Nucleosynthesis of 56Ni in wind-driven Supernova Explosions and Constraints on the Central Engine of Gamma-Ray Bursts

0.28 MB·English

by  Keiichi Maeda

Mon.Not.R.Astron.Soc.000,1–10(2008) Printed21January2009 (MNLATEXstylefilev2.2)
Nucleosynthesis of 56Ni in wind-driven Supernova
Explosions and Constraints on the Central Engine of
Gamma-Ray Bursts
9
0
0 Keiichi Maeda1⋆ and Nozomu Tominaga2
2
1Institute for the Physics and Mathematics of the Universe (IPMU), Universityof Tokyo, Kashiwano-ha 5-1-5,
n Kashiwa-shi, Chiba 277-8568, Japan
a 2Divisionof Optical and Infrared Astronomy, National Astronomical Observatory of Japan, 2-21-1 Osawa,
J
Mitaka, Tokyo181-8588, Japan

5

]
E
H
ABSTRACT
.
h Theoretically expected natures of a supernova driven by a wind/jet are discussed.
p Approximate analytical formulations are derived to clarify basic physical processes
- involved in the wind/jet-driven explosions, and it is shown that the explosion prop-
o
r erties are characterized by the energy injection rate (E˙iso) and the mass injection
st rate(M˙iso).ToexplainobservationsofSN1998bwassociatedwithGamma-RayBurst
[a (GRB)980425,thefollowingconditionsarerequired:E˙isoM˙iso &1051 ergM⊙ s−2 and
E˙iso & 2×1052 erg s−1 (if the wind Lorentz factor Γw ∼ 1) or E˙iso & 7×1052 erg
1 s−1 (if Γw ≫1). In SN 1998bw,56Ni (∼0.4M⊙) is probably produced in the shocked
v stellarmantle,notinthewind.TheexpectednaturesofSNe,e.g.,ejected56Nimasses
0 and ejecta masses, vary depending on E˙iso and M˙iso. The sequence of the SN prop-
1
4 erties from high E˙iso and M˙iso to low E˙iso and M˙iso is the following: SN 1998bw-like
0 – intermediate case – low mass ejecta (.1M⊙) where 56Ni is from the wind – whole
. collapse.ThisdiversitymayexplainthediversityofsupernovaeassociatedwithGRBs.
1
Our result can be used to constrain natures of the wind/jet, which is linked to the
0
central engine of GRBs, by studying properties of the associated supernovae.
9
0
Key words: gamma-ray:bursts – supernovae: general – supernovae: individual (SN
:
v 1998bw) – nuclear reactions, nucleosynthesis, abundances.
i
X
r
a
1 INTRODUCTION nearbyGRB-SNedetected in thisway are found to besim-
ilartooneanother.Thecategory includesGRB980425/SN
Gamma-Ray Bursts (GRBs) are energetic cosmological
1998bw (the proto-typical GRB-SN; Galama et al. 1998),
events, emitting & 1051 ergs in γ-ray. A leading model for
GRB 030329/SN 2003dh (Hjorth et al. 2003; Kawabata et
thecentral engine of GRBs is the formation of a black hole
al.2003;Mathesonetal.2003;Staneketal.2003),andGRB
(BH) and an accretion disk, following the gravitational col-
031203/SN 2003lw (Malesani et al. 2004; Thomsen et al.
lapse of a massive star whose main sequence mass (Mms) 2004). Optical observations of these GRB-SNe are well ex-
is at least as large as 25M⊙ (for reviews, see Woosley & plainedbyanexplosionofacarbon-oxygen(CO)star,which
Bloom 2006; Nomoto et al. 2007). A relativistic flow gen-
has evolved from a massive star (Mms 40M⊙) and has
erated by neutrinoannihilation (Woosley 1993; MacFadyen ∼
lost its H- and He-envelopes during the hydrostatic evolu-
& Woosley 1999) or magnetic activity (Brown et al. 2000;
tionary phase (Iwamoto et al. 1998; Woosley et al. 1999;
Proga et al. 2003) is proposed totrigger a GRB.
Nakamura et al. 2001a; Mazzali et al. 2003, 2006). The ki-
novaAe (lSinNkebIect)weheans (baeecnlasesstoafb)liGshReBdsoabnsedrvTaytpioenaIcllys.upTehre- neticenergy(EK)oftheexpansionislarge,E51 ≡EK/1051
ergs & 10 (note that E51 1 for canonical supernovae).
most convincing cases for the supernovae associated with They eject 0.3 0.7M⊙∼of 56Ni (which powers the SN
GRBs(hereafterGRB-SNe)havebeenprovidedbyspectro- luminosityb∼ythed−ecaychain56Ni Co Fe).Hereafter,
scopicdetectionofsupernovafeaturesinanopticalafterglow the mass of 56Ni is denoted by M(5→6Ni). R→ecently, another
of a GRB or at the position consistent with a GRB. Three
example of the association has been reported (Della Valle
et al. 2008; Soderberg et al. 2008), i.e., GRB 081007/SN Ic
⋆ E-mail:[email protected]
2 K. Maeda and N. Tominaga
2008hw, while the observed properties of this SN have not 2 MODELS
been modeled yet.
Weconsiderasituationthatasupernovaexplosionisdriven
Despitethesimilaritywithinthewellstudiedcasesmen- by outflow of materials (e.g., a disk wind; MacFadyen &
tionedabove,GRB-SNedoseemtohavediverseproperties. Woosley1999)fromthevicinityofacentralremnant(likely
Peakmagnitudesofso-calledsupernovabumpsseeninGRB a black hole). This energy input can be different from the
optical afterglows show diversity (Zeh, Klose, & Hartmann relativistic jet producing a GRB; The wind can either be
2004;Woosley&Bloom2006),highlightedbysub-luminous relativisticornon-relativisticatitsinjectionfromthecentral
(possible) SNe in GRBs 040924 and 041006 (Soderberg et remnant into thesurroundingstellar mantle.
al. 2006). A few GRBs show no evidence for the supernova A schematic picture of the problem considered in this
bump (Hjorth et al. 2000; Price et al. 2003). Non-detection paper (and most of the past numerical studies) is shown
of SN features in two nearby GRBs 060505 and 060614 has in Figure 1. Throughout this paper, we adopt 25M⊙ and
been reported,placing theupperlimit tobrightness ofpos- 40M⊙progenitormodelsfromNomoto&Hashimoto(1988).
sible underlying SNe 100 times fainter than SN 1998bw Following the treatment of the past numerical studies, the
∼
(Della Valle et al. 2006; Fynbo et al. 2006; Gal-Yam et al. wind/jet (we hereafter frequently call it simply a wind) is
2006). injected byhand at a certain radius (Rw in this paper).
In spite of the observational constraints, the explosion
mechanism of GRBs and GRB-SNe are still unknown. In 2.1 Properties of the wind/jet
particular, how properties of the central engine are related
In a one-dimensional hydrodynamic problem, three inde-
to the bulk expansion of the stellar materials, observed as
pendent variables must be specified (two for thermody-
a supernova mainly in visual light, is still under debate. A
namic variables and one for a hydrodynamic variable) as
possibility is that a supernova is induced by a disk wind
initial/boundary conditions. For simplicity, we assume that
generated byviscous heating (MacFadyen & Woosley 1999;
the wind is dominated by the kinetic energy. This does not
Narayan, Piran, & Kumar2001).
drastically defeat our results, since it is expected that the
Although there were numerical calculations for the thermal energy deposited at the root of the wind, near the
wind/jet-driven explosions (Khokhlov et al. 1999; Mac- central remnant, is quickly converted to the kinetic energy
Fadyen, Woosley, & Heger 2001; Maeda & Nomoto 2003; wellbelowRw.Thankstothissimplification,onlytwoinde-
Nagatakietal.2003;Maeda2004,Nagataki,Mizuta,&Sato pendent variables at Rw are required to determine the hy-
2006; Tominaga 2007a; Tominaga et al. 2007b; and Tomi- drodynamicevolutionofthesystem.Thechoiceoftheinde-
naga2009),ithasnotbeenclarifiedwhatfundamentallyde- pendentvariables can bearbitrary,thuswetaketheenergy
termines the properties of resulting SNe and in what ways. injection rate (E˙w) and the mass injection rate (M˙w). The
Also, the numerical investigations have been restricted in Lorentzfactorofthewind/jetattheinjection(atRw)isex-
theparameter space. pressedasafunctionofthesetwoindependentvariables,i.e.,
Γw 1+E˙w/(M˙wc2),wherecisthespeedoflight.Hereafter,
∼
Aiming toovercome theseproblems, thisstudyiscom- quantitiesexpressingthepropertiesofthewindatinjection
plementary to the past numerical studies. Our goal in this (atRw)aredenotedbythesubscript”w”.Trivially,thewind
paper is to express theoretically expected features of SNe is initially highly relativistic only if E˙w,51/M˙w,⊙ 1000,
resulting from the wind/jet-driven explosion, as a function where E˙w,51 E˙w/(1051 erg s−1) and M˙w,⊙ M≫˙w/(M⊙
of rates of the mass and energy (M˙w and E˙w, where the s−1). ≡ ≡
subscript”w”denotes”wind”,orM˙iso andE˙iso,referringto Ouranalysis is based on a one-dimensional radial flow;
theisotropicequivalentvalues)generatedandinjectedfrom The collimation of the wind is taken into account with a
the central system (i.e., a black hole plus a disk) into the geometrical factor fΩ(6 1). fΩ relates the wind intrinsic
surrounding stellar mantle (in this paper, stellar ”mantle” properties and isotropic equivalents as
refers to the stellar materials above the central remnant,
i.e.,theoutermostlayeroftheFecore,andtheSi-andCO- E˙w = fΩE˙iso , (1)
layers). M˙w = fΩM˙iso , (2)
Our strategy is the following. (1) We first clarify what Ew = fΩEiso , (3)
are main ingredients of the wind/jet-driven SN explosion where Ew is the intrinsic total energy (i.e., the injected en-
( 2).Wedevelopasimplifieddescriptionfortheshockprop- ergyintegrated overtime),andthesubscript”iso” refers to
§
agation and nucleosynthesis in the explosion ( 3). We es- isotropic equivalent values. fΩ is a measure of the collima-
pecially focus on the production of 56Ni, addr§essing how tionangleofthewindatRw:Whenthewindisaspherically
twoproposedsitesforthe56Niproduction,ashockedstellar symmetric flow, fΩ = 1. The wind is more narrowly colli-
mantle and a disk wind (MacFadyen & Woosley 1999; see mated at the injection for smaller fΩ.
Maeda & Nomoto 2003 for a review), can be distinguished. For thetemporal history of thewind properties, we fo-
(2)WethencomparetheresultswithobservationsofSNeas- cusonthesituationthatthewindisinjectedatRw,withE˙w
sociatedwithaGRB,inordertoconstrainM˙iso andE˙iso in and M˙w constant in time for t < tw, and then terminated
these SNe ( 4). The required values for M˙iso and Eiso then at t=tw, so that
§
should be regarded as conditions that any models for the
central engine should satisfy. In other word, by constrain- Ew =E˙wtw (or equivalently,Eiso =E˙isotw) . (4)
ing M˙iso and E˙iso, we aim to provide useful constraints in Some of the following results, however, do not rely on this
studyingtheproperties of thecentral engine of GRBs. assumption.
Nucleosynthesis of 56Ni in wind-driven Supernova Explosions 3
Figure 1. A schematic picture of a wind/jet-driven SN explosion. (a) A situation before the launch of the outward shock wave. The
wind/jet is assumed to be continuously injected from the central remnant withproperties denoted by the subscript”w” at radius Rw.
When materials initially at Mr in the presupernova progenitor fall to Rw, their properties are denoted by the subscript ”m”. At this
point, the mass of the central remnant (the mass within Rw) is Mr. (b) A situation after the launch of the outward shock wave. The
shock wave arrives at radius Rsw, and temperature behind the shock is denoted by T. 56Ni is produced both in the shocked stellar
materialsbetween Rsw andRw andwithinthewind/jetitself.
2.2 Processes involved difficulttofollowthedynamicsoftheshockwavepropagat-
ing into the stellar mantle; Thus, it has not been yet clear
2.2.1 Collapse to Explosion
howtheamountof56Nisynthesizedinthewindisconnected
Thefirstfunctionwehavetoconsideristhedynamicaleffect to properties of the progenitor star and of the resulting su-
( 3.1) ofthewind tothecollapsing stellar mantle;theover- pernova.
§
lying stellar mantle continues to collapse onto the central
remnant, and the wind does not always have a sufficiently
large momentum to overcome the ram pressure of the in-
2.2.3 Geometry of the ejecta
falling materials (Fig. 1a). The importance of the dynam-
ical effect to determine the outcome of a wind/jet-driven Finally, the properties of the wind affect the shape of the
supernova explosion was pointed out by Maeda (2004) and supernovaejecta,astheshockwavepropagatesthroughthe
numericallyexaminedbyMaeda(2004)andTominagaetal. stellar mantle. This has been directly examined in the past
(2007b).Somenumericalcalculations haveincludedthedy- numericalstudies,butrestricted intheparameterspace. In
namicaleffectself-consistently(e.g.,Maeda&Nomoto2003; thispaper,wederiveasimpleestimateoftheshape,interms
Maeda2004;Tominaga etal.2007b; Tominaga2009), while of theproperties of the wind ( 3.3).
§
others have not (e.g., Maeda et al. 2002; Nagataki et al.
2003).
3 WIND/JET-DRIVEN EXPLOSIONS
2.2.2 Production of 56Ni
3.1 Collapse to Explosion
Once the shock wave is launched, it propagates outward
into the stellar mantle (Fig. 1b). The kinetic energy is now In this section, we examine dynamical effect of the wind
converted to the thermal energy following the shock wave on the collapsing stellar mantle (Fig. 1a). We follow analy-
propagation.Thetemperaturebehindtheshockwaveisini- sis similar to that given by Fryer & M´esz´aros (2003) for a
tially so high that the nuclear burning converts the initial standard neutrino-driven delayed explosion for (canonical)
stellar composition (mostly oxygen and silicon) mainly to supernovae.
56Ni( 3.2.1). Thetemperaturedecreases as theshock wave We evaluate outcome of the interaction between the
moves§outward;oncethetemperaturesdropsbelow 5 109 wind and the infalling materials at radius Rw 8 107
K, then the efficiency of the production of 56Ni d∼ecr×eases cm (for Mms = 40M⊙) and 1.2 108 cm (Mms∼= 25×M⊙),
×
rapidly. where the presupernova enclosed mass (Mr) is 1.4M⊙. As
Atthesametime,thetemperatureoftheinjectedwind time goes by, materials initially at larger M collapses to
r
itself may also be sufficiently high so that a fraction of this the radius Rw, and add to the mass of the central rem-
material may be converted to 56Ni ( 3.2.2). In most of the nant(MBH).Duringawholeperiodbeforethelaunchofthe
past numerical studies, the producti§on of 56Ni in shocked shock, the temporal evolution of the system is thus speci-
stellar mantle has been investigated in detail. The ejection fiedbythecentralremnantmassMBH whichmonotonically
of 56Ni in the wind has been examined by a different kind increases as a function of time (i.e., larger MBH for later
ofnumericalsimulationsinvolvingtheinnermostpartofthe time).
collapsing star(e.g., MacFadyen&Woosley 1999; Nagataki Ifthetrajectoryoftheinfallingmaterialsisthatoffree
et al. 2007). However, in such simulations, it is practically fall, the density (ρm) and velocity (vm) of the infalling ma-
4 K. Maeda and N. Tominaga
8
10 10
107 T
9
6
10 7.5
5
3] 10
-
m
4
c 10 5

g
[ 3
ρ 10 T
9
2
10 2.5
1
10
0
10 0
8 9 10 Figure3.TherequirementforE˙isotoinitiatetheexplosionasa
10 10 10 function of MBH. For Γw ∼1, the case withM˙iso =0.1M⊙ s−1
R [cm] isshownforpresentation(notethattheproductE˙isoM˙isoshould
exceedthevaluecorrespondingPmifΓw∼1;equation10).Three
horizontal dashed lines at E˙iso,51 = 50, 10, and 7 are shown
Figure 2. Density Structure (left vertical axis label) following for illustrating purpose (arrows indicate the position where the
the gravitational collapse, at MBH = 2.4M⊙ (thick-solid line), outward shock wave is launched); Let us assume Mms =40M⊙,
3p.o4sMts⊙ho(cmketdeimumpe-rsaotluidr)e,(aTn9d≡13TM/⊙10(9thKin;-rsioglhidt)v.eArtlsicoaslhaoxwisnliasbtehle) Γshwoc∼k1waanvdeMis˙ilsaou=nc0h.e1dMa⊙t tsh−e1.inIfteE˙risseoc,5ti1on=s5b0eotwre1e0n,tthheeoSuit-waanrdd
aansdafEu˙niscoti=on1o0ft(hdeasshheodcklrinaed)iuosr,f1or(MdoBttHed=l2in.4eM).⊙T,hEeispor,o51ge=nit3o0r, tChOe-mlayoemrsenattumMBcHan∼n2o.t4Mex⊙ce.eOdnthteheinoftahlleirngharnadm, ipfrEe˙sissuo,r5e1u=nt7il,
model (Mms = 40M⊙) is from Nomoto & Hashimoto (1988). nearly the whole CO star collapses to the central remnant at
The position within which the postshock temperature is above MBH∼13M⊙.
5×109 K (horizontal thin-dashed line), for MBH = 2.4M⊙, is
markedbytheopencircle(E˙iso,51=10)andbytheopensquare
(E˙iso,51=1).
lows.
E˙w AwΓw(Γw 1)ρwvwc2 ,and (8)
terials at Rw are written as M˙w ∼= AwΓwρwvw−, (9)
ρm ∼ ρpresn(cid:16)RRprwesn(cid:17)3/2 , (5) where Aw is the area subtended by the wind. The asymp-
2GM 1/2 totic expressions for the momentum balance are derived by
vm r . (6) substitutingequations (8) and (9) into equation (7);
∼ (cid:16) Rw (cid:17)
The subscript ”presn” is used for the pre-collapse initial E˙wM˙w fΩ E˙isoM˙iso > 1 Awρmg¯GMBH
values for the material at the mass coordinate Mr. The p ≡ p √2 Rw
subscript ”m” refers to quantities of the same material (at for Γw 1 , (10)
∼
Msnar)pwshhoetnsiotfctohlleapdseenssittoyRstwru(cFtuigruerefo1rad).iffFeirgeunrteM2BsHhofwosr E˙w ≡ fΩE˙iso>cAwρmg¯GRMwBH
Mms =40M⊙. for Γw 1 . (11)
Theoutwardshockwaveislaunchediftherampressure ≫
of the wind (Pw) overcomes that of the infalling materials TheRHS’sofequations(10)and(11),exceptforAwde-
(Pm), i.e., pendingonfΩ,arecompletelydeterminedbytheprogenitor
structure,asafunctionofMBH(Fig.2).Therequirementfor
Pw ∼Γw2ρwvw2 >Pm =ρmvm2 ≡ρmg¯GRMwBH . (7) aE˙swaafnudncMt˙iowninoftMerBmHs(oif.eE.˙,iasosaanfdunMc˙tisioonisosfhtoimwen).inHFeriegaufrteer3,
Here MBH is the mass of the BH when the material at Mr we use a notation Xn X/10n in CGS unit.
(= MBH) collapses to Rw. Although a numerical constant If Γw 1, it is n≡ecessary that the product E˙isoM˙iso
∼
g¯= 2 in the above estimate, we find g¯ = 1/2 yields a bet- shouldbelargerthanaspecificvaluegivenbyequation(10),
ter representation of a set of numerical simulations (Maeda which corresponds to Pm as a function of MBH, in order to
2004; Tominaga et al. 2007b). Weassume g¯=1/2 through- initiate the explosion. For Γw 1, E˙iso should exceed the
≫
out the paper. valuegiven by equation (11), and therequirement becomes
Equation(7)can beexpressedintermsofE˙w andM˙w, independent from M˙iso. In other ward, once the temporal
which are related toother properties of thewind/jet as fol- evolutionofE˙iso andM˙iso isgiven,theoutwardshockwave
Nucleosynthesis of 56Ni in wind-driven Supernova Explosions 5
islaunchedwhenE˙isoM˙iso (orE˙isoifΓw 1)ishigherthan
≫
thespecific value representing Pm (Fig. 3).
Although Pm overall decreases as a function of MBH
following the density decrease, jumps are seen at the edges
of the characteristic hydrostatic burning layers. Assuming
freefall, Pm ρpresnRp3/re2snMBH. Within each layer, ρpresn
∝
drops slowly as a function of MBH and Rpresn, making Pm
nearly constant as a function of MBH. On the other hand,
at the edges, ρpresn decreases suddenly, leading to rapidly
decreasingPm.Asaresult,localminimumvaluesappearat
theintersectionofdifferentlayers(Fig.3).Itinfersthatthe
explosionislikelyinitiatedatoneoftheseintersections,but
not within each layer.
This behavior of Pm provides an interesting implica-
tion. For illustration purpose, let us take the case with
Mms = 40M⊙ and Γw 1, assuming that E˙iso,51 and
M˙iso are constant in time∼. If E˙iso,51M˙iso,⊙ & 1, then the
explosion is initiated below the Fe/Si interface (MBH <
2.4M⊙). If 0.8 < E˙iso,51M˙iso,⊙ < 1, the explosion position Figure4.Theisotropicmassof56Nisynthesizedintheshocked
jEu˙imso,p5s1Mt˙oisot,h⊙e.Si/0C.8O, ailnmtoesrtfawceho(lMeBCHO∼lay3e.r4Mco⊙ll)a.pTsehseonn,tiof Mstemllsar=m4a0nMtl⊙e ((M13T.89M>5⊙; sCoOlidstcaorn)t.oTurhse),dfaosrheMdBaHsla=nt2.l4inMe⊙shaownds
the central remnant. Thus, we conclude that a small dif- E˙iso,51=E˙tw,51 (Eiso)asisdefinedbyequation (14).Notethat
ference of the wind properties can lead to totally different thepossiblevalueofE˙iso that leadstoMBH=2.4M⊙ isafunc-
outcome, with thecritical valueE˙iso,51M˙iso,⊙ 0.8 1(for tion of M˙iso (see the main text). Three cases are shown for il-
Γ(pFewnig∼d.s31o)).norMEm˙iss,o,a51nd∼i7s0sm(foarlleΓrwfo≫r a1)l.esTshmeacsrsiti∼vicealprvoa−glueneidtoer- ilΓtus≫itsra1rte,iqotuhnie,reardesqsutuhmiraeitndgEv˙iatshlou,e5e1tfeo∼mrpE7˙oi0sroa(liolsythinceodrwnepsitseaenndttehnwetionfurdotmwinajMred˙citssiooh,noa:cnkId-f
wave cannot be launched at MBH = 2.4M⊙.) If Γw ∼ 1, it is
requiredthattheproductE˙isoM˙isoshouldexceedacertainvalue,
i.e.,E˙isoM˙iso ∼1forthe outward shockwave beinglaunched at
3.2 Production of 56Ni MBH=2.4M⊙(e.g.,E˙iso,51∼10ifM˙iso,⊙=0.1,andE˙iso,51∼1
ifM˙iso,⊙∼1).
Oncetherampressureofthewind(Pw)overcomestheram
pressure of the infalling material (Pm), an outward shock
wave is launched and explosive nucleosynthesis takes place.
Iwdniasvcthueisspsaspesrscoetsdioutnch,triwoouengeohxfat5mh6Neinisetineplrlaaordstumeclltaainortnlmeoaf(nF5t6ilgeN.hi1eabas)t.tehdWebesyhfiotrhcsket T9∼(cid:26) 25.47EE˙i1si1/so/o4,65,521RRs−s−ww3,1/8,/843ρ¯16/12 ffoorr tt<>ttww ., (13)
shockwave(§3.2.1),thenwecommentonproductionof56Ni This expression explicitly utilizes the assumption that E˙iso
within thewind/jet (§3.2.2). is constant in time for t6tw and zero afterward.
SinceMBH increasesmonotonically withtime,theden-
sity structure of the stellar mantle is specified for given
3.2.1 Shocked stellar mantle MBH (Fig.2).Therefore,wecanestimatetheenclosedmass
within a sphere in which the post-shock materials attain a
Theshockedmantlecanbecomethepredominantsiteforthe certain temperature (Fig. 2; see also figure 7 of Maeda &
56Ni synthesis, if the shock wave sweeps up a large amount Nomoto 2003). 56Ni is synthesized in the region (MT9>5)
of the stellar mantle, at least ∼0.1M⊙. This inevitably re- where T9 > 5. Figure 4 shows MT9>5 [= M(56Ni) /fΩ; the
sults in a non-relativistic shock wave, even if the wind/jet isotropic equivalent value for M(56Ni)] for MBH = 2.4M⊙
at its emergence from the central system at Rw is highly and Mms =40M⊙.
relativistic.Letusdenotetheaveragevelocityandthemass MT9>5 depends not only on Eiso and E˙iso, but also on
of the expanding material swept up by the shock wave, by MBH. On the other hand, as concluded in 3.1, there is a
Γshock and Mshock, respectively,then conditionintermsofE˙iso andM˙iso,toinitiat§etheexplosion
Γshock ∼1+6×10−4ME˙swho,5ck1,t⊙ , (12) tathotagotibvEte˙aniisnoM,5M1BMHB˙:HisTo,=h⊙is2&.i4nMd1i⊙r(efocatrnlydΓrwMelamtse1s=)M˙o4ir0soMtth⊙oa,tMiEtB˙iHisso.,r5Ie1nqo&uridr7eed0r
∼
at time t (in second). In order to accelerate 0.1M⊙ of (for Γw 1). Vertical dotted lines in Figure 4 explicitly
the stellar mantle materials to Γshock = 100, t∼he explosion describe≫the requirement: (1) for Γ 1, E˙iso,51 = 70, (2)
energy is required to behigher than 1055 ergs for Γ 1 and M˙iso,⊙ =0.1, E˙iso,51 =≫10, and (3) for Γ 1
We can thususe non-relativistic approximation for the and M˙∼iso,⊙ =1, E˙iso,51 =1. ∼
postshock temperature (T) as a function of the radius of Interestingly, the dependence of M(56Ni) on E˙iso and
the shock wave (Rsw). For a radiation-dominated fireball Eiso (Fig.4)changesthebehaviorataspecificaslantlinein
expandingintoauniformmediumwiththedensityρ¯(Maeda theE˙iso Eiso plane.Thelineisobtained byequalizingthe
−
& Nomoto 2003), two RHS’sof equations (13) and definedas
6 K. Maeda and N. Tominaga
E˙tw,51(Eiso)≡(cid:16)2T.99(cid:17)−10/3ρ¯−61/2Eiso,512/3 , (14) αα--rriicchh ffrreeeezzeeoouutt,islecaovninsigstmenatinwlyith4Hneu,mneortic5a6lNcai.lcTuhlaetisotnrsonogf
the collapsing star (Nagataki et al. 2007). In what follows,
wtioitnhoTf956=N5i.,Fi.oer.,gtihveenteEmispoe,rMat(u5r6eNni)eciesslsaarrgyerfofrorthlaerpgerrodE˙uicso- we take M(56Ni) =0.2Mw in the wind which is typical for
as long as E˙iso6E˙tw, but saturated for E˙iso >E˙tw. the strong α-rich freezeout (e.g., Pruet et al. 2004; but see
The dividing energy injection rate, E˙tw(Eiso), appears, §5 for further discussion on theuncertainty).
becauseofthedifferentbehaviorofthepost-shocktempera-
turebeforeandaftert=tw(Maeda2004).Themassof56Ni 3.3 Geometry of the ejecta
isdeterminedbytheradiusoftheoutwardshockwavewhen
the temperature drops down to T9 = 5. If E˙iso 6 E˙tw (for Thegeometryofthebulksupernovamaterialsprovideause-
given Eiso), then the temperature behind the shock wave ful constraint on the model, since recently more and more
drops down to T9 = 5 during the phase when the energy observationaldatahavebeenavailabletoaddresstheejecta
source is still active (i.e., t < tw). Thus, larger E˙iso corre- geometry (see Wang & Wheeler 2008 for a review of spec-
spondstothelargeramountofenergycontainedbehindthe tropolarimetry; for recent spectroscopy, see Maeda et al.
shock wave, leading to the larger radius of the shock wave 2008; Modjaz et al. 2008). Here we give a rough estimate
when the condition T9 =5 is satisfied (e.g., see Fig. 2). On ofhowthegeometrydependsonthewindparameters.Note
the other hand, the condition E˙iso > E˙tw corresponds to that thegeometry of thebulksupernovamaterials is differ-
the case in which all the energy from the central source is entfromthegeometryofthewindattheinjectionmeasured
liberated in short time scale such that the temperature is byfΩ; Even if the wind is initially collimated (fΩ <1), the
still high (T9 >5) at t=tw. The following evolution of the bulk expansion of the stellar mantle, as is induced by the
shockwave,whichdeterminesMT9>5,iscontrolled byEiso, wind, can beless collimated or even quasi-spherical.
not by E˙iso. The shock breakout time (tsb) of the wind/jet is esti-
It has been suggested that M(56Ni) is larger for larger mated by
tEhiesoe(nee.grg.,yNisakgaemneurraateedtaaln.d20l0ib1ebr)a,tbedutatlhmisoshtopldrsomtrputelyon(liy.ei.f, tsb ∼ RCO/vshock , (17)
tMw(→56N0i)adnodesE˙nisoot→dep∞en).dFoonrtehxeamtoptlael,einfeEr˙gisyo,5E1is=o (1o,rtEhwen) vshock ≡ (cid:18)(MC2OE˙isotMsbBH)(cid:19)1/2 . (18)
−
(as long as Eiso,51 &0.2) (Fig. 4).
HereRCO and MCO are theradius and themass of theCO
star. For Mms = 40M⊙, these values are RCO 1010 cm
∼
3.2.2 56Ni in the wind/jet and MCO ∼13.8M⊙.
Asthewind/jetpushesthestellarmantle,thewind/jet
Mass of the wind is loses its energy by depositing its energy into the surround-
Mw = ME˙˙wwEw ∼ c2(ΓEww−1) , (15) eitnhngresor.guygIfhotfthhteeheepnrweorigngedyn/iitjneotjreascurteiroftnarcaeisn(stRfeeCrrmrOe)idn,aattoleatdrhgebeesfftroearlcelatiriotnmboarfneattlhkees.
whichmustbesmallerthanthemassoftheaccretedmateri- Thisresultsinaquasi-sphericalexplosionofthebulkofthe
als,i.e.,Mw <MBH 1.4M⊙.Composition ofthematerials stellar mantle, even if the wind/jet is initially collimated.
−
withinthediskwindislargelyuncertain,becauseitdepends This happensif thefollowing condition is satisfied.
onthethermalhistoryofthewindinthevicinityofthecen-
otrfatlhreemconmanptos(iit.eio.,nwoefllthbeelowwinRdwm).atTehriealdreeeqpuuirnedsenrustmanerdiicnagl tw ∼ EE˙iissoo <tsb ∼(cid:18)RC2O(M2CE˙Ois−o MBH)(cid:19)1/3 . (19)
calculations or analytic investigation of the innermost part
Ontheotherhand,iftw >tsb,thenalargeamountofmate-
of the collapsing star (e.g., MacFadyen & Woosley 1999;
rialsareejectedtowardthejetdirectionascomparedtothe
Pruet, Thompson, & Hoffman 2004; Nagataki et al. 2007).
equatorialdirection,resultinginastronglyjettedexplosion.
However, it is still possible to make a rough estimate,
in order to see in what regions in the E˙iso M˙iso plane
−
the disk wind is important. Assuming that the energy of
4 COMPARISON TO OBSERVATIONS
the wind is initially thermal energy-dominated near the
central remnant, the typical entropy of the wind material The expected SN properties can be expressed by E˙iso and
sst≡antS/a(nkdBa/tmomu)ic(wuhneitremkaBssa,nredspmeuctaivreelyt)heisBworlittztmenanbnycE˙oinso- pM˙airsaomaentdersthaeretesmppecoirfiaeldev(oMlumtsio,nEoisfotohresEe,w,onacnedthfΩe)o.tFhoerr
and M˙iso as follows: comparisontotheobservationpresentedbelow,weexamine
s∼(cid:26) 1212E(˙Ei˙3si/os4,o5,251M˙iso,⊙)3/8 ffoorr ΓΓww ≫∼11,. (16) ittsihmeaessfiimotrupatlte<isotntwceaaxsnea,dmizinneerwodhaiinfcthemrEwo˙iasstrodoa(fni.dteh.,eME˙pisirsooeva=iroeuE˙scisoonntuwsmt)a.enTrtihciainsl
We find that s 1 in the parameter range of our in- studies, except for MacFadyen et al. (2001) and Maeda &
≫
terest; s can be as small as about unity, only when either Nomoto (2003) who examined the case where the energy
E˙isoM˙iso or E˙iso is very small, but in such a case the whole injectedbythejet (wind)isconnectedtotheaccretion rate
star collapses onto the central remnant ( 3.1). Thus, when to thecentral remnant.
E˙iso and M˙iso are large enough to resu§lt in a supernova First,themomentumbalancedetermineswhentheout-
explosion, the wind material should experience the strong ward shock wave is launched, yielding MBH at this time as
Nucleosynthesis of 56Ni in wind-driven Supernova Explosions 7
afunctionofE˙iso andM˙iso ( 3.1;Figure3).Then,theanal- 4.2 Diversity of GRB-SNe
ysis presented in 3.2.1 gives§us the mass of 56Ni, as this is
given as a functio§n of MBH, E˙iso, and Eiso (Figure 4). The Figure5bshowstheexpectedcharacteristicsofGRB-SNefor
massof 56Niin thedisk canberoughlyevaluated usingthe Eiso=30,fΩ =0.3,andMms =40M⊙.Avarietyoffeatures
are predicted for the wind-driven supernovae depending on
result of 3.2.2 (equation 15). Finally, the typical geometri-
(ca3l.f3e;aetquur§eatcioann 1b9e).derived as a function of E˙iso, Eiso, MBH E˙isoIafnEd˙isMo˙,5is1oM.˙iso,⊙ &1 (Γw ∼1) or E˙iso,51 &70 (Γw ≫1),
§ then the resulting SNe should be similar to SN 1998bw in
Using the expressions derived in 3, we characterize
properties of a SN explosion as a functi§on of E˙iso and M˙iso theejectedmass.Thefeaturesarefurtherdividedasfollows
(regions A, B, and C in Figure 5b):
for a given progenitor model (Mms). Other parameters are
fΩ,Ew (ortw),butthesecanbesetwithoutlargeambiguity (A) SN 1998bw-like – GRB-SNe 1998bw, 2003dh,
by observations for some GRB-SNeof special interest. and 2003lw: In the parameter region A, the wind-driven
supernovaissimilartoGRB-SN1998bw,invirtuallyallthe
observedcharacteristics.Theluminosityissimilartothatof
4.1 SN 1998bw: the origin of 56Ni SN 1998bw, since M(56Ni) ∼0.3−0.6M⊙.
This category can account for GRB-SNe 1998bw, 2003dh,
SN Ic 1998bw is a prototypical SN associated with a GRB. and 2003lw. The region is relatively large in terms of E˙iso
For SN 1998bw, intensive observational data in the optical and M˙iso, and the features of supernovae are insensitive to
wavelength are available: modeling these observations (by E˙iso and M˙iso; This can explain why these three GRB-SNe
one-dimensional radiation transfercalculations; Iwamotoet are similar in theoptical properties.
al. 1998; Woosley et al. 1999; Nakamura et al. 2001a) sug- (B)Sub-luminous SN1998bw-like–GRBs 040924
geststhatEiso 30andMms 40M⊙.Addingtothis,there and041006?: IntheparameterregionB,theexpectedsu-
∼ ∼
existsintensivestudyforthisobjectusingmulti-dimensional pernovaeare similar to SN 1998bw in theejected mass and
radiation transfer calculations (Maeda 2006a; Maeda et al. the geometry, but the difference is seen in M(56Ni). This
2006bc; Tanaka et al. 2007). The study suggests that the indicates that thesupernovae has similar optical properties
intrinsicexplosionenergyissmallerthantheisotropicvalue withSN1998bw,withthediversityintheluminosity,cover-
abpyparofxaicmtoartioonf.about 3, inferring that fΩ ∼ 0.3 is a good ibnegca∼us0e.,3i−n1thtiismpeasrathmaettoerfSreNgi1o9n9,8Eb˙iwso.iTshsimsadlilveerrtshitaynaEr˙istwes,
and FMigmusr=e 54a0Msh⊙o.wHsetrhee, wreesutrltyftoorcEonissot,5r1ai=n t3h0e,pfrΩop=er0ti.e3s, Ea˙nisdotihsulsarMge(r56tNhai)nisE˙dtwepienndreegnitononAE,˙iasond(§3th.2u.1s)M[n(o5t6eNti)haist
of the wind generated by the activity of the central engine independent from E˙iso unlike region B]. This parameter re-
of SN 1998bw, i.e., E˙iso and M˙iso (or equivalently, E˙w and gion B can account for sub-luminous SNe found as bumps
M˙w). For SN 1998bw, observations constrain three quanti- inopticalafterglows ofsomeGRBs(e.g.,GRBs040924 and
ties, theejecta mass, M(56Ni), and the shape of the ejecta: 041006).
(1)TheejectacontainalargeamountoftheCO-coremate- (C)BipolarSN1998bw-like: Intheparameterregion
rials, so that MBH <10M⊙, as inferred by the optical light C, a resulting SN is similar to SN 1998bw in its luminosity
curve and spectra ( 3.1). (2) M(56Ni) 0.4M⊙ to explain and ejecta mass, but the ejecta are more highly beamed
§ ∼
itspeakluminosity( 3.2).(3)Theejectaaresuggestedtobe than SN 1998bw. We havenever directly observed such pe-
§
aspherical, but still a large amount of materials are ejected culiarGRB-SNe.IfsuchaGRB-SNisobservedinfuture,it
intotheequatorialdirectionasinferredespeciallybyspectra willprovideastrongevidenceofthediversepropertyofthe
at 1 year since the explosion (Patat et al. 2001; Mazzali central engine, in terms of E˙iso and M˙iso.
et a∼l. 2001; Maeda et al. 2002). This indicates that the ex- Here, 56Ni is mainly produced within the wind materials.
panding supernova ejecta are quasi-spherical, rather than Thebipolarexplosionresultsfromthecentralenergysource
extremely bipolar ( 3.3) being active for a long time period (c.f., equation 19; e.g.,
The three con§ditions are simultaneously satisfied if low E˙iso smaller than E˙w). As such, (1) the mass of 56Ni
E˙iso,51M˙iso,⊙ & 1 and E˙iso,51 & 20 (for Γw 1), or producedwithintheshockedstellarmantleissmall,and(2)
E˙iso,51 >70 (for Γw 1) (region A in Figure 5b)∼. In these thewindshouldbemassivetoinitiatetheexplosion(topro-
cases, 56Ni is predom≫inantly produced at the shocked stel- videthesufficiently large momentumin thenon-relativistic
lar mantles. The wind origin for 56Ni is disfavored for SN regime). As a result, the wind contribution dominates over
1998bw. The wind contribution exceeds the shocked man- theshockedstellar mantlein theproduction of 56Ni for the
tle contribution only if E˙iso,51 . 5 and M˙iso,⊙ & 0.2 (i.e., (extremely) bipolar explosion case.
rpeagriaomneCterosfrFesiugultrsein5bt)w./Htsobw>ev1era,ntdhiasncoemssbenintaiatliloynboipfotlhaer thenItfhEe˙irsoe,s5u1lMt˙inisgo,⊙SN.e1sh(oΓuwld∼in1t)rionrsiEc˙ailsloy,5d1i.ffe7re0n(tΓfwro≫mS1N),
explosion, as is inconsistent with theobservation.
1998bwintheejectedmass( 3.1;regionsDandEinFigure
largeInM˙sihsoorat,ntdhelawrginedtwsh;osueledebqeumataiossniv1e5a)n,dinloonrgd-elirvetdo(pi.reo.-, b5be)t.rAigtgetrheed.low end of E˙iso o§r M˙iso, a supernova can never
duce a large amount of 56Ni within the wind (as large as
0.4M⊙). However, such an explosion with a long-lived (D) Bipolar SN with the ejected mass . 1M⊙ –
∼
energy injection results in the extremely bipolar explosion, GRBs 060505 and 060614?: In the parameter region
since the outward shock wave can reach the stellar sur- D,only asmall amountof materials (<1M⊙)nearthesur-
face before the energy injection is terminated: This argues face of the CO layer are ejected. The supernova ejecta are
against thewind-origin of 56Ni in SN 1998bw. essentially bipolar in this case. 56Ni is mainly originated in
8 K. Maeda and N. Tominaga
Figure 5. (a) Estimate of M(56Ni) in the E˙iso−M˙iso plane. The progenitor is the 13.8M⊙ CO star (Mms = 40M⊙), taken from
Nomoto&Hashimoto(1988).TheothertwoparametersaresetasEiso,51=30andfΩ=0.3,representingthebrightGRB-SN1998bw.
M(56Ni)/M⊙isshownfortheshockedstellarmantle(redcontours)andthewind(bluecontours).MBH/M⊙isshownbytheblack-dotted
contours.Theshadedregiondenotedby”Forbidden”isexcludedbytherequirementthattheejectedmassmustnotexceedtheavailable
accreted mass budget. The dark shaded region denoted by ”Collapse” corresponds to whole collapse of the CO star. The green curve
showsthelinewheretw∼tsb.(b)ExpectedcharacteristicsofGRB-SNeforMms=40M⊙.Ineachregion,”L”denotearoughestimate
oftheSNpeakluminositynormalizedbythatofSN1998bw.
thewind.Thus,M(56Ni) M˙w/E˙w,rangingfrom M(56Ni) properties can potentially explain the diversity of super-
< 10−5M⊙ to 0.3M⊙.∝The expected luminosity is thus novaeassociated with GRBs.
diverse, depend∼ing on E˙iso and M˙iso. This may correspond Further diversity can arise from different progenitors
tonon-detectionofsupernovafeaturesinGRBs060505 and (Mms). Irrespective of Mms, the expected features of the
060614 (which may also beexplained by region E below). wind-driven supernovae can be categorized as we did for
(E)Nosupernova–GRBs060505and060614?: In Mms =40M⊙ ( 4), i.e., regions A – E. The positions of the
§
theparameterregionE,thewindinjectioncanneversetthe boundaries between different regions, as well as maximum
outwardshockwave.Thiscasecorrespondstowholecollapse M(56Ni) at the high end of E˙iso and M˙iso, are dependent
of the progenitor CO star without a SN. on Mms. For Mms = 25M⊙, the boundary between regions
A and D is located at E˙iso,51M˙iso,⊙ 0.05 (for Γw 1) or
E˙iso,51 20(forΓw 1),smallertha∼nforMms =40∼M⊙ by
∼ ≫
5 CONCLUSIONS AND DISCUSSION about one order of magnitude (Fig. 3). Using fΩ =0.3 and
Eiso,51 =30,maximumM(56Ni)is 0.24M⊙evenifMBH =
∼
In this paper, we discussed theoretically expected charac- 1.4M⊙. Thus, Mms =25M⊙ neveryield bright SN 1998bw-
teristics of a supernova driven by the wind/jet. We found likeSNe.WethereforeconfirmedthattheprototypicalGRB-
that the resulting supernova features can be categorized as SN 1998bw and the similar SNe 2003dh and 2003lw must
afunctionofE˙iso andM˙iso.Thus,itispossibletoconstrain have originated from a massive progenitor (Mms ∼ 40M⊙)
thenatureofthecentralengineofGRBsbyobservationsof from thenucleosynthesis argument.
associated supernovae. Also,thediversityarisingfrom Mms maybeimportant
Resultsofthepastnumericalstudiescanbeunderstood toexplain thesub-luminoussupernovaepossibly associated
as a limiting case. For example, Nakamura et al. (2001b) with GRBs 040924 and 041006. These can be explained
showedthatM(56Ni)islargerforlarger Eiso.Inthispaper, by Mms 40M⊙ with smaller E˙iso and M˙iso than for SN
∼
wehaveclarifiedthatitholdstrueonlyforthelimitingcase 1998bw.Alternatively,itisalsopossiblethatthecentralen-
where tw 0 and thus E˙iso . For an another exam- gine provides E˙iso and M˙iso similar to SN 1998bw, but the
ple, Tomin→aga et al. (2007b) →sho∞wed that E˙iso determines progenitor mass is smaller than SN 1998bw (i.e., region A
M(56Ni). In this paper, we have shown that this behavior for Mms = 25M⊙). It is thus important to derive Mms for
appearsasalimitingcasewhereΓ 1,andthedependence these sub-luminous cases, by spectroscopic and light curve
≫
is different when Γ 1. modeling, to distinguish thesepossibilities.
∼
For SN 1998bw, we find that observations are repro- On the other hand, the diversity in terms of Mms is
ducedonlyifE˙iso,51M˙iso,⊙ &1andE˙iso,51 &20(ifΓw 1), not important in interpreting no detection of supernovae
or E˙iso,51 &70 (if Γw 1) (region A in Fig. 5b). We∼favor associated with GRBs 060505 and 060614. If they are the
≫
theshockedstellarmantleasthemainsiteoftheproduction outcomeofthecorecollapseofamassivestar,theproperties
of 56Ni. of the wind/jet should fall in the low end of E˙w and M˙w
Furthermore,theobserveddiversityofsupernovaeasso- (region E), or region D with M˙iso,⊙ . 10−3. If this is not
ciated (or not associated) with GRBs can be accounted for the case, we should have detected associated supernovae.
by the diverse properties of the wind from the central rem- NotethatTominagaetal.(2007b)showedthatthewind/jet-
nant (E˙iso and M˙iso). It is shown that the different wind drivenexplosion with small E˙iso ”can” accountforthenon-
Nucleosynthesis of 56Ni in wind-driven Supernova Explosions 9
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ACKNOWLEDGMENTS
press (astro-ph/0702472)
ThisresearchhasbeensupportedbyWorldPremierInterna- Patat, F. et al. 2001, ApJ,555, 900
tional Research Center Initiative (WPI Initiative), MEXT, Price P.A., et al. 2003, ApJ, 584, 931
Japan, and partly by the Grant-in-Aid for Young Scien- ProgaD.,MacFadyenA.I.,ArmitageP.J.,Begelman M.C.
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thankKen’ichiNomoto, HideyukiUmeda,Masaru Shibata, 1006
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Soderberg A.M., et al. 2008, GCN Circ., 8662
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