Quark-Novae in Low-Mass X-ray Binaries with massive neutron stars: A universal model for short-hard Gamma-Ray Bursts

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by  Rachid Ouyed

Draftversion January18,2011
PreprinttypesetusingLATEXstyleemulateapjv.11/26/04
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QUARK-NOVAE IN LOW-MASS X-RAY BINARIES WITH MASSIVE NEUTRON STARS:
A UNIVERSAL MODEL FOR SHORT-HARD GAMMA-RAY BURSTS
Rachid Ouyed1, Jan Staff2, and Prashanth Jaikumar3,4
DepartmentofPhysicsandAstronomy,UniversityofCalgary,2500UniversityDriveNW,Calgary,Alberta,T2N1N4Canada
DepartmentofPhysicsandAstronomy,LouisianaStateUniversity,202NicholsonHall,TowerDr.,BatonRouge,LA70803-4001, USA
DepartmentofPhysicsandAstronomy,CaliforniaStateUniversityLongBeach,1250BellflowerBlvd.,LongBeachCA90840and
InstituteofMathematicalSciences, CITCampus,Taramani,Chennai 600113,India
Draft version January 18, 2011
1
1
ABSTRACT
0
2 We showthat severalfeatures reminiscentofshort-hardGamma-rayBursts (GRBs) arisenaturally
n when Quark-Novae occur in low-mass X-ray binaries born with massive neutron stars (≥ 1.6M⊙)
and harboring a circumbinary disk. Near the end of the first accretion phase, conditions are just
a
J right for the explosive conversion of the neutron star to a quark star (Quark-Nova). In our model,
the subsequent interaction of material from the neutron star’s ejected crust with the circumbinary
6
disk explains the duration, variability and near-universal nature of the prompt emission in short-
1
hard GRBs. We also describe a statistical approach to ejecta break-up and collision to obtain the
] photon spectrum in our model, which turns out remarkably similar to the empirical Band function
E (Band et al. 1993). We apply the model to the fluence and spectrum of GRB 000727, GRB 000218,
H and GRB980706A obtaining excellent fits. Extended emission (spectrum and duration) is explained
by shock-heating and ablation of the white dwarf by the highly energetic ejecta. Depending on the
.
h orbitalseparationwhentheQuark-Novaoccurs,weisolateinterestingregimeswithinourmodelwhen
p
bothpromptandextendedemissioncanoccur. Wefindthatthespectrumcancarrysignaturestypical
-
o of Type Ib/c SNe although these should appear less luminous than normal type Ib/c SNe. Late X-
r ray activity is due to accretion onto the quark star as well as its spin-down luminosity. Afterglow
t activity arise from the expanding shell of material from the shock-heated expanding circumbinary
s
a disk. We findacorrelationbetweenthe durationandspectrumofshort-hardGRBsaswellasmodest
[ hard-to-soft time evolution of the peak energy.

2 Subject headings: Stars: evolution, stars: binary, stars: neutron, supernovae: general, gamma-ray
v burst: general
8
7
3 1. INTRODUCTION nism than long bursts and that neutron star-black hole
3 Intensive sampling of gamma-ray burst (GRB) light (NS-BH) ordouble neutronstar (NS-NS) mergersmight
4. curves by Swift has revealed that at least in some short be at play (Bloom&Prochaska 2006). But how does
one obtain extended emission in this case? Extended
0 GRBs (henceforth shGRBs), prompt emission is fol-
accretion episodes in NS-BH mergers have been pro-
0 lowed by softer, extended emission lasting tens to hun-
posed (Barthelmy et al. 2005), while Rosswog (2006)
1 dreds of seconds. For example, the lightcurve of GRB
suggest that some debris may be launched during the
: 050709 (Villasenor et al. 2005) has a short-hard pulse
v merger process, which would fall back later to power
T 0.2sandalong-softpulseT 130s;GRB050724
i 90 ∼ 90 ∼ flares at late times. Alternatively, disk fragmentation
X (Barthelmy et al. 2005) has prominent emission lasting
(Perna et al. 2006) or magnetic field barrier near the
for 3 s followed by a long, soft, less prominent emis-
r
a sion∼peaking at 100 s after the trigger, while XRT accretor (Proga & Zhang 2006) may induce intermit-
∼ tent accretion that power the flares. Other types of
observations reveal strong flare-like activities within the
mergers such as a white dwarf-neutron star (WD-NS)
firsthundredsofseconds. InGRB060614,thelightcurve
merger (King et al. 2007) have been advanced to inter-
has a short-hard episode followed by an extended soft
pretshGRBs. Dai et al.(2006)arguethatthefinalprod-
emission component with strong spectral evolution. On
uct of a NS-NS merger may be a heavy, differentially-
average, the principal properties of the intense prompt
rotatingNS,whosepost-mergermagneticactivitywould
component (< 2 s) of short bursts with and without
give rise to flares following the merger events. The ‘off
extended emission (EE) are indistinguishable, suggest-
axis collapsar’ models (Lazzati et al. 2010) and the mil-
ing that short bursts are “universal” (Norris & Gehrels
lisecond magnetar model (Metzger et al. 2008) offer ex-
2008). While the origin of such extended emission (seen
planations for why the extended emission from short
roughly in a quarter of Swift shGRBs) is debated, it can
GRBs may resemble the prompt emission from long
provide some clues to the identity of the elusive mecha-
GRBs. However,itisunclearhowthismodelcanexplain
nism of shGRBs.
the apparent different galactic environments in which
TheassociationofsomeshGRBswithearlytypegalax-
shortandlongGRBs arefound. Inbothtypes ofbursts,
ies that have low star formation rates suggests that
the photonspectrumis wellfittedby the Bandfunction,
short bursts arise from a different progenitor mecha-
hinting at a kind of universality in the underlying phys-
Electronicaddress: [email protected] ical model.
2 Ouyed et al.
Here, we will explore an alternative model based on This paper is organized as follows: In 2 we describe
§
a Quark-nova explosion occurring in a low mass X-ray the LMXB parameters and give a brief account of the
binary (LMXB). Such an explosion can happen if the Quark-Nova (henceforth QN) and the relativistic ejecta
neutron star, in its spin-down evolution, reaches the it produces. In 3, we describe characteristicsof the cir-
§
quark deconfinement density and subsequently under- cumbinarydisk(CD).Theinteractionbetweentheejecta
goes a phase transitionto the more stable strangequark from the QN and the CD is studied in 4 where we de-
§
matter phase (Itoh 1970; Bodmer 1971; Witten 1984), scribethe energeticsandspectralfeaturesof the prompt
resulting in a conversion front that propagates towards emission. Using our model, we perform fits to observed
the surface in the detonative regime. This hypothesis GRBs that bring out the origin of the empirical Band
is motivated by recent work on numerical simulations of function. We also derive the duration of the prompt
the conversion of neutron matter to strange quark mat- emission in our model and predict an optical counter-
ter (Niebergal et al. 2010b). Previous works that have part. Extended emission arisingfrom ablating the white
focused either exclusively on conversionthroughstrange dwarf is addressed in 5. Late X-ray activity ascribed
§
matter diffusion (eg. Olinto 1987; Horvath&Benvenuto to the spin-down evolution of the quark star and accre-
1988) or use pure hydrodynamics (eg.Drago et al. 2007; tion from trapped debris is explained in 6 along with
§
Cho et al. 1994) do not find detonation, only deflagra- afterglows from the shocked CD. 7 contains our main
§
tion. However, in the most recent work on this mat- conclusions.
ter (Niebergal et al. 2010b), we have analyzed the issue
numericallyincluding bothreaction-diffusionand hydro- 2. QUARK-NOVAINANS-WDLMXB
dynamicsconsistently,albeitwithina1Dapproximation.
Staff et al. (2006), using standard equations of state
Wealsoincludedforthefirsttime,theeffectsofneutrino
(EOS) of neutron-rich matter, considered the likely NS
cooling, finding numerical evidence for laminar speeds
candidates that could reach quark deconfinement den-
approaching 0.04c (much higher than previous works; c
sity in their core; the fiducial value was taken to be
is the speed of light) and a possible wrinkling instability 5ρ , where ρ 2.7 1014g/cc is nuclear saturation
in higher dimensions. As argued recently by Horvath ∼ 0 0 ∼ ×
density. To reach 5ρ , a NS with gravitational mass
(2010),wrinklescauseturbulence inthe conversionfront ∼ 0
MG 1.8M⊙(usingtheAPREOS;Akmal et al.(1998))
that serve as a platform for detonation. If this is borne ∼
is required.
outbymoresophisticatedsimulations,wewillhavefound
QuarkdeconfinementinthecoreoftheNScanhappen
potentially a new engine for GRBs that is capable of ex-
in two ways: (i) right after the neutronstar is formed, if
plaining several features of the GRB light curve, as we
the above criteria is satisfied. This is not relevant to the
show in this work. This exciting possibility, combined
model presented, but could be important for superlumi-
with the problems faced by more conventional scenarios
nous supernovae (Ouyed et al. (2009, 2010)); (ii) If the
described in the previous paragraph, should make our
above criteria is not met, the neutron star can accrete
model worth exploring as a possible option.
mass from a companion and reach the critical mass for
WenotethatNiebergal et al.(2010b)isalreadyanim-
deconfinementandsubsequentQNexplosion. Thisisthe
portantstepforwardinresolvingsomeofthe issueswith
scenario considered in our model.
uncertainties in the dynamics of the Quark-Nova. The
equation of state of quark matter, and hence the puta-
2.1. The LMXB evolution
tive quark deconfinement density is also unconstrained,
but recently O¨zel & Psaltis (2009) have argued that ac- We begin with the standard LMXB evolution model
curatemeasurementofmassandradiusforaminimumof (Verbunt 1993), where a system with a relatively long
three neutron stars will provide strong constraints up to orbitalperiod( 1day)producesarecycledpulsarwhen
∼
several times nuclear saturation density. While we must the subgiant progenitor of the white dwarf overflows
awaitfuture observations of neutron star parameters for its Roche lobe. In the first accretion phase, mass and
the resolution of some of these issues on the underlying angular momentum losses shorten the orbital period.
Quark-Novadynamicstobemoreconfidentinourmodel Once this first accretionphase ceases, the orbital period
assumptions,weneverthelessadoptaworking hypothesis continues to decay due to gravitational wave radiation.
in this paper that the detonative regime is indeed the Eventually, the orbit decays enough for the white dwarf
likely result of a quark-hadron phase transition inside a itselftooverflowitsRochelobe,commencingthesecond,
neutronstar. Fromthispointon,wewillfollowstandard ultrashortorbitalperiod(<1hr)LMXBphase. Herewe
arguments to demonstrate that a Quark-Nova occurring are mainly concerned with the first recycling phase.
inanLMXBfollowingtheendofthefirstaccretionphase Theorbitalseparationofthebinary,a,evolvesaccord-
leads to features (prompt emission, extended emission ing to Kepler’s third law
and flaring) reminiscent of those discussed above in the
context of short GRBs as observed by Swift. The same a=2.28 109cm(1+q)1/3 MT 1/3 Porb. 2/3 , (1)
engineatplayinsideacollapsarcannaturallyexplainthe × M⊙ 60 s
(cid:18) (cid:19) (cid:18) (cid:19)
manysimilaritiesbetweenshortandlongdurationGRBs,
thus exhibiting universality between the two classes. In whereq =M /M ,andM =M +M thetotal
WD NS T NS WD
theend,weobtainmanyobservedfeaturesofshortGRBs mass of the system. The mass of the WD, M , is con-
WD
beginning with our single hypothesis of the detonative strained by the expected core mass of the Roche Lobe
transition,implying that shGRBs couldwellprovide the filling progenitor in the LMXB phase, MWD 0.15M⊙.
∼
astrophysicalevidence for the Quark-nova,andthat this We adopt MWD 0.1M⊙ representing the WD’s fidu-
∼
mechanism deserves detailed study. cial value near the end of the first accretion phase. We
will make use of the WD mass-radius relation (Savonije
Quark-Novae in Low-Mass X-ray Binaries 3
(1983)) 2.3. The Quark-Nova ejecta
RRW⊙D =0.028(1+X)5/3MW−1D/,30.1 , (2) QuUanrlkik-neovSauepeisrnnovoate,feanseiubtlrei,noa-sdrnievuentrimnoasssareejecttriaopnpeind
inside a hot and dense expanding quark core, once
with MWD,0.1 being the white dwarf mass in units of it grows to more than 2 km (Ker¨anen et al. 2005).
0.1M⊙. HereafterwetakeX =0(the Hydrogenfraction In (Ker¨anen et al. 2005)∼, we used a typical inelas-
in the WD), since we assume a pure He WD near the tic neutrino-nucleon cross-section to show that these
end of the first accretion phase. (electron-) neutrinos are absorbed in the neutron-rich
At the end of the first recycling process (i.e. first ac- crust of the neutron star. The corresponding diffusion
cretion phase), the neutron star will reach an equilib- timescale out of the conversion front is 0.1 seconds.
riumperiod(Bhattacharya&vanden Heuvel1991)which Although the neutrinos have a similar lu∼minosity as in
is approximated by the Keplerian orbital period at the supernovae(using the per species number for the latter),
Alfv´en radius (Ghosh & Lamb 1992): there is an important difference in the outcome for mass
ejection. Unlikethecaseofasupernova,whereneutrinos
6/7 16/7 −5/7 −3/7 can drive a wind from the surface of the proto-neutron
P 2msB R M m˙ , (3)
NS,eq. ∼ NS,9 NS,6 NS,1.4 NS,Edd star, the neutron-rich crust is much too dense for these
where B , R and M are the neutron star neutrinostotransferkinetic energyeffectively (the grav-
NS,9 NS,6 NS,1.4
surface magnetic dipole field, radius and mass in units itational potential well is much deeper). Rather, they
of 109 G, 106 cm and 1.4 solar mass respectively. The heat up the crust to temperatures of 1 MeV. In com-
∼
accretion rate onto the neutron star, m˙ , is given in parisontothephoton-drivenexplosion(Vogt et al.2004)
NS
units of the Eddington limited accretion rate M˙ whichwehaveinvokedhere,energydepositionbyneutri-
Edd
10−8M⊙yr−1 abovewhich the radiationpressuregene∼r- nos is less than 10% of the total energy budget. There-
ated by accretion will stop the accretion flow. fore, it is not expected to lead to much baryon-loading
Neutron stars as massive as 1.8M⊙ are not easy to of the photon fireball. Mass ejection due to core bounce
produce in LMXBs even for in∼itially high mass of the is alsounlikely unless the quarkcore is verycompact(1-
donor star, unless they were already born as relatively 2) km. A more promising alternative is mass ejection
massiveneutron stars(e.g. Lin et al.(2010)). Assuming from an expanding thermal fireball that is a direct con-
up to 0.2M⊙ can be accreted (which will spin-up the sequenceofdense,hotquarkmatterproducedasaresult
NSto∼millisecondperiods)1 by the endofthe firstaccre- of the conversion from neutron matter(Vogt et al. 2004;
tion phase, this puts a constraint on the minimum mass Ouyed et al. 2010). The fact that more than 50% of the
of the NS at birth of 1.6M⊙. Therefore a prolonged gravitational and conversion energy is released as pho-
phase of accretionat th∼e Eddingtonrate for >107 years tons is unique to the QN (Ouyed et al. 2005). It allows
is needed. Thus LMXBs with NSs born massive and for ejecta with kinetic energy easily exceeding 1052 ergs.
accreting near the Eddington limit are the most likely Depending onthe conversionefficiency of photonenergy
systems to experience a QN explosion near the end of to kinetic energy of the ejecta (the neutron star’s out-
the first recycling phase. Figure 1 (upper panel) is a ermost layers), up to 10−2M⊙ of neutron-rich material
schematicrepresentationoftheLMXBsystemattheend canbe ejected at nearly relativistic speeds. Calculations
of the first accretion phase just before the QN goes off. of mass ejection in the QN scenario accounting for en-
ergytransfer to the crust, give estimates ofejected mass
2.2. The Quark-Nova of 10−5M⊙-10−2M⊙ (Ker¨anen et al. 2005). The average
Lorentz factor of the QN ejecta is,
IntheQuark-novapicture(Ouyedetal. 2002),thereis
anexplosivephasetransitiontothemorecompact(u,d,s) ηE EKE
QN QN,52
Γ = 10 , (4)
quark phase, and the gravitational potential energy is QN MQNc2 ∼ MQN,−3.3
released partly into outward propagating shock waves
from the supersonic motion of the (u,d,s) conversion whereEQKNE,52 =η0.1EQN,53 is thekinetic energyofthe
front. The temperature of the quark core thus formed ejecta in units of 1052 erg. Here η 0.1 is the efficiency
risesquicklyto10-20MeVsincethecollapseisadiabatic ofenergytransferfromtheQNtota∼lenergy( 1053 ergs
rather than isothermal (Gentile et al. 1993). As men- from gravitational and phase conversion ene∼rgy) to the
tioned in the introduction, a complete dynamical treat- ejecta’s kinetic energy. The QN ejecta mass, MQN,−3.3
ment of the QN, including multi-dimensional neutrino is given in units of 5 10−4M⊙ which we adopt as our
transport and hydrodynamics is only in the preliminary fiducial value. ×
stages (Niebergal et al. 2010b) but results based on cal- There are two main sources of uncertainty to calcu-
culations incorporating the most physics suggest that a lating the Lorentz factor (the neutrino contribution, we
detonation is possible. The important outcome of the havearguedabove,is notmorethan10%andis unlikely
QN, as we now explain, is that chunks of the neutron to change much even with improvedcalculations of neu-
star’scrustalmatter, richiniron-groupelements,canbe trinotransportindensematter). Thefirstisthethermal-
ejectedfromthesurfaceoftheneutronstaratrelativistic to-kineticconversionefficiencyηineq.(4),whichwehave
speeds. taken as 0.1 (e.g. Frank et al. (2002)). Fixing this num-
ber is equivalent to fixing the amount of mass ejected.
1 The spin-up of a slowly rotating neutron star with We are at present working on reducing this uncertainty
rmioodmenPtNS,oeqf. in=ertia2π/IΩNS ∼to 2themseqwuoiluibldriumrequsiprein mpases- bynumericalmodeling ofthe hydrodynamicsofthe con-
accretion of at least ∼ INSΩNS/(GMNSRNS)1/2 ∼ version (from neutron to quark matter) front, including
0.2M⊙(Bhattacharya&van denHeuvel 1991). particle diffusion and advective forces into account (see
4 Ouyed et al.
Fig. 1.— Upper Panel: A LMXB system which consists of a neutron star, a low mass white dwarf (MWD ∼0.1M⊙ in our model)
and a circumbinary disk near the end of the first accretion era. Lower Panel: Near the end of the first accretion phase, the neutron
star experiences a QN explosion ejecting the neutron star crust and leaving behind a quark star. The QN chunks are ejected during the
QNexplosionhittingthecircumbinarydiskmaterialatdifferentlatitudes. BelowacriticalheightzCD,vap (fromthemidplaneofthedisk),
the disk density is sufficiently high that the QN chunks are shocked above evaporation temperatures (circles with horizontal stripes). At
heightsgreaterthanzCD,vap thedensityislowenoughfortheQNchunkstosurvivetheshockandemitintheoptical(circleswithvertical
stripes). A portion of the QN chunks will collide with the white dwarf causing it to partially or fully ablate, depending on the NS-WD
separationandthewhitedwarfmass,whentheQNexplosionoccurs.
Niebergal et al. 2010b). Once these simulations are ex- or liquid phase) The typical chunk size at birth was cal-
tended beyond 1D, a comparison of the thermal energy culated to be ∆rchunk,b 186 cmψ−3rb,7ρb,8 where ψ is
density of the conversion front versus the pressure ex- the ironbreakingstrain∼inunits of10−3 (the ironbreak-
ertedonthe surrounding matter shouldprovide a better ingstrainis 10−3forsolidironand 0.1fortheliquid
constraint on η. phase), r ∼is the radius in units of 10∼7 cm when break-
b,7
The second source of uncertainty is the EoS and com- ing starts and ρ is the ejecta density in units of 108 g
b,8
position of the high density matter that is ejected. Our cm−3atbreakup(see 3.4andTable1inOuyed & Leahy
estimateforthefiducialvalueofejectedmass(andhence 2009). Thechunk’sre§stlengthlr ismuchlargerthan
Γ ) is based upon a fit to the EoS of the neutron star chunk
QN its width ∆r 200 cm for the solid phase (which
chunk,b
crust,specificallytheBPSequationofstatefortheouter ∼
we consider in this paper) or equal to its width for the
crust (Baym et al. 1971) which is matched to a higher
liquid phase (Ouyed & Leahy 2009). In the solid case,
density equation of state for the inner crust. The equa-
the chunks thus resemble “iron needles” moving parallel
tion of state for the outer crust is reasonablyrobust but
to their long axis. In the observer’s frame the ratio of
canbeimprovedbyafewpercentbytakingintoaccount
length to width will be contracted by 1/Γ .
QN
morerecentmodelsbasedonquantummoleculardynam-
Even beyond the break-up radius, the chunks remain
ical (QMD) simulations of the nature of the crust post-
in contact with each other within the relativistically ex-
accretion (eg. Horowitz & Kadau 2009); astrophysical
panding ejecta as a whole. This is because the pieces
phenomena such as quasi-periodic oscillations (QPOs)
expand in volume, filling up the space between them,
can also provide constraints (eg. Watts & Strohmayer
and causing the density of each piece to continuously
2007). The higher density of the inner crust is how-
decrease, until they reach the zero pressure iron-density
ever poorly constrained, since it involves very neutron- (ρ ), at radius r 2 109 cm, at which point they
Fe sep
richnucleiandpossiblysome“pasta”phases. Weexpect ∼ ×
stopexpanding. From Ouyed&Leahy(2009),the typical
thatupcomingexperimentswithrare-isotopebeams(eg. chunk size at r is ∆r 105 cm while its length
sep chunk
FRIB; http://www.jinaweb.org/ria/) will provide better ∼
is of the order of 106 cm for the solid ejecta case. As
constraints in the high-density regime.
thechunkscontinuetoexpandradiallyoutwardstoara-
Ouyed & Leahy (2009) studied the relativistically ex-
dius r >r , we can associate with them a filling factor
pandingshellofiron-richejectaemittedfromtheneutron sep
f =r2 /r2.
star crust and found that it breaks up into numerous chunk sep
chunks because of lateral expansion forces (107 to 1011 We will now motivate the statistical model of ejecta
break-up. Investigations of the fragment-size distribu-
chunksdependingonwhethertheejectacoolsinthesolid
Quark-Novae in Low-Mass X-ray Binaries 5
tion (FSD) that results from dynamic brittle fragmenta-
tion show that fragmentationis initiated by randomnu-
104
cleation of cracks that are unstable against side-branch
formation (˚Astr¨om et al. 2004). The initial cracks and
side branches individually form an exponential and a 100
scale invariant contribution, respectively. Both merge
teoxpyrieesldsedthienrethsueltginengefrraalgfmoremntnsi(zse)distsr−ibαuft1i(osn/sw1h)icwhitihs (cid:144)NNc 1
∝
α = (2D 1)/D (D being the Euclidean dimension of
− 0.01
the space), f a scaling function (an exponential) that
1
is independent of s for fragments smaller than s and
1 1
decays rapidly for s > s (˚Astr¨om 2006). Here s = λD 10-4
1 1
where λ is the penetration depth of a branching crack.
Adopting this model, the emergent picture for the rel- 10-6
ativistically expanding shell of QN ejecta is as follows: 0.001 0.01 0.1 1
cracks are initially nucleated at random and more or s(cid:144)s
0
less uncorrelated locations. From these locations, cracks
Fig. 2.—Log-LogplotoftheQuark-Novanormalizedfragments
propagate in different directions. The main cracks form sizedistributionforD=2. Thedottedtopcurveisforβ=0(chunk
largefragmentswithanexponentialPoissondistribution distributionispurelydefinedbythemaincracks)whilethedashed
with a typical size determined by ∆r . The size (i.e. bottom curve is for β=1 (chunk distribution is purely defined by
chunk thebranchingcracks). Thethreesolidcurves,fromtoptobottom,
mass) of the Poisson-process fragments will be limited
areforβ=0.9,0.99,0.999andillustratethecontributionfromthe
or reduced by the creation of small-size fragments (s1) twosizedistributionsasdescribedinequation(5).
around each crack by the side-branching process. The
final fragment size distribution becomes (˚Astr¨om 2006): equation (5) are equal, which we call, ∆r . For s <∆r
t t
FSD is dominated by the side branching. As can be
−2D−1 seen in Figure 2 the transition size becomes smaller,
n(s)=(1 β) s D exp 2D s (5) i.e. mt/mchunk << 1 as β → 1. As we show later in
− s − s 4.1, this statistical model plays a key role in explaining
(cid:18) 0(cid:19) (cid:18) 0(cid:19) §
the empirically observed Band function of GRB spectra
(s1/D+s1/D)D
+βexp 0 , (Band et al. (1993)).
"− s0 #
3. THECIRCUMBINARYDISK
where s =∆r in our model.
0 chunk Circumbinary disks (CDs) could accompany the for-
Theoretical and experimental investigations
mation and evolution of binaries. These CDs are either
(Sharon et al. 1995; Sharon & Fineberg 1996) have
theremainsoffallbackdisksproducedinthe supernovae
shown that a dynamic instability controls a crack’s
that formed the compact object (e.g. Wang et al. 2006;
advance when its velocity exceeds a critical velocity
Cordes & Shannon 2008), or material injected into cir-
of 0.36v where v is the Rayleigh wave speed in the
R R cumbinary orbits during the process of mass loss by the
material(aRayleighwaveisasurfaceacousticwavethat
Roche lobe-filling companions (Dubus et al. 2002). The
travels on solids). Beyond 0.36v the mean acceleration
R possibility that the CB disk is the remnant of the com-
of the crack dynamics change dramatically. At that
monenvelopeevolutionphaseofbinariescannotberuled
point,themeanaccelerationofthecrackdrops,thecrack
out either. Due to the tiny CD mass that would re-
velocity starts to osciallateleading to the so-calledYoffe
sult, even after billions of years, such disks would be
instability (Yoffe 1951) which leads to side-branching
difficult to detect in the optical and infrared. Nev-
(e.g. Buehler & Gao 2006). Numerical solutions of
ertheless, the detection of excess mid-infrared flux in
the model for v > 0.36v exhibit the occurence of
R few LMXBs in quiescence suggests that dusty circumbi-
these branching events where the main crack sprouts
nary material might be present around some LMXBs
side branches. The spacing between these branches is
(Muno & Mauerhan2006)(seealsoBowler(2010)). This
a function of the amount of dissipation. The above
remainstobeconfirmed. Hereweconsider(andassume)
equation assumes that crack branches propagate easily
the case in which the binary system is surrounded by
with large penetration depth given by λ = ∆r1/D . stable circumbinary disk/material continuously replen-
chunk
Shown in Figure 2 is the FSD for two-dimensional ished during the process of (near-Eddington) accretion.
fragmentation with D = 2 (i.e. α = 1.5)2. There are As we have said, building massive NSs in LMXBs would
two natural scales in our model, the typical fragment require near Eddington accretion rates. This, we argue,
scale ∆rchunk (and thus mass mchunk) and, the scale couldfavortheformationandthesustainmentoftheCD.
at which the two terms/contributions in the RHS of There are different possible configurations for the CD
such as a flat or flared disk. Here, we assume the re-
2 A special case is, however, when a two-dimensional object plenished disk to be a geometrically thin, viscous Kep-
(i.e. a thin plate and/or closed shells) is fragmented in a three-
lerian circumbinary disk. Is scale height, H , at the in-
dimensional space. Models revealed that the branching-merging in
picture for two-dimensional fragmentation may be valid for the ner edge, Rin, is Hin 0.03Rin (Belle et al. 2004). The
∼
initialstagesoffragmentation,butfragmentationmaycontinueas disks could lie as close to the center of mass of the bina-
a cascade of breakups beyond the limit of maximum fragmenta- ries as R 1.7a, at which point they would be tidally
tion through branching and merging of cracks. It was found that in ≃
truncated (Taam et al. 2003; Dubus et al. 2004). The
atlaterstagesadditionalfragmentswereformedwithapower-law
FSDthathadα∼1.17(Linnaetal.2004). dependenceoftheverticallyintegratedsurfacemassden-
6 Ouyed et al.
sity,ΣonradiusR,istakenfrom(Taam & Spruit2001), the solid angle extended by the critical surface defined
particularly Figure 8 in that paper, from which one can by ρ>ρ is then
CD,vap.
approximate Σ Σ (R /R). The mass of the disk is
∼ in in 2πR 2z H
then Ω = in CD,vap. 4.8 in 0.15 . (11)
c 4πR2 ∼ R ∼
Rout in in
MCD= Σ(R)RdR∼ΣinRi2n(Rout/Rin−1) (6) The amount of QN ejecta material contained within Ωc
ZRin is MQN,c =ΩcMQN 7.5 10−5M⊙MQN,−3.3.
≃1.5×10−6M⊙Σin,5a210ζCD,out The time it takes≃a gi×ven chunk to be completely
eroded is
where ζ =(R /R 1) . For a disk that is
CD,out out,12 in,10
verticallyisothermal,atagivenr−adiusR,theequationof lr lr a1/2
habydovroesytaietlidcse(qBueillilbereiutmal.co2m00b4i)n:ed with the radial profile tvap. = 2vcshΓun2QkN ∼4.3×10−6 s Σch1in/u,2n5kΓ,53QN1,010 , (12)
Σ R z2 wherevs = (kBTc/2mH)isthespeedatwhichtheheat
in in
ρ = exp . (7) wave crosses the chunk. The chunk’s Thompson opti-
CD H√2π R −2H2 p
(cid:18) (cid:19) (cid:18) CD(cid:19) cal depth, 1/(nchunkσT), is much less than the typical
chunk size,∼∆r 105 cm (the chunk’s density is
Unlike a Shakura-Sunyaev disk, in CDs, the opening chunk
about 103 g cm−3 b∼y the time they reachthe CD; see
angle, H/R, is expected to be roughly constant (here
∼
5). During their extremely brief life as solid entities in
0.03). RecallingthatR 1.7a,theabovethenyields
in §
∼ ≃ theCD,mostofthe chunkswillemitasblackbodiesata
ρ ρ Rin 2exp z2 , (8) rFaotreaLtcyhupnikca=l cAhuchnukn,kσLTc4hunk wh1e0r4e0 Aercghusn−k1∼Tπ4∆rwc2hhuenrke.
CD ∼ in R −2H2 chunk ∼ c,30
(cid:18) (cid:19) (cid:18) CD(cid:19) the chunk’s temperature is giveninunits of30 keV.The
where ρ 10−4 g cm−3 (Σ /a ) and Σ = totalluminosity in blackbody emissioncan be estimated
in ∼ in,5 10 in,5 byrecallingthatthetotalareaextendedbythechunksis
Σin/105 g cm2. The columndensity asseenata viewing 4πa2 so that L Ω 4πa2σT4 1049 erg s−1T4 .
angle90o−i=θview =zview/R,whereiistheinclination ∼ThecorrespondingBiBso∼tropcicblackbcod∼ycontributionfrc,o3m0
angle of the system is
all the chunks is L = L /Ω 1050 erg s−1. An
BB,iso. BB c
∼
observer would see a blackbody peaking at Γ T .
QN chunk
1 R A givenchunk will continue to plow CD materialeven
NCD= ρCDdR (9) after complete vaporization and erosion since the va-
µ m
CD H ZRin porized material will retain the directional motion and
6 1029Σ exp θv2iew , size distribution of the incoming chunks. The vaporized
∼ × 5 − 2θ2 chunkmaterialwilleventuallyenteradecelerationphase
(cid:18) c (cid:19)
once it plows a mass of m /Γ (e.g. Rhoads
chunk QN
where mH is the proton mass and µCD 1.2 is the 1997) in CD material. T∼his puts a constraint on the
∼
mean molecular weight for the CD material which we minimum mass of the CD disk necessary to stop the
assume has a solar composition. QN ejecta within Ω . It is M = M /Γ
c CD,min QN,c QN
4. PROMPTEMISSION 7.5×10−6M⊙MQN,−3.3. In our model, this translate∼s
to a minimum disk size R = R given by
out,min stop
thWeyeimfirpsatctadthdereCssDt.hTehfeatcehuonfksthsehoQckNthcheuCnkDs mwahteen- TζChDis,stcorpit≃icaRlsrtaodp/iuRsinis∼im5p0o0rtManQtNf,o−r3.t3h/e(Σkiinn,5eam210aΓtiQcsN,a1n0)d.
rial, and in turn, undergo a reverse shock that heats dynamics of a given chunk in the CD. Assuming a CD
them to a temperature found from pressure balance withR >R impliesthatmostofthevaporizedQN
out stop
Pchunk = PCD,sh.. The pressure in the shocked CD ma- ejecta will lose its momentum and kinetic energy to the
terial,PCD,sh.,is foundfromthe jumpconditions(Russo CD material before it reaches the outer edge of the disk
1988; Iwamoto 1989). We get, at R=Rin, (see 4.2).
§
Σin,5Γ2QN,10 z2 4.1. Spectrum: Origin of the Band Function
k T 30 keV exp . (10)
B chunk ∼ a −2H2
10 (cid:18) in(cid:19) Forthepromptemission,weisolatetwobroadregimes
(in addition to the blackbody emission from the heated
Noting that non-degenerate iron will vaporize if heated
chunks) in our model depending on whether the shock
to 0.3 eV (CRC-Press 2005, for vaporization temper-
≥ temperature of the chunks exceeds the nuclear dissocia-
ature of iron at normal density), the rapid fall-off of
tion temperature (binding energy) for iron or not.
the CD density given by eq.(8) implies that the vapor-
ization is already efficently accomplished by interaction
4.1.1. Regime 1 - MeV photons, transient Fe lines
with the inner edge of the disk. At the inner edge of
the CD, we can define a critical disk height, zCD,vap. Upon impact, the chunks are heated to temperatures
(and corresponding density ρCD,vap.), below which the above 0.3 eV but below 8.8 MeV, in which case they
chunks will start vaporizing upon impact (see Fig. 1 for vaporize without dissociation. Prompt emission in our
a schematic illustration of the process). This critical model is tied to the shocked CD material, whose tem-
dheenigshittyisρ zCD,vap. ∼5.84.8H1i0n−7wigthcma−c3o.rrIetspfoolnlodwinsgtChaDt ppleorwatsutrheroTuCgDh,sthcaenCbDewehstiliemvaatpedoraizsinfoglilnowtos:aThohtesctrheuanmk
CD,vap.
∼ ×
Quark-Novae in Low-Mass X-ray Binaries 7
of gaseous particles which retain the directional motion QN ejecta. We assume a one-to-one relationship be-
of the incoming chunks. In this case, in the observer’s tween the temperature of the shocked CD material and
frame, the resulting radiation energy; in this case, the prompt
photon energy E will inherit the fragmentation distri-
γ
bution. Although a full calculation of the non-thermal
ρ dM
ρobCsD. ΓQNmchunkc2 ∼ µ CDm,plowkBTCD,s. (13) spectrum of the photons is beyond the scope of this
chunk CD H work, we test this idea by performing fits to some ob-
We have dM =A ρ dR where served GRB spectra by assuming that the photon lumi-
CD,plow chunk CD CD,obs.
Achunk is the area of the chunk and dRCD,obs. is the nosity (νF(ν) = Eγ2n(Eγ)) follows the Weibull distribu-
distance traveled by the vaporized chunk as measured tion given by (16),
by an observer. The above equation is valid as long as
dMCD,plow < mchunk/ΓQN. The chunk’s rest mass is E δ+2 (E /E )δ+1
mchunk = ρrchunkAchunklcrhunk with ρochbus.nk = ΓQNρrchunk. Eγ2n(Eγ)=Aγ Eγ exp − γδ+01 (17)
Replacing in eqn.(13) yields (cid:18) 0(cid:19) (cid:20) (cid:21)
where A is a normalization factor and E the average
γ 0
lr photon energy in the distribution. The peak of the dis-
k T µ m c2 chunk . (14)
B CD,s ∼ CD H × dR tribution occurs at
CD,obs
1
A limit on the observed temperature is then, Epeak =(2+δ)1+δ E0 . (18)
lr In our model, the assumed one-to-one correspondence
kBTCD,s >10keVµCD chunk,5 , (15) between TCD,s and Eγ implies that Epeak evolution in
× dRCD,obs,10 time is given by eq.(14). The fits to GRB 000218, GRB
where dR c dt 1010 cm to account for a 000727, and GRB 980706A are shown in Figures 3&4.
CD,obs ∼ × ≈ In reality the spectrum evolution will be related not
typical GRB duration dt 0.3 s (see 4.3). The
GRB ∼ § just to the chunk distribution but also the vaporitza-
temperature given above is the peak temperature after
tionrate. We expectthe softeningofthe spectrumto be
the chunk has plowed through most of the CD. Higher
caused mostly by the erosion of the fast moving chunks
peak temperatures occur in the early stagesof the plow-
whichwillbeheatedthemostandthusradiatethemost.
ing process.
The totalnumber ofchunks shouldalsodecreaseintime
Eqn.(14) is illuminating - it implies a direct propor-
with smaller chunks most likely disappearing (i.e. fully
tion between the temperature of the shocked material
eroded) first. The case of GRB 000727 is an interest-
and the linear size of the chunk. Since the linear size is
ing one since its observedE increases in time. However
distributedaccordingto Eqn.(5), the temperature ofthe 0
notethedataisforshorterintegrationtimesthanforthe
shockedCD materialwill alsodisplay the samedistribu-
other two candidates (see Table 1). In our model, there
tion.
is a minimum accumulation time (presumably of the or-
It is interesting to note that several GRBs show spec-
derofmilliseconds)beforethefullmassdistribution(i.e.
tra(moreprecisely,νF(ν)curves)thatappeartofollowa
theBandspectrum)buildsup. Theexcellentfitssuggest
log-normaldistribution(Band et al.1993). Accordingto
that the underlying physical picture relating the spec-
Brown&Wohletz (1995), fragmentation of a massive ob-
trum to the ejecta fragment size or mass distribution is
ject that proceeds by formation and branching of cracks
probably correct. In reality, the fragmentation process
results in a distribution of masses (per unit logarithm
may be more complicated (e.g. eq.(5)), but the Weibull
in mass) that follows the Weibull distribution (Weibull
distribution appears to be an excellent approximation.
1939),whichlooksalmostidenticaltothelog-normaldis-
There are specific predictions for this component
tribution. The Weibull distributionin mass canbe writ-
within our model: (i) The peak photon energy (related
ten as (Brown&Wohletz 1995)
to lr ) should evolve in time according to equation
chunk
(14); (ii) The prompt emission is composed of contribu-
m δ+2 (m/m )δ+1 tions from 107-1011 “blobs” of emitting CD material
m2n(m)=Nm exp 1 (16) ∼
1 m − δ+1 plowed and heated by the relativistic vaporized chunks
(cid:18) 1(cid:19) (cid:20) (cid:21) material. Agivenpulseinourmodelconsistsofdifferent
where δ = D/3 is a fractal dimension and m1 a mass “blobls”ofheated CD materialemitting simultaneously.
−
relatedtothe averagefragmentmassinthe distribution. Thus a pulse should also carry the signature of the dis-
The peak of the distribution (called the most proba- tributionin massof the chunks (i.e. the Bandfunction);
ble mass; in our case mchunk) is given by mpeak/m1 = (iii) finally, the prompt emission (regime 1) should have
(2 + δ)1+1δ; N is a normalization factor related to the signatures of iron-group elements (Fe, Co, Ni) brought
total number of fragments. Note the formal similarity in by the chunks, but this should be transient since the
to the first term in eqn.(5), derived by ˚Astr¨om (2006), vaporized material will cool rapidly.
who based his expression on a more complicated frag-
4.1.2. Regime 2 - GeV photons, no Fe lines
mentation situation involving the competition between
side-branching and cracking, which leads to the second Chunks heated above 8.8 MeV will be dissociatedinto
term in eqn.(5). nucleons and lose their identity by mixing with the CD
Therefore, we put forward the intriguing suggestion matter. From Eqn.(10), this regime could occur for sys-
that the GRB spectrum arises due to photons from the tems with ΓQN >100 (i.e. with MQN <5 10−5M⊙).
×
shock-heated CD material whose temperature distribu- The mixed CD and QN material will have a common
tionisinheritedfromafragmentationdistributionofthe temperature and will have some neutron excess (since
8 Ouyed et al.
555555 000000
111111000000 111111000000
))))))
111111
------
111111))))))111111000000444444 eVeVeVeVeVeV111111000000------111111
------ kkkkkk
ssssss
111111
-2-2-2-2-2-2 111111000000333333 ------ssssss111111000000------222222
mmmmmm
cccccc 222222
------
rgrgrgrgrgrg111111000000222222 cmcmcmcmcmcm111111000000------333333
eeeeee ((((((
((((((
ssssss
FFFFFFνννννν111111000000111111 onononononon111111000000------444444
νννννν tttttt
oooooo
hhhhhh
111111000000000000 pppppp111111000000------555555
111111 222222 333333 111111 222222 333333
111111000000 111111000000 111111000000 111111000000 111111000000 111111000000
EEEEEE ((((((kkkkkkeeeeeeVVVVVV)))))) EEEEEE ((((((kkkkkkeeeeeeVVVVVV))))))
555555 111111
111111000000 111111000000
))))))
111111
------
-1-1-1-1-1-1))))))111111000000444444 keVkeVkeVkeVkeVkeV 111111000000000000
ssssss
222222 -1-1-1-1-1-1 111111000000------111111
------ ssssss
mmmmmm
cccccc 333333 222222 ------222222
111111000000 ------ 111111000000
gggggg mmmmmm
rrrrrr cccccc
(e(e(e(e(e(e ( ( ( ( ( (111111000000------333333
νννννν111111000000222222 nsnsnsnsnsns
FFFFFF oooooo ------444444
νννννν tttttt111111000000
oooooo
hhhhhh
111111000000111111 pppppp111111000000------555555
111111 222222 333333 111111 222222 333333
111111000000 111111000000 111111000000 111111000000 111111000000 111111000000
EEEEEE ((((((kkkkkkeeeeeeVVVVVV)))))) EEEEEE ((((((kkkkkkeeeeeeVVVVVV))))))
555555 000000
111111000000 111111000000
))))))
111111
------
111111))))))111111000000444444 eVeVeVeVeVeV111111000000------111111
------ kkkkkk
ssssss
111111
-2-2-2-2-2-2 111111000000333333 ------ssssss111111000000------222222
mmmmmm
cccccc 222222
------
rgrgrgrgrgrg111111000000222222 cmcmcmcmcmcm111111000000------333333
eeeeee ((((((
((((((
ssssss
FFFFFFνννννν111111000000111111 onononononon111111000000------444444
νννννν tttttt
oooooo
hhhhhh
111111000000000000 pppppp111111000000------555555
111111 222222 333333 111111 222222 333333
111111000000 111111000000 111111000000 111111000000 111111000000 111111000000
EEEEEE ((((((kkkkkkeeeeeeVVVVVV)))))) EEEEEE ((((((kkkkkkeeeeeeVVVVVV))))))
Fig. 3.—Modelfit(solidcurves)toobservedphotonspectrum(νFν) Fig. 4.— Model fit (solid curves) to observed photon flux n(Eγ)
withdatafromMazetsetal.(2004)forGRB000218(toppanel),GRB withdatafromMazets etal.(2004)forGRB000218(toppanel),GRB
000727 (middle panel), and GRB 980706A (bottom panel). The top 000727 (middle panel), and GRB 980706A (bottom panel). The top
and bottom data points, in each panel, are for different integration and bottom data points, in each panel, are for different integration
time(seeTable1). time(seeTable1).
the QN ejecta inherits its composition from the neutron production,processessuchasp+p p+p+π0,n+p
star crust) with a mean molecular weight µ 1. We p+p+π−,p+p p+n+π+ can→occur. Chargedpio→n
∼ →
expect a given chunk to instantly transfer a significant decay produces electrons and positrons which can anni-
fractionofits kinetic energyinto heating m /Γ of hilate to form photons, and neutron pions can produce
chunk QN
CD material. The resulting thermal energy per nucleon photons by π0 2γ. The luminosity can be estimated
→
is then from
k T 100 GeV Γ2 . (19)
B nucl. ∼ QN,10 ρ ρ
chunk CD
L E 2πR H l cσ (20)
For such energies which exceed the threshold for pion ∼h πi m2 in CD chunk pp
H
Quark-Novae in Low-Mass X-ray Binaries 9
The energy deposited by the QN ejecta (into
TABLE 1 ∼
M /Γ of CD material) gives us an estimate of the
Model fitstoobserved photon spectrum (GRB000218, QN,c QN
GRB000727,andGRB980706A)using eq.(17). The fitsare prompt GRB emission. The total thermal energy pro-
performedforobservationsattwodifferentintegration duced which we assume to be roughly equal to the to-
time,tint.. tal GRB energy is E Γ (M /Γ )c2
GRB QN QN,c QN
∼ × ∼
GRB tint. δ E0 (keV) Epeak/E0 Aγ 1.5×1050 erg MQN,−3.3 where we have used MQN,c =
Ω M . The effective isotropic energy in our model is
000218 0-256ms -0.2 520 2.1 0.164 c QN
256ms-5.888s -0.5 420 2.3 0.027 thus EGRB,iso. EGRB/Ωc 1051 erg MQN,−3.3. Ob-
∼ ∼
000727 0-128ms -0.33 73 2.2 3.2 served short GRBs also have E up to 1051erg
GRB,iso
256ms-768ms -0.65 80 2.4 5.0 (Nakar 2007).
980706A 0-256ms -0.39 760 2.2 0.2 In our model, the disk only subtends an angle 15%
256ms-8.448s -0.67 580 2.4 0.009 of the sky. Thus, at most 1051 ergs of the QN ∼kinetic
energy1052 ergintersectst∼he disktobe radiated. Itwas
2 1049ergs/sΣ l a shownby(Zhang et al.2007)thattheradiativeefficiency
in,5 chunk,5 10
∼ × (E /(E +E )) couldvary froma few percents to more
γ γ K
wherewehaveassumedatypicalcross-sectionforppcol- than 90% depending on how the kinetic energy is calcu-
lisions in the tens of GeV range of σpp 10−25 cm2 (i.e. lated. Thesestudiesarebasedonkeyassumptionsabout
∼
100 millibarns, Blattnig et al. 2000) and a typical pion (i) the unknown shock parameters; (ii) the required de-
energy Eπ 150 MeV (threshold production). tailed afterglow modeling with its own limitation and
h i∼
We offer specific predictions for this high-energy com- (iii) assumes the fraction of the electron energy in the
ponent within our model: (i) We expect this component internal energy of the shock to be 10%. Nevertheless let
also to show traces of the chunk size distribution and us assume an efficiency of 0.1 for short GRBs which
partly follow a log-normal-like distribution. However, is also the number derived∼in an independent study by
chunks heated to extreme temperatures (>> 1 GeV) (Berger 2007). A 10% radiative efficiency would imply
will start to move away from the band function; (ii) it an E 1050 erg in true energy release corresponding
γ
should have no signatures of Iron-group elements since to an is∼otropic energy release of E 1051 erg. This
γ,iso.
theejectaisdissociatedcompletelytonucleons;(iii)GeV number could be higher for disks with∼higher solid an-
and higher energy photons should be observed only in gle (e.g. flared CDs) and/or for QN ejecta with higher
systems with ΓQN >100. kinetic energy.
For the two regimes isolated above, the CD mate-
rial between photons and the observermust be optically 4.3. Duration
thin. The radiation will scatter most off of free elec-
In our model, emission duration is related to the time
trons in the ionized CD material. The Compton opti-
it takes the QN ejecta to be stopped by its interaction
cal depth is N σ 2.7 Σ exp θ2 /2θ2 where
CD C ∼ × 5 − view c with the CD:
σ = (3/8)σ ln(x)/x2 is the Compton cross-section,
σTC = 6.652×T10−25 cm2 the Thom(cid:0)pson cross-(cid:1)section, tGRB ≃ (Rst2ocpΓ−2 Rin) ∼1.4 s Σ MaQN,Γ−33.3 . (21)
andxistheratiobetweenthepromptphotonenergyand QN in,5 10 QN,10
the electron rest-mass which is 1 GeV/(mec2) >> 1. The time-variability can vary from microseconds to a
∼
The shocked matter is therefore Compton-optically thin fraction of a second depending on the number of chunks
to the high-energy photons. For the low energy pho- (out of the 107-1011) that emit simultaneously at a
ton regime, the Thomson optical depth is too high and given time. ∼
thisshouldnegativelyaffecttheabilitytoseetheshocked
chunksortheshockeddiskgassincethedepositedenergy 4.4. The optical counterpart
should thermalize over the whole disk. The disk geom-
TheQNchunkshittingtheCDinregionswhereρ <
CD
etry and vertical scale height, in our model, is assumed
ρ (i.e. at heights z > z ; see Fig. 1) will
CD,vap. CD,vap.
tobethesimplestwithH/R=constant. Howeverother
not be fully vaporized, nor do they expand significantly.
disk configurations are possible, such as a CD with a
Rather, they pass through the CD, effectively punctur-
flared surface; H R3/2. This is a great advantage ing it. During this interaction the temperature of a QN
∝
since a large solid angle is available for the chunks to be chunk is determined by shock heating, and will not ex-
able to plow at an angle into increasing smaller density ceed 3 eV Γ as seen by an observer; thus emis-
QN,10
and thus be able to radiate in the optically thin regime. sion w∼ould be in the optical band. The observed opti-
Specifically,forsuchaflareddisk/atmospheremostofthe cal emission is related to shock efficiency with emitted
chunks would be flying throughthe larger volume of the energy E (0.3 eV M /(µ m )) Γ2
disk/atmosphere so most energy would be deposited in Opt. ≃ × QN,Opt. e H × QN ∼
the region that is dense enough to significantly slow the 1041 ergs Γ2QN,10MQN,−3.3 where µe = 2 is the mean
weight per electron and M = Ω M with
chunks. Uphighintheflareddisk/atmosphereregionthe QN,Opt. Opt. QN
Ω =(z z )/R 0.3H /R with the
densityissolowthatthechunksflythroughunimpeded. Opt. CD,Opt.− CD,vap. in ∼ in in
maximum height z is calculated from eq.(10) us-
In summary, a H R3/2 would make a large visible im- CD,Opt.
∝ ing the lower limit of the optical band. Our model pre-
pact surface available with a a longer lived black body
dicts opticalemissionto appear simultaneously with the
from the optically thick part and a short lived brighter
promptemissionsincethechunkspenetrateallregionsof
part from flared region of the disk.
thediskatthesametime. Theopticalandpromptemis-
sion are both composed of short pulses from the impact
4.2. Energetics
of the chunks striking the CD.
10 Ouyed et al.
4.5. Afterglow activity time” (R /c) (ρ /ρ )0.5 (R /c) << 1 s
WD WD QN WD
∼ × ∼
(Klein et al. 1994). Thus it is reasonableto assume that
The innermost CD material acts as a buffer for the
all of the intercepted kinetic energy will go into heating,
dissociated and pulverized chunks. The cross-sectional
rather than shocking on and flowing around, the WD.
area of the chunks increases rapidly sweeping up more
CD mass, which leads to a run-away process. As they
5.1. Energetics
expand, they start increasing the covering factor of the
blast. Ifenoughchunksgetdissociateda covering/filling The total thermal energy deposited by the chunks im-
factor of unity would eventually be reached. Energy- pacting the WD is ΩWDEQKN, given by
Momentumconservationimpliesthatthemixedejectais
ejected with speed, vf, given by E 9.3 1049 erg EK M−2/3 a−2 , (23)
th ∼ × QN,52 WD,0.1 10
Γ2 1
β Γ = QN− MQN,c Γ , (22) whereΩWD =RW2 D/(4a2)isthe solidanglesubtended
f f q1+ MMQCND,c ∼ MQN,c+MCD QN berygythoefWthDe.WIfDEth exceeds the gravitationalbinding en-
where β = v /c. Depending on M /M , the
f f QN,c CD
mixed ejecta resembles a massive shell which expands
E =GM2 /R 1.4 1048 erg M7/3 (24)
at mildly relativistic speeds. External shocks with the G,WD WD WD ∼ × WD,0.1
circumstellar matter will slow down further the mixed
then the WD would be ablated as a result of the QN
ejecta and should produce an afterglow, which could
explosion. The condition above imposes a nearness re-
lasts for months. This is similar to afterglow activity in
striction on the WD as
the internal-externalshock model of GRBs (Piran 2001)
exceptthatthepromptemissioniscausedbythechunks (EKE )1/2
rather than by internal shocks. Additional afterglow a<a =8.2 1010 cm QN,52 . (25)
activity can also occur if the non-dissociated chunks th. × M3/2
WD,0.1
interact with matter beyond the outer edge of the CD.
For systems with a < a , one should observe both
th.
prompt and extended emission.
The temperature per nucleon of the ablated WD will
be an important quantity that determines the spectrum
To summarize, the QN naturally provides the rel-
of the extended emission. It can be estimated as
ativistic “bullets” required by some GRB models
(Heinz&Begelman1999;Umeda2000). However,thesur-
µ EKE
roundingenvironmentandemissionregionis verydiffer- WD,4/3 QN,52
k T 133 keV , (26)
entinourmodel. Theinteractionbetweentherelativistic B WD ∼ a2 M5/3
10 WD,0.1
QN chunks and the CD (as well as the chunks’composi-
tion)isthe keyto the successesofourmodel-the origin where µWD is the meanmolecular weight in units of 4/3
oftheBandfunction,aswellasenergetics,variabilityand (forapureHeliumWD).Thisimplies thatWDmaterial
durationwhichareconsistentwithobservations. Thein- is ejected at speeds
teractionbetweentheQNchunksandtheWDisanother
feature of our model which we explore next. EKE 1/2
5. EXTENDEDEMISSION Vth.,ejec. ∼3×103 km s−1a QMN,55/26 , (27)
Thus far, we have focused on the interaction between 10 WD,0.1
the chunks of ejecta and the CD as a means to explain 5.2. Spectrum
the prompt emission. We now investigate the interac-
Nuclear burning timescales are much shorter than dy-
tionbetweenthechunksandtheWD.Asweshowbelow,
namical timescales. Depending on T , various nuclear
the WD can be ablated by the highly energetic chunks, WD
burning processes are expected to take place. As for the
leading to properties reminiscent of the extended emis-
case of prompt emission, we can isolate the following in-
sion (EE) observedin some shGRBs. The density of the
teresting regimes:
QN ejecta at a distance r from the point of explosion is
given by eq.(B3) in Ouyed&Leahy (2009). A combina-
5.2.1. Regime 1: T >0.26 MeV, Ni production
tion of mass conservation and thermal spreading of the WD
QNejectathickness,∆r,givesρ 103 g cm−3 ρ From equation (26), this temperature regime corre-
QN 0,14
∼ ×
∆r /(r9/4M1/4 )wherethedensityatejectionpoint sponds to
0,4 10 QN,−3.3
o1c(tmr9h9∆−e53)rs.u(Trfofahrceae1enjoienfMcttuaQhneNtithNs∼icSokf)5n,1e×ρ0ss041,ac0itm−se4.gjMeiTvc⊙ehtineoencjinehicustuna∆nk;isrtD’0sda∼oteftn1as10i0et4t1y4caamlgt. a10 <aNi =0.7 µ1W/D2,M4/W35E/D6QK,0NE.1,521/2 , (28)
0,4
the WD ∼location (r = a) is then 103 g cm−3 ρ In this range, the WD temperature TWD TSi 3
∼ 0,14 × 109 K, the temperature for Silicon burning.≥As a r∼esul×t,
9/4 1/4
∆r /(a M ) which is of the order of the WD
0,4 10 QN,−3.3 α-burning can occur all the way up to Nickel-56. The
density ρWD ∼ 6.3 × 103 g cm−3 MW2D,0.1. We thus sequenceofα-burningfromHeliumthroughtoNickelwill
expect the QN induced shock to pass through the release on average 7-8 MeV per nucleon, resulting in an
WD rather than flowing around since the “crushing energyrelease 2 1050 M erg,whichisenough
WD,0.1
∼ × ×