 10
P(∆t)= R0 +∞ . (5) unt 8
λf(λ)dλ o
0 c
WeadoptedthemodelfoRr f(λ)providedbyLietal.(2014) (t) [ 6
in their eq. (5), which has been proposed to fit the WTD l 4
obtained for solar X–ray flares and solar energetic particle
events, 2
f(λ) = Aλ- αexp(- βλ), (6)
0
with α and β free parametersand A is a normalizationterm 10 20 30 40 50 60 70 80 90
(0≤ α<2). This model generalizes several other models Time [s]
whichhadbeenputforwardin thesamecontext(Wheatland
¯ FIG.1.—Exampleoftime–varyingPoissonprocesswithvariablerateλ(t)
2000;Aschwanden&McTiernan2010). Themeanrateλis (dashed) assuming α=1 and β=0 in Eq. (6) and a constant one (solid)
withthesamemeanvalues. Squaresandverticalbarsinthetopregionmark
+∞
λ¯ = λf(λ)dλ=Aβα- 2Γ(2- α), (7) pthreoncoournrecsepdocnlduisntgeriezvaetniotntiomfetshethvaatriwabelreecgaesneeoravteerdthaescaoncsotnasnetqouneencise.cleTahr.e
Z
0
where Γ() is the gammafunction. From Equations(5-6) the 100
correspondingWTDis BAT
10-1 BATSE
GBM
P(∆t)=(2- α)β2- α(β+∆t)- (3- α), (8) 10-2
ai.ned. it+∞isnPo(∆rmt)adli(z∆edt)li=ke1.aTphreorbeaabrieliotynldyetnwsiotyfrfeuenpcatiroanm(eptderfs),, -1t) [s] 1100--43
0
α,wRhichdeterminesthelevelofclusterization,andthechar- DP( 10-5
acteristicWTβatwhichtheWTDbreaks:at∆t≫β,Eq.(8) 10-6
becomesapower–lawwithanindexof(3- α).
Equation(6)naturallygivesrisetoclusterization,i.e. time 10-7
intervalscharacterizedbyanintenseactivitywithahighrate 10-8
10-1 100 101 102 103 104 105
ofpeaks(highλ),interspersedwithquiescentperiods,during
D t [s]
whichthe rate dropssignificantly(low λ). The largerα, the
shallower the power–lawregime at long WTs, and the more FIG.2.—WTDsofγ–raypulsesoftheBAT(crosses),BATSE(circles),and
GBM(squares) samples together with their corresponding bestfitmodels.
clustered the time profile (Aschwanden&McTiernan 2010;
Errorbarsarethe(normalized)squarerootofcountsandarejustindicative.
Aschwandenetal. 2014). The details of how clustered the
time profile lookslike, in particularhow the shotrate varies basedonPoissonstatistics,whichisessentiallytheCstatistic
withtime,aredirectlydescribedbyEq.(6). Atagivenaver- (Cash1979)andholdsexactlyeveninthelowcountregime.
¯
agerateλ,byincreasingαthevarianceofλincreasescorre- WeuseditinthecontextofaBayesianMarkovChainMonte
spondingly,i.e. theshotratevariesmorewildly. Thismeans Carloapproach.ThedetailsarereportedinAppendixA.
deviatingmoreandmorefromtheconstant–ratecase,thusen-
hancingthe clusteringcharacter. Figure 1 illustrates the dif- 3. RESULTS
ferencebetweenatime–varyingprocesslike(α=1,β=0)in Table1reportsthebestfitparametersforalloftheWTDs
Eq.(6)andaconstantonesharingthesamemeanrateovera weconsidered. InallcasesthemodelofEq.(8)providesan
100–stimewindow. Thetemporalsequenceofeventsforthe acceptable description. The lowest confidence level is that
formerisevidentlymoreclusteredthanthatofthelatterand, of BATSE (3.0%), still comparable with nominal 5% usu-
inspiteofthetypicalfluctuationsofaPoissonpointprocess, allyadoptedasathreshold. TheBATSEsampleisthelargest
tracks the behavior of λ(t). It is worth nothing that, in gen- (6560WTs), sothehighstatisticalsensitivityislikelytoen-
eral, in a Poisson process individual events are independent hancesmalldeviationsfromthemodel.
ofeachotherand,assuch,havenomemoryoftheeventsthat Figure 2 displays the WTDs for the γ–ray peak samples
occurredearlier,regardlessofwhethertheexpectedrateλis only. Apart from the GBM, whose power–law index is sig-
constantortime–dependent.Thedifferenceinsteadliesinthe nificantly steeper, the BAT and BATSE samples are fit with
observedaverageratesasafunctionoftime,sonoton(a)but comparable indices, 2.06+0.10 and 1.76±0.04, respectively.
- 0.09
on(b):dependingonwhetherλ(t)varieseitherinadetermin- Thisis remarkable,giventhe differentkindofdetectors, en-
isticway,orrandomlywith/withoutmemory,theaveragerate ergy passbands, different GRB populations each instrument
inheritsthecorrespondingdegreeofcorrelation. ismostlysensitiveto(Band2006). ThesoftandhardBATSE
The WTDs we wanted to model are characterized by rare sampleshavethe same index,showingno significantdepen-
longWTs,wherethecountstatisticsissolowthatonecannot dence on the energy channel. We investigated the reasons
use a simple χ2 minimizationto fit the expecteddistribution for the steeper WTD of the GBM set as follows: the dearth
of Eq. (8) to the counts collected in each bin. On the other of long WTs is likely due to the shorter scanned time inter-
side,mergingthebinssoastohaveenoughcountslosesinfor- vals, mostly from - 30 to 300 s (Sect. 2.1.3). We therefore
mationandresolution. Wethereforedevisedalog–likelihood truncated the light curves of the Swift/BAT set and revised
WaitingtimesinGRBs 5
Our results may be summarized in four fundamental as-
TABLE1
pects:
BESTFITPARAMETERSOFTHEMODELINEQ.(8)OBTAINEDFOR
DIFFERENTWTDS.SIZEISTHENUMBEROFWTS.
1. γ–raypeaksandX–rayflaresarecompatiblewithbeing
Sample Size α β PLindex CL differentaspectsofthesamestochasticprocess,which
(s) (=3- α) (%)
goesonaftertheendoftheGRBitselfandspansmore
BGBBBBBBAAAAAAABTTTTTTTM--StSSXXrEEEuzn13c24 161514554189838645315520569249 111...00011211.....4989763448445±±±+-+-+-+-+- 0000000000000..........0111110011...0009067566701455 66162161........5759723236392336+-+-+-+-+-+-+-+- 1011001100001100................29411162321152742844963839864022 111...22211788.....6120326566265±±±+-+-+-+-+- 0000000000000..........1011110011...0000976657610455 231755376686........24054356 2. rtfaatsihaohrncyearoymnprpbotrefieyot(rahvicsnkneee;tsroseohskarr,empridnnteuehecdldraesesinoefloas)fafeftaGmGtrnmeeRdRronasBrBtlgeobrsnneedwiaaigtinrulifei(tgfdzhq;eeaourQteniiinenolTsytntcsisflemreaonosneftms)d;tfWhraeeenaTqsocsauhtmhebneoeertttwhstwhteeoariectnhnthhotaγhausn–e-t
3. GRBs with several (≥8) γ–ray pulses are unlikely to
the WT selection accordingly. The results are reported in
exhibit X–ray flares after the end of the promptemis-
Table 1 as BATtrunc set, which includes 1445 WTs. Com-
sion;
pared with the original BAT set, the WTD of the truncated
data becomes steeper, from 2.06+- 00..1009 to 2.22+- 00..1165, i.e. com- 4. γ–raypeaksandX–rayflarestendtoclusterinmuchthe
patible with the GBM value within uncertainties. Hence we samewaythatsolarflares,energeticparticleevents,and
interprettheslightlysteepervalueoftheGBMsetastheresult CMEsdo,eventhoughtheprocessesmaybedifferent.
ofshortertimeprofileswhichdisfavorlongWTs.
The lognormal nature of the WTD originally claimed
Figure 3 displays the BAT-X set (squares) together with
(Li&Fenimore 1996) has recently been shown to be possi-
the corresponding best fit model. The power–law index is
1.66+0.07, i.e. compatiblewithBATSEsets withinuncertain- bly an artifact of the peak detection algorithms in the short
- 0.06 WT end (.1 s), where peaks significantly overlap and can
ties. Whatismore, mergingX–rayflaresdid notchangethe
hardlybeseparated(Baldeschi&Guidorzi2015). We found
stochasticnature exhibitedby the WTD, butextendedits dy-
that the long value (> few s) tail needs no more to be de-
namicalrange by at least one order of magnitudewith WTs
scribed as the sum of a lognormaltail and a power–law ex-
aslongas105s. Acommonstochasticmodelisfoundtowell
cess due to the presence of QTs, that were interpreted as
describe the WTD observed across more than five orders of
a different component (Nakar&Piran 2002; Quilliganetal.
magnitude.
2002; Drago&Pagliara 2007). This apparent diversity also
A similar result is obtained when one restricts to the
concernstheso–calledprecursors(Lazzati2005;Burlonetal.
known–redshiftsample BAT-Xz in the GRB rest–frame(cir-
cles in Fig. 3): here the power–law index is 1.55+0.11, i.e. 2008,2009;Charisietal.2014),whicharenothingbutemis-
- 0.10 sion periods that are less intense than the following activity
somewhat shallower. The rest–frame characteristic time is
from which they are separated by a quiescent time. Our re-
significantlyshorterbecauseofcosmicdilation,1.3sinstead
sults(1.and2.)showthatallkindsofWTs,includingprecur-
oftheobserver–framevaluesof6–7s.
sors, can be described within a common stochastic process,
We also searched for possible correlations between WTs
andthisholdsallthewayuptolateX–rayflares,thuspoint-
and peak intensities and between WTs and peak fluences of
ingtowardsacommonmechanism,whichkeepsonworking
adjacent pulses, but found none. Finally, we repeated the
duringandaftertheendofthepromptγ–rayemission,before
analysis for various subsets of GRBs, by requiring a mini-
theafterglowemissionduetotheinteractionwiththeexternal
mum numberof of peaksper GRB and found no significant
mediumtakesover.
difference.
Another question concerns the break at low values in the
WTD modeled in terms of the characteristic WT β: is it an
3.1. γ–raypeaksvs. X–rayflares
intrinsicpropertyorisitentirelyduetothelowefficiencyat
Figure 4 shows the distribution of the number of γ–ray short values of the peak detection algorithm? The capabil-
peaks per GRB for two different classes of GRBs, depend- ityofseparatingoverlappingstructuresdependsonanumber
ingonwhethertheirsubsequentX–rayemissioncontainsX– ofvariables,suchastheratiobetweenWT andpeakwidths,
ray flares. Surprisingly, it is found that almost all GRBs onintensities, andontemporalstructures. While thedropat
(23/25) with N ≥ 8 γ–ray peaks have no X–ray flares, al- ∆t . 0.5 s is certainly due to the algorithm efficiency, the
though the two groups have comparable size, 131 and 163 break itself modeled with β is more complex: β is shorter
GRBswithandwithoutflares,respectively.Thetwodistribu- at harder energies (Table 1). A given pulse has a narrower
tionsareunlikelytoshareacommonpopulationofevents: a temporalstructure at harder energies(Fenimoreetal. 1995),
Kolmogorov–Smirnov(KS) test yieldsa mere0.21%proba- whereasinthesoftestenergychannelsthereisaslow–varying
bility,i.e. theyaredifferentwith3.1σ(Gaussian)confidence. component (Vetereetal. 2006). The presence of such soft
Wevisuallyinspectedeachoftheseγ–andX–raylightcurves componentmighthinderthepeakidentificationinsomecase,
and found only one case of a flareless GRB, whose X–ray so we examinedthe lightcurvesin the harderchannels. Vi-
light curve exhibited some low–level flaring activity which sual inspection suggests that the paucity of subsecond WTs
didnotpassthe90%–confidencethresholdintheflaresample withrespecttothepower–lawextrapolationisrealandisun-
selection (Sect. 2.1.4). Therefore, GRBs with many pulses likelytobeamereartifactofthepeakidentificationprocess.
areunlikelytoexhibitflaresinthefollowingdecliningX–ray In addition, minimum pulse widths observed in GRB pro-
emission. filestypicallyareintherange0.1–1s(Fenimoreetal.1995;
Norrisetal. 1996; Marguttietal. 2011). The MEPSA effi-
4. DISCUSSION ciencyisabove10%forsuchpulsewidths,forWTs>0.5s,
6 Guidorzietal.
0
10
BAT-X
-1
10 BAT-Xz (rest-frame)
-2
10
-3
10
]
1
-s 10-4
[

)
t -5
10
D(
P
-6
10
-7
10
-8
10
-9
10
-1 0 1 2 3 4 5 6
10 10 10 10 10 10 10 10
D
t [s]
FIG.3.—WTDsofγ–raypulsesandX–rayflaresofBAT-X(squares)andofBAT-Xz(circles)setstogetherwiththeircorrespondingbestfitmodels.Errorbars
arethe(normalized)squarerootofcountsandarejustindicative.
ing disk being fragmented as the source of the stochastic
100
with X-ray flares process which is responsible for the prompt γ–ray emis-
no X-ray flares sion (Woosley 1993; MacFadyen&Woosley 1999) as well
as for the subsequent X–ray flare activity (Kingetal. 2005;
Pernaetal. 2006; Kumaretal. 2008; Cannizzo&Gehrels
2009;Gengetal. 2013). We brieflysummarizethe basic in-
y
nc 10 gredientsof themodel, whichare thenthoroughlydescribed
ue intheremainingpartofthisSection:
q
e
r • anumberoffragmentsareindependentlyaccretedwith
F
thesame,constant,probabilityperunittime;
• thenumberofavailablefragmentsisobviouslydecreas-
1 ingwithtime;thisnaturallyleadstoatime–dependent
Poisson process whose mean accretion rate decreases
withtime;
0 5 10 15 20 25 30 35 40 45
Number of g -ray peaks • atthebeginning,ifthemeanrateistoohigh(λ>1/β),
accretion becomes inefficient and is suppressed by a
FIG.4.—Distributionofnumberofγ–raypeaksperGRBfortwodistinct factorofexp(- λβ);
subsetsoftheSwiftBAT-Xsample,dependingonthepresence/lackofflares
inthefollowingX–rayemission. Almostall(23/25)GRBswithatleast8
• for some (∼30%) GRBs the reservoirof fragmentsis
γ–raypeakshavenoX–rayflares. AKStestyieldsacommonpopulation
probabilityof0.21%. splitintotwoseparategroupssharingthesameindivid-
ualaccretionprobabilityperunittime,butwiththesec-
andformeasuredsignaltonoiseratios>5(Guidorzi2015).
ondgroupbecomingavailableonlyatlatertimes(e.g.,
Itisthereforeunlikelythatthealgorithmefficiencyisentirely
thelategroupcouldbeidentifiedwiththeouterpartof
responsible for the observed exponential cutoff observed in
theaccretiondisk).
thelowendoftheWTDs.
Let us assume that the disk or the inner part of it has been
4.1. Asimpletoymodel splitintoanumberoffragments,eachofwhichhasthesame
We devised a very simple toy model to explore more givenprobabilityofaccretingperunittime,independentlyof
in detail how a time–dependent Poisson process like the theothers. Theprobabilityforagivenfragmenttosurviveup
one of Eq. (6) could be obtained in a GRB engine. For toagiventimet isproportionaltoexp(- t/τ),whereτ isthe
the sake of clarity, suppose each pulse marks the accre- mean accretion time for each fragment. The total expected
tion of a single fragment of an hyperaccreting disk. Actu- ratescalesasthenumberoffragmentsthatarestillavailable,
ally, the idea behind this model is more general and only λ=|N˙(t)|=(N0/τ)exp(- t/τ)=N(t)/τ. Theanalogous f(λ)
deals with the sequence of bursty emission episodes and toEq.(6)isfoundas
their probability of occurring within a given time; how-
dλ
ever, hereafter we refer to the model of an hyperaccret- f(λ)dλ ∝|dt|=τ , (9)
λ
WaitingtimesinGRBs 7
101 D t-2 10-1 BAT-X
Toy Model
-2
10
1] 100 10-3
-DP(t) [s 10-1 time -1t) [s] 1100--54
DP( -6
10
-2
10
-7
time 10
-8
10
-3
10
0.01 0.1 1 10 10-9
-1 0 1 2 3 4 5
D t [s] 10 10 10 10 10 10 10
D t [s]
FIG.5.—Seriesofexponential WTDs(thinsolid)ofasequenceofcon-
stantPoissonprocesseswithprogressivelyincreasinge–folding.Atinterme- FIG.6.— WTDfor the Swift BAT-Xsample (squares) compared with a
diatevaluesthemeanWTD(thicksolid)scalesasapower–lawwithindex2, simulatedsampleof903WTsderivedfromatoymodel(circles).Theshaded
shownforcomparison(dashed). intervaliswherethepeaksearchalgorithmefficiencydrops.
which correspondsto the α=1 case in Eq. (6) at λ≪1/β.
Ratherthanacontinuouslyvarying,λ(t)ofthismodelisde- accreting by some mechanism connected with the accretion
scribedmoreexactlybyapiecewisePoissonprocesslikethe rateitself. Forinstance,thisself–regulatingmechanismcould
one of Eqs. (2-3), where λ =i/τ is the expected rate when be driven by the magnetic field (e.g., Proga&Zhang 2006;
i
i fragmentsare left over. All terms have equal weightsφ’s, Uzdensky&MacFadyen2006;Bernardinietal.2013),which
i
since each piecewise constant process contributes one WT. isknowntohaveacomplexroleinaccretionprocessesofut-
TheresultingWTDisthusgivenbyEq.(2), terly different objects such as T Tauri stars (Stephensetal.
2014). However, we cannot provide a more specific and
1 N0 i physical justification for the exponential character of this
P(∆t) = e- i∆t/τ , (10)
self–quenchingmechanism, which is thereforead–hoc in its
N τ
0 X
i=1 presentformulation.
whichcanalsobeexpressedas, Weassumedthelogarithmicaverageand1-σscatterofthe
BAT-X WTD, 16.8 s and a multiplicative scatter of 7.1, to
x N0xN0+1- (N0+1)xN0+1 generatethevaluesforτ foreachsimulatedburst. Thenum-
P(x) = , (11)
(cid:2) N τ(1- x)2 (cid:3) ber of generated peaks in each simulated curve was taken
0
from the observed distribution and was augmented by 20%
where x=e- ∆t/τ. In Figure 5 an example of such WTD is toensurethatthedetectedpeakswereenough(sincesomeare
shown,withN =20initialfragments,τ =1s. Astimegoes missedbythealgorithm). Thepeaktimesforeachsimulated
0
by, λ(t) decreases and the e–folding of the individual expo- curvewererandomlygeneratedfromanexponentialdistribu-
nentialWTDs(thinsolid)increasecorrespondingly. Asare- tionwithe–foldingτ,i.e. independentlyfromeachother. To
sult, the total WTD (thick solid) show a power–law regime mimicthedropinthepeakdetectionefficiencyatshortWTs
with index 2 at intermediate values of ∆t. At ∆t .τ/N aswellasthemechanismmentionedaboveaboutthesuppres-
0
the total WTD is dominated by the initial exponential with sionathighrates,weoverlookedeachpeakoccurringwithin
e–folding τ/N , when fragments are all available. This 0.5 s of the previous one, while through a binomial we as-
0
agrees with the result that the intrinsic (i.e., corrected for signedeach∆t aprobabilityexp(- β/∆t)ofbeingobserved,
the algorithm efficiency) WTD at short values is likely ex- whereβwassettothefittedvalueoftherealWTD(Table1).
ponential,thatis,compatiblewithaconstantPoissonprocess To obtain a good match with the observed WTD over the
(Baldeschi&Guidorzi2015). same range we had to make a further assumption: we as-
Thusfar,withreferencetoEq.(6)ourmodelimplicitlyas- sumedthatforafractionofGRBs(∼30%)randomlyselected
sumed β =0 (see Eq. 9). However, in our attempt to repro- through a binomial, the disk is fragmented equally in two
ducetheobservedWTDwiththepiecewiseconstantprocess groups,thefirstofwhichisavailableforbeingaccretedfrom
ofEq. (10) failed to modelthe observedbreakat ∆t ∼1/β. thebeginning(t =0),whilethesecondonebecomesavailable
0
So we required that, when the expected rate becomes com- fromt =50τ on,whereτ isthecommonmeanaccretiontime
0
parable or higher than 1/β, the number of observed WTs for each individualfragmentfromt =t . The reason behind
0
is suppressed by a factor of exp(- λβ) with respect to our thisistheobservationoftwosimilarbunchesofγ–raypeaks
model. This can be interpreted as if, when the number N interspersedwithalongquiescenttime(uptoseveraltensof
offragmentsthatcanbereadilyaccretedissuchthattheex- seconds)forasmallfractionofGRBs. Physically,thiscould
pected rate is λ=N/τ &1/β, the overall process becomes betheresultofanouterpartofthediskbeingaccretedatlater
inefficient and the rate is suppressed by exp(- λβ). In other times with respect to the inner one or, more in general, de-
words, the number of WTs shorter than β is smaller than layedadditionalenergyreservoirbecomingavailableforlate
whatis expectedfromEq. (9). Thisintroducessome degree internaldissipation,withminimumvariabilitytimescalecom-
of memory in the initial stages of the accretion process: as parable with that of the early prompt emission (Fan&Wei
long as the number of fragmentsready to be accreted is too 2005; Lazzati&Perna 2007; Trojaetal. 2014). While the
high (τ/N .β) some of them are temporarily halted from choice of the fraction of such GRBs and the duration of the
8 Guidorzietal.
afterglowcontinuumwhichovershinespossibleX–rayflares,
80
Toy whichwouldthengounseen.
70 BAT-X Overall,weassumedadirectconnectionbetweenemission
and observed times of the peaks. Within the context of in-
60
ternal shocks the observed time profile of both prompt γ–
y 50 rays and late X–ray flares is strictly connected to the emis-
c
n
e sionhistory(Kobayashietal.1997;Maxham&Zhang2009).
u 40
q Shouldthis notbe the case, little could be inferredabouton
e
Fr 30 emissiontimes-andpotentiallythetimesofindividualaccre-
tionepisodes-fromthestudyoftheobservedWTD.However,
20
thisconnectionbecomesmorecomplexduetothevarietyof
10 Lorentz factors associated with the wind of shells colliding
with each other. Even though the intrinsic duration of the
0
10-3 10-2 10-1 100 101 102 103 GRBengineactivitymaydifferbyafactorofafewfromthe
D t /D t observedone(Gao&Mészáros2014),onaveragethetempo-
i+1 i ralsequenceofmutualcollisionsbetweenrandomly–assigned
Lorentz–factorshells shouldtrack theemission time history.
FIG.7.— Distribution of the ratio of adjacent WTsof the Swift BAT-X The nature of a given WTD is not altered as a whole when
sample(solid)andofthetoy–modelsample(shaded).
onepasses from the emission to the observedtimes: in fact,
quiescenceperiodaresomewhatarbitrary,thegoodmatchbe- eachshellhasaLorentzfactor-whichisinprinciplewhatcan
tweensimulatedandrealWTDdoesnotdependcruciallyon maketheobservedWTsverydifferentfromtheemittedones-
them. Overall, the goal here is just to show the plausibility thatis independentoftheWTs precedingandfollowingthat
oftheessentialpropertiesofthismodel,whichcanreproduce shell. Thisstatisticalindependenceensuresthattheobserved
theobservedpropertiesinspiteofthesimpleassumptions.We WTDkeepsmemoryontheemissiontimedistribution. Only
endedupwithasetof903simulatedWTs,whosedistribution atlate times, whenthe averageLorentzfactoris expectedto
iscomparedwiththerealoneinFig.6. systematically decrease and the statistical independentchar-
We further tested the consequences of this toy model by acterlikelybeginstofail, longWTsarelikelytobeaffected
studyingthe distributionof the ratio between adjacentWTs. asaconsequence.
WhileWTDsdescribehowWTsdistributeasawhole,losing
informationontheirtemporalsequence,theratiodistribution 4.2. Solaractivity: analogiesanddifferences
focuses on that. We therefore selected from the BAT-X as It is remarkable and intriguing that WTDs of both solar
wellasfromthesimulatedsampletheGRBswith≥3peaks, eruptiveevents(X–rayflares,radiostorms,high–energypar-
soastohaveatleasttwoWTs,andderivedthetwodistribu- ticle events, CMEs) and of GRBs can be modeled with the
tionsshowninFig.7. AKStestbetweenthetwosetsyields same kind of time-dependent Poisson process. The power–
43%probabilitythattheyweredrawnfromacommonpopu- lawcharacterizationoftheWTDheavytailmustnotbeover-
lation. Logarithmicmean and dispersion for the real (simu- interpretedfromamathematicalviewpoint,sincepower–laws
lated)dataareµ=0.14andσ=0.72(µ=0.09andσ=0.81). are,ingeneral,whatoneendsupwithwhendealingwiththe
Similar results are obtained adopting other non–parametric sumofindependentheavy–tailedvariables. Itworksmuchin
tests,suchastheWilcoxon–Mann–Whitneyorthemoresen- the same way thata normaldistributionis the final outcome
sitive Epps–Singleton one (Epps&Singleton 1986), respec- of the sum of independent finite–variance variables. In ad-
tivelyyielding45%and9%probability.Interestingly,simply dition, claiming that data are power–law distributed is con-
replacingEq.(9)withaconstantPoissonprocessandapply- trived whenever the explored range does not cover at least
ing the very same following steps, one ends up with a nar- two decades (Stumpf&Porter 2012). In this sense, invok-
rower and more zero-centeredlogarithmic ratio distribution, ing a SOC–driven mechanism for GRBs purely based on
µ= 0.013 and σ = 0.32, which is rejected with high confi- the power–law character of the WTD, and possibly of the
dence(p–value<2×10- 16)fromaKStest. Thismeansthat energy distribution too, as suggested for X–ray flares from
foraconstantprocesstheratiois,onaverage,closerto1,and GRBs (Wang&Dai 2013) or from otherblack hole systems
islessscatteredarounditthanrealdata. Thecompatibilityof (Wangetal.2014),isastretchedinterpretationofthedata,as
theratiodistributionpredictedbythetoymodelwiththereal weexplainbelow.
oneprovesthatthetemporalsequenceofWTsiscompatible Thesameorverysimilarpower–lawindicesimplythatboth
withanevolvingPoissonprocessandis incompatiblewitha processes have very similar degrees of clusterization, with
constant one on long timescales. In particular, X–ray flares analogousswings between intense and low–activity periods,
are nothing but some of the last fragmentsthat are left over apart from temporal rescaling (seconds for GRBs, hours for
andwhichareaccretedonlongtimescales,whentheratede- theSun). However,onehastobecarefulextendingthisanal-
creasesinagranularway,followingtheverysamestochastic ogytoacommonphysicalmechanism.Overall,thereisafun-
process ruling the accretion of the earliest ones. Hence, no damental difference in terms of dynamical systems between
correlationistobeexpectedbetweenγ–raypromptemission GRB inner engines during core collapse and the Sun: for
duration(T ) andX–rayflarestimes, inagreementwithob- the latter, the regions where eruptive phenomena take place
90
servations(Liangetal.2006). Finally,theresultofSect.3.1 continuously receive energy, which is then released through
can be easily explained: GRBs with many γ–ray peaks ac- avalanches,thusmakingtheSOCinterpretationplausible(al-
cretefragmentsrapidlywithrelativelyshortτ, sothatatlate though alternatives based on MHD turbulence seem equally
time very few or none at all are left over for X–ray flares. compatiblewithobservations).Instead,GRBenginesaresys-
Thesameresultcouldbeexplaineddifferentlythough:multi– tems which start with a very-far-from-equilibrium configu-
peaked GRBs could have on average a brighter early X–ray ration, evolve very fast using up all of the available energy,
WaitingtimesinGRBs 9
which-nomatterhowmuch-islimited. AGRBinnerengine unificationundera commonprocessof all differentkindsof
cannotreturntoitsoriginalconfiguration,itgoesthroughan waitingtimesinGRBs(shortinterpulsetimes,longquiescent
obviouslyirreversible evolution,whereasthisisnotthecase times, time intervalsfollowingprecursors,time intervalsbe-
for the solar active regionsoversufficiently longtimescales. tweenthe endofthe γ–raypromptemission andsubsequent
For this reason, one needs not invoke SOC mechanisms re- X–rayflares)isanewresult.
lated to accretion disks, in particular there is no need for a AnothernoteworthyresultisthatGRBswithmany(≥8)γ–
mechanismliketheoneproposedtoexplain1/f fluctuations raypulsesareunlikely(3.1σconfidenceinGaussianunits)to
inblackholepowerspectra(Mineshigeetal.1994). exhibitX–rayflaresintheirsubsequentearlyX–rayemission.
Therefore,asimpletime–varyingPoissonprocessexplains This resultis naturallyexplainedin the contextof the time–
thesecularevolutionofthemeanrateofbursts/flaresaswell varyingPoissonprocess:manypulsesobservedintheprompt
as the stochastic independentcharacter of individual energy of a given GRB are indicative of a relatively shortmean ac-
release episodes. This model disregards the physical origin cretion time for a single disk fragment. Consequently, most
offragmentationandhowenergyisdistributedamongdiffer- of the available fragments are consumed during the prompt
entfragments:thus,inprincipleitiscompatiblewithvarious phase,withveryfewornoneatallleftoverforthesubsequent
physicaldrivers, such as gravitational(Pernaetal. 2006), or phase.
magnetorotationalinstability(Proga&Zhang2006). In the lightof the irreversible evolutionof GRB inner en-
gines,theinterpretationofatime-varyingPoissonprocessap-
5. CONCLUSIONS pears to be simple and reasonable: the secular evolution of
In this paper we studied the waiting time distributions the expected rate of events is naturally linked to the energy
of GRB γ–ray pulses in three catalogs, CGRO/BATSE, reservoirbeinggraduallyusedup,whereasthestochastically
Fermi/GBM, and Swift/BAT. For the latter, for the first time independentaccretionofindividualfragmentsisexplainedby
we merged γ–ray pulses and X–ray flares detected with thePoissoniancharacteroftheprocess.
Swift/XRT belonging to the same GRBs, and for a subsam- Although self–organized criticality models naturally pre-
ple the same analysis was carried out in the source rest– dict power–law tailed distributions of waiting times and en-
frame. WefoundthatallWTDscanbedescribedintermsof ergy,drawinguponthiskindofdynamicsforGRBsmightbe
a common time–varyingPoisson process that rules different premature. Otherequallyplausiblealternatives,suchasfully
waiting time intervals, which thus far in the literature have developedMHDturbulence,canexplainthesameproperties,
been treated differently: specifically, we showed that short as it was suggested forthe solar case. Possible evidencefor
WTs (.1 s), long quiescent times (&10 s) all the way up turbulenceinGRBs hasalsobeensuggestedfromtheanaly-
to late time X–ray flares are the manifestationof a common sis of powerdensity spectra (Beloborodovetal. 1998, 2000;
stochasticprocess. GRBWTDsexhibitheavytailswhichare Guidorzietal.2012;Dichiaraetal.2013). Yet,wefindthata
modeledwith power–lawsover4–5decadesintime within- simpletime–varyingPoissonprocesssuchasthatofasystem
dices in the range1.7–2.1, dependingon the relative weight graduallyusingupalltheavailablepiecesalreadyprovidesa
of late time events, such as X–ray flares, in each GRB sam- remarkablyaccuratedescription.
ple. Because of the ubiquitous nature of power–laws (cen- The energy distribution, which was beyond the scope of
trallimittheoremforheavy–taileddistributions),thecharac- thispaper,willhelptofurtherconstrainthestochasticprocess
ter of WTDs must not be imbued with a mystical sense or andpossiblyclarify whethermorecomplexdynamicalmod-
overinterpreted as evidence for a universal process. In this els,suchasSOCorMHDturbulence,aretobeconsidered.
sense, the similarity of the WTD power–lawindex with that
ofsolareruptivephenomena,suchasflaresandcoronalmass
Wearegratefultotheanonymousrefereeforaconstructive
ejections, proves nothing but a similar degree of clusteriza-
andinsightfulreview. PRINMIURprojecton“GammaRay
tion in time. Nonetheless, it is remarkablethat the WTD of
Bursts: from progenitorsto physics of the prompt emission
γ–raypulsesandthatof X–rayflares notonlyhavecompat-
process”,P.I.F.Frontera(Prot. 2009ERC3HT)isacknowl-
iblepower–lawindicesbuttheyjoinandextendthedynami-
edged.
cal range for a commonsample of GRBs. All this pointsto
a common stochastic process ruling both phenomena. The
APPENDIX
LOG–LIKELIHOODTOFITTHEDISTRIBUTIONS
LettheWTDconsistofN logarithmicallyspacedbins,eachcollectingC counts. Let∆t and∆t bethelowerandupper
i i,1 i,2
bounds of the i–th bin (i=1,...,N). Integrating Eq. (8) within this time interval yields the corresponding expected counts,
E(α,β),whereweexplicitlymeantthatitdependsonthemodelparameters:
i
E(α,β) = C ∆ti,2P(∆t)d(∆t) = C β2- α (β+∆t )α- 2- (β+∆t )α- 2 (A1)
i tot Z∆ti,1 tot h i,1 i,2 i
whereC = N C,andβisafunctionofbothmodelparameters(Sect.2.2). TheprobabilityofC countsisruledbythePoisson
tot i=1 i i
distributionwPhereEiistheexpectedvalue,
ECi
P(C|α,β) = e- Ei i , (A2)
i C!
i
10 Guidorzietal.
wherethedependenceonmodelparametersisimplicitthroughE. Thetotalprobabilityisthus
i
N ECi
P(C|α,β) = e- Ei i , (A3)
C!
Y i
i=1
whereCisthesetofobservedcountsperbin{C}(i=1,...,N). Thecorrespondingnegativelog–likelihoodistherefore
i
N N
L(α,β) = - log(P(C|α,β)) = E +log(C!)- C logE . (A4)
i i i i i
X X(cid:16) (cid:17)
i=1 i=1
We determine the best fit model parametersand their uncertainties in the Bayesian context. The posterior probability density
functionoftheparametersforagivenobserveddistributionC,is(Bayestheorem)
P(C|α,β)P(α,β)
P(α,β|C)= , (A5)
P(C)
wherethefirstterminthenumeratoroftheright-handsideofeq.(A5)isthelikelihoodfunctionofEq.(A3),P(α,β)istheprior
distributionofthemodelparameters,andthedenominatoristhenormalizationterm. We assumedauniformpriordistribution,
sincenoaprioriinformationisavailableonthemodelparameters.ThemodeoftheposteriorprobabilityofEq.(A5)istherefore
foundbyminimizingEq.(A4).
We estimate the posterior density of the model parameters through a Markov Chain Monte Carlo (MCMC) algorithm such
as the random–walk Metropolis–Hastings in the implementation of the R9 package MHADAPTIVE10 (v.1.1-8). We initially
approximate the posterior using a bivariate normal distribution centred on the mode and with covariance matrix obtained by
minimizingEq.(A4). ForeachWTDwegenerate5.1×104setsofsimulatedmodelparametersandretainoneevery5MCMC
iterationsafter excludingthe first 1000. The remaining104 sets of parametersare thereforeused to approximatethe posterior
density. Finally,oncethe bestfit modelparametersaredetermined,the bivariateposteriordistributionof(α,β) is sampledvia
MCMCsimulations,whichyieldexpectedvalueand90%confidenceintervalsforeachofthem.
Asamatteroffact, sincethebinsinthelowendofdistributionarestronglyaffectedbythepoorefficiencyof MEPSA, these
aretobeignored.Inpractice,onehastoreplaceinEq.(A1)C withC′ = k2 C,wherek andk arethefirstandlastbinsto
tot tot i=k1 i 1 2
beconsidered.Inaddition,thesameE inEq.(A1)hastobefurtherdividedPbyarenormalizingfactorsothatitbecomes,
i
(β+∆t )α- 2- (β+∆t )α- 2
E(α,β) = C′ i,1 i,2 . (A6)
i tot (β+∆t )α- 2- (β+∆t )α- 2
k1,1 k2,2
FortheWTDsdiscussedinthepresentpaperweconsidered∆t≥0.5s(∆t≥0.2s)intheobserver(source)restframe.
ToassessthegoodnessofthefitforagivenWTD,weuseeachsetofsimulatedvaluesfor(α,β)togenerateasmanysynthetic
WTDs from the the posterior predictive distribution. Hence for a given observed WTD, we directly calculate 104 synthetic
realizations of the same WTD. For each of these WTDs we calculate the negative log–likelihood with Eq. (A4) and derive a
correspondingdistributionofvalues,againstwhichthevalueobtainedfromtherealWTDischecked. Thiscomparisondirectly
providesaconfidencelevelofmodellingtheobservedWTDintermsofthebestfitmodelofEq.(8).
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