A Beaming-Independent Estimate of the Energy Distribution of Long Gamma-Ray Bursts: Initial Results and Future Prospects

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by  Isaac Shivvers

DRAFTVERSIONJANUARY5,2011
PreprinttypesetusingLATEXstyleemulateapjv.03/07/07
ABEAMING-INDEPENDENTESTIMATEOF THEENERGY DISTRIBUTION OF LONG GAMMA-RAY BURSTS: INITIAL
RESULTS AND FUTUREPROSPECTS
I.SHIVVERS1&E.BERGER1
DraftversionJanuary5,2011
ABSTRACT
We present single-epoch radio afterglow observations of 24 long-duration gamma-ray burst (GRB) on a
1 timescaleof&100daftertheburst. Theseobservationstracetheafterglowevolutionwhentheblastwavehas
1 deceleratedtomildly-ornon-relativisticvelocitiesandhasroughlyisotropized.Weinferbeaming-independent
0 kineticenergiesusingtheSedov-Taylorself-similarsolution,andfindamedianvalueforthesampleofdetected
2 bursts of about 7×1051 erg, with a 90% confidence range of 1.1×1050- 3.3×1053 erg. Both the median
and 90% confidence range are somewhat larger than the results of multi-wavelength, multi-epoch afterglow
n
modeling(includinglargebeamingcorrections),andthedistributionofbeaming-correctedγ-rayenergies.This
a
J is due to bursts in our sample with only a single-frequencyobservationfor which we can onlydeterminean
upperboundonthepeakofthesynchrotronspectrum.Thislimitationleadstoawiderrangeofallowedenergies
3
than for bursts with a well-measured spectral peak. Our study indicates that single-epoch centimeter-band

] observationscoveringthespectralpeakonatimescale ofδt ∼1 yrcanprovidea robustestimate ofthetotal
E kinetic energydistribution with a small investmentof telescope time. The substantial increase in bandwidth
H ofthe EVLA (upto 8GHz simultaneouslywith fullcoverageat1- 40GHz)will providetheopportunityto
estimatethekineticenergydistributionofGRBswithonlyafewhoursofdataperburst.
.
h
Subjectheadings:gamma-rays:bursts
p
-
o
r 1. INTRODUCTION ergies of ∼ 1052 erg (Cenkoetal. 2010b,a). The existence
t of these highly energetic bursts depends at least in part on
s The energy budget of gamma-ray bursts (GRBs) provides
a the ability to correctly infer their large beamingcorrections.
fundamentalinsightintothenatureoftheexplosions,there-
[ Indeed, the inference of jet opening angles from breaks in
sultingejecta properties,andtheidentityofthecentralcom-
the afterglow light curves has become controversial in re-
1 pact remnant (“engine”). While the isotropic-equivalent γ-
centyearsduetoconflictingtrendsinopticalandX-raylight
v ray energy (E ) can be easily determined from a mea-
3 surement of thγe,isbourst fluence and redshift, a complete ac- curves(Liangetal. 2008; Racusinetal. 2009). Similarly, in
0 countingof the energybudgetrequiresdetailedobservations somecasesatwo-componentjethasbeeninferred,withanar-
6 of the afterglow emission. The afterglow observations pro- rowcoredominatingtheγ-rayemissionandawidercompo-
0 nentdominatingtheafterglowemission(Bergeretal.2003b;
vide a measure of the isotropic-equivalentblastwave kinetic
. Racusinetal.2008). Numericalsimulationssuggestthatoff-
1 energy (EK,iso), as well as the explosion geometry (quanti- axisviewinganglescanalsoleadtoshallowbreaksthatmay
0 fied by a jet openingangle, θ ). The resulting beaming cor-
1 rections, f- 1 ≡ 1- cos(θ ), cajn be substantial, approaching bemissedormis-interpreted(vanEertenetal.2010).
1 three orderbs of magnitudje in some cases (Frailetal. 2001; Inadditiontopotentialdifficultieswiththeinferenceof fb,
: theγ-rayandkineticenergiesmeasuredfromtheearlyafter-
v Panaitescu&Kumar 2002; Bergeretal. 2003a; Bloometal.
glowemissiononlypertaintotherelativisticejecta. Theexis-
Xi 2003). To properly determine EK,iso and fb it is essential to tenceof a substantialcomponentofmildlyrelativistic ejecta
observe the afterglowsfrom radio to X-rays over timescales
canonlybedeterminedfromobservationsatlatetimeswhen
r ofhourstoweeks,clearlyachallengingtask. Thisisparticu-
a suchputativematerialcanrefreshtheforwardshock.Clearly,
larlyaproblemforthesubsetof“dark”GRBs forwhichthe
the existence of substantial energy in a slow ejecta compo-
lackofdetectedopticalemission,orlargeextinction,prevent
nent will place crucial constraints on the activity lifetime of
adeterminationofE andlikely f (e.g.,Bergeretal.2002;
K,iso b thecentralengine.
Piroetal.2002).
Suchlate-timeobservationsalsohavetheaddedadvantage
Over the past decade detailed afterglow observations
thattheyprobetheblastwavewhenithasdeceleratedtonon-
have been obtained at a great cost of telescope time for
relativistic velocitiesand henceroughlyapproachesisotropy
about 20 long-duration GRBs, with the basic result that
(Frailetal.2000;Livio&Waxman2000). Thisallowsusto
the beaming corrections are large and diverse, leading to
use the well-established Sedov-Taylor self-similar solution,
typical true energies of E ∼ E ∼ 1051 erg (Frailetal.
γ K withnegligiblebeamingcorrections,toestimate thetotalki-
2001;Panaitescu&Kumar2001,2002;Bergeretal.2003a,b;
neticenergyofboththedeceleratedejectaandanyadditional
Bloometal. 2003). More recently, it has been recognized
initially non-relativistic material. Since the peak of the af-
that some nearby long GRBs have much lower energies,
terglow spectrum on these timescales is located in the radio
E ∼ 1049- 1050 erg, and appear to be quasi-isotropic
iso band,thelackof opticalafterglowemission(e.g., dueto ex-
(Kulkarnietal. 1998; Soderbergetal. 2004b, 2006). Simi-
tinction) does not have an effect on the ability to determine
larly,someburstsappeartohavelargebeaming-correcteden-
E .
K
This approach was first exploited by Frailetal. (2000) to
1Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cam-
bridge,MA02138 modelthelate-timeradioafterglowemissionofGRB970508
(atδt&100d)fromwhichthekineticenergywasinferredto
2
beE ∼5×1050 erg. Bergeretal.(2004)usedthesameap- the case of a uniform density medium3. For the typi-
K
proachtomodeltheradioafterglowemissionofGRB980703 cal expected parameters of long GRBs, the initially colli-
ontimescalesof&40d,andtore-modelGRB970508. They mated blastwave approaches spherical symmetry and decel-
found kinetic energiesof E ∼3×1051 erg for both bursts. erates to non-relativistic velocity on similar timescales, t ≈
K s
Finally, Frailetal. (2005) modeled the radio emission from 150(E /n )1/4t1/4 d and t ≈40(E /n )1/4t1/4 d,
GRB030329at δt &50 d and foundE ∼1051 erg. Only 3 K,iso,52 e j,d NR K,iso,52 e j,d
K respectively(Livio&Waxman2000);here,n isthecircum-
bursts have been studied in this fashion so far because only burst density in units of cm- 3 and t is the “ejet break” time
thoseeventshavewell-sampledradiolightcurvesontherele- atwhichthejetbeginsto expandsidjeways(i.e., Γ(t )∼θ- 1,
vanttimescalesofδt&100d. j j
where Γ is the bulk Lorentz factor). In this paper we as-
However,thekineticenergycanstillbeestimatedusingthe
sumethattheblastwavehastransitionedtothenon-relativistic
samemethodologyevenfromfragmentarylate-timeradioob-
isotropic phase by the time of our observations and subse-
servations. Such an approach will naturally result in larger
quentlycheckforself-consistency.
uncertainties for each burst, but it can be applied to a much
The blastwave dynamics in the non-relativistic phase are
largersampleofevents. Herewepresentsuchananalysisfor
described by the Sedov-Taylor self-similar solution with
24 long-duration GRBs with radio observations at &100 d,
butwith only1- 3data points(at1.4 to 8.5GHz) perburst. r(t)∝ (ESTt2/n)1/5. To calculate the synchrotron emission
Using these observations we infer robust ranges for the ki- emergingfromtheshock-heatedmaterial,wemaketheusual
neticenergyofeachburstandforthepopulationasawhole. assumptions: (i) the electrons are accelerated to a power-
The plan of the paper is as follows. The radio observations law energy distribution, N(γ)∝ γ- p for γ > γm, where γm
are summarized in §2. The model for synchrotronemission is the minimum Lorentz factor; (ii) the value of p is 2.2 as
fromaSedov-Taylorblastwave,andthevariousassumptions inferredfromseveralbursts(e.g.,Panaitescu&Kumar2001,
weemployarepresentedin§3. In§4wedetailtheresulting 2002; Yostetal. 2003); and (iii) the energy densities in the
kineticenergiesandtherangefortheoverallsample,andwe magnetic field and electrons are constant fractions (ǫB and
compare these results to multi-wavelength analyses of early ǫe,respectively)oftheshockenergydensity. Accountingfor
afterglows in §5. We conclude with a discussion of future synchrotronemissivityandself-absorption,andincludingthe
prospects. appropriateredshifttransformations,thefluxobservedatfre-
quencyνandtimetisgivenby(Frailetal.2000;Bergeretal.
2004):
2. RADIODATA
We use radio observations of 24 long GRBs at δt & 100 Fν=F0(t/t0)αF[(1+z)ν]5/2(1- e- τν)f3(ν/νm)f2- 1(ν/νm), (1)
d since on those timescales the blastwave is expectedto be-
wheretheopticaldepthisgivenby:
comenon-relativisticandroughlyisotropic(Livio&Waxman
2000), and the peak of the afterglow emission is at or be- τν =τ0(t/t0)ατ[(1+z)ν]- (p+4)/2f2(ν/νm), (2)
low the centimeter band. This has been confirmed with de-
the synchrotron peak frequency, corresponding to electrons
taileddatainthecaseofGRBs970508,980703,and030329
withγ=γ ,isgivenby:
m
(Frailetal. 2000; Bergeretal. 2004; Frailetal. 2005). We
restricttheanalysistoGRBswithaknownredshiftandwith ν =ν (t/t )αm(1+z)- 1, (3)
m 0 0
early-timedetections,whichforthecaseofasingledetection
andthefunction f (x)isgivenby
l
orupperlimitallowustoinferthatthepeakofthespectrum
x
hastransitionedbelowourobservingfrequency. f (x)= F(y)y(p- l)/2dy, (4)
The observations are primarily from the Very Large Ar- l Z
0
ray(VLA2),withtheexceptionofGRBs980425and011121
where F(y) is an integration over Bessel functions
which were observed with the Australia Telescope Com-
(Rybicki&Lightman1979). Thetemporalindicesinthecase
pact Array (ATCA). The data were obtained between 1997 ofauniformdensitymediumareα =11/10,α =1- 3p/2,
and 2009 as part of a long-term GRB radio program (e.g., andα =- 3. ThenormalizationsaFresuchthatFτ andτ are
Frailetal.2003a). m 0 0
thefluxdensityandopticaldepthatafrequencyofν =1Hz
Forthepurposeofouranalysis,weseparatetheburstsinto
att =t , andν isthesynchrotronpeakfrequencyintherest
threecategoriesbasedonthequalityofthedata. InGroupA 0 0
frame of burst at t =t . Furthermore, the synchrotron self-
are3burstswithlate-timedetectionsatmultiplefrequencies 0
absorptionfrequency,ν ,isdefinedbytheconditionτ (ν )=
thatconstrainthepeakofthesynchrotronspectrum(thesame a ν a
1.
three bursts that have been studied in detail by Frailetal.
We fit this synchrotron model to our radio data using F ,
2000;Bergeretal.2004;Frailetal.2005).InGroupBare11 0
τ , and ν as free parameters. Since we have no a priori
burstswith single-frequencydetections, whileGroupC con- 0 0
knowledgeabouttheexpectedvaluesofthesynchrotronspec-
sists of 10 GRBs with late-time non-detections. The VLA
trumparametersweassumethattheyfollowaflatdistribution
measurements and relevant burst properties are listed in ta-
in log-space. We note that anyfurtherassumptionaboutthe
ble1.
distributionoftheseparameterswillonlyservetorestrictthe
resultingenergydistributions, andwe thereforeconsiderour
3. SYNCHROTRONEMISSIONFROMANON-RELATIVISTIC assumedflatdistributiontobeconservative. Forthedetected
BLASTWAVE
objects (GroupsA and B) we retain all solutionsthat repro-
Our modeling of the radio data follows the method- duce the measured flux density within the error bars, while
ology of Frailetal. (2000) and Bergeretal. (2004) for
3SinceweuseasingleepochofobservationsforeachGRBourinferred
2TheNationalRadioAstronomyObservatoryisafacilityoftheNational density can beeasily converted to amassloss rate forthe case ofawind
Science Foundation operated under cooperative agreement by Associated medium. Thedifferenceindynamicalevolutionbetweenthesetwomodels
Universities,Inc. doesnothaveaneffectinthiscase.
3
for the non-detectionswe use 3σ as an upperbound. More- is well-determined, lead to the best constraints in this two-
over,fortheburstsinGroupA,ν isconstrainedbythemulti- dimensionalphasespace. Indeed,asshownforGRB980703
0
frequency observations and no additional constraints are re- (Figure 1), the allowed range of energies for all solutions
quired. However, for the bursts in Groups B and C, which thatreproducethe observedflux density is about1048- 1053
haveonlyasingle-frequencyobservation,werequirethatboth erg, while the solutions that also satisfy the requirements
ν andν havevaluesbelowtheobservingfrequencysincethe that(E +E ).E /2spanamuchnarrowerrangeofabout
m a B e ST
lightcurvesare alwaysdecliningat the time of ourobserva- 1051- 1052 erg, with a roughly log-normal distribution cen-
tions4. teredonlog(E )≈51.6.
ST
Using the allowed ranges of F0, τ0, and ν0 we determine Theburstswithonlysingle-frequencyobservations(Group
thesetofrelevantphysicalparameters: ne, γm,andB,where B)coveramuchlargerareaintheEST- nephase-spacesince
Bisthemagneticfieldstrength. Theradiusoftheblastwave, only an upper bound can be placed on ν and ν . To fur-
m a
r, remainsunconstrained(e.g.,Frailetal. 2000;Bergeretal. ther constrain the energy we place an additional conserva-
2004): tive constraint on the density of n < 100 cm- 3, motivated
B=11.7(p+2)- 2F0-,-252(r17/dL,28)4 G, (5) by the results of detailed broad-beand modeling that show
n ∼ 0.1- 10 cm- 3 (e.g., Panaitescu&Kumar 2001, 2002;
γm=6.7(p+2)F0,- 52ν01,/92(r17/dL,28)- 2, (6) Yeostetal.2003). Giventhe anti-correlationbetweendensity
and energy, our conservative limits lead to a wider range of
ne=3.6×1010cnη1F03,- 52ν0(1,9- p)/2τ0,32r1- 71(r17/dL,28)- 6 cm- 3, allowed energies than if we chose a limit of ne <10 cm- 3.
(7) AnexampleofthisadditionalconstraintforaGroupBburst
c =(1.67×103)- p(5.4×102)(1- p)/2(p+2)2/(p- 1) (8) (GRB010921)isshowninFigure1. Wedonotplacealower
n
bound on the density, since for the phase-space of allowed
wheredL,28 istheluminositydistanceinunitsof1028 cmas- solutions this would not lead to a significant change in the
sumingthestandardcosmologicalparameters(H =71kms- 1 energydistribution. We stress that beyondplacing an upper
0
Mpc- 1,ΩM =0.27,andΩΛ=0.73),andη isthereciprocalof boundonne noconstraintshavebeenplacedonthedistribu-
thethicknessoftheemittingshell. tionsof eithern orβ since bothare inferred,andnotinput,
e
The unknown radius of the blastwave can be con- parametersinourmodel.
strained by introducing a relationship between E = As noted above, we do not consider the energies for the
ST
n m (r/1.05)5[t /(1+z)]- 2 and the energy in the electrons bursts in Group C since the upper limits generally allow a
e p NR
and the magnetic field. We use the condition that at most widerangeofsolutionsthatarenotconsistentwiththeSedov-
half of the blastwave energy is available for accelerating Taylorformulation.Thisindicatesthatthelimitsaregenerally
electrons and producingthe magnetic field, i.e., (E +E ). not deep enough to provide a meaningful constraint on the
B e
E /2. The total energy in the accelerated electrons is E = energy. Future deep radio observations may provide much
ST e
[(p- 1)/(p- 2)]n γ c2V, while the energy in the magnetic betterconstraints(seebelow).
e m
field is E = B2V/8π, where V = 4πr2/η is the volume of Theresultingenergyprobabilitydistributionsforthebursts
B
the synchrotron emitting shell. The energy budget is min- inGroupAandGroupBareshowninFigure3. Themedian
imized near equipartition (i.e., E ≈ E ), and we use this energy and 90% confidence range (i.e., 5- 95% of the dis-
e B
constraint to determine the minimum required energy (e.g., tribution)for each burstare listed in Table 2. We includein
Frailetal.2000;Bergeretal.2004);thisconclusionwasver- theserangesthesmallsubsetofsolutionsthatleadtoβvalues
ified with radio interferometric measurements of the size of inslightexcessof1sincetheseareatmostmildlyrelativistic
GRB030329(Tayloretal.2004;Frailetal.2005). andfurthermoredonotsignificantlychangethedistributions
Finally, using the inferred radius for each possible solu- (Figure3). Wefindthatvaryingtheelectronpowerlawindex
tion, we require for self-consistency that β .1, where β = overtherangep=2.1- 2.5(e.g.,Curranetal.2008)leadstoa
2r(1+z)/5ct. Theresultingβ distributionsareshowninFig- changeinthemedianenergyofonly0.1- 0.2dex(compared
ure2. Theseresultsindicatethatmostofthedetectedbursts toourfiducialvalueof p=2.2),withlargervaluesof plead-
obey the self-consistency requirement, although we reject ingtolowermedianenergies.Similarly,varyingthemagnetic
GRBs 000926, 020819, and 021004 from the analysis since energyfractionawayfromequipartitiontoǫB=0.1and0.01,
&50%oftheirallowedsolutionsleadtorelativisticvelocities leads to an increae in the median energy of about 0.25 and
at the time of the observations. This does not rule out that 0.5dex,respectively. Bothoftheseeffectsaremuchsmaller
theSedov-Taylorsolutionis applicable,butsimplyindicates than the overallspread in energyfor each burst, but they do
thatadditionalobservationsare requiredto narrowdownthe produceminorsystematictrends.
rangeofallowedsolutions. Wefurthermorefindthatthebulk Thecombineddistributionforthesubsetof11burstswhose
oftheupperlimitsdonotruleoutrelativisticexpansion,and solutions are generally self-consistent is shown in Figure 4.
wethereforedonotusetheselimitsintheenergydistribution Themedianand90%confidencerangesare7×1051ergand
analysisbelow. 1.1×1050- 3.3×1053erg.
5. DISCUSSIONANDCONCLUSIONS
4. THEDISTRIBUTIONOFGRBKINETICENERGIES
The key results of our analysis are that the median en-
The resultingsolutionsfor each burstcan be castin terms
ergy for the 11 bursts with self-consistent solutions is E ≈
ofatwo-dimensionalparameterspaceinn versusE . Thus, K
e ST 7×1051 erg,whilethe90%confidencerangeis1.1×1050-
thereisadegeneracybetweenthetwoparameters,inthesense
3.3×1053 erg. The median value is about a factor of 3
thatlargerdensitiesleadtolowerenergies.Clearly,thebursts
times higher than previous calorimetric measurements for
inGroupA, forwhichthepeakofthesynchrotronspectrum
GRBs 970508, 980703, and 030329, for which energies of
4 The opposite case of either νm or νa being larger than the observing 3×1051,3×1051,and1051erg,respectively,weredetermined
frequencyleadstorisinglightcurves. (Bergeretal.2004;Frailetal.2005).
4
Similarly, the inferred energies are somewhat larger than straintrequiresadditionalassumptionsaboutthecircumburst
the distributions of beaming-correctedγ-ray and kinetic en- density and results in a much wider energy range. Indeed,
ergies inferred from broad-band multi-epoch studies (Fig- thisisthekeyreasonforthewiderrangeofallowedhighen-
ure5). Fromvarioussuchanalyses,themedianγ-rayenergy ergy solutions (&1052 erg) compared to the results for E
γ
is hE i≈8×1050 erg (Frailetal. 2001; Bloometal. 2003; and E (Figure 5). Observations of GRBs 970508, 980703,
γ K
Friedman&Bloom2005),whilethemediankineticenergyis and030329demonstratethatthespectralpeakistypicallylo-
hE i≈5×1050 erg (e.g., Panaitescu&Kumar 2001, 2002; cated at ∼few GHz on a timescale of ∼150 d. Thus, ob-
K
Yostetal. 2003); see Figure 5. Inboth cases the 90%range servations in the 1- 10 GHz range on a timescale of ∼few
spansabout2.5ordersofmagnitude,somewhatnarrowerthan hundreddaysshouldallowustodeterminethepeakfluxand
our inferred 90% confidence range for E . The extension frequency. Thiswillinturnprovideanenergyestimatewith
ST
to larger energies found in our analysis mainly reflects the asimilarlevelofprecisiontotheresultsofearly-timebroad-
lackofspectralpeakdeterminationsfortheburstswithsingle- bandmodeling.
frequencyobservations(see Figure 3). These large energies Thisisafortuitousconclusionsincewiththefullfrequency
can be generally eliminated with a measurementof the syn- coverageoftheExpandedVLA(EVLA)itwillsoonbepos-
chrotron peak in the GHz frequency range (e.g., Group A sibletocoverthisentirerangeinafewhoursofobservations
bursts;Figure5). toasensitivitythatisaboutanorderofmagnitudebetterthan
In the context of our results we note that recent numeri- the VLA. As we demonstrated here, such a modest invest-
cal work by Zhang&MacFadyen (2009) led these authors mentofobservingtime(2- 3hoursperburst)canyieldaro-
to conclude that the timescale to reach isotropy is ∼102 yr bust estimate of the GRB energy distribution, regardless of
rather than ∼1 yr as indicated by the analytic formulation theabilitytomeasurejetopeningangles. Pursuingtheseob-
of Livio&Waxman (2000) which we follow here. As a re- servationsforallburstswithameasuredredshiftwillrequire
sult, they note that using the Sedov-Taylorformulationmay only∼50- 100hrofEVLAtimeperyear. Indeed,withsuch
leadtoanerroneousestimateofthekineticenergy.However, observations we should be able to constrain the energy dis-
inspectionof theresultingpotentialdisrepanciesrevealsthat tribution to a comparable level as existing studies within a
thiseffectisatmostafactorof2aslongasself-consistency single year given that about 30 GRBs with known redshifts
between the inferred energy and density and the transition just from 2009 are now available for EVLA observations(a
to the Sedov-Taylor phase is ensured (see their Figure 10). similar sample is available from 2008 bursts). In the longer
Thediscrepanciesbecomelargerifthewrongtimescaleisas- term,thelargenumberofobjectswillallowustotesttheen-
sumedforthetransitiontonon-relativisticexpansion,butthis ergy distribution as a function of redshift, at least over the
quantityisnotafreeparameter.Indeed,ourdistributionsofβ range z∼1- 3 where the bulk of the detected bursts occur
valuespointtoself-consistencyformostbursts,andallowus (Bergeretal.2005;Jakobssonetal.2006). Similarly,thisap-
torejectobjectsthatarepotentiallystillrelativistic.Sincethe proachwillbeparticularlyusefulforburststhatlackdetailed
potentialsystematicuncertaintyofabouta factorof2is sig- opticalorX-raylightcurvesduetoobservationalconstraints
nificantly smaller than the overall spread in allowed energy ordustextinction,andforburstswithcontroversialestimates
foreachburst,wedonotconsiderthistobeanobstacletoour ofthejetopeningangles.
analysis,ortofutureworkontheenergyscaleusinglate-time
radiomeasurements.
As clearlydemonstratedin Figure3, the mostconstrained
energy determinations require a measurement of the syn- We thank Dale Frail and Eli Waxman for helpful discus-
chrotronspectralpeak(GroupA);theabsenceofsuchacon- sionsandcommentsonthemanuscript.
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5
TABLE1
LATE-TIMERADIOAFTERGLOWMEASUREMENTS
GRB z δt ν Fνa Ref.
(d) (GHz) (µJy)
970508 0.835 117.55 8.46 355±47 Frailetal.(2000)
117.55 4.86 425±57
117.55 1.43 206±63
970828 0.958 157.99 8.46 <51 Djorgovskietal.(2001)
980425 0.0085 248.20 8.70 700±200 Kulkarnietal.(1998)
980703 0.966 143.79 8.46 110±20 Frailetal.(2003b)
143.79 4.86 146±24
134.85 1.43 99±25
990506 1.307 141.23 8.46 <75 Tayloretal.(2000)
991208 0.706 291.58 8.46 51±15 Galamaetal.(2003)
000210 0.846 108.37 8.46 <78 Frailetal.(2003a)
000301C 2.030 506.10 8.46 39±11 Bergeretal.(2000)
000418 1.118 405.76 8.46 38±11 Bergeretal.(2001)
000911 1.058 125.78 8.46 <54 Priceetal.(2002)
000926 2.066 257.43 8.46 75±21 Harrisonetal.(2001)
010222 1.477 206.63 8.46 <42 Frailetal.(2003a)
010921 0.451 225.42 8.46 52±15 Frailetal.(2003a)
011121 0.362 132.07 8.70 <141 Frailetal.(2003a)
020819 0.411 126.45 8.46 79±25 Jakobssonetal.(2005)
021004 2.329 140.24 8.46 94±16 Frailetal.(2003a)
030226 1.986 113.85 1.43 <117 Frailetal.(2003a)
030329 0.168 135.48 8.46 1525±56 Frailetal.(2005)
129.57 4.86 1955±62
129.58 1.43 1276±56
031203 0.105 137.15 8.46 426±37 Soderbergetal.(2004b)
050416A 0.654 182.28 8.46 <114 Soderbergetal.(2006)
070125 1.547 341.96 8.46 64±18 Chandraetal.(2008)
070612A 0.617 488.54 8.46 101±39 Frailetal.(2003a)
090323 3.570 131.18 8.46 <81 Cenkoetal.(2010a)
090902B 1.822 199.16 8.46 <48 Cenkoetal.(2010a)
NOTE.—aLimitsare3σ.
6
TABLE2
GRBENERGIESINFERREDFROMCALORIMETRY
GRB hlog(EST)i log(EST),90a log(Eγ)b log(EK)c
(erg) (erg) (erg) (erg)
GroupA
970508 51.8 51.3- 52.5 50.6 51.3
980703 51.6 51.1- 52.2 51.0 51.5
030329 51.3 50.9- 51.8 49.9 50.4
GroupB
980425 49.4 48.9- 50.1 47.8 ∼50
991208d 51.9 51.0- 53.4 51.2 50.4
52.1 51.1- 53.6
000301C 52.4 51.8- 53.5 50.9 50.5
52.6 51.8- 53.9
000418 50.6 49.8- 51.6 51.7 51.5
000926 52.0 51.6- 52.8 51.2 51.2
52.7 51.8- 54.3
010921 51.6 50.6- 53.1 <51.2 ···
51.9 50.7- 53.7
020819 51.3 50.4- 52.6 <51.8 ···
51.8 50.6- 53.6
021004 51.8 51.4- 52.3 50.9 ···
52.8 51.7- 54.5
031203 51.1 50.2- 52.5 49.5 49.2
51.5 50.3- 52.5
070125 52.2 51.6- 53.2 52.4 51.2
52.6 51.7- 54.0
070612A 52.1 51.6- 53.2 <52.0 ···
NOTE. —a Thisisthe90%confidencerangefortheenergyofeach
burst.
bValuesforEγaretakenfromFrailetal.(2001),Bloometal.(2003),and
Friedman&Bloom(2005).
cValuesforEK aretakenfromPanaitescu&Kumar(2002),Bergeretal.
(2003b), Yostetal. (2003), Soderbergetal. (2004b), Soderbergetal.
(2004a), Soderbergetal. (2006), Cenkoetal. (2010b), and Cenkoetal.
(2010a).
dThefirstlineisforastrictcut-offofβ<1,whilethesecondlineallows
asmallfractionofsolutionwithβslightlylargerthan1(Figure3).
7
FIG. 1.—Electronnumberdensity plottedagainstkinetic energyfortworepresentative cases. Thelightgrayregions indicate thephase-space thatleads
to a predicted flux density in agreement with the observed values. The medium gray regions encompass the subset of solutions that satisfy the condition
(EB+Ee).EST/2.Theblackregionsmarksthesubsetofsolutionsthatsatisfyβ<1intheSedov-Taylorframework. Left:GroupAburstwithawell-defined
spectral peak. Right: GroupBburstwithasinglefrequency detection forwhichweusetheadditional limitthatn<100cm- 3. Thisfigurehighlights the
significantadvantageofmeasuringthespectralpeak.
8
0.6 0.6 0.6 0.7
970508 980703 030329 980425
0.5 0.5 0.5 0.6
0.5
0.4 0.4 0.4
Fraction0.3 Fraction0.3 Fraction0.3 Fraction00..34
0.2 0.2 0.2
0.2
0.1 0.1 0.1 0.1
0 0 0 0
−1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9
log[b] log[b] log[b] log[b]
0.7 0.7 0.7 0.7
991208 000301C 000418 000926
0.6 0.6 0.6 0.6
0.5 0.5 0.5 0.5
Fraction00..34 Fraction00..34 Fraction00..34 Fraction00..34
0.2 0.2 0.2 0.2
0.1 0.1 0.1 0.1
0 0 0 0
−1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9
log[b] log[b] log[b] log[b]
0.7 0.7 0.7 0.7
010921 020819 021004 031203
0.6 0.6 0.6 0.6
0.5 0.5 0.5 0.5
Fraction00..34 Fraction00..34 Fraction00..34 Fraction00..34
0.2 0.2 0.2 0.2
0.1 0.1 0.1 0.1
0 0 0 0
−1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9
log[b] log[b] log[b] log[b]
0.7 0.7 0.4 0.4
070125 070612A 970828 990506
0.6 0.6 0.35 0.35
0.5 0.5 0.3 0.3
Fraction00..34 Fraction00..34 Fraction000..12.255 Fraction000..12.255
0.2 0.2 0.1 0.1
0.1 0.1 0.05 0.05
0 0 0 0
−1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9
log[b] log[b] log[b] log[b]
0.4 0.4 0.4 0.4
000210 000911 010222 011121
0.35 0.35 0.35 0.35
0.3 0.3 0.3 0.3
Fraction000..12.255 Fraction000..12.255 Fraction000..12.255 Fraction000..12.255
0.1 0.1 0.1 0.1
0.05 0.05 0.05 0.05
0 0 0 0
−1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9
log[b] log[b] log[b] log[b]
0.4 0.4 0.4 0.4
030226 050416A 090323 090902B
0.35 0.35 0.35 0.35
0.3 0.3 0.3 0.3
Fraction000..12.255 Fraction000..12.255 Fraction000..12.255 Fraction000..12.255
0.1 0.1 0.1 0.1
0.05 0.05 0.05 0.05
0 0 0 0
−1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9 −1.2−0.9−0.6−0.3 0 0.3 0.6 0.9
log[b] log[b] log[b] log[b]
FIG. 2.—Normalizedhistogramsofinferredexpansionvelocity, β≡v/c, atthetimeofourobservations. Avalueof.1(vertical lines)isrequiredfor
self-consistency,andthisisindeedthecaseforthebulkoftheacceptablesolutions.Notethatthescalesforthethreegroupsaredifferent.
9
0.4 0.4 0.4 0.4
0.35 970508 0.35 980703 0.35 030329 0.35 980425
0.3 0.3 0.3 0.3
Fraction000..12.255 Fraction000..12.255 Fraction000..12.255 Fraction000..12.255
0.1 0.1 0.1 0.1
0.05 0.05 0.05 0.05
0 0 0 0
47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55
log[E] (erg) log[E] (erg) log[E] (erg) log[E] (erg)
0.4 0.4 0.4 0.4
0.35 991208 000301C 0.35 000418 0.35 000926
0.3 0.3 0.3 0.3
Fraction000..12.255 Fraction0.2 Fraction000..12.255 Fraction000..12.255
0.1 0.1 0.1 0.1
0.05 0.05 0.05
0 0 0 0
47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55
log[E] (erg) log[E] (erg) log[E] (erg) log[E] (erg)
0.4 0.4 0.4 0.4
0.35 010921 0.35 020819 0.35 021004 0.35 031203
0.3 0.3 0.3 0.3
Fraction000..12.255 Fraction000..12.255 Fraction000..12.255 Fraction000..12.255
0.1 0.1 0.1 0.1
0.05 0.05 0.05 0.05
0 0 0 0
47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55
log[E] (erg) log[E] (erg) log[E] (erg) log[E] (erg)
0.4 0.4
0.35 070125 0.35 070612A
0.3 0.3
Fraction000..12.255 Fraction000..12.255
0.1 0.1
0.05 0.05
0 0
47 48 49 50 51 52 53 54 55 47 48 49 50 51 52 53 54 55
log[E] (erg) log[E] (erg)
FIG.3.—NormalizedhistogramsofGRBenergiescalculatedusingtheSedov-Taylorsolution.ThelightgrayhistogramsforsomeGroupBburstsindicatethe
subsetofsolutionswithastrictcut-offofβ<1. GRBs000926,020819,and021004arerejectedfromoursamplesincethebulkoftheirsolutionsstillleadto
relativisticexpansion.
10
0.15
0.14
0.13
0.12
0.11
0.1
0.09
n
o 0.08
i
t
c
a 0.07
r
F
0.06
0.05
0.04
0.03
0.02
0.01
0
47 48 49 50 51 52 53 54 55
log[E] (erg)
FIG.4.—NormalizeddistributionofGRBkineticenergiescalculatedusingtheSedov-Taylorsolutionforthesampleof11burstswithself-consistentsolutions
(Figure3).Themedianand90%confidencerangearemarkedbyahorizontalbar.