Table Of ContentAPJACCEPTED2006DECEMBER30
PreprinttypesetusingLATEXstyleemulateapjv.2/19/04
LOW-LUMINOSITYGRB060218:ACOLLAPSARJETFROMANEUTRONSTAR,LEAVINGAMAGNETARASA
REMNANT?
KENJITOMA1,KUNIHITOIOKA1,TAKANORISAKAMOTO2,3,ANDTAKASHINAKAMURA1
ApJaccepted2006December30
ABSTRACT
The gamma-ray burst (GRB) 060218 has 105 times lower luminosity than typical long GRBs, and is
∼
associated with a supernova (SN). The radio afterglow displays no jet break, so that this burst might arise
fromamildly-relativisticsphericaloutflowproducedbytheSNshocksweepingthestellarsurface. Sincethis
modelisenergeticallydifficult,weproposethattheradioafterglowisproducedbyanon-relativisticphaseof
7 aninitiallycollimatedoutflow(jet). Ourjetmodelissupportedbythedetectionofopticallinearpolarization
0 intheSNcomponent. Wealsoshowanalyticallythatthejetcanpenetrateaprogenitorstar. Weanalyzedthe
0 observationaldata of the prompt emission of this burst and obtained a smooth power-law light curve which
2 mightlastlongerthan106 s. ThisbehaviorcontrastswiththelongintermittentactivitieswiththeX-rayflares
n oftypicalGRBs, implyingthatthe centralengineof thisburstisdifferentfromthose of typicalGRBs. This
a argumentisconsistentwiththeanalysisoftheSNcomponentofthisburst,whichsuggeststhattheprogenitor
J starwaslessmassiveandcollapsedtoaneutronstarinsteadofablackhole. Thecollimation-correctedevent
8 rate of such low-luminosityGRBs is estimated to be 10 times higherthan that of typical long GRBs, and
they might form a different GRB population: low-lum∼inosity GRBs are produced by mildly-relativistic jets
2 fromneutronstars at the collapsesof massivestars, while typicallongGRBs by highly-relativisticjets from
v black holes. We suggest that the central engine of GRB 060218 is a pulsar (or a magnetar) with the initial
7 rotation period P 10 ms and the magnetic field B 1016 G. A giant flare from the magnetar might be
0
6 ∼ ∼
observedinfuture.
8
Subjectheadings:gammarays:bursts—gammarays:theory—supernovae:general
0
1
6
1. INTRODUCTION thiseventisoneofthelongestbursts.
0
/ The gamma-ray burst (GRB) 060218 is the second near-
h (2) The isotropic-equivalent luminosity of the prompt
p est event (z = 0.033) and it is spectroscopically associ- emission is extremely low 1047 erg s- 1, which is
- ated with the supernova (SN) 2006aj (Modjazetal. 2006; about105 timeslowerthant∼hoseoftypicalcosmolog-
o Sollermanetal. 2006; Pianetal. 2006; Mazzalietal. 2006;
ical GRBs. The isotropic-equivalent energy, extrap-
tr Mirabaletal. 2006; Cobbetal. 2006; Ferreroetal. 2006). olated to the 1- 104 keV band in the central engine
s Such a long GRB/SN association is common and this event frame,isE 6 1049erg.
a further supports the established picture that all long GRBs γ,iso≃ ×
: are related to the deaths of massive stars (for recent re-
v (3) Thespectrumofthepromptemissionisquitesoftcom-
i views,Woosley&Bloom2006;Mészáros2006;Piran2005; pared to those of typical bright GRBs, and the aver-
X Zhang&Mészáros2004). Inthemostpopularmodeloflong
agedspectralpeakenergyinthecentralengineframeis
r GRBs, the so-called collapsar model, the core of a massive 4.9keV.(NotethatthiseventobeystheAmaticorre-
a
starcollapsestoablackholeoraneutronstar,whichdrivesa ≃
lation,despitetheotherpeculiarities. See§2formore
highlyrelativisticjet,breakingoutthestarandmakingaGRB
details.)
(Woosley1993;MacFadyen&Woosley1999). Therelativis-
ticspeedoftheoutflowisrequiredforthenon-thermalprompt (4) ThermalcomponentsaredetectedintheX-raybandat
emission(Lithwick&Sari2001,andreferencestherein).The t.104sandintheUV/opticalbandat104.t.105s,
collimationoftheoutflowisstronglysuggestedbyabreakin while other GRBs do not exhibit such a clear thermal
theafterglow,sinceitisproducedbythesidewaysexpansion component.
ofthejet(e.g.,Rhoads1999;Sarietal. 1999;Harrisonetal.
1999). (5) TheX-ray,UV/optical,andradioafterglowsshowchro-
However, the prompt and afterglow emission of GRB matic features. The X-rayafterglow(att &104 s) de-
060218 have many peculiarities (Campanaetal. 2006; cays with a standard temporal slope, but has a spec-
Soderbergetal. 2006; Ghisellinietal. 2006b; Butler 2006; trummuchsteeperthanthoseoftypicalGRBX-rayaf-
Liangetal.2006c): terglows. The UV/optical afterglow is quite dim and
dominatedbythethermalcomponentandtheSNcom-
(1) Thedurationofthepromptnon-thermalemissioninthe ponent. Only the radio afterglow seems rather typ-
highenergyband(15- 150keV)isδt 103s,andthus ical and explainable in the standard external shock
∼
synchrotronmodel(Sarietal. 1998; Rees&Mészáros
1 Department of Physics, Kyoto University, Kyoto 606-8502, Japan; 1992;Paczyn´ski&Rhoads1993),butdoesnotshowa
[email protected] jetbreakuntilt 22days.
2 NASA - Goddard Space Flight Center, Greenbelt, Maryland 20771, ≃
USA;[email protected]
Although such a low-luminosity event looks rare, the in-
3NationalResearchCouncil,2101ConstitutionAveNW,WashingtonDC
20418,USA trinsic event rate could be very high RLL 102 Gpc- 3 yr- 1
∼
2 Tomaetal.
compared with the local rate of typical long GRBs deduced scope(BAT;Barthelmyetal.2005)andtheX-RayTelescope
from the BATSE data, R 1 Gpc- 3 yr- 1 (Soderbergetal. (XRT; Burrowsetal. 2005a) of Swift satellite (Gehrelsetal.
LG
2006; Pianetal. 2006; Cobb∼etal. 2006; Liangetal. 2006b; 2004). We found that the non-thermal component of the
Guettaetal. 2005). For this reason, it has been actively promptemission of GRB 060218may be fitted by the Band
debated whether low-luminosity GRBs form a new GRB functionwiththespectralparameterssimilartothoseoftyp-
populationand whether they have intrinsically differentout- ical GRBs. Furthermore, we found that the light curve of
flowmechanismsandemissionmechanisms(seeStaneketal. thenon-thermalcomponentevolvessmoothlyasapower-law
2006;Ghisellinietal.2006a;Kanekoetal.2006;Amatietal. functionoftime,anditmightlastsolongdurationastocon-
2006; Strattaetal. 2006; Daietal. 2006; Wangetal. 2006). necttotheanomalousX-rayafterglowdetectedupto 106s.
∼
It has been also argued that the high energy neutrino back- This behavior contrasts with the long intermittent activities
groundfromthe low-luminosityGRBs couldbe comparable withtheX-rayflaresoftypicalGRBs,whichmightbeasign
withorlargerthanthatfromtypicallongGRBs(Muraseetal. ofthedifferenceofthecentralengines.Thisisalsosuggested
2006;Gupta&Zhang2006). by the analysis of the SN componentof this event. The SN
ForGRB060218,ithasbeenwidelysuggestedthattheout- spectrum has less broad lines than those of other GRB-SNe
flow is spherical since the radio afterglow has no jet break andlacksoxygenlines,andthelightcurveevolvessomewhat
(Soderbergetal.2006;Fanetal.2006). Iftrue,thecollapsar faster. All these facts indicate that the total kinetic energy
modelcannotbeappliedtothiseventbecausetheoutflowbe- ESNandtheejectedmassMejarebothlessthanthoseofother
comesnon-relativisticbyloadingallthematterofaprogenitor GRB-SNe. Mazzalietal. (2006) performeda detailed mod-
star. Therelativisticsphericaloutflowmightbeproducedby elingofthespectraandlightcurveoftheSNcomponentand
theoutermostpartsofthestellarenvelopethatthe SNshock obtainedESN 2 1051ergandMej 2M⊙. Thentheyhave
≃ × ≃
accelerateswhenpropagatingthroughthesteepdensitygradi- suggestedthatthe progenitorstarof thisburstwaslessmas-
entnearthestellarsurface(Colgate1974;Matzner&McKee sivethanthoseoftypicallongGRBsandcollapsedtoaneu-
1999;Tanetal.2001). However,Tanetal.(2001)hasshown tronstarinsteadofablackhole.
that the energy 1048 erg is transferred to mildly relativis- Ourgoalisshowingthatthe low-luminosityGRB 060218
tic material whe∼n the kinetic energy of the SN is E hasapromptnon-thermalemissionwithatypicalBandspec-
SN
1052erg.ForGRB060218,E isestimatedas 2 1051er∼g trumandmayoriginatefromastandardcollapsarjetpossibly
SN
(Mazzalietal. 2006), so that it is quite unli≃kely×that the driven by a neutron star. We will not discuss the emission
promptnon-thermalemissionwithE 6 1049ergispro- mechanismofthepromptnon-thermalemissionandthether-
γ,iso
ducedbythistypeoftheoutflow(seeal≃soM×atzer2003). Li mal emissions. This paper is organized as follows. In § 2,
(2006) has also shown that the energy of the thermal com- weshowtheresultsofouranalysisoftheBATandXRTdata
ponents&1049 ergistoolargetobe explainedbytheshock andsuggestseveralimplicationsforthenatureoftheprompt
breakoutintheunderlyingSN. non-thermalemission. In §3, we displayajetmodelofthis
Inthispaper, we showthatGRB 060218can beproduced event,anddiscusswhetherthejetcanmakeaholeinthestar
bythe standardcollapsarjet model. We showthatthe avail- in§4.Asummaryanddiscussionaregivenin§5.SinceGRB
ableradiodatamaybeinterpretedasanon-relativisticphase 060218isveryclose(z=0.033),wewillneglectthecosmo-
of the external shock after the jet break within the standard logicaleffect,i.e.,wesetz=0,forsimplicitythroughoutthis
model(e.g.,Frailetal.2000;Livio&Waxman2000). Wear- paper.
gue that the outflow with an initial opening angle θ0 0.3 2. THEPROMPTNON-THERMALEMISSION
andLorentzfactorΓ 5canproducethesynchrotron≃radia-
0≃ 2.1. BATandXRTdataanalysis
tionwhichexplainstheradioafterglowandiscompatiblewith
the UV/opticaland X-ray afterglow(§ 3). We also examine WeperformedtheanalysisoftheBATandXRTdatausing
whether such a wide and weak jet can penetrate a progeni- thestandardSwiftsoftwarepackage(HEAsoft6.0.4)andthe
torstarbyextendinganalyticalconsiderationsofthecollapsar CALDB 2005-11-28. The detector plane histogram (DPH)
modelbyMatzer(2003)(§4). datawereusedtoextractthespectraofBAT.Beforeproduc-
Remarkably, the detection of opticallinear polarizationin ing the spectra, we applied baterebin to rebin the DPH
the SN component of this event has been recently reported data using the most accurate non-linear energy correction.
(Gorosabeletal. 2006). This observation strongly supports Thedetectormapfordisablingnoisydetectorsintheanalysis
ourargumentsthatGRB060218arisesfromajet. wascreatedbybathotpix. Themask-weightingmapwas
Within the jet scenario, there is a possibility that createdbybatmaskwtimgusingtheopticalafterglowposi-
low-luminosity GRBs arise as typical cosmological tion.Withincludingthedetectormapandthemask-weighting
GRB jets viewed off-axis (e.g., Yamazakietal. 2003; map to batbinevt, 13 BAT spectra were extracted from
Ramirez-Ruizetal. 2005; Granotetal. 2005; Tomaetal. the DPH file which contains the data just after the space-
2005). The relativistic jet emits γ-rays into the expansion craft slew. The BAT response matrices were generated by
direction through the beaming effect. Thus, if the jet is batdrmgen. The systematic error vectors were applied to
viewed off-axis, the γ-ray flux is strongly suppressed. This theBAT spectrausingbatphasyserr. FortheXRT data,
scenario leads to similar event rates of typical GRBs and weobtainedthecleanedeventfileinthewindowtiming(WT)
low-luminosity GRBs, which seems inconsistent with the mode from the Swift HEASARC Archive. The foreground
observations (Cobbetal. 2006). Aside from this statistical was excluded by the box region of 1.2′ 0.6′. The back-
×
argument, we also examine the off-axis scenario for GRB groundregionwasextractedbythesameregionsize,butout-
060218 and conclude that this scenario is unlikely for this sideoftheforegroundsourceregion.13XRTforegroundand
event because an unrealistically high γ-ray efficiency is backgroundspectrausingthesametimeintervalsoftheBAT
required. spectrawereextracted.Theauxiliaryresponsefilewasgener-
In addition, we analyzed the data of the Burst Alert Tele- atedbyxrtmkarf.TheXRTspectrawerebinnedtocontain
aminimumof20photonsforeachspectralbin.
AcollapsarjetmodelofGRB060218 3
TheXRTandBATspectraldatawereanalyzedjointlywith Wefindthatthepromptnon-thermalemissionofthisevent
XSPEC11.3.2. Theenergyrangesusedintheanalysiswere has spectral properties similar to those of the prompt γ-ray
0.5–10keVand14.0–150keVforXRTandBAT,respectively. emissionsofGRBsdetectedsofar. Thelow-andhigh-energy
We multiplied the constant factor to the spectral model to photon indices α and β do not deviate significantly from
B B
take into account the calibration uncertainty in the response - 1.0and - 2.5,respectively(Figure1). Thesevaluesare
≃ ≃
matrices of the instruments. As reported by the several au- quitetypicalforthepromptemissionsofGRBs(Preeceetal.
thors(Kanekoetal.2006;Campanaetal.2006;Butler2006; 2000). Thus the prompt non-thermal emission may be cat-
Liangetal.2006c),theblackbodycomponentwasnecessary egorizedinto X-ray Flashes (XRFs), which are the transient
toobtainagoodfitforthelowenergypartoftheXRTspec- events with properties similar to GRBs except lower spec-
trum.Thus,weperformedthefittingwithanadditionalblack- tralpeakenergiesandsmallerfluences(e.g.,Sakamotoetal.
body (BB) componentto an absorbed Band function and an 2005). The averaged spectral peak energy 4.9 keV and
absorbed power-law times exponential cutoff (CPL) model. the isotropic-equivalent energy E 6 ≃1049 erg of the
γ,iso
≃ ×
For the purpose of this paper, we are interested in the spec- non-thermal emission obey the well-known Amati correla-
tralparametersofthenon-thermalcomponent. Toobtainthe tion, which is satisfied by the GRBs and XRFs with known
wellconstrainedspectralparametersofthenon-thermalcom- redshifts except for outliers such as GRB 980425 and GRB
ponent, we fixed the parameters of the absorption (N ) and 031203 (Amatietal. 2002, 2006). Moreover, it has been
H
theBBcomponenttotheirbestfitvaluesofeachfit,andthen reported that this event obeys the lag-luminosity correlation
calculate the uncertainties of the spectral parameters of the (Liangetal.2006d;Gehrelsetal.2006).
non-thermalcomponent.TheresultsareshowninTable1and We mayconfirmthatthespeedofthe emittingregionwas
2. Themostofthespectrawerewellfittedwithanabsorbed close to the light speed. For the emission radius r , the
0
CPLmodelwithanadditionalBBcomponentasreportedby conditionofthe transparencyforthe Comptonscatteringoff
Kanekoetal.(2006)andCampanaetal.(2006).However,we the electrons associated with baryons is given in the non-
founda significantimprovementin a fitwiththe Bandfunc- relativisticformulationby
tionforthelastfourtimeintervalsfrom1550sto2732safter
σ E
theBATtrigger. Thedifferencesinχ2 betweentheCPLand τ T iso <1, (1)
≃ 2πm c2β 2r 2
Bandfitwere9.6,11.7,17.0,and13.8in1degreeoffreedom p 0 0
for these spectra. Based on this result, we decided to adopt where E is the isotropic-equivalent kinetic energy of the
iso
the spectral parameters derived by the Band function for all outflow,σ istheThomsoncrosssection,andm isthepro-
T p
timeintervals. Thetemperatureofthe BBcomponentvaries ton mass. Here we assume thatthe outflowis dominatedby
slightlybetween0.12and0.29keV. Thisbehavioris consis- the kinetic energy, and E is larger than the photon energy
iso
tentwiththeresultoftheotheranalyses(Kanekoetal.2006; E . Estimating that r cβ δt and adopting the values
γ,iso 0 0
Campanaetal.2006;Butler2006). E 6 1049 erg and δ∼t 3000 s, we obtain β >0.9.
γ,iso 0
Figure1showstheresultof thetemporalvariationsof the This c≃orre×sponds to the Lor≃entz factor Γ > 2. For the
0
low-energy and high-energyphoton index, αB and βB. Fig- relativistic wind, the condition of the transparency is given
ure2showsthelightcurveinthe15- 150keVbandandthe by τ σ L /(8πm c3Γ 3r ) < 1 (Daigne&Mochkovitch
T iso p 0 0
spectralpeakenergyE asafunctionoftime. Atthefluxde- ≃
p 2002;Mészáros2006), whereL isthe isotropic-equivalent
iso
cayphase(t&500s),thelightcurveandspectralpeakenergy
kinetic luminosity of the wind. Estimating that L >
arewelldescribedbypower-lawfunctionsoftime,F t- 2.0 iso
andEp∝t- 1.6(dottedlines). ∝ Efaγc,tisoor/.δ(tSainncdert0h∼emΓa0x2cimδtu,mwephoobttoaninennoerlgiymditeotencttehdeiLsolroewnetzr
Investigatingthe consistency of the BAT light curvespre-
than the electron’s rest energy m c2, there is no limit on
sentedbyusandCampanaetal.(2006),wefoundsystematic e
the Lorentz factor from the photon annihilation or from the
differences especially for t >1000 s. Although both of the
Comptonscatteringoffpair-producedelectronsandpositrons
BAT light curves are based on the joint spectral analysis of
(Kanekoetal. 2006; Lithwick&Sari 2001).) Therefore the
the BAT and XRT data, Campanaetal. (2006) use the CPL
expansion speed of the outflow should be close to the light
modelwhile we use the Band function. We noticed that the
speed,althoughitisnotnecessarilyultra-relativistic.
fluxesofthelightcurveofCampanaetal.(2006)att>1000s
Allthesefactsshowthatthepromptnon-thermalemission
areinthelevelof 10mCrab.BATcandetecttheCrabneb-
∼ of this event is a typical GRB (or XRF), except that its lu-
ulaat6σ levelin1sexposure. IftheBATsensitivitycanbe
minosity is extremely low and it is accompaniedby the soft
scaledasasquarerootoftheexposuretime(Markwardtetal.
X-raythermalcomponent. Thisisoneofourmotivationsfor
2005), we need the exposure time of 104 s to detect the
∼ considering that GRB 060218 arises from a jet like typical
10 mCrab source assuming the Crab-like spectrum. On the
GRBs in § 3 and 4. In the rest part of this section, we fo-
other hand, the fluxes of our light curve are in the level of
cus on the decay phase of the promptnon-thermalemission
100mCrabatthelate phase. Thisflux levelisreasonable
∼ tosuggestthatitsemissionprocessdoesnotstopabruptlybut
todetectin 100sexposurebyBAT. Thus,webelievethat
∼ decaysslowlytoconnecttothedetectedanomalousX-rayaf-
thereisasystematicerrorbyusingtheCPLmodelintheBAT
terglow.
fluxcalculationpresentedbyCampanaetal.(2006). Wealso
believethatthisinvestigationstrengthensourconclusionthat 2.2.2. Thedecayphase
theBandfunctionisthebestrepresentedspectralmodelofthe
The decay phase of the prompt non-thermal emission is
non-thermalemissionofGRB060218.
well described by F(t) t- 2.0 and E (t) t- 1.6, which im-
p
∝ ∝
2.2. Implicationsforthenatureofthepromptnon-thermal pliesahardness-intensitycorrelationF ∝Ep1.3. Suchacorre-
emission lation,F E ζ,canbeseeninthedecayphasesoftheGRB
p
∝
prompt emissions observed by BATSE, which has a broad
2.2.1. Overallspectralproperties
distribution of the power-law indices ζ, with 0.6 . ζ . 3
4 Tomaetal.
(Borgonovo&Ryde 2001; Ryde&Petrosian 2002). Swift but to slowly decaying emission processes. The first possi-
satellitehassucceededinobservingthedecayphasesofmany bility of the decaying emission is the synchrotron emission
GRBpromptemissionsmoredeeplythantheobservationsin from the external shock (Rees&Mészáros 1992; Sarietal.
thepre-Swiftera.Theresultisthatthedistributionofthetem- 1998). Since the high-energyrange of the spectrum will be
poral indices of the flux is also broad; - 5.α.- 1, where above the cooling frequency, the high-energy photon index
F tα (O’Brienetal. 2006). Thus the flux decay of the β - 2.5correspondstotheenergydistributionindexofthe
B
∝ ≃
prompt non-thermal emission of GRB 060218 is relatively accelerated electrons p 3. Then the flux should decay as
shallow. F(t) t12- 43p t- 1.75.Th≃echaracteristicfrequencyνmevolves
First, we argue that the decay phase is not attributed ast- 3∝/2. The∼se temporalbehaviorsare not inconsistentwith
to the kinematical effect due to the curvature of the emit-
theresultsofouranalysis.However,thestandardmodelgives
ted region which ceased the emission process suddenly.
muchlowercharacteristicfrequencyν thantheobservedone
m
Since the emitting shell should have a curvature, the ob- 1018 Hz at the decelerationtimet 500 s with the X-ray
server receives the radiation far from the line of sight af- ∼luminosityL 1047 ergs- 1. Theref≃orethismodelisnotso
X
ter the cessation of the emission process. The region at ∼
appealing.
higher latitude from the line of sight has a lower veloc-
The second possibility is thatthe decayingemission is at-
ity towards the observer, so that the emission becomes
tributed to the central engine activity. The decaying flux in
dimmer and softer progressively because of the relativistic 15- 150keV range, if extrapolatedas F(t) t- 2.0, becomes
beaming effect. This effect is the so-called curvature ef- 10- 11ergs- 1att 104s. Thisfluxiscom∝parabletothatof
fect, and have been widely studied for the decay phases of ∼thedetectedanomal∼ousX-rayafterglowin0.3- 10keVrange,
GRB prompt emissions (Fenimoreetal. 1996; Sari&Piran decaying as F t- 1.1 with the photon index β - 3.2,
X X
1997; Kumar&Panaitescu 2000; Ryde&Petrosian 2002; ∝ ≃
which cannot be explained within the external shock syn-
Dermer 2004). Swift observations have suggested that the
chrotronmodel(Campanaetal.2006;Soderbergetal.2006).
decay phases of GRB prompt emissions with the tem-
If the anomalous X-ray afterglow is a continuation of the
poral indices α < - 2 are due to the curvature effect
prompt non-thermal emission (i.e., the X-ray emission de-
(Liangetal.2006a; Zhangetal. 2006a;Lazzati&Begelman tectedatt&104sisnotanafterglowbutthedecayingprompt
2006;Yamazakietal.2006;Kobayashi&Zhang2006).
emission), the prompt non-thermal emission of this burst is
Suppose that the shell moving with a bulk Lorentz factor very long (>106 s) and it is necessary that the photon in-
Γ0 ceasestheemissionataradiusr0 andatatimet0 andthat dex varies from β - 2.5 to β - 3.2. Ghisellinietal.
B X
the line of sight is within the shell, i.e., the observer views ≃ ≃
(2006b) have also suggestedsuch a scenario and shownthat
the shell on-axis. The comoving frequencyν′ is boosted to
a synchrotron inverse-Compton model could reproduce the
ν=ν′ intheobserver’sframe,where =[Γ (1- β cosθ)]- 1
0 0 UV/optical thermal component as well as the prompt non-
D D
istheDopplerfactor. Becausetheobservedtimeisgivenby
thermal emission and the anomalous X-ray afterglow. Here
t=t0- r0cosθ/c,theDopplerfactorisrelatedtotheobserved we note that the flux decays steeper than t- 1 after the peak
timeby
time t 500 s, so that the time-integrated radiation energy
r
= 0 (t- t )- 1, (2) doesno≃tincreasesomuchafterthepeaktime. Thusthetotal
D cβ Γ r
0 0 radiationenergyofthepromptnon-thermalemissionremains
twhheecreentrtr≡alte0n-grin0/e(.cTβ0h)uisstthheetdimepea-rrteusroelvtiemdesopfecthtreaslhpeelalkfreonm- EγT,ishoe≃ref6o×re1t0h4e9eenrgg.ine of this event could be active for at
ergyevolvesas least 106 s. Recent detailed observations of GRB X-ray af-
E (t- t )- 1. (3) terglows have revealed that a number of bursts show erratic
p r
∝D∝ X-rayflares,whichmaybelinkedtotherestartingsofthecen-
The received flux in a given time interval dt is Fνdt tralengineafterthepromptemissionphase(e.g.,Zhangetal.
jν′′D2dΩ, where jν′′ isthecomovingsurfacebrightness,an∝d 2006a; Burrowsetal. 2005b; Iokaetal. 2005). Since some
thesolidangleoftheemittingregionisrelatedtotheobserved X-ray flares are discovered 105 s after the trigger, the en-
time intervalby dΩ=2πd(cosθ) dt (Rybicki&Lightman ginesoftypicalGRBscould∼bealsoactivefor&105s. How-
∝
1979;Granotetal. 1999;Ioka&Nakamura2001). Thusthe ever,theyarespikyandintermittentactivities,whilethisevent
receivedfluxisestimatedby showsthesmoothandpower-lawactivity. We speculatethat
this indicates that the engine of this event is different from
Fν ∝ jν′′D2∝(ν′)1+βBD2∝ν1+βBD1- βB ∝ν1+βB(t- tr)- 1+βB, those of typical GRBs which are believedto be black holes.
(4)
Although this is just a speculation, it agrees with the analy-
bwehcearuesweethaess1u5m- e1t5h0ehkiegVh-beannedrgiysBabanodvesptheectrpuemakjeν′n′e∝rgνy′1E+βB. sis of the SN component of this event, which suggests that
p the progenitorstar was less massive and collapsed to a neu-
WiththeobservedvalueβB - 2.5,wehaveF(t) (t- tr)- 3.5 tronstarinsteadofablackhole(Mazzalietal.2006).Further
and Ep(t) (t- tr)- 1. To re≃produce the tempora∝l index ob- discussionwillbepresentedin§5.
∝
tained from our analysis, t <0 is required for F(t), while
r
in contrast, t >0 is required for E (t). Therefore F(t) and 3. AJETMODELOFTHERADIOAFTERGLOW
r p
E (t)cannotbefittedsimultaneouslybythismodel. Onecan The afterglow of this burst shows chromatic light curves.
p
alsoconsiderthestructuredjetmodelin whichthe spectrum TheUV/opticalafterglowisquitedimanddominatedbythe
andbrightnessvaryintheangulardirection(Rossietal.2002; thermal component and the SN component (Campanaetal.
Zhang&Mészáros 2002; Yamazakietal. 2006; Dyksetal. 2006; Ghisellinietal. 2006b). The X-ray afterglow has a
2006). However,theshallowerfluxdecayrequiresajetwith spectrum much steeper than those of typical GRB X-ray af-
brighterrim,whichseemsimplausible. terglows (Soderbergetal. 2006; Butler 2006), and it could
Thus the emission process is attributed not to the curva- be a continuation of the prompt non-thermal emission (see
ture effect of the shell which ceases the emission abruptly, § 2.2.2). Only the radio afterglow seems rather typical and
AcollapsarjetmodelofGRB060218 5
canbeexplainedbythe standardexternalshocksynchrotron mediumn 102 cm- 3,andtheratiosoftheacceleratedelec-
∼
model (Soderbergetal. 2006; Fanetal. 2006). In this sec- tronsenergyandthemagneticenergytotheshockedthermal
tion,weshowthattheavailableradiodatamayarisefroman energyǫ 10- 2 andǫ 10- 3,respectively. Withthesepa-
e B
∼ ∼
initially collimated outflow. In the spherical outflow model, rameters, the Lorentz factor of the outflow is estimated as
it seems difficult for a SN shock to provide the relativistic Γ 2 (t/1 day)- 3/8. If the opening angle of the outflow is
≈
sphericalejectawithsufficientenergy,asstatedin§1. Inthe θ 1,theoutflowwillbegintoexpandsidewaysatt 6days
0
≤ ≤
jetmodel,takingintoaccounttheUV/opticalandX-rayafter- andtheradiolightcurveswillbreak(Rhoads1999).However,
glows,wewillconstraintheinitialopeningangleandLorentz theactuallightcurvedoesnotshowsuchabreak,sothatitis
factorofthejetasθ 0.3andΓ 5,respectively.Wealso concludedthatθ >1inthismodel,i.e.,theoutflowisspher-
0 0 0
≃ ≃
showthatthe off-axisscenarioin the jetmodelis less likely ical.Inthiscase,theoutflowbecomesnon-relativisticatt 6
because it requires higher γ-ray efficiency than the on-axis days,andtheradiofluxvariesfromt- 0.85tot- 1.1. Thisisc≃on-
scenario. sistentwiththeavailableradiolightcurves(Fanetal.2006).
The parameters E 1048 erg, n 102 cm- 3, ǫ
k,iso e
3.1. Theradioafterglow 10- 1, and ǫ 10- 1 also∼satisfy the abov∼e three constrain∼ts
B
For 2- 22 daysthe afterglowflux at 8.46GHz is well de- and fit the r∼adio data (Soderbergetal. 2006). However,
tectedandshowsthepower-lawdecayast- 0.85. Thespectrum since the isotropic-equivalent γ-ray energy is E 6
γ,iso
≃ ×
att 5dayshasapeakbetween1.43GHzand4.86GHz,and 1049 erg, the inferred γ-ray efficiency is extremely high;
this≃peakcan be interpretedas the self-absorptionfrequency η E /(E +E ) 98%,andthusthisparameterset
γ γ,iso γ,iso k,iso
νa in the standard model. This peak would not be the typi- isim≡plausible. ∼
calsynchrotronfrequencyν becauseofthesmall1.43GHz The relativistic spherical outflow is not consistent with
m
flux.Thusthe8.46GHzismostlikelyinthefrequencyrange the standard collapsar scenario, in which a relativistic col-
νm<νa<ν<νc,whereνcisthecoolingfrequency. limated outflow digs a narrow hole in the progenitor star
In the standard model, the temporal index of the flux at without loading so much stellar matter (Woosley 1993;
each frequency range can be calculated for several cases; MacFadyen&Woosley1999). Thusonemayconsiderasce-
forarelativistic/non-relativisticblastwaveexpandingintocir- narioin which a sphericaloutflow is accelerated to a mildly
cumburst medium of constant/wind-profile density, or for a relativistic speed as the SN shock sweeps the stellar enve-
sideways expandingrelativistic blastwave. The temporalin- lope with steep density gradient (Matzner&McKee 1999;
dex of the flux at the frequencyrange νm <νa <ν <νc for Tanetal. 2001). However, it seems impossible that the re-
eachcaseis- 3p+3 (relativisticblastwave,constantdensity), quiredkinetic energyE 1050 ergis transferredinto the
4 4 k,iso∼
- 3p+21 (non-relativisticblastwave,constantdensity),- 3p+ relativistic ejecta by the SN with the total kinetic energy
12(rela1ti0visticblastwave,wind-profiledensity),- 7p+5 (n4on- ≃1051erg(Mazzalietal.2006),asdiscussedin§1. Forthis
r4elativisticblastwave,wind-profiledensity),and- 6p(sid6eways reason,weconcludethatthefirstpossibilityisquiteunlikely.
expanding), where p is the index of the energy distribution
function of the accelerated electrons (e.g., Livio&Waxman 3.1.2. Non-relativisticblastwavemodel
2000;Sarietal.1998,1999;Chevalier&Li1999;Frailetal. Now we investigate the second possibility; the radio af-
2000). Foreachcasetoreproducetheobservedtemporalin- terglow may arise from the non-relativistic phase of an ini-
dex - 0.85, p is required to be 2.1,2.0,1.5,1.4, and 0.9, tially jetted outflow with p 2.0 expanding into the con-
respectively. Thevalues p&2ar≃etypicalfortheGRBafter- stant density medium. The≃relativistic jet is decelerated
glows(e.g.,Panaitescu&Kumar2002;Yostetal.2003),and and subsequently expands sideways, and finally becomes
forthisreason,wefocusonthefirsttwopossibilities(i.e.,the non-relativistic at a certain transition time t . The transi-
s
constantdensitycases). Note, however,thatthepossibilities tion time is estimated by requiring that the speed of the
p<2 (i.e., the wind-profile density cases and the sideways Sedov-Taylor blastwave is close to the light speed, i.e., β =
expandingcase)cannotberuledout. 2[E /(nm t3)]1/5 1, where E is the kinetic energyofthe
Forthetwopossibilitiesofthestandardmodel,theavailable 5c k p ∼ k
sphericalblastwave(seeLivio&Waxman2000):
radiodatarequiresthefollowingthreeconditions:
(a) Thespectrumpeaksatνa∼4×109Hzatt≃5days. ts≃7.5×106s(cid:18)Ekn,51(cid:19)1/3. (5)
(b) Thefluxat22.5GHzis 0.25mJyatt 3days.
∼ ≃ Here(andhereafter)wehaveadoptedthenotationQ=10xQ
(c) The coolingfrequencyis ν 5 1015 Hz so that the x
c ≤ × incgsunits. Sincetheavailable8.46GHzlightcurvedoesnot
synchrotron spectrum of the external shock electrons showthesidewaysexpandingrelativisticphaset- p t- 2,we
doesnotdominatethedetectedanomalousX-rayafter- ∼
requiret .2days.
glow. s
Three constraints on the physical parameters of the after-
glow are given by the conditions (a)- (c) in the first part of
3.1.1. Relativisticblastwavemodel
thissubsection(§3.1). AssumingthattheminimumLorentz
Fanetal. (2006) have modeled the first possibility, i.e., factor of the accelerated electrons is γ 10- 1β2ǫ m /m ,
m e p e
thesynchrotronspectrumfromtherelativisticblastwavewith wemayderivethefollowingthreeconstrai≃ntsonthephysical
p 2.1expandingintotheconstantdensitymedium,andde- parameters:
≃
rived three constraints on the physical parameters of the af-
terglowfromtheabovethreeconditions(a)- (c)(theirEqua- ǫe,- 11/3ǫB,- 21/3Ek,511/3n1/3 1, (6)
gtieosntsed(8a)re- t(h1e0)i)s.otrTohpeics-aetqiusfivyainlegntpakrianmeteicteernsetrhgeyyohfatvheesouugt-- ǫe,- 1ǫB,- 20.75Ek,511.3n0.45∼3∼×10- 3, (7)
flowEk,iso 1050 erg,thenumberdensityofthecircumburst ǫB,- 21.5Ek,510.6n0.9 0.4. (8)
∼ ≥
6 Tomaetal.
The results are similar to those derived in the relativistic jetisdeterminedby
blastwave model (see § 3.1.1; Fanetal. 2006). These con-
straintsandtherequirementoft .2 105saresatisfiedby t 1/2
Ek≃2×1048erg,n≃102cm- 3,sǫe≃×10- 1,andǫB≃10- 1. θ0(cid:18)tsj(cid:19) ≃1. (9)
Then the isotropic-equivalent kinetic energy before side-
ways expansion is estimated by E = 2E /θ 2 4 Whent 2 105 s andt 2 104 s are adopted, we ob-
k,iso k 0 s j
1048 θ0- 2 erg. To obtain a reasonable γ-ray effi≃cienc×y tainθ0≃≃0.3.×Ifθ0 issmalle≃r,th×eratiots/tj islarger. Atthe
η = E /(E +E ) . 0.5 (Lloyd-Ronning&Zhang Blandford-McKeephase,theLorentzfactorofthejetevolves
2γ004;Ioγk,isaoetaγl.,i2so006;kF,isaon&Piran2006;Granotetal.2006; as Γ t- 3/8, and finally becomes Γ θ - 1 at the jet-break
0
∝ ≃
Zhangetal.2006b),theopeningangleθ0.0.3isfavorable. timetj. ThustheinitialLorentzfactorofthejetisdetermined
Inthefollowingsection,wewillfurtherconstraintheopening by
angleandalsotheLorentzfactorofthejetusingtheobserved t - 3/8
UV/opticalandX-rayemission. Γ j θ - 1. (10)
0(cid:18)t (cid:19) ≃ 0
dec
Whent 2 104 sandt 6 103sareadopted,weob-
j dec
tain Γ ≃5.×If Γ is larger,≃the×ratio t /t is larger. The
0 0 j dec
≃
3.2. ConstraintsfromtheUV/opticalandX-rayafterglow upperboundof the opticaldata doesnotallow the valuesof
In our model(E 2 1048 erg,n 102 cm- 3,ǫ 10- 1, ts/tj andtj/tdec largerthanthosewehaveadopted,andthere-
and ǫ 10- 1), thke≃cha×racteristic par≃ameters at t e ≃5 days foreitdoesnotallowthesmallerθ0 andthelargerΓ0. Since
are caBlc≃ulated as F 2 10- 25 erg cm- 2 s≃- 1 Hz- 1, theconsiderationabouttheγ-rayefficiencyrequiresθ0.0.3
ν,max ∼ × (see§3.1.2),theopeningangleisrestrictedtoanarrowrange
1νm013≃H5z.× 1T0h6usHtzh,e νoapt≃ical4 ×νFν109fluxHza,t athnids tνimc e≃is7 × aroInunadsθu0m≃m0a.r3y., we have obtained a jet modelfor the radio
1015Fν,max(νc/νm)- 0.5(1015/νc)- 1≃4×10- 15ergcm- 2 s- 1.∼If afterglow of GRB 060218, with θ0 0.3, Γ0 5, Ek,iso
wetracebacktoearliertimesthroughthestandardjetmodel, ≃ ≃ ≃
4 1049 erg, and η 0.6, which is compatible with the
the optical flux gets much higher than this value (see Fig- × γ ≃
UV/optical and X-ray data. The collimation-corrected en-
ure 3). The detectedopticalflux at 104- 105 s is dominated
by the thermal component with νFν ∼ 10- 11 erg cm- 2 s- 1 esurgmyinogftthhaetjtehteidsuErajt≃ion(Eoγf,tishoe+pErok,misop)θt0n2o/n4-t≃he1rm04a8leermgi.ssAiosn-
(Ghisellinietal. 2006b; Campanaetal. 2006). The require-
δt 103 sistheactivetimeofthecentralengineδT,thelu-
mentthatthesynchrotronfluxfromourjetshouldnotexceed mi∼nosityofthejetisestimatedbyL E /δT 1045ergs- 1.
thisfluxgivesconstraintsontheinitialopeningangleθ0 and j∼ j ∼
theinitialLorentzfactorΓ ofthejet.
0
Theexternalshockofthejetprogressivelyexperiencesthe 3.3. Off-axisscenario
Blandford-McKee evolution, the sideways expanding evo- Intheaboveargumentsin§2and3,wehaveassumedthat
lution, and the Sedov-Taylor evolution. For each evolu- theobserverviewsthe jetfromthe on-axisdirectionandthe
tion phase the characteristic frequencies of the synchrotron luminosityofthepromptnon-thermalemissionisintrinsically
spectrum with p 2.0 vary as νm t- 1.5,t- 2, and t- 3; low. ItispossiblethatthelowluminosityofGRB060218is
ν (> ν ) t- 2/3,t≃- 0.5, and t- 2/3; ν (<∝ν ) t0,t- 0.2, and attributed to a typical bright GRB jet viewed off-axis (e.g.,
a m a m
t1.2; and ν∝ t- 0.5,t0, and t- 0.2 (e.g., Sarieta∝l. 1998, 1999; Yamazakietal.2003;Ramirez-Ruizetal.2005;Granotetal.
c
Livio&Wax∝man2000). Ifwetracebacktot 104s,ν ,ν , 2005; Tomaetal. 2005). However, we find that the off-axis
m a
∼
and ν all remain lower than the optical band. Thus for scenario is less likely than the on-axisscenario since higher
c
t&104s,theopticalbandisattherangeν>ν ,andtheopti- γ-rayefficiencyisrequired.
c
calenergyfluxevolvesast- 1 attheBlandford-McKeephase, Since the outflow is spherical when the radio afterglow is
t- 2 at the sideways expanding phase, and t- 1 at the Sedov- observed,theoff-axisscenarioisthesameastheon-axisone
Taylor phase. With these behaviors of the optical flux from inthisphase. Thustheisotropic-equivalentkineticenergyis
the external shock, we can suggest a following possible jet determinedbyE =2E /θ 2 4 1048θ - 2erg.Intheoff-
k,iso k 0 0
scenario (see Figure 3 for a schematic picture of the optical axis scenario, the isotropic-equ≃iva×lent γ-ray energy E
γ,iso,on
lightcurve). The external shock of the jet is decelerated at measured if viewed on-axis is larger than the observed one
tadnecd≃sh6ift×st1o0t3hes,Sbeedgoivn-sTtaoyeloxrpeaxnpdasnisdieownaaytstatt2j ≃120×5s1.0T4hse, Ehiγg,ihsoer≃th6a×n1th0a49t eersgti.mSaotetdhewγi-trhaiynetfhfiecoienn-cayxiissescxepneacrtieod(teo.gb.e,
s
peakopticalfluxisνF 3 10- 12,whichis≃less×thantheob- Tomaetal.2006).
ν
≃ ×
servedone,sothattheopticalafterglowestimatedwithinour To obtain the smaller γ-ray efficiency, we could consider
jetmodelisnotinconsistentwiththeavailableopticaldata. θ <0.3. However, the smaller θ leads to even higher γ-
0 0
Since the X-ray band is also above ν and p 2.0, the rayefficiency,aswe willsee below. Thesmallerθ requires
c 0
X-ray νF flux is comparable to the optical one.≃Thus this t <104 s, andthentheopticalflux,ifreceivedfromtheon-
ν j
external shock emission does not overwhelm the observed axisdirection,exceedstheconstraintfromthedetectedoptical
anomalous X-ray afterglow shown in Campanaetal. (2006) afterglow (see Figure 3). To escape the constraint, the syn-
andSoderbergetal.(2006). chrotronflux fromthe off-axisjet shouldpeak at some time
Now we give the reasonable values of θ and Γ . At t aftert . The peak occurswhenthe line ofsight entersthe
0 0 v j
the sideways expansion phase, the opening angle of the jet relativisticbeamingconeoftheemissionfromtheedgeofthe
evolvesasθ=Γ- 1 t1/2(Rhoads1999;Sarietal.1999),and sidewaysexpandingjet,i.e.,θv θ+Γ- 1 2Γ- 1. TheLorentz
finally becomes θ∝1 at the transition time t to the Sedov- factorvariesasΓ t- 1/2inthe∼sideways≃expansionphaseand
s
Taylorexpansionp≃hase. Thustheinitialopeningangleofthe becomesΓ 1at∝t ,andthusthefollowingequationissatis-
s
∼
AcollapsarjetmodelofGRB060218 7
fied: ofthemomentumfluxintheframeofthejethead,
1 t 1/2
θ s 1. (11) w Γ 2Γ 2(β - β )2+p =w Γ 2β 2+p . (13)
2 v(cid:18)t (cid:19) ≃ j j h j h j a h h a
v
Herethesubscripts j andhdescribethejetandthejethead,
Since tv should be greater than 104 s, we obtain θv & 0.4. respectively, and w( e+p), p, and e are the enthalpy, the
Theisotropic-equivalentγ-rayenergymeasuredifviewedon- ≡
pressure,andtheenergydensityincludingrestenergydensity
w1ax0ei4s7oibΓst0ar6oinuegrEgh,lywehsetri>emwa1tee0d4h7abθvye-E6uγse,eisrdog,o.θnvF≈>or[θΓθ00.(θ<Wv-0it.θh30,Γ)]t0h6Ei≥sγ,lθies0oa-d>1s, ρρcaIc2f,2Γraersepisgeocstoiigdvneailfipycp.arFonoxtlriymsltaaetrligloaenrrset.hnvaneloupneitsy,,ppa≪canρabce2naengdlewcate=d
γ,iso,on 0 0 j j
tohigherγ-rayefficiencythantheon-axisscenario;ηγ >0.7. inthe left-handside ofEquation(13), andwe obtain(βh- 1-
Thereforewe may conclude that the off-axisscenario is un- 1)- 2=w Γ 2/(ρ c2)=L /(4πr2ρ c3),whereatthelastequal-
likelyforthisevent. j j a iso a
ityweintroducetheisotropic-equivalentluminosityL ofthe
iso
4. ACOLLAPSARMODELOFGRB060218 jet. Forthejettodrivetheexplosionofthestar,thisequation
should be satisfied when r=R, where R is the radius of the
moIndetlhtehaptriesvcioonussissteecnttiowni,thwtheehaavvaeilaobblteaainfteedrgalopwodssaitbal.eOjuert progenitorstar,sothat(βh- 1- 1)- 2≈10- 3Liso,51(R/R⊙)- 2ρa- 1.
Thisfactorismuchlessthanunityfortheparametersoftyp-
jetmodelrequiresanopeningangleθ 0.3,whichissome-
0≃ ical GRBs (and so for the low-luminosity GRBs), so that
what larger than those of typical cosmological GRBs (e.g.,
the jet head is found to be non-relativistic. Thus, with the
see Ghirlandaetal. 2004), and an extremely small luminos-
ity L 1045 erg s- 1, which is about105 times smaller than collimation-correctedluminosityofthejetLj=wjΓj2cπr2θ2,
j ∼ weobtain
those of typical cosmological GRBs. Then one may won-
L 1/2
der whether such a wide and low-luminosity jet can pene- j
β . (14)
trate a progenitorstar. It is possible that a jet becomesnon- h≈(cid:18)πr2θ2ρ c3(cid:19)
a
relativisticifitloadssomuchstellarmatterwithalargecross
Wetakeρ M/(4πR3/3)forsimplicity,whereMisthetotal
section.Matzer(2003)hasdiscussedanalyticallyseveralcon- a
∼
massoftheprogenitorstar.
ditionsformakinga holeinthe collapsarmodel. We extend
his theoretical considerations to conclude that an adiabatic
4.2. Cocoonstructure
coldjetisexcluded,butanon-adiabatichotjetisappropriate
for this event. (Our argumentcan be also applied to typical Thecocoondrivesashockintothestellarenvelope,which
GRB jets, and some wide GRB jets also would need to be expands non-relativistically at the velocity βc given by the
hot.) transverse balance of the cocoonpressure and the ram pres-
sureofthestellarenvelope,
4.1. Themotionofajethead
p =ρ c2β 2. (15)
Thestandardcollapsarmodelassumesthatablackholeora c a c
neutronstarwithanaccretiondiskisformedafteranironcore The pressure of the cocoon p is roughly constant away
c
of the massive progenitor star collapses and that the system from the jet head, and is approximated by E /(3V) with
in c
producesajet(Woosley1993;MacFadyen&Woosley1999). the thermal energy E deposited in the cocoon and the co-
in
Since the free-fall timescale of a stellar envelope is longer coon volumeV . Since the jet head is non-relativistic, most
c
than,orcomparablewith,thecrossingtimescaleforthejetto of the kinetic energy of the jet gets thermalized through
propagatewithintheprogenitorstarandtohititssurface,the the reverse shock. The cocoon energy is the energy caught
stellarenvelopewoulddisturbtheprogressofthejet. by the jet head, which is calculated by E = rL dr
Consider the progressof a relativistic jet outward through L r . The cocoon length is given by r in cβR0t ajscβlhon≃g
the stellar envelope. Two shocks form: a reverse shock re- jcβh ≃ h
as β > β is satisfied, and the cocoon width is given by
ducing the jet speed and increasing its internal energy; and h c
R cβ t.SothecocoonvolumeisroughlyV (π/3)R 2r
a forward shock that propagates into the surroundingstellar c c c c
≃ ≃ ≃
envelopegivingitinternalenergy. Asaresult,therearefour (π/3)r3βc2/βh2.Thereforethecocoonpressurecanbewritten
distinctregionsin this system: the propagatingjet; the head byp L β /(πr2cβ 2).Equation(15)givesusthefollowing
c j h c
≃
ofthejetlyingbetweenthetwoshocks;thestellarenvelope; equations:
anda cocoonconsistingofshockedjetandshockedambient
material(seeFigure2ofMatzer2003).Thissystemissimilar Lj 1/4
β = β , (16)
toclassicaldoubleradiosources(Begelman&Cioffi1989). c (cid:18)πr2ρ c3 h(cid:19)
a
Two conditionsare requiredfor a jet to break out the star
relativistically. First, the Lorentz factor of the jet head Γ Lj 1/2
h p = ρ cβ , (17)
should be smaller than the inverse of the opening angle of c (cid:18)πr2 a h(cid:19)
the jet θ. If so, the shocked material may escape sideways
which describe the velocity and the pressure of the cocoon
to form the cocoon, and the jet may avoid baryon loading.
withtheparametersofthejetandthestellarenvelopewithin
Second,thespeedofthejetheadshouldbelargerthanthatof
theassumptionofβ >β .
theexpandingcocoon, h c
βh>βc. (12) 4.3. Coldjet
If this condition is violated, the cocoon expands spherically Here we examine an adiabatic jet that propagatesballisti-
aroundthe jet and finally explodethe star, producinga non- callyin theprogenitorstar (see also Mészáros&Rees 2001;
relativisticdensesphericaloutflow.Whentheabovetwocon- Waxman&Mészáros 2003). An important point is that the
ditionsaresatisfied,wemayconsiderthelongitudinalbalance openingangleand the Lorentzfactorof the coldballistic jet
8 Tomaetal.
do not change after exiting the progenitor star. If the open- theenginehastostopsuddenlyjustafterbreakingoutthestar,
ingangleandLorentzfactorofthejetatthebreakoutareθ andthisseemsunlikely.
br
and Γ , they are also θ =θ and Γ =Γ after exiting the Forthehotjet, p ρ c2 andw =4p aregoodapproxi-
br 0 br 0 br j j j j
≫
star. Incontrast,ahotjetwhichpropagatesnon-adiabatically mations,andthusp =L /(4Γ 2cπr2θ2).Beforethebreakout,
j j j
by the interactionbetween the jet and the cocoonand keeps theopeningangleofthehotjetisdeterminedbythetransverse
dominated by the internal energy changes its opening angle pressurebalancebetweenthejetandthecocoon:
andLorentzfactorthroughthefreeexpansionoutsidethestar.
Thehotjetwillbediscussedinthenextsubsection. pj=pc. (20)
For the jet to drive an explosion, the consistency β >β
h c UsingEquations(14)and(17),weobtain
shouldbesatisfied. FromEquations(14)and(16),thisleads
totheconstraintontheopeningangleofthejet, L 1/6
θ= j Γ - 4/3, (21)
Lj 1/6 (cid:18)256πr2ρac3(cid:19) j
θ< . (18)
(cid:18)πr2ρac3(cid:19) which satisfies the consistency of βh > βc (Equation (18)).
ForGRB060218,adoptingthevaluesL 1045 ergs- 1,R
For GRB 060218, the underlying SN is Type Ic, which j
implies that the progenitor star is a C/O Wolf-Rayet star 1011 cm, ρa 1 g cm- 3, and Γj 3, we∼obtain the openin∼g
(e.g., Mazzalietal. 2006). Its radius and mass are typically angleattheb∼reakouttimeasθbr ≃3 10- 3.
∼ ×
R 1011 cm and M 1034 g, respectively. Then the av- The velocities of the jet head (Equation (14)) and the co-
era∼ged mass density ρ∼ is 1 g cm- 3. For the luminosity coon(Equation(16))arecalculatedbyusingEquation(21),
a
L 1045ergs- 1,theopeni∼ngangleofthejetatthebreakout
(i.je∼.,whenr=R)isconstrainedtoθbr<0.03,sothatθ0<0.03 β = 16Lj 1/3Γ 4/3, (22)
within the cold jet scenario. The valueof the openingangle h (cid:18)πr2ρ c3(cid:19) j
a
suggestedinourjetmodel(θ 0.3)violatesthisconstraint.
Thereforethecoldjetscenari0o≃isexcludedforthisevent. β = 2Lj 1/3Γ 1/3. (23)
Awidecoldjetmaybeallowedifthejetisbeinglaunched c (cid:18)πr2ρac3(cid:19) j
longafterthecocoonexplodesthestar,sincetheambientden-
Theratioofthetwovelocitiesisβ /β =2Γ . Then,ifΓ is
sity ρ dropsthroughthe expansionin Equation(18). How- h c j j
a close to unity, we have β β and hence the cocoon ex-
everthestarmustexpandfromRc∼1011cmto∼1017cmin pands quasi-spherically. Tch∼is shituation is in a good agree-
orderthatθ 0.3isallowed,whichisunlikely.
0≃ ment with recent series of numerical simulations performed
byMizutaetal.(2006). Theyhavearguedthatthemorphol-
4.4. Hotjet
ogyoftheexplosiondependsontheLorentzfactorΓ ofthe
j,0
Next we consider the possibility that the jet is dominated jet given at the inner boundary: when the Lorentz factor is
bytheinternalenergythroughoutthepropagationinthepro- high(Γ &3), thehighpressurecocooncollimatestheout-
j,0
genitorstar (see also Lazzati&Begelman2005). Atthe jet- flow to form a narrow,relativistic jet; and when the Lorentz
cocoonboundary,obliqueshocksandshearinstabilitiesmay factorislow,onthecontrary,theoutflowisnotcollimatedand
occur and dissipate the kinetic energy of the jet. This sit- expandsquasi-spherically.
uation can be seen in several numerical simulations of the The breakout timescale is determined by t = R/(cβ ),
br h,br
collapsar model (Aloyetal. 2000; Zhangetal. 2003, 2004; whereβ isthevelocityofthejethead(Equation(22))ob-
h,br
Umedaetal.2005;Mizutaetal.2006). tainedatr=R:
In contrast to the cold jet, the hot jet expandsfreely after
ebxyittihnegtLhoerestnatrz. fAacstaorreΓsuj,eltx, tjhuestobpeefnoirnegeaxnigtilnegisthdeetsetramr.ineInd tbr≃(cid:18)1π6ρLaj(cid:19)- 1/3R5/3Γj- 4/3. (24)
thecomovingframeofthehotmaterialmovingoutwardrela-
tivisticallywiththeLorentzfactorΓj,ex,thehotmaterialwill ForGRB060218,weobtaintbr∼300s,whichisareasonable
expandfreely and finally get the Lorentzfactor Γ′ =e/ρ c2. value. Thehotjetofthiseventtakestbr 300stobreakout
j the progenitor star and lasts further δT∼ 103 s to eject the
Inthelaboratoryframe,thematerialwillbebeamedintothe
∼
materialproducingthepromptemission.
openingangle
θ Γ - 1, (19)
0∼ j,ex 4.4.2. Apossibleadditionalwideningeffect
and the Lorentz factor of the material is given by Γ
0 After the breakout time, the opening angle of the jet be-
Γ′Γ (1+β′β ) 2Γ′Γ . Note that Γ θ 2Γ′ >1∼is
satisjfi,eexd. j,ex ∼ j,ex 0 0 ∼ f3ore1e0x-i3ti,ngwthhiechstiasrmdeitgehrmtbineceodmbeywthideerprthesasnutrheebvaallauneceθbrb∼e-
ForGRB060218,Γj,ex∼θ0- 1≃3isrequiredtoreproduce tw×eenthejetandcocoon.Thecocoonexpandsfreelyoutward
θ 0.3. For the initial Lorentz factor of the jet exiting the
0 from the star after the breakouttime, and finally the cocoon
star≃tobeΓ 5,Γ′=e/ρ c2 1- 2isfavorable. Therefore
0≃ j ∼ pressurebecomesnegligibleinR/(c/√3) 10s. Thecocoon
the jet of this burst may originate from a mildly hot jet in ∼
shockthatwillsweepthestellarenvelopeisnotsostrong,and
whichtheinternalenergyeiscomparablewiththerestenergy
theenvelopewouldremaincold.Afterthecocoondisappears,
ρ c2beforeexitingthestar.
j thejetopeninganglewouldbedeterminedbythe transverse
balancebetweenthejetpressureandtherampressureofthe
4.4.1. Thebreakouttimescale
stellarenvelopematter.Thebalanceequationinthecomoving
Letusestimatethebreakouttimescale. Weexpectthatthe frameofthepropagatingjetis
breakouttimescaleisnotmuchlargerthantheactivetimeof
thecentralengineafterthe breakoutδT 103 s. Otherwise, p =(Γ ρ )(rθ˙)2, (25)
j j a
∼
AcollapsarjetmodelofGRB060218 9
where r is the radial coordinate in the laboratory frame. A AnumberofSwiftGRBsshowlate-timeX-rayflares,which
Lorentz factor appearingin the right-handside is due to the may be attributed to the long intermittent activities of their
Lorentz transformation. Substituting p =L /(4Γ 2πr2θ2c), centralengines(e.g.,Zhangetal.2006a;Iokaetal.2005). In
j j j
wefindthatθgraduallyincreasessincethejetpushesthestel- contrast, the engine of this burst might has the long power-
lar envelope. Let L and Γ be constant and integrate over lawactivity.Thenitispossiblethattheengineofthisburstis
j j
time,andweobtain differentfromthoseoftypicalGRBswhicharebelievedtobe
blackholes. Mazzalietal.(2006) haveperformeda detailed
L 1/4
θ= j r- 1Γ - 3/4δT1/2. (26) modeling of the spectra and light curve of the SN compo-
j
(cid:18)πρac(cid:19) nent and argued that the progenitor star of this event had a
For GRB 060218, if we adopt the values L 1045 erg s- 1, smallermassthanotherGRB-SNe,suggestingthataneutron
j
ρ 1gcm- 3,r=R 1011 cm,Γ 3,andδ∼T 103 s,we starratherthanablackholewasformedasthecentralengine
a j
∼ ∼ ≃ ∼ of the jet. The intrinsic rate of such low-luminosity GRBs
obtainθ 0.1. Thisislessthantheopeningangleafterfree
∼ would be larger than the local rate of typical cosmologi-
expansionθ 0.3inEquation(19),sothatthewideningef-
0
≃ calGRBs(Soderbergetal.2006;Pianetal.2006;Cobbetal.
fectisnotimportantinthisevent.Insomecasesthewidening
2006;Liangetal.2006b). Forthesereasons,low-luminosity
effectcoulddominatethefreeexpansioneffect.
GRBs might be a distinct GRB population involving neu-
5. SUMMARYANDDISCUSSION tronstarengines. We speculatethatmassiveprogenitorstars
We have investigated whether GRB 060218 arises from a formblack holesat the core collapse which producehighly-
collimatedjet. So far the lack of the jetbreakhasled to the relativistic jets making high-luminosity GRBs with strong
interpretation that the outflow of this event is spherical, and spikypromptemissionsandflares,whilelessmassiveprogen-
thereby the outflow is not the standard collapsar jet but the itorstarsformneutronstarswhichproducemildly-relativistic
outermostpartsofthestellarenvelopethatthe SNshockac- jetsmakinglow-luminosityGRBswithweaksmoothprompt
celeratestoamildlyrelativisticspeed(Soderbergetal.2006; emissions.
Fanetal. 2006). However, we have shown that the avail- If the opening angles of the low-luminosity GRBs are
ableradiodatamaybeinterpretedasanon-relativisticphase around 0.3,thetruerateofthelow-luminosityGRBswith
of an initially collimated outflow within the standard exter- abeami∼ngcorrectionisR 103Gpc- 3 yr- 1. Thelocalrate
LL
nalshocksynchrotronmodel, andthatthe jetmodelwith an of Type Ibc SNe is R 1∼04 Gpc- 3 yr- 1 (Soderbergetal.
initial opening angle θ0 0.3, Lorentz factor Γ0 5, and a 2006; Dahlenetal. 20S0N4;∼Cappellaroetal. 1999). Then the
collimation-corrected lum≃inosity Lj 1045 erg c≃an explain low-luminosityGRBsmightbecreatedat 10%rateofType
the radiodata and is compatiblewith∼the UV/opticalandX- Ibc SNe. By comparison, the collimatio∼n-corrected rate of
raydata. Thismodelismorenaturalthantheinitiallyspheri- the typical cosmological GRBs is 1% of that of Type Ibc
caloutflowmodel,becauseinthelattermodel,therelativistic SNe. Notethatitisalsosuggestedt∼hatthebirthrateofGalac-
ejectaforthepromptandafterglowemissionwith 1050erg ticmagnetarsis 10%ofSNrate(Kouveliotouetal.1998).
could not be producedby the underlyingSN of th≃e total ki- Now if the ne∼utron star loses its rotational energy mainly
netic energy 1051 erg (Mazzalietal. 2006). Furthermore, through magnetic dipole radiation, and a fraction of energy
≃
the jet model is supported by the recent report of the detec- f is transferred to the jet, then the luminosity of the jet is
tionofopticallinearpolarizationintheSNcomponentofthis approximatedby(Shapiro&Teukolsky1983)
event (Gorosabeletal. 2006). We also show that the jet of
t - 2
this eventcan penetrate the progenitorstar by extendingthe L fL 1+ , (27)
j 0
analytical considerations by Matzer (2003). The jet would ∼ (cid:16) τ(cid:17)
berelativisticallyhotintheprogenitorstarandhenceexpand where
freelyoutsidethestarintotherelativelywideopeningangle. B2R 6Ω 4
Theoff-axisscenariowithinthejetmodelisunlikelybecause L0= 6sc3 0 ∼1047ergs- 1B162Rs,66P0,- 2- 4, (28)
itrequiresanunrealisticallyhighγ-rayefficiency.
coWrreicthtedthγe-rjaeyteonpeergnyinogftahnisgleeveθn0ti≃sE0γ.3≃, 3th×e1c0o4l8liemrga,tiaonnd- τ = B2R3cs63ΩI 02 ∼103sI45B16- 2Rs,6- 6P0,- 22. (29)
it makes this eventconsistentwith the Ghirlandacorrelation
Here B is the magnetic field strength at the pole, R is the
(Ghirlandaetal.2004;Ghisellinietal.2006a). neutron star radius, Ω is the initial angular frequencsy, P =
For the prompt emission, we have analyzed the BAT and 2π/Ω is the initial r0otation period, and I is the mom0ent
XRT data and found that the non-thermalcomponentof the 0
of inertia of the neutron star. The characteristic spin-down
promptemissionmaybefittedbytheBandfunctionsimilarly
timescale τ might be the peak time t 103 s of the prompt
tootherGRBs. Thelow-andhigh-energyphotonindicesare ∼
non-thermal emission, and the temporal decay of the spin-
walhmicohstacroenqsutaitnettwypitihcaαlBfe≃atu-re1saonfdGβRBB≃s.-T2h.5e,1r5es- p1e5c0tivkeelVy, dthoewpnrolummpitnonsointy-thaeftremraτl,eLmjis∝siot-n2,flmuxig,hFt agrte-e2.0w,itshhothwant oinf
flux of the prompt non-thermal emission shows a relatively
Figure 2. Assuming f 10- 2, we obtain∝B 1016 G and
shallowdecay,andweshowthatthisdecayisnotdueto the
P 10msinorderthat∼L 1045 ergs- 1 and∼τ 103 sare
curvatureeffectoftheemittingshellthatceasestheemission 0 j
processsuddenly.Thedecayofthenon-thermalemissionmay rep∼roduced. In this case, the∼rotationalenergyIΩ∼2/2 would
directly connect to the anomalous X-ray afterglow detected be overwhelmed by the magnetic energy (B2/8π)(4πRs3/3)
up to t 106 s. If this is correct, the central engine might at t 104 s. This timescale is comparable to the transition
beactiv≃efor&106 s. Insummary,the low-luminosityGRB time∼ofthe decay indexof the promptnon-thermalemission
060218has a typicalpromptnon-thermalemission and may fromF t- 2.0 toF t- 1.1. Ifthisscenariowastrue,thero-
X
originate from a standard collapsar jet possibly driven by a tationpe∝riodwoulde∝volveasP 10(t/τ)1/2msandbecome
∼
long-actingcentralengine. P 1satt 1yr. Ifagiantflareoccursatthismagnetarlike
∼ ∼
10 Tomaetal.
softgamma-rayrepeater(SGR)1806-20(Hurleyetal.2005), Within the hot jet model discussed in § 4, the energy de-
thiswouldbeaclearevidenceforourproposal. posited into the cocoon by the breakout time is estimated
Theemissionmechanismofthepromptnon-thermalemis- by L t 3 1047 erg, which is smaller than the energy of
j br
∼ ×
sionremainsunclear.Ghisellinietal.(2006b)havedrawnthe the detected thermal emission &1049 erg. The rest energy
de-absorbedνFν spectrumfromtheSwiftUltraViolet/Optical in the cocoon is estimated by Mc2θ 2/4 1049 erg, so that
br
Telescope (UVOT) data and arguedthat the extrapolationof thecocoonexpansionvelocitybecomesβ∼ 10- 1outsidethe
the non-thermal spectrum with a constant low-energy pho- c∼
star. Thecocoonshellcatchesupwith the afterglowshellat
ton index to the optical band joins with the detected ther-
t r /cβ 107 s,andthusthecocoonenergywillnotaf-
mal optical flux. This implies that the non-thermal spec- ∼ dec c∼
fecttheradioafterglow.Thethermalcomponentsaredetected
trum does not extend below 1015 Hz. If this cutoff is
∼ intheX-raybandatt .104 s andin theUV/opticalbandat
due to the synchrotron self-absorption in the emitting shell
104.t.105 s. Toexplainsuchabehaviorisaninteresting
with the isotropic kinetic energy L 1047 erg s- 1 and the
bulk Lorentz factor Γ 5, the emisioss∼ion radius of the non- futureproblem.
0
thermal emission is est≃imated as r 1012 cm. It is less
0
∼
than the deceleration radius of the afterglow shell estimated
byr 4Γ 2ct 2 1016cminourjetmodel. Thisim- We appreciate the comments from the referee. We thank
dec 0 dec
≃ ≃ ×
pliesthatthepromptnon-thermalemissionmaybeproduced A. Mizuta, K. Murase, and N. Kawanaka for useful discus-
bytheinternaldissipationofthejet. Ghisellinietal.(2006b) sions. ThisworkissupportedinpartbyGrant-in-Aidforthe
haveshownthatasynchrotroninverse-Comptonmodelcould 21stCentury COE “Center for Diversity and Universalityin
reproduce the prompt non-thermal emission, the anomalous Physics”fromtheMinistryofEducation,Culture,Sports,Sci-
X-rayafterglow,andtheUV/opticalthermalcomponentwith enceandTechnology(MEXT)ofJapan. K.T.wassupported
using parameters similar to those we suggest in this paper, bytheJSPSResearchFellowshipforYoungScientists,grant
Γ =5,θ =0.2,andr =7 1011cm. 182666.K.I.wassupportedbytheGrant-in-Aid(18740147)
0 0 0
×
The origins of the thermal components of this event are from the MEXT of Japan. T. S. was supported by an ap-
also unclear. A possible candidate is the emission from pointmentoftheNASAPostdoctoralProgramattheGoddard
the expanding cocoon as suggested by Fanetal. (2006) and Space Flight Center, administered by Oak Ridge Associated
Ghisellinietal. (2006b) (see also Ramirez-Ruizetal. 2002). UniversitiesthroughacontractwithNASA.
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