Constraining the Structure of GRB Jets Through the log(N)-log(S) Distribution

0.23 MB·English

by  Dafne Guetta

DRAFTVERSIONFEBRUARY2,2008
PreprinttypesetusingLATEXstyleemulateapjv.3/25/03
CONSTRAININGTHESTRUCTUREOFGRBJETSTHROUGHTHElogN- logSDISTRIBUTION
DAFNEGUETTA1,2,JONATHANGRANOT3,4 ANDMITCHELLC.BEGELMAN2
DraftversionFebruary2,2008
ABSTRACT
A generalformalism is developedfor calculating the luminosityfunctionand the expectednumberN of ob-
servedGRBsaboveapeakphotonfluxSforanarbitraryGRBjetstructure. Thisformalismdirectlyprovides
thetrueGRBrateforanyjetmodel,insteadoffirstcalculatingtheGRBrateassumingisotropicemissionand
thenintroducinga‘correctionfactor’toaccountforeffectsoftheGRBjetstructure,aswasdoneinprevious
works. Weapplyittotheuniformjet(UJ)anduniversalstructuredjet(USJ)modelsforthestructureofGRB
5 jetsandperformfitstotheobservedlogN- logSdistributionfromtheGUSBADcatalogwhichcontains2204
0
BATSEbursts. We allowforascatterinthepeakluminosityLforagivenjethalf-openingangleθ (viewing
0 j
angleθ )intheUJ(USJ)model,whichisimpliedbyobservations.Acoreangleθ andanouteredgeatθ
2 obs c max
areintroducedforthestructuredjet,andafiniterangeofopeninganglesθ θ θ isassumedforthe
min j max
n uniformjets. Theefficiencyforproducingγ-rays,ǫ ,andtheenergypersolid≤angl≤einthejet,ǫ,areallowed
γ
a tovarywithθ (θ )intheUJ(USJ)model,ǫ θ- bandǫ θ- a. Wefindthatasinglepower-lawluminosity
J j obs γ ∝ ∝
functionprovidesa goodfit tothe data. Sucha luminosityfunctionarisesnaturallyin the USJmodel,while
7 intheUJmodelitimpliesa power-lawprobabilitydistributionforθ , P(θ ) θ- q. Thevalueofqcannotbe
directlydeterminedfromthefittotheobservedlogN- logSdistributjion,anjd∝anajdditionalassumptiononthe
2
valueofa orb is required. Alternatively,anindependentestimate of thetrue GRB ratewouldenableoneto
v
3 determinea,bandq. Theimpliedvaluesofθc (orθmin)andθmax areclosetothecurrentobservationallimits.
6 ThetrueGRBratefortheUSJmodelisfoundtobeRGRB(z=0)=0.86-+00..0154Gpc- 3yr- 1 (1σ)whilefortheUJ
0 modelitishigherbyafactor f(q)whichstronglydependsontheunknownvalueofq.
7 Subjectheadings:gammarays: bursts—gamma-rays: theory— cosmology: observations— ISM:jets and
0 outflows
4
0
/ 1. INTRODUCTION Kumar&Panaitescu 2000; Moderski,Sikora&Bulik 2000;
h
Panaitescu&Mt’eszt’aros 1999; Rhoads 1997, 1999;
p There are several lines of evidence in favor of jets in
Sari,Piran&Halpern 1999), where the energy per solid
- gamma-ray bursts (GRBs). For GRBs with known redshift
o angle ǫ and the initial Lorentz factor Γ are uniform within
z,thefluencecanbeusedtodeterminethetotalenergyoutput 0
r some finite half-opening angle θ , and sharply drop outside
st iEnγ-ratyhsatawsseurmeiinngfesrprhederiicnatlhsiysmwmayetrsyo,mEeγt,iimso.esThapepvraolaucehseodf, of θj, and (2) the universal stjructured jet (USJ) model
a γ,iso (Lipunov,Postnov&Prokhorov2001;Rossi,Lazzati&Rees
v: hanigdhiennoenrgeiecsasaere(vGeRryBh9a9rd10to23re)ceovnecnileexwceitehdperdoMge⊙nict2o.rmSuocdh- 2002; Zhang&Mészáros 2002), where ǫ and Γ0 vary
i smoothlywiththeangleθ fromthejetsymmetryaxis. Inthe
X els involving stellar mass objects. A non-spherical, colli-
UJ model the different values of the jet break time t in the
mated outflow (i.e. a jet) can significantly reduce the total j
r afterglowlightcurvearise mainlydueto differentθ (andto
a energyoutputinγ-rayscomparedtoEγ,iso,sinceinthiscase alesserextentduetodifferentambientdensities). InjtheUSJ
the γ-raysare emitted only into a small fraction, f 1, of
b ≪ model,allGRBjetsareintrinsicallyidentical,andthediffer-
the total solid angle. A more direct (and probably the best
ent values of t arise mainly due to differentviewing angles
so far) line of evidence in favor of jets in GRBs is from j
θ fromthe jetaxis[in fact, the expressionfort is similar
achromaticbreaksintheafterglowlightcurves(Rhoads1997, obs j
tothatforauniformjetwithǫ ǫ(θ )andθ θ ]. The
1999;Sari,Piran&Halpern1999). observed correlation t E-→1 (Boblosom,Fraji→l&oKbsulkarni
Despite the large progress in GRB research since the dis- j ∝ γ,iso
coveryoftheafterglowemissioninearly1997,the structure 2003; Frailetal. 2001) implies a roughly constant true
of GRB jets is still an open question. This is a particularly energy E between different GRB jets in the UJ model, and
interesting and important question, since it affects the total ǫ θ- 2 outside of some core angle θc in the USJ model
energyoutputandeventrateofGRBs,aswellastherequire- (R∝ossi,Lazzati&Rees 2002; Zhang&Mészáros 2002).5
mentsfromthecentralenginethatacceleratesandcollimates The probability distribution of jet half-opening angles be-
theserelativisticjets. tween different GRBs in the UJ model, P(θj), is not given
The leading models for the jet structure are: (1) a-prioribythemodel,andisafreefunctiontobedetermined
the uniform jet (UJ) model (Granotetal. 2001, 2002; byobservations.
Severalmethodshavebeenusedsofarinordertoconstrain
1 RacahInstituteforPhysics,TheHebrewUniversity, Jerusalem91904, thestructureofGRBjetsandhelpdistinguishbetweentheUJ
Israel;[email protected] model and the USJ model. The afterglow light curves are
2 JILA, 440 UCB, University of Colorado, Boulder, CO 80309; similar for the UJ and USJ models, but nevertheless some
[email protected]
3InstituteforAdvancedStudy,OldenLane,Princeton,NJ08540
5 Thelatterisobtainedassumingthattheefficiencyinproducingγ-rays,
4 KIPAC, Stanford University, P.O. Box 20450, MS29, Stanford, CA
94309;[email protected] ǫγ,doesnotdependonθ. Welaterexaminetheconsequences ofrelaxing
thisassumption.
2
differences still exist which might help distinguish between θ θ θ , andinanalogytotheUSJmodelwechose
min j max
them(Granot&Kumar2003;Kumar&Granot2003). After- ǫγ(θj≤)=Θ≤(θmax- θj)Θ(θj- θmin)ǫγ,0(θj/θmin)- bUJ.
glow light curves also constrain the jet structure in the USJ Theenergyper solid angle is also assumed to behaveas a
model,asdiscussedin§3. Thismethodrequiresaverygood powerlaw,ǫ(θ)=Θ(θmax- θ)ǫ0min[1,(θ/θc)- aUSJ]intheUSJ
anddensemonitoringoftheafterglowlightcurves,especially model and ǫ(θ)= Θ(θmax- θj)Θ(θj- θmin)ǫ0 (θj/θmin)- aUJ in
nearthejetbreaktimetj. Anotherpossiblewaytodistinguish theUJmodel. Thepower-lawindexesaandbcanbediffer-
betweenthesetwomodelsforthejetstructureisthroughthe entbetweentheUSJandUJmodels(asisemphasizedbythe
differentexpectedevolutionofthelinear polarizationduring differentsubscriptforthetwomodels),andwealsoconsider
the afterglow(Rossietal. 2004). However,some difficulties differentvaluesforaandbwithineachmodel.
andcomplicationsexistin thismethod, like the poorquality Theexternalmassdensityistakentobeapower-lawinthe
ofmostpolarizationlightcurves,andapossibleorderedmag- distance R from the central source, ρ =AR- k. For a con-
ext
neticfieldcomponentthatmightcausepolarizationthatisnot stantefficiencyǫ (b=0)theobservedcorrelationt E- 1
relatedtothejetstructure(Granot&Königl2003). Thedis- (Bloom,Frail&Kγulkarni2003;Frailetal.2001)imjp∝liesγa,is=o
tribution of viewing angles θobs, inferred from the observed 2(Rossi,Lazzati&Rees2002;Zhang&Mészáros2002). In
valuesoftj,hasbeenusedtoargueinfavoroftheUSJmodel thefollowing,weallowbtovaryandfindthejointconstraint
(Perna,Sari&Frail 2003). However, when the known red- thattheFrailetal.(2001)relationputsonaandb.
shifts of the same sample of GRBs are also taken into ac- TheafterglowemissionbecomesprominentwhentheGRB
count,thereisaverypooragreementwiththepredictionsof ejecta sweeps enough of the external medium to decelerate
the USJ model (Nakar,Granot&Guetta 2004). In order to significantly. After this time most of the energy is in the
reachstrongconclusionsusing thismethod, a largeanduni- shocked external medium, and energy conservation implies
formsampleofGRBswithknownredshiftsisneeded,likethe
oneexpectedfromSwift. ǫ Γ2µc2=Γ2Ac2R3- k/(3- k), (1)
InthispaperweusetheobservedlogN- logS distribution ≈
(the number N of GRBs above a limiting peak photon flux whereµistheswept-uprestmasspersolidangle.
S) in order to constrain the jet structure. A somewhat sim- In the UJ model Γ(tj) 1/θj (Rhoads 1997;
∼
ilar analysis was done by Guetta,Piran&Waxman (2004). Sari,Piran&Halpern 1999) while in the USJ model
They found that the observed logN- logS distribution rules Γ(tj) 1/θobs (Rossi, Lazzati & Rees 2002; Zhang &
∼
out the USJ model, due to the paucity of GRBs with small Mészáros2002)and the emission aroundtj is dominatedby
peak fluxes compared to the prediction of the USJ model, materialclosetoourlineofsight. Togetherwiththerelation
and fit a double power-law luminosity function for the UJ R 4Γ2ct,wehave
∼
model. Firmanietal. (2004) tried to constrain the redshift 1 (3- k)ǫ(θ) 1/(3- k)
evolutionoftheGRBrateandluminosityfunction,usingthe t θ2(4- k)/(3- k) θ[2(4- k)- a]/(3- k) , (2)
logN- logS distribution (as well as the redshift distribution j≈ 4c Ac2 ∝
(cid:20) (cid:21)
derivedfromtheluminosity-variabilityrelation).
where θ = θ for the UJ model and θ = θ for the USJ
j obs
This paper improves upon previous works by: (1) allow-
model. Now, in order to satisfy the observed correlation6,
ingtheefficiencyinproducingγ-raystobeafunctionofthe t E- 1 θa+b,wemusthave
angle θ, ǫγ =ǫγ(θ), and varying ǫ(θ) accordingly, using the j∝ γ,iso∝
Frailetal.(2001)relation,(2)includinganinternaldispersion 2(4- k)- a
inthepeakisotropicequivalentluminosityLatanygivenan- λ≡a+b= 3- k . (3)
gleθ,(3)introducinganinnercoreangleθ andanouteredge
c
Thereforeifweallowtheefficiencyandtheenergypersolid
θ in the USJ model, (4) using a largerand more uniform
max angle to vary in this way, we obtain the condition b=(2-
GRBsample.
a)(4- k)/(3- k)ora=2- b(3- k)/(4- k)inordertoreproduce
In §2 we allow ǫ and ǫ to vary with θ, and derive the
γ
theobservedcorrelationofFrailetal.(2001).
constraintsonthe power-lawindexesthatare impliedby the
For the USJ model, interesting constraints on the power-
Frailetal.(2001)relation. Theformalismforcalculatingthe
lawindexa oftheenergypersolidanglehavebeenderived
luminosityfunctionandtheobservedGRBrateasafunction USJ
fromtheshapeoftheafterglowlightcurve(Granot&Kumar
ofthelimitingfluxfordifferentjetstructuresisderivedin§3.
In§4and§5wecomparetheobservedlogN- logSdistribu- 2003; Kumar&Granot2003). For aUSJ.1.5 the changein
thetemporaldecayindexacrossthejetbreakistoosmallcom-
tiontothepredictionsoftheUSJandUJmodels,respectively.
pared to observations, while for a & 2.5 there is a very
Ourresultsarediscussedin§6. USJ
pronouncedflatteningofthelightcurvebeforethejetbreak,
2. THEENERGYANDGAMMA-RAYEFFICIENCYDISTRIBUTIONS whichisnotobserved.Thelatterfeaturearisessincetheinner
The efficiency for producing γ-rays, ǫ , is taken to be a partsof the jet near the core, where ǫ is the largest, become
γ
function of the angle θ from the jet symmetry axis in the visible. This explains why this feature becomes more pro-
USJ model, and a function of the jet half-opening angle θj nouncedfor aUSJ >2, where mostof the energyin the jetis
in the UJ model. For convenience we assume a power- concentratedatsmallangles,nearthecore. Thereisnosharp
law dependence on θ in the USJ model, ǫ (θ) = Θ(θ - borderline where the light curves change abruptly. Instead,
γ max
θ)ǫγ,0min[1,(θ/θc)- bUSJ], where Θ(x) is the Heaviside step ttohgeelitghhert,ctuhrevoesbscehravnegdesshmapoeotohflythweiathftethrgelpoawralmigehttecruarUvSeJ.cAonl--
function. Anouteredgeatθ hasbeenintroduced,aswell
asacoreangle,θ &1/Γ max5 10- 4rad,whichisneeded strainstheparameteraUSJtobeintherange1.5.aUSJ.2.5.
c max
inordertoavoidadivergenc∼eat×θ=0. HereΓ 2000is
the maximum value of the Lorentz factor to wmhaixch∼the fire- 6WehaveEγ- 1,iso∝θa+bsinceEγ,iso(θ)=4πǫ(θ)ǫγ(θ).Thepeakisotropic
ballcanbeaccelerated(Guetta,Spada&Waxman2001). For equivalentluminosityisgivenbyL=Eγ,iso/T whereT istheeffectivedura-
tionoftheGRB,whichisassumedtobeuncorrelatedwithLsothatLhasthe
the UJ model P(θj) is restricted to a finite range of values, sameθ-dependenceasEγ,iso
3
Itisimportanttostressthatthisconstraintappliesonlytothe universal structured jet model where7 P(Lθ) = δ[L- L(θ)]
USJmodelandnottotheUJmodel,sinceintheformeraUSJ and L(θ) = L0(θ/θ0)- λ. Using Eq. (4) this| implies P(L) =
determinesthestructureofanindividualjet(andthereforeaf- (θ /λL )(L/L )- 1- 1/λsinθ(L),whereθ(L)=θ (L/L )- 1/λ,and
0 0 0 0 0
fects the light curves) while in the latter the structure of an forθ(L) 1itreducesto P(L) (θ2/λL )(L/L )- 1- 2/λ. Al-
individual jet is fixed and aUJ only affects how the uniform ternative≪ly,onemayassume8P(L≈)=C0 L- η0,wher0eη=1+2/λ
ǫchangesbetweendifferentjetsofdifferenthalf-openingan- 0
glesθj. andC0∼λ- 1θ02L0η- 1 isdeterminedbythenormalizationcon-
One might also try to constrain the power-law index b of dition, P(L)dL=1.
thegamma-rayefficiencyǫ fromobservations. Thiswould, AnimportantpointtostresshereisthatP(L)representsthe
γ
of course, apply to both the USJ model and the UJ model. averageRover all possible viewing angles. Even if the jet is
The way in which this has been done so far is by taking assumedtohaveasharpouteredgeatsomefiniteangleθmax,
ǫγ =Eγ,iso/(Eγ,iso+Ek,iso) where Ek,iso is the initial value of with some probability distribution P∗(L) for θ <θmax, such
the isotropic equivalent kinetic energy. However, it is hard that P∗(L)=(1- cosθ )- 1 θmaxdθsinθP(Lθ) and P(Lθ >
toevaluateEk,iso veryaccurately. Itisusuallyevaluatedfrom θj)=δ(L),thenP(L)=m(a1x- cos0θmax)P∗(L)+c|osθmaxδ(L).|
afterglow observations several hours to days after the GRB, If the GRB jet structure iRs not universal, and P(Lθ) de-
andcanprovideanestimateaccuratetowithinafactorof 2 pends on additional parameters that describe the jet|struc-
∼
or so for the kinetic energy at that time. An additional and ture, then Eq. (4) needs to be averaged over these parame-
potentiallylargeruncertaintyarisessinceatearlytimesthere tersin ordertoobtaintheGRB luminosityfunction. Forex-
isfastcoolingandradiativelossesmightreducetheinitialki- ample, for a uniform jet of half-opening angle θ , we have
j
neticenergybyuptoanorderofmagnitudeorso. Thiscanbe P(Lθ,θ )=Θ(θ - θ)P∗(Lθ )+Θ(θ- θ )δ(L)whereP∗(Lθ )
j j j j j
takenintoaccount,butintroducesanadditionaluncertaintyin isth|eprobabilitydistributi|onofLinsidethejet(atθ<θ )|for
j
thevalueofEk,isothatisestimatedinthisway. a given value of θj. In this case, if P(θj) is the probability
Panaitescu&Kumar (2001, 2002) evaluated Ek,iso from a distributionforθj,thentheGRBluminosityfunctionisgiven
fittothebroadbandafterglowdatafortendifferentGRBsand by
obtained ǫ &0.5 for most of these GRBs and ǫ &0.1 for
γ γ π/2 π/2
allofthem. Thereseemstobenoparticularcorrelationwith P(L)= P(θ )dθ P(θ)dθP(Lθ,θ )
j j j
θ, but due to the small number of bursts and the reasonably Z0 Z0 |
large uncertainty in Ek,iso there might still be some intrinsic π/2
correlation. Lloyd-Ronning&Zhang (2004) used the X-ray = P(θ )dθ (1- cosθ )P∗(Lθ )+cosθ δ(L) .(5)
j j j j j
|
luminosityat10hrtoestimatethekineticenergyatthattime Z0
andusedasimpleanalyticexpression(Sari1997)toaccount InanalogywiththeUSJ(cid:2)model,weassumefortheUJm(cid:3)odel
fortheradiativelosses. Inthisway theyestimatedǫγ for17 that ǫ=ǫ0(θj/θmin)- aUJ and ǫγ =ǫγ,0(θj/θmin)- bUJ. Note that
GRBs and one X-ray flash (XRF 020903). Their Figure 6 sinceǫ=Eiso/4π≈E/2πθ2j thisimpliesthatthe trueenergy
suggests that ǫγ decreases with θ, although they claim that is not necessarily constant, E θ2- aUJ. In order to imitate a
thisisnotstatisticallysignificant.Sincemostoftheestimates ∝ j
structuredjetwithauniformcore,wechose
forǫγ are&0.5andareforsmallvaluesofθ,andsinceǫγ<1 P(θ )=Θ(θ - θ )Θ(θ - θ )C(q)θ- q+B(q)δ(θ - θ ),
by definition, this suggests b&0 (otherwise we would have j j min max j j j min
(6)
ǫ >1 at large θ, which is impossible). Moreover,Figure 6
owγfeLalsosyudm-Reotnhnatin0g.&bZ.ha1n,gt(h2e0n0E4)qs.u(g3g)ewstosuthldatim0p.lyb1..215..If w[1h-erBe(C1)(]q/)l=n([θ1m-axB/(θqm)]in()1f-roqm)/(tθhm1e-aqxn-orθmm1-ianql)izfoatrioqn6=1Pan(θdjC)d(θ1)j==
a.2,2.λ.2.25fork=0,and1.5.a.2, 2.λ.2.5 1.ForP∗(Lθj)=δ[L- L(θj)]whereL(θj)=Lmin(θj/θmax)- λUJ
fork=2. andL =L|(θ ),thisimplies R
min max
Cθ1- q
3. LUMINOSITYFUNCTIONFORDIFFERENTJETSTRUCTURES P(L)=Θ(L- L )Θ(L - L) max [1- cosθ (L)]
min max j
λ L
Forsimplicity,weassumethattheemissionfromtheGRB UJ min
jet is axially symmetric, so that the observedluminosity de- L - 1- (1- q)/λUJ
+B(1- cosθ )δ(L- L )
pendsonlyontheviewingangleθfromthejetaxis,anddoes × L min max
notdependon theazimuthalangleφ. Itisconvenientto de- (cid:18) min(cid:19)
fiprnoebtahbeilpitryobfaobrialintyisdoistrtoripbiuctieoqnuPiv(aLl|eθn)t,lwuhmeirneoPsi(tLy|θb)edtwLeisenthLe + Bcosθmin+C θmaxdθθ- qcosθ δ(L)
andL+dL,whenviewingthejetfromanangleθ. Naturally, " Zθmin #
wemusthave P(Lθ)dL=1foranyvalueofθ. Theproba- Cθ3- q L - 1- (3- q)/λUJ
bilityofviewingthe|jetfromananglebetweenθandθ+dθis Θ(L- Lmin)Θ(Lmax- L) max
≈ 2λ L L
R π/2 UJ min(cid:18) min(cid:19)
simplyP(θ)dθ=sinθdθ,and 0 P(θ)dθ=1.Averagingover θ2
theviewingangleθweobtaintheoverallprobabilitydistribu- +B minδ(L- L )
max
tionforL, R 2
θ2 C(θ3- q- θ3- q)
π/2 1 + 1- B min- max min δ(L), (7)
P(L)= P(θ)P(Lθ)dθ= P(Lθ)dcosθ. (4) " 2 2 (3- q) #
| |
Z0 Z0
For a universal GRB jet structure, i.e., if all GRB jets have 7Thisisthestandarduniversalstructuredjetmodel(Rossi,Lazzati&Rees
2002;Zhang&Mészáros2002)forλ=2,andassumingnoscatterinLfora
thesameintrinsicproperties,thenEq.(4)representstheGRB
givenθ.
luminosityfunction. 8ThisisapurepowerlawinLforallθ,insteadofjustforθ≪1aswe
As a simple and very useful example, let us consider a hadbefore.
4
termsinEq. (10)fortheUSJmodelbyafactorof
P∗(L)=Θ(L- Lmin)Θ(Lmax- L)λCUθJLm1-maqxin(cid:18)LmLin(cid:19)- 1- (1- q)/λUJ f(q)=(cid:18)θθmmainx(cid:19)1- q+((31-- qq))"1- (cid:18)θθmmainx(cid:19)1- q#
+Bδ(L- Lmax), (8) (3- q)/(1- q) (q<1),
whIenrethθej(cLa)s=eθomfaxth(Le/sLtmruinc)t-u1r/eλdUJ.jet discussed above, with a ≈([2/(q- 1)](θ /θ )q- 1 (q>1), (14)
max min
sharpouteredgeatsomefiniteangleθ ,andauniformcore
max where f(1)=1+2ln(θ /θ ). Thisfactor f isthe ratioof
withinsomeangleθ ,wehave max min
c thetrueGRBratefortheUJmodeltothetrueGRBrateforthe
P(Lθ)=Θ(θ - θ)δ(L- L )+Θ(θ- θ )δ(L) USJmodelthatcorrespondstothesameluminosityfunction
c max max
|
+Θ(θ- θ )Θ(θ - θ)δ[L- L(θ)], (9) (i.e. Eqs. 7and10). Figure1 shows f(q)forthreedifferent
c max
valuesofθ /θ . Thefactthat f(q)>1meansthatalarger
whereL(θ)=Lmin(θ/θmax)- λUSJ andLmin=L(θmax).Therefore, numberofmunaxiformminjetsisneeded,comparedtostructuredjets,
in order to reproduce the same luminosity function and the
θ sinθ(L) L - 1- 1/λUSJ ˙
P(L)=Θ(Lmax- L)Θ(L- Lmin) mλaUxSJLmin (cid:18)Lmin(cid:19) ssaammeeNo˙bGsReBrvtheedirnattreinosficGGRRBBs,rNatGeRpBe.rTuhniistcmoemaonvsitnhgatvofolurmthee,
+(1- cosθc)δ(L- Lmax)+δ(L)cosθmax RGRB(z),andspecificallyRGRB(z=0),islargerbyafactorof
θ2 L - 1- 2/λUSJ f fortheUJmodelcomparedtotheUSJmodel.
Θ(L - L)Θ(L- L ) max (10) Thesamefactor f isobtainedfromtheratiooftheenergy
max min
≈ λUSJLmin(cid:18)Lmin(cid:19) outputingamma-raysofasinglestructuredjettotheaverage
θ2 θ2 energyoutputingamma-raysofauniformjet. Thisratiomust
+ cδ(L- Lmax)+ 1- max δ(L), equaltheratiooftheintrinsicratessincethetotalenergyout-
2 2
(cid:18) (cid:19) putingamma-raysperunittimeperunitvolumemustbethe
Θ(L - L)Θ(L- L ) L - 1- 1/λUSJ same(Guetta,Piran&Waxman2004).
P∗(L)= (1- comsaθxmax)θm- 1axλUSmJLinminsinθ(L)(cid:18)Lmin(cid:19) forWqe>ca1n.seTehitshactanf b∼e1unfdoerrqsto<od1aasndfolflo∼ws(.θmSaixn/cθemimn)oq-s1t
(1- cosθ )
+ c δ(L- L ) ofthesolidangleofthestructuredjetis nearθmax, the num-
(1- cosθmax) max ber of uniform jets with θj θmax should be comparable to
∼
2 L - 1- 2/λUSJ the number of structured jets. Therefore, for q <1 where
Θ(L - L)Θ(L- L ) mostoftheuniformjetshaveθ θ wehave f 1,while
≈ max min λUSJLmin(cid:18)Lmin(cid:19) for q > 1 where most of the ju∼nifomramx jets have ∼θj θmin,
+ θθc 2δ(L- Lmax), (11) jfet∼sw(θitmhaxθ/jθminθ)mqa-x1.is roughly the inverse of the frac∼tion of
(cid:18) max(cid:19) We have ∼demonstrated that the luminosity function of the
swhhoeurledθb(eL.)N=oθtemtahx(aLt/thLemcino)-e1f/fiλcUiSeJnatnindthLe∞mifinPrs∗t(tLe)rdmLin=t1heasexi-t UprSiaJtemcohdoeilcceaonfbPe(iθmj)i.tatTehdebiymtphleieUdJtrmueodGelRwBitrhatteheRaGpRpBr(oz)-
R ˙
pressionforP(L)includesθmax onlythroughthecombination forthesameobservedGRBrateNGRBwouldalwaysbelarger
θ L1/λ=θ [L(θ )]1/λ,whichisindependentofθ (for (byafactor f)fortheUJmodel. FortheUJmodel,theterm
amfiaxxedminnormmaalxizatiomnaxofL)sinceL(θ) θ- λ. max Bδ(θj- θmin)inP(θj)whichproducestheterm∝δ(L- Lmax)in
∝ P(L)issomewhatartificial,andwasintroducedonlytomake
Bycomparingequations(7)and(10)onecanalsoseethata
similarluminosityfunctioncanbeobtainedfordifferentval-
uesofλ=a+b,aslongasλ /λ =(3- q)/2. Inthiscase,
UJ USJ 106
whenq=1thenλUJ=λUSJ. Forthesameθmax,Lmin andLmax 106
pwleiehsaθvce/6 Lθmmianx/=L(mθimni=n/6(θθmmaaxx)/(1θ-mqi)n/)2λ(UwJ =he(θremaθxc/θ<c)θλmUSinJ wfohricqh<im1- 105 110045 qq mmaaxx//qq mmiinn==110010.5
andθc>θminforq>1).Inorderfortheluminosityfunctions f103 q max/q min=10
tobethesamealsoforthecoreofthestructuredjet,whichis 104 102
representedbytheterm δ(L- L ),theratioofthistermto 101
max
theothertermsshouldb∝ethesameforequations(7)and(10). f103 10100−1 100
Thusweobtain q
θ1- q θ1- q- θ1- q - 1 102
C(q)= min + max min , (12)
"(3- q) (1- q) # 101
B(q)= 1+(3- q) θmax 1- q- 1 - 1 , (13) 10−02 −1 0 1 2 3 4
(1- q) θ q
( "(cid:18) min(cid:19) #)
whereB(1)=C(1)/2=[1+2ln(θ /θ )]- 1andC(q)/B(q)= FIG. 1.— Theratio f (Eq. 14)ofthetrueGRBratefortheUJmodelto
max min thetrueGRBratefortheUSJmodel,giventhesameobservedGRBrateand
(3- q)θq- 1. luminosityfunction (i.e., thesamelogN- logSdistribution), asafunction
min
Thenormalizationof thefirst twotermsin Eq. (7) forthe oftheslopeqoftheprobabilitydistributionfortheopeningangleθj ofthe
UJmodelissmallerthanthatofthe(corresponding)firsttwo uniformjet,P(θj)∝θ-jqforθmin<θj<θmax.Seetextfordetails.
5
a completeanalogywith P(L)forthe USJmodel. Theexact Theluminosityfunctioninthiscasecanbeobtainedusing
formof the cutoffnear θ is notclear and probablywould equations(4)and(15). Itisimportanttonotethattheintegral
min
not have a very large effect on the logN- logS distribution, inEq.(4)mustbedoneoverallpossibleviewingangles,even
makingithardtodistinguishbetweenthetwomodelsinthis ifthejetisassumedtohaveacoreangle,θ ,andmaximalan-
c
way.Itisalsoimportanttokeepinmindthattheobservational gle,θ . Infact, togetherwiththepower-lawindex(λ)and
max
constraintsontheparameterafromtheshapeoftheafterglow normalization(θλL ), theothertwo intrinsic parametersthat
0 0
lightcurve, namely1.5.a .2.5, applyonly to the USJ defineapower-lawuniversalstructuredjetaretheanglesofits
USJ
modelandnottotheUJmodel. outeredge(θ )andinnercore(θ ). FortheUJmodelout-
max c
Theformoftheluminosityfunctiongiveninequations(7) linedabove,θ andθ arereplacedbyθ andθ which
or(10)isvalidonlyifθλL(θ) const,whereθ=θ forthe determinethemraaxngeofcpossibleθ values,mwaxhereP(mθin) θ- q
∼ obs j j ∝ j
USJmodelandθ=θj fortheUJmodel. Howwellthiscon- alsodependsonanadditionalparameterq.
dition is satisfied may be tested by using the values of both TheobservedrateofGRBs(overtheentiresky)withapeak
L and θ that were estimated from observations for a sam- photonfluxgreaterthanSisgivenby:
ple of GRBs. We considera sub-sampleof 19 GRBs out of
the Bloom,Frail&Kulkarni (2003) sample, for which there N˙ (>S)= P(L)dL zmax(L,S)RGRB(z)dV(z)dz, (16)
is both a known redshift and an estimate for θ. There are GRB (1+z) dz
Z Z0
larger uncertainties in the determination of the peak lumi- whereP(L)isgivenbyequations(4)and(15),zistheredshift
nosity,9 as the bursts were detected by different instruments of the GRB, z (L,S) is the maximalredshiftfromwhich a
max
with different temporal and spectral sensitivities. Follow- GRBwithluminosityLandpeakfluxScanbedetected,and
ingGuetta,Piran&Waxman(2004)wehaveextrapolatedthe R (z)isthe(true)GRBrateperunitcomovingvolumeV.11
GRBfluxestotheBATSErange(50keV- 300keV)usingthe ThGeRBfactor(1+z)- 1accountsforthecosmologicaltimedilation
methoddescribedinSethi&Bhargavi(2001).Inouranalysis anddV(z)/dzisthecomovingvolumeelement.
wehaveusedthemedianvalueofthespectralphotonindexin
the 50keV- 300keV bandfor the longburstssample, - 1.6, 4. CONSTRAINTSONTHELUMINOSITYFUNCTIONOFTHEUSJ
as found by Schmidt (2001). The redshifts and fluxes were MODEL
takenfromthetablegiveninVanPutten&Regimbau(2003). We consider all the GRBs in the GUSBAD catalog
We calculate the values of θλL for such a sample and for (Schmidt 2004), which lists 2204 GRBs detected at a time
different values of λ within the range allowed for the USJ scale of 1024ms. This catalog contains all the long GRBs
modelfromtheconstraintsgivenabove(namely,1.5.aUSJ. (T90 > 2 sec; Kouveliotouetal. 1993), detected while the
2.5andEq. 3): 1.83.λ.2.17fork=0,and1.5.λ.2.5 BATSEon-boardtrigger(Paciesasetal.1999)wassetfor5.5
fork=2. Thereforeoverall1.5<λ<2.5. Theresultofthis σ over background in at least two detectors, in the energy
analysisis that the distributionof θλL(θ) is notquite a delta range 50- 300 keV. Using this sample we estimate the ra-
function,andthereissomedispersionaroundthemeanvalue. tioC /C for each burst, whereC is the countrate in
max min max
Thisdispersionisreasonablyfitbyalog-normaldistribution, thesecondbrightestilluminateddetectorandC isthemin-
min
imumdetectablerate. Wefind V/V =0.335 0.007.
max
1 [lnL- lnL(θ)]2 We consider also the Rowhan-Robiinson SFR±(RR-SFR;
P(Lθ)= exp - , (15)
| Lσ√2π× ( 2σ2 ) Rowan-Robinson1999)thatcanbefittedwiththeexpression
100.75z z<1,
with L(θ)=L0(θ/θ0)- λ, where θ0λL0 is the average value of RGRB(z)=ρ0× 100.75 z>1, (17)
θλL(θ)fortheobservedsample,andσ isthestandarddevia- (cid:26)
whereρ =R (z=0)isthetrueGRBrateperunitcomoving
tionofln[θλL(θ)],whichisdeterminedbyafittothedisper- 0 GRB
volume at z=0. Throughoutthe paper we use the following
sion of the observedsample. This distribution approachesa
deltafunction(i.e. withnoscatter)forσ 0. cosmologicalparameters:(ΩM,ΩΛ,h)=(0.3,0.7,0.7).
We performed fits of θλL to a sampl→e of 19 bursts from ObjectswithluminosityLobservedbyBATSEwithaflux
limit S are detectable to a maximumredshiftz (L,S ).
Bloom,Frail&Kulkarni (2003) which were observed by lim max lim
The limiting flux has a distribution G(S ) that can be ob-
BATSE and Beppo-SAX. For λ = (1.5,2,2.5,3) we ob- lim
tained θλL = (15,5.8,2.2,1.1) 1049 ergs- 1 and σ = tained from the distribution of Cmin of the GUSBAD cata-
0 0 × obs log. Considering five main representative intervals we ob-
(1.2,1.2,1.3,1.5). Thesevaluesofσ canbethoughtofas
obs tainthat30%,30%,10%,20%,and10%ofthesamplehave
upperlimitsonσ(i.e. σ.σobs),sincetheobservedscatterin S 0.23, 0.25, 0.22, 0.26, and 0.27phcm- 2s- 1, respec-
θλL(θ)includesboththeintrinsicscatter(whichshouldbere- lim ∼
tively.Therefore,wehave
flectedinσ)andadditionalscatterduetomeasurementerrors
˙
inθ(whichproducessomescatterσerr),whichcanbeashigh NGRB(>S)=
as tens of percent. Bloom,Frail&Kulkarni(2003) obtain a zmax[L,max(S,Slim)]R (z)dV(z)
factorof 2.2forthe scatter in ǫθ2, implyingthatσlnθ .0.4, P(L)dL G(Slim)dSlim (G1R+Bz) dz dz
and therefore σerr .0.4λ. 10 Assuming that the measure- Z Z Z0
ment errors in θ are uncorrelated with the intrinsic scatter S zmax(L,S)R (z)dV(z)
GRB
itnherθeλfLo(rθe)(,1.t0h,is0.i9m,0p.l8ie,s0.σ9o2)b.s ≈σ.σ2(+1.σ2,e2r1r..2,1σ.23,+1(.05.)4fλo)r2aλn=d =Z P(L)dL"Z0 G(Slim)dSlimZ0 (1+z) dz dz
(1.5,2,2.5,3). ∞ zmax(L,Slim)R (z)dV(z)
+ G(S )dS GRB dz ,(18)
lim lim (1+z) dz
9Weusethepeakluminositysinceitisusuallythemostrelevantquantity ZS Z0 #
forthetriggeringofthedifferentdetectors.
10ThisissincethedominantmeasurementerrorinθλLcomesfromθλ, 11NotethatRGRB(z)isthetrueGRBrate,anditisobtaineddirectlyfrom
whileLismeasuredmoreaccurately. thisformalismwithoutanyneedfora‘correctionfactor’etc.
6
less steep than r- 4 can develop a shocked layer near the jet
TABLE1. FITSTOTHElogN- logSDISTRIBUTION boundary,duetothelossofcausalcontactbetweenthejetin-
teriorandwall. Ifthisshockisresponsibleforthegamma-ray
P(L) model RGRB(z=0) goodness emission,itcouldleadtoradiativeefficiencyincreasingwith
model parameters [Gpc- 3yr- 1] offit θ.
splaionwwgleer λσθθcmU=a=SxJ00==..500+-2500..7+-..97500+-±..0000..012505.2 5.04.+-8116..80+- 00(..U1045J(,UqS=J)1) χP((219==.1d010..o17σ..2f0))35 2dmetaOchtreinessatwhsoeefosuoǫwltdh.ietiIhrmfhpθλal,nydwb,hia>fipcthph0re,iosila.uecmm.hoaeinrsgeo2asc.mi5otymntfhsauiis-snrtceawtnyiotoeunwflfidiimcthfipeavndliocieryrsecaλǫtγUkeSt=Jsht>ai2-t
γ USJ
fisλPi.xUnLegS.dJl+e θθλσcmU=a=SxJ00==..800+-2300.5+-(..42fi00±..x0001e05d.)05 5.00.+-7105..38+- 00(..U0076J(,UqS=J)1) χP((121=0=.40d16..48oσ.5.6f)5)7 dluemWnsieintoypsepirtrfyoofifrumlene.cdtitownowfihtiscthoctahnedaraitsaeuesiitnhgeraisninthgeleUpSowJmer-oldaewl
double α=0.6±0.1 χ2=10.57 orintheUJmodel. Inthefirstfitweletallfourfreeparam-
σlpaow=w+0er Lβ5=1=0.18.+-600+-..2100..86† (1U0J.7,q±=2.13) P((10=0.80d6..6oσ0.f)8) pvetaaelrruasemvoeaftreyλr:sfiλtoxUeSvdJa,raθytc:,λθθUmS,aJxθ=an2daanσnd.dIanσll.othwTeehsdeectshoeencdorenfimdtawfiinteiwnhgaesltdhptreheree-
c max
formedsince a value of λ =2 is expectedin the simplest
USJ
NOTE. —Thebestfitparameterswiththeir1σconfidenceintervalsare versionoftheUSJmodelwherethegamma-rayefficiencyǫ
showntogetherwiththeimpliedtrueGRBrateandthegoodnessoffit. The γ
isconstant(b =0)andthereforea =λ =2becauseof
confidenceintervalsaretheprojectionontotherelevantparameteraxisofthe USJ USJ USJ
regioninthemulti-dimensionalparameterspacearoundtheglobalminimum the Frail relation (Eq. 3). Therefore, it is interesting to test
χ2 ofχ2where∆χ2=χ2- χ2 <1. whetherthissimplestversionofthetheUSJmodelisconsis-
min min
†HereL51=L∗/(1051ergs- 1). tentwiththeobservedlogN- logSdistribution.
Inordertoassignaχ2 valuetothefitwedividedthe2204
GRBsintheGUSBADcataloginto14binsaccordingtotheir
valueofS,thepeakphotonflux. Thefirst11binsareequally
where P(L) is given by equation (4). For the USJ model
spacedinlogS. Intheremaining3bins,whichcorrespondto
P(Lθ > θ ) = δ(L) while P(Lθ < θ ) is given by
max max the highest values of S, we chose a larger range of S values
L(1m5in)|(θwmiitnh/θLm(aθx))-=λULSJm.aAx×simmiilna[r1d,i(sθp/e|θrmsiion)n- λisUSiJn]trwohdeurceedLimnaxth=e shoavaesrteoahsoavneabaltelePaositssNomninst=at4is0tiGcsR.BSsinicneetahcehobviner,ailnlonrodremratlo-
UJmodel.ThepredictedlogN- logSdistributiondependson izationisanadditionalfreeparameterinourfits,thenumber
thevaluesofλ , θ ,θ , σ andthenormalizationθλUSJL , ofdegreesoffreedom(d.o.f)inourfitsisν=14- (4+1)=9
USJ c max 0 0
wherethelastparameterisdeterminedthroughafittoobser- inourfirstfit,andν=10inoursecondfit.
vationsandisnotconsideredtobeafreeparameter.Thescat- TheresultsofthefitsarepresentedinTable4andinFigures
terσisconstrainedbyobservations,(1.0,0.9,0.8,0.9).σ. 2and3. Whenλisfreetovaryweobtainabestfitvalueof
(1.2,1.2,1.3,1.5)forλUSJ=(1.5,2,2.5,3),butwasallowed λUSJ=2.9-+00..52, however,whenwefixλUSJ =2westillgetan
tovaryoverawiderrangewhenperformingthefittothedata. acceptablefit. ThissuggeststhatalthoughvaluesofλUSJ>2
The smallestobservedvaluesof θ providean upperlimiton arepreferredbythedata,avalueofλUSJ=2isstillpossible.
θcofθc.0.05whilethelargestobservedvaluesofθprovide On the other hand, values of λUSJ <2 become increasingly
a lower limit on θmax of θmax &0.5. For the USJ model the hard to reconcile with the data. Values of λUSJ >2, which
shape of the afterglowlight curvesimplies1.5.a .2.5
USJ
andtherefore1.5.λ .2.5,however,sincethislimitdoes
USJ
not apply to the UJ model and since we wanted to properly
check the consistency of the USJ with the data we allowed 3.5
λUSJtovaryoverawiderrangewhenperformingthefittothe observed
UJ
data. 3 best fit USJ
Combining the constraint from the Frail relation (Eq. 3) best fit USJ l=2
with the limit on λ from the luminosity function allows us 2.5
separately to constrain a and b. From Eq. (3) we have
a=2(4- k)- (3- k)λ=8- 3λ for k=0 and 4- λ for k=2. 2
N)
SincefortheUSJmodel1.5.aUSJ.2.5fromtheafterglow og(
lightcurves, this implies 11/6.λ .13/6 for k =0 and l1.5
USJ
1.5.λ .2.5 for k=2. For k=0, λ must be close to 2
USJ
and b=(4- k)(λ- 2)falls in the range - 2/3.b.2/3. for 1
k=2thisimplies- 1.b.1. Asdiscussedin§2, directes-
timatesofǫ suggest0.b.1(thisappliesbothtotheUSJ 0.5
γ
modelandtotheUJmodel). Thedifferentconstraintsonthe
parametersaandbaresummarizedinFig. 5. 00 0.5 1 1.5 2 2.5 3
IftheluminosityfunctionforcesλUSJ tovalueslessthan2, log(CM/Cm)
thenbmustbenegativeandifλUSJapproaches1.5thiswould FIG.2.— Theobservedcumulativelog10N- log10(Cmax/Cmin)distribution
favor an externalmedium with a k=2 density profile. Note takenfromtheGUSBADcatalog(solidstep-likeline)comparedtothepre-
that b<0 corresponds to the dissipation efficiency increas- dictedlog10N- log10(P/Plim)distributionsforourbestfitmodels: asingle
power-lawluminosityfunction(relevantforboththeUSJandtheUJmodels),
ing with angle, as would be expectedif it is associated with
bothwhenlettingallparametersvary(dashesline)andwhenfixingλUSJ=2
a shear layer well outside the jet core. We note that highly (dottedline),andabrokenpower-lawluminosityfunction(solidline;relevant
relativisticjetspassingthroughanexternalpressuregradient onlyfortheUJmodel).
7
2.8 2.6
2.6 2.4 observed
best USJ
2.2
2.4
2
log[N(C/C)]Mm2.22 oUbbbeeJsssett rffiivtt eUUdSSJJ l=2 log[N(C/C)]Mm11..68
1.8
1.4
1.6 1.2
1.4 1
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
log(CM/Cm) log(CM/Cm)
FIG. 3.— The same as Figure 2 but for the differential distribution: FIG.4.— ThesameasFigure3butforadifferentbinningoflog10(P/Plim):
log10(P/Plim)isdividedinto14bins,thefirst11ofequalsize,andthere- 26binsandNmin=15.
maining3withvaryingsizeschosensuchthatthenumberofbustsperbinin
theobservedsampleisatleastNmin=40(inordertohavereasonablestatis-
ticssothatthePoissonerrorwillnottobetoolarge),andNistheobserved
(circles)ortheoretical(lines;onlythevaluesatthecenterofeachbincount) theexactchoiceofbinninghassomeeffectonourresults,this
numberofburstsineachbin.Theedgesofthebinsarealsoplotted. effectisrathersmall.
Inordertoestimatetheeffectofourchoiceforthestarfor-
mationrate(ormoreaccuratelytheGRBratethatisassumed
to follow the star formation rate up to an unknown normal-
arepreferredbythedata, correspondtob>0(agamma-ray ization constant) we repeated our fit with the original bin-
efficiency ǫγ that decreases with θ) and for the USJ model ning(14bins)butwitha differentstar formationrate: SFR2
λUSJ ≈2.5 favors k=2 (i.e., a stellar wind external density fromPorciani&Madau(2001). Weobtained: λUSJ=2.9+- 00..34,
profile). σ=0.5 0.5,θ =0.04 0.005andθ =0.9 0.2. While
Thefirstfitgivesσ=0.5+- 00..75 whichindicatesthatthisfitis changin±gthestacrformati±onratehasalamragxereffec±tthanchang-
notverysensitivetothevalueofσ. Italsoincludestherange ingthebinning,theparametervaluesforthetwodifferentstar
0.8.σ .1.3 for λUSJ =2.5 and 0.9.σ .1.5 for λUSJ = formation rates that we considered are still consistent with
2.9thatare impliedby observations. Incontrast, the second eachotherwithintheir1σ confidenceintervals. Forthenew
fit with a fixed λUSJ =2 gives σ =0.8+- 00..42 which requires a starformationrateweobtainaslightlylowertrueGRBrate:
positivevalueofσinordertogetareasonablefittothedata. R (z=0)=0.65+0.05Gpc- 3yr- 1.
GRB - 0.17
Thebestfitvaluesofθ andθ areslightlyhigherforthe
c max
firstfit,butareratherclosebetweenthetwofits. Thebestfit 5. THELUMINOSITYFUNCTIONFORTHEUNIFORMJETMODEL
value of θc =0.05-+00..00105 for a free λ and θc =0.03-+00..00015 for a The two fits that were discussed in §4 for the USJ model
fixedλUSJ =2 are close to the upperlimit ofθc .0.05from still apply to the UJ model with θ θ and λ
c min USJ
observations,andtheallowedconfidenceregionsdonotallow λ 2/(3- q). The second substitutio→n indicates a dege→n-
UJ
valuesmuchsmaller(bymorethanafactorof 2)thanthis ×
eracy where the luminosity function P(L) and therefore the
∼
limit. Thebestfitvaluesofθmax=0.7 0.2fora freeλand logN- logS distribution would be the same for UJ models
±
θmax =0.5 0.05 for a fixed λUSJ =2 are close to the lower withthesamevalueofλ /(3- q)=(a +b )/(3- q). This
± UJ UJ UJ
limit of θmax &0.5 from observationsand are not consistent degeneracy does not enable us to determine P(θ ), which is
j
withθmax=π/2. parameterized by q from a fit to the observed logN- logS
The true GRB rate that is implied from our fits is rather distribution. Such a fit can only determine η=1+2/λ =
close to the value of RGRB(z=0) 0.5Gpc- 3yr- 1 that was 1+(3- q)/λ where P(L) L- η. The degeneracymigUhStJbe
∼ UJ
found by Perna,Sari&Frail (2003). We obtain a slightly ∝
brokenifwecouldestimatethetrueGRBrateinanindepen-
larger rate for the free λ fit compared to the fixed λUSJ =2 dentway.ThisissincethetrueGRBratefortheUSJmodelis
fit, but the difference is very small. The rates we obtain for determinedfromthefittotheobservedlogN- logSdistribu-
theUSJmodelarelowerbyafactorof 300comparedtothe
tion,andthecorrectionfactorfortheUJmodeldependsonly
∼
estimateofFrailetal.(2001)fortheUJmodel.
onq(seeEq.14andFigure1). Thusanindependentestimate
Finally, since the exact choice for the number and sizes
ofthetrueGRBratewouldenablethedeterminationofqand
of the bins used in our fit is somewhat arbitrary and might
wouldthereforeconstrainP(θ ).
j
havesomeeffectonourresults,weestimatethiseffectmore Thevalueofλ=a+bisthesamefortheUJandUSJmodels
quantitativelyby repeatingour fit for a single power law lu-
if and only if q=1. In fact, the Frail relation as manifested
minosityfunctionwithalargernumberofbins,26insteadof
in Eq. (3) implies that for q=1 and a given value of k, it
14 (see Fig. 4). The best fit parameter values remained the
is enough that either a or b be the same for the two models
sameasfortheoriginalbinning(14bins)uptothesignificant in order for λ= a+b to be the same. Add to this the fact
digits shown in Table 4, while the 1σ confidence intervals that for q=1 there is an equal number of uniform jets per
aslnigdhθtmlyaxc=ha0n.g7e+- d00..:21.λTUhSJe=G2R.9B+- 00r..a24t,eσis=s0li.g5h+- t00l..y74,hθicg=he0r,.0R5G±RB0(.z0=5 tlhoegamriothsmt ‘incaitnutrearlv’avlailnueθjf,oarnqd. qH=ow1emveirg,hwtebeemcopnhsaidsiezreedthaast
0)=0.95+0.25Gpc- 3yr- 1, but well within the 1σ confidence thereisnophysicalbasisforcomparingpairsofmodelswith
- 0.22
intervalfortheoriginalbinning.Thisdemonstratesthatwhile λ =λ ,andthereforenoapriorreasonforchoosingq=1.
USJ UJ
8
3 suggeststhatabrokenpower-lawisnotreallynecessary,and
l l l l
=2 =2.4 =2.9 =3.1 a single power-law can provide a reasonable fit to the data
2.5 k=0 even for σ =0 (as we obtain for the first fit from the previ-
oussection,withdifferentL andL whichcorrespondto
k=2 differentθ andθ ). min max
max c
2
a 6. DISCUSSION
We have developed a formalism that enables us to cal-
1.5
culate the theoretical logN- logS distribution for any GRB
jet structure, and directly provides the true GRB rate from
1 a fit to the observed logN- logS distribution. This is a
more straightforward approach compared to previous works
(Guetta,Piran&Waxman 2004; Schmidt 2001) which first
0.5
−1 −0.5 0 0.5 1 1.5 2 calculatedthe GRB rate assuming an isotropicemission and
b
then introduced a ‘correction factor’ in order to account for
FIG.5.—Asummaryoftheconstraintsonthepowerlawindexesaandbof the effects of the GRB jet structure. We have applied this
thekineticenergypersolidangleandthegamma-rayefficiency,respectively: formalismtotheuniformjet(UJ)modelandtotheuniversal
ǫ∝θ- aandǫγ∝θ- b,whereθ=θobsfortheUSJmodelandθ=θj forthe structuredjet(USJ)modelandperformedfitstotheGUSBAD
UJmodel. Theconstraint fromthe Frail et al. relation (Eq. 3)is shown
bothforaconstant densityexternal medium(k=0)andforastellar wind catalogwhichincludes2204BATSEbursts. Ouranalysisim-
environment(k=2). Alsoshownarethevaluesofλ=a+bforthebestfit proves on previous works by: (1) allowing the efficiency in
value(λUSJ=2.9)andthe1σconfidenceinterval(2.4<λUSJ<3.1)forthe producingγ-raystovarywithθ,(2)includinganinternaldis-
USJmodel,aswellasλ=2whichisstillacceptable. Theshadedregions
persioninthepeakluminosityLatanygivenθ,(3)introduc-
showtheconstraints1.5.aUSJ.2.5fromtheafterglowlightcurves,which
appliesonlytotheUSJmodel,and0.b.1fromestimatesofthegamma- inganinnercoreangleθc andanouteredgeθmax intheUSJ
rayefficiency,whichappliestoboththeUSJmodelandtheUJmodel. The model,and(4)usingalargerandmoreuniformGRBsample.
intersectionofthesetwoshadedregionsisindicated. TheresultsofourfitsaresummarizedinTable4andFigures
2and3.
A power law luminosity function has also been fit-
ted to the the logN - logS distribution in some previ-
Theonlymeaningfulcomparisonistoobservations.
ous works (Hakkilaetal. 1996; Loredo&Wasserman 1998;
Thediscussionin§4aboutthevaluesofλ , σ, θ and
USJ max Stern,Tikhomirova&Svensson 2002, hereafter STS02). Of
θ forthe USJ modelare still valid forthe UJ model, where
θc is replaced by θ and λ is replaced by 2λ /(3- q). theseworks,itismostusefultocompareourresultstothose
c min USJ UJ ofSTS02,sincetheirassumptionsaretheclosesttoours:sim-
The true GRB rate for the UJ model is higher than that for
the USJ model for the same fit to the observed logN- logS ilar to us, they assumed (ΩM,ΩΛ) = (0.3,0.7) and that the
GRB rate follows the star formation rate. It is important to
distributionbyafactorof f(q)whichisgiveninEq. (14)and
keepinmind,however,thatSTS02usedadifferentGRBsam-
showninFigure1. InTable4weprovidethevaluesforq=1
ple. STS02obtainedapowerlawluminosityfunctionatlow
asanexample.ThetrueGRBratethatweobtainforq=1isa
factorof 6- 6.6lowerthanthatofGuetta,Piran&Waxman luminosities,withanindexofη=1.4,whereP(L)∝L- η.This
(2004)an≈dafactorof 45- 50smallerthanthatofFrailetal. isratherclosetothepowerlawindexofη=1.7(orbetween
≈ 1.6 and 1.8 at 1σ) that we obtain over the whole luminos-
(2001). However,since theGRBratethatisobtainedinthis
ityrangeforasinglepowerlawluminosityfunction.Itisalso
way has a strong dependence on the value of q (see Figure
closetothepowerlawindexofη=1.6(orbetween1.5and1.7
1), the rates for q=1 are not very meaningful(as discussed
at1σ)thatweobtainforlowluminositieswhenusingabro-
inthepreviousparagraph). Aconstantefficiency(b=0)and
kenpowerlawluminosityfunction. Also,forasinglepower
aconstantenergy(a=2)implyλ =2which,togetherwith
thebestfitvalueofλ 2.9,givUeJ q=3- 2λ /λ 1.6. law,wehaveamaximalluminosity(correspondingtothecore
USJ≈ UJ USJ≈ angleθ intheUSJmodelortheminimaljethalf-openingan-
Forthebestfitparametervaluesthiswouldimply f(q) 14.4 c
andatrueGRBrateofR (z=0) 12.3Gpc- 3yr- 1.≈ gleθminintheUJmodel)withandexponentialtail[duetothe
GRB ≈ assumedscatteraroundthemeanvalueofL(θ)],whichisnot
It is remarkable that we obtain a good fit to the data for
verydifferentfromanexponentialcutoff(ordecline)athigh
the UJ modelfor a single power-lawdistribution of jet half-
luminosities,thatwasfoundtobeaviableoptionbySTS02.
openingangles,P(θ ),andascatterσ inthepeakluminosity
j Thus, we concludethat our results are consistent with those
for a given θ which is consistent with observations. Previ-
j ofSTS02.
ousworksusedabrokenpower-lawluminosityfunctionP(L)
Themaindifferencesbetweenourworkandpreviousworks
whichcorrespondstoabrokenpower-lawP(θ ),andnoscat-
j intheliteraturewhichaimedconstrainingtheGRBluminosity
ter (σ = 0). For this reason, we also performed a fit with
functionusingthepeakfluxdistributionare: (i)thedifferent
σ=0andabrokenpowerpower-lawluminosityfunctionP(L)
samplethatweuse,and(ii)thefactthatweconsideradiffer-
withtheparameterizationofGuetta,Piran&Waxman(2004),
where12P(L) L- 1- αforL<L∗andP(L) L- 1- β forL>L∗. ential peak flux distribution instead of the cumulative distri-
∝ ∝ bution,sincerandomerrorspropagateinanunknownwayin
TheresultsofthisfitareshowninTable4andFigures2and
3. We geta goodfit whereα=0.6 0.1andβ =0.8+0.2 are the cumulative distribution. Moreover, differently from pre-
± - 0.1 viousauthors,weconsiderthepossibilitythatthejetisstruc-
consistentwithhavingthesamevalue,α β 0.7.Thislast
≈ ≈ turedandderivethecorrespondingluminosityfunctionwhich
value implies η 1.7, which is consistent with our best fit
value of λ =2≈.9 that implies η =1+2/λ 1.7. This in turn puts constraintson the jet structure throughthe fit to
USJ USJ ≈ thelogN- logSdistribution.
12Herewedefineαandβwithaminussignwithrespecttotheirdefinition Thevaluesweobtainforθc intheUSJmodelorθmininthe
inGuetta,Piran&Waxman(2004). UJmodelareclosetotheupperlimitof 0.05thatisimplied
∼
9
by the smallest values of θ that are inferred from afterglow onetobreakthisdegeneracy.
observations. Furthermore,theyarenotconsistentwithzero. Alternatively, one can make an additional assumption on
Th0e.5vaflruoemofafθtmeragxltohwatowbeseorvbatatiionniss,calnodseistonothtecolonwsiesrtelnimtwitiothf aanUdJ oarcboUnJ.staTnhteentreuregyenceorrgryesipnonthdesjteot sacale=s2a.s EIf∝wθe2j-aasU-J
∼ UJ
π/2.Therefore,wefindthatincludingθc(orθmin)andθmaxis sume a constant energy, the Frail relation (Eq. 3) implies
requiredinordertoobtainagoodfittothedata. a constant efficiency (b =0), and vice versa. In this case
We fit the observeddistribution of θλL(θ) (where L is the λ = 2 and the best fitUJvalue of η = 1+2/λ 1.7 im-
UJ USJ
peak luminosity), for a sample of 19 GRBs with a known plies q 1.6. For the best fit parameter values t≈his would
redshift z and an estimate for θ, to a log-normal distribu- imply f≈(q) 14.4 and a true GRB rate of R (z = 0)
GRB
tion with a standard deviation σ. We found the fit to a 12.3Gpc- 3y≈r- 1. This last value is a factor of 2.7 low≈er
log-normal distribution to be acceptable. The implied val- than that of Guetta,Piran&Waxman (2004) and≈a factor of
ues of σ are (1.0,0.9,0.8,0.9).σ.(1.2,1.2,1.3,1.5) for 20 smaller than thatof Frailetal. (2001). It is also a fac-
λ=(1.5,2,2.5,3),wherewehavetakenintoaccountthatpart ≈tor of 4.9 103 smaller than the rate of SNe Type Ib/c,
oftheobserveddispersionmightariseduetoerrorinthees- R ≈(z=0)× 6 104Gpc- 3yr- 1.
SNIb/c
timatedvalueofθ. Thisisconsistentwiththewiderangeof For a = 2≈and×b = 0, the 1 σ confidence interval of
possibleσvaluesweobtaininourfitforafreeλ,σ=0.5-+00..57, 2.4<λUUSJJ<3.1inλUUJSJ correspondsto1.3<q<1.7,9.5<
and with the smaller range of values for our fit with a fixed f(q)<16.5,andR (z=0)= (7.7- 16.5)Gpc- 3yr- 1. These
λUSJ=2,σ=0.8+- 00..42. values of q are noGtRvBery far∼from q = 1 for which there is
TheFrailetal. (2001) relationconstrainsthe valuesofthe an equal number of jets per logarithmic interval in θ . It
j
powerlawindexesaandboftheenergypersolidangleinthe implies that there are more jets with θ θ compared to
j min
jet(ǫ)andthegamma-rayefficiency(ǫγ),respectively(seeEq. θ θ , by a factor of (θ /θ )q-∼1 2.1- 9.1where
3). For the USJ modelthe observedshapesof the afterglow j ∼ max ∼ max min ∼
9.θ /θ .23(seeTable4). Ifweallowthevalueofei-
lightcurvesimply1.5.a .2.5(Granot&Kumar2003; max min
USJ thera orb tovary,theymustbothvarytogetherinorderto
Kumar&Granot2003),whichinturnimplies11/6.λ . UJ UJ
USJ satisfytheFrailrelation(Eq. 3). Thiswouldcausethevalues
13/6and- 2/3.b .2/3fork=0or1.5.λ .2.5and
USJ USJ ofq, f(q)andthetrueGRBratefortheUJmodeltoallvary
- 1.b .1fork=2. Asdiscussedin§2,directestimates
USJ accordingly. Thestrongdependenceof f(q)onq(seeFigure
ofǫ fromafterglowobservationssuggest0.b.1.Ourbest
γ 1)impliesthatthiscanpotentiallyincreasethetrueGRBrate
fitvalueofλUSJ=2.9+- 00..25favorastellarwindexternaldensity byalargefactor. Thefactthat f(q)>1forallqimpliesthat
profile(k=2)andanefficiencythatdecreaseswithθ(b>0). thetrueGRBratefortheUSJmodel,whichisgiveninTable
However,λUSJ =2, whichcorrespondsto b=0anddoesnot 4,providesalowerlimitfortheUJmodel.
constrainthevalueofk,stillprovidesanacceptablefittothe
data. TheconstraintsonaandbaresummarizedinFig. 5.
For the UJ model we find a degeneracy in the luminosity
function: P(L) L- η where η = 1+(3- q)/λ and λ = Wethanktherefereeforusefulcommentswhichhelpedim-
UJ UJ
a +b . Since∝a fit to the observed logN- logS distribu- provethepaper. D.G.thankstheWeizmanninstituteforthe
UJ UJ
tion can only constrain the luminosity function P(L), in our pleasant hospitality. This work was supported by the RTN
case it can only provide the value of η. However, this con- “GRBs - Enigma and a Tool" and by an ISF grant for a Is-
strainsonlythevalueof(3- q)/λ =(3- q)/(a +b ),and raeli Center for High Energy Astrophysics (D. G.), by the
UJ UJ UJ
thereforestilldoesnotenableustodeterminethevaluesqand W. M. Keck foundation and by US Department of Energy
λ separately.AnindependentestimateofthetrueGRBrate undercontractDE-AC03-76SF00515(J.G.)andNSFgrants
UJ
wouldconstrain f(q)andthereforeq,andwouldthusenable AST-0307502(D.G.andM.C.B.)andPHY-0070928(J.G.).
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