LMC X-3 May Be a Relic of a GRB Similar to Cosmological GRBs

0.21 MB·English

by  G. E. Brown

Draftversion February2,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
LMC X-3 MAY BE A RELIC OF A GRB SIMILAR TO COSMOLOGICAL GRBS
Gerald E. Brown
DepartmentofPhysicsandAstronomy,StateUniversityofNewYork,StonyBrook,NY11794, USA.
Chang-Hwan Lee
Department ofPhysics,PusanNationalUniversity,Busan609-735,Korea.
and
Enrique Moreno M´endez
8 DepartmentofPhysicsandAstronomy,StateUniversityofNewYork,StonyBrook,NY11794, USA.
0 Draft versionFebruary 2, 2008
0
ABSTRACT
2
The present scenario for high-luminosity long γ-ray bursts is strongly influenced by the paper of
n
Fruchter et al. (2006). Whereas the main contention of this paper that these GRBs occur in low-
a
metallicityirregulargalaxiesisbasedonaconsiderablecollectionofobservationalresultsandalthough
J
themainthesisisdoubtlesscorrect,thepaperdoesnotexplainthedynamicsthatproducessuchGRBs
3
and much of the discussionnot directly concerning the main thesis is wrong. We propose a dynamics

and elucidate how the Fruchter et al. (2006) results may be tested, in our neighborhood in the LMC,
]
h suggesting that LMC X-3 is a relic of a high luminosity explosion, probably accompanied by a GRB
p and hypernova explosion. The way to test our suggestion is to measure the system velocity of the
- present black hole. We correct errors of the Fruchter et al. paper in stellar evolution, so that the
o
study of GRBs is consistent with it. We show that the subluminous GRB 060218 had a low-mass
r
t black hole as central engine.
s
a Subject headings: binaries: close — gamma rays: bursts — black hole physics — supernovae: general
[ — X-rays: binaries

1
v 1. INTRODUCTION respectively. Thenatalrotationalenergywas430bethes
7 (one bethe= 1051 ergs) and the final (measured) energy
7 Fruchter et al. (2006) collect an important amount of
isindistinguishable,withinerrors,fromthenatalenergy.
4 data on long γ-ray bursts. They propose that the long-
Brown et al. (2000) had reconstructedthe GRB and hy-
0 duration γ-ray bursts are associated with the most ex-
pernova explosion for Nova Sco. The nature of the ex-
. tremely massive stars and that they may be restricted
1 plosioncouldbereconstructedfromthedonor,whichac-
to galaxies of limited chemical evolution. Also that long
0 ceptedanumberofα-particlenuclei,especially32Swhich
γ-ray bursts are relatively rare in galaxies such as our
8
isspecialforhypernovaexplosions,butrareinsupernova
0 own Milky Way.
explosions.
: In recent papers (Moreno M´endez et al. 2007;
v Thus, from the near equality of the natal and present
Brown, et al. 2007) we have developed the Blandford-
i rotational energies, only a few percent of the available
X Znajek model of GRBs into quantitative calculations
rotational energy could have been accepted in the GRB
of the angular momentum energy that can be delivered
r and hypernova explosion.
a for GRBs and hypernova explosions. This was possible
Moreno M´endez et al.(2007)andBrown, et al.(2007)
because Lee et al. (2002) showed how to work out the
showed from population syntheses that the soft X-ray
Kerr parameters of the rotating black holes. We showed
transient sources were sufficient in number to account
thatthe GalacticsoftX-raytransientsourceswererelics
for all of the subluminous bursts in our neighborhood.
of Galactic explosions and constructed the energy of all
Whataboutthecosmologicalburstswhicharethelong
15 of the known Galactic sources.
γ-ray bursts considered by Fruchter et al. (2006)? The
It turns out that most of the Galactic sources under-
WoosleyCollapsarmodelwasinventedtodescribethese.
went subluminous GRBs, not because they did not pos-
In terms of numbers these are only a few percent of the
sess sufficient rotational energy, but mostly because the
subluminousbursts(Liang et al.2007). Wewillnottake
rotational energy was so large that it destroyed the ac-
issuewiththeWoosleyCollapsarmodeldescribingthese,
cretion disk so quickly that the central engine was dis-
because our binary evolution begins by the donor spin-
mantled before the GRB could properly develop. We
ning the He star, progenitor of the black hole, to what-
give as examples the transient sources Nova Sco (GRO
J1655−40)andIlLupi(4U1543−47)forwhichLee et al. ever angular momentum is needed. Then the donor de-
couples, acting only as a witness to the explosion, with
(2002) had predicted Kerr parameters of a = 0.8 and
⋆
the He star collapsing into a black hole in the same way
which were measured, by Shafee et al. (2006), at the
presenttime,tobea =0.65−0.75anda =0.75−0.85, as in the Woosley Collapsar model, the tidal locking be-
⋆ ⋆
tween the donor and the He star being transferred to
Electronicaddress: GEB:[email protected] one between the donor and the black hole. From the
Electronicaddress: CHL:[email protected] latter, Lee et al. (2002) predicted the Kerr parameter of
Electronicaddress: EMM:[email protected]
2 Brown, Lee, Moreno M´endez
the black hole. GRB 060218/SN2006aj;namely that the centralengine
ThemainpointdevelopedbyBrown, et al.(2007)was was a black hole, not the magnetar conjectured by most
that the rotational energy of the binary is roughly in- of the authors, and that the black hole was one of min-
versely proportional to the mass of the donor. This fol- imum black hole mass with an ∼ (18−20)M⊙ ZAMS
lows from Kepler’s law and from the fact that in Case (ZeroAge MainSequence)progenitor. We alsocorrecta
C mass transfer (following the He burning), the initial number of errors in Galactic black hole evolution in the
a s (distance between the giant and the companion) of Fruchter et al. (2006) article.
i
the binaries are roughly equal. The fact that, accord-
ingtoFruchter et al.(2006),thelongγ-rayburstsarein 2. EVOLUTIONOFBLACKHOLESINOURGALAXY
low-metallicity galaxies, does not elucidate the dynam- Webeginbyexpandingontheevolutionarydiscussions
ics which produce the long bursts. The dynamics result in Brown et al. (2001). The history of black hole evo-
from the fact that low-metallicity galaxies tend to have lution in binaries (which is the only place where black
stars of higher mass than Galactic. The higher mass of holescouldbestudiedindetail)wasthatwhatevermass,
the donors slows the binary down sufficiently that the withinreasonablelimits,oneproposed,thebinarywould
rotationalenergy can be accepted by the central engine. turn out to have had a neutron star rather than black
In other words, the question of energy is a “Goldilocks” hole as compact object. The first clear explanation of
one. It must be not too much, because in that case the this was given by Brown et al. (2001); namely, the evo-
central engine will be dismantled, and not too little, be- lution of black holes in our Galaxy depends upon bina-
cause that would only be sufficient for a subluminous rity. Namely, in Case A or B mass transfer (mass trans-
burst, but for a long highluminosityγ-rayburstit must fer while the giant star is in main sequence or red giant
be just right. stage) the strong winds in our Galaxy blow off sufficient
We have learned enough about Kerr parameters from ofthe“naked”Heenvelopesothattheremainingcoreof
ourcalculationsandfromtheSmithsonian-Harvardmea- metals was too light to evolve into a black hole; rather,
surementsto constructa “guesstimate”. Namely,we be- it would end up as a neutron star.
lieve that LMC X−3 underwent the closest (in energy) OnlyinCaseCmasstransfer,ifmasstransferfollowing
explosion in our Galaxy, to a cosmological GRB. (Of He burning were carried out, would the remaining core
course,onecansaythatLMC X−3is notinourGalaxy, have the possibility of evolving into a black hole. Now,
but in the LMC.) justwhatthelimitisforthelowestZAMSmassstarthat
We believe that in its ∼ 1/3 solar metallicity will evolve into a black hole is determined by what we
(Russell & Dopita 1990), it tends towards the low- call the Woosley Ansatz. In our opinion this is one of
metallicity stars considered by Fruchter et al. (2006), so the mostpowerfuldevelopments instellarevolution. We
that at the leastit has somewherebetween Galactic and will combine this “Ansatz” with the Bethe et al. (1979)
the low metallicity favored there. There is uncertainty considerations of entropy in the Fe core.
in the masses, but Davis et al. (2006) have a value of The Woosley Ansatz basically divides the burning
a⋆ ≃ 0.26 for the present Kerr parameter. They took of 12C into low energy, T ∼ 20keV burning through
7M⊙ as the massofthe blackhole, whichwouldimply a the 12C+α →16O process and the T ∼ 80keV
donorof∼4M⊙ fromthelowerendofthemeasurements 12C+12C→24Mg etc. process. The 12C is produced by
of Cowley et al. (1983). We take these to be representa- α+8Be⋆ →12C;i.e., essentiallythroughα+α+α→12C.
tive; other investigators have found other masses, so we This is a three body process, going as the square of the
suggest our evolution as only a possible one. density,ρ2. The 12C is burnedinto16O bythe twobody
All of the binaries in Brown, et al. (2007) and process 12C+α →16O+γ, which goes as ρ. As long as
Moreno M´endez et al. (2007) which had so much energy 12C is present, the latter reaction will take place. How-
that they dismantled the black hole accretion disk had ever,withincreasingM ,thedensitydecreases. The
ZAMS
donor masses of (1−2)M⊙, so the donor in LMC X−3 entropy, which goes inversely with the density, is known
is at least double those masses. The donor mass is close toincreasewithM . Therefore,therewillbeavalue
ZAMS
to the ∼ 5M⊙ that Moreno M´endez et al. (2007) esti- ofMZAMSwherethe12Cisremovedbythe12C+α→16O
mated would give the energy of a cosmological GRB. In as rapidly as it is formed by the α+α+α process. At
anycase,LMCX−3istheclosest“nearby”relicofbina- this value of M the 12C+12C→24Mg etc. shuts off,
ZAMS
riessimilartotheprogenitorsofcosmologicalGRBs. We because there is no 12C; actually, it shuts off once the
can,therefore,useit asanexampleto tryto reconstruct central carbon abundance is less than ∼ 15% because
theexplosionandmodeltheenergyoftheexplosion. We there is notenoughcarbonfor convective(steady)burn-
estimate the mass loss in the explosion, finding it to be ing. At this point the burning processes are all of low
substantial. It is likely that the estimated system veloc- temperature,T ∼20keV.At this pointthe metallicity is
ity canbe measured,at least the radialcomponent of it, close to zero, independent of the average metallicity of
which should test our prediction. the star.
In this paper we wish to also summarize results of Now, this temperature is too low for neutrino-pairs
earlier calculations which have a direct bearing on the to carry off appreciable energy and entropy (the rel-
5 papers in the 31 August 2006 Nature (Young 2006; ativistic ν, ν¯ pair cross section goes as T11 power),
Campana et al. 2006; Pian et al. 2006; Soderberg et al. whereas copious amounts of entropy were carried off
2006; Mazzali et al. 2006), in order to show that useful by 12C+12C→24Mg etc. What happens to the entropy
evolutions of black holes have been carried out in the which increases with increasing M ?
ZAMS
past, of which the 119 astronomerswho signed these ar- Bethe et al. (1979) showed that the entropy per nu-
ticles were unaware. We show that these previous cal- cleon in the Fe core of a star in advanced burning was
culations have a direct bearing on the measurements of ∼ 1 in units of k . The only place the increase in the
B
LMC X-3 as a relic of GRBs 3
0.30
2.0
0.25
ndance0.20 M]sun 1.8 MPC
Abu s [
on 0.15 mas
Carb ore 1.6
ntral 0.10 Fe c B A+AB M
Ce NS
B1913+16
1.4 naked
0.05
0.00
10 15 20 25 30 35 40 10 15 20 25 30 35 40
M [M ] M [M ]
ZAMS sun ZAMS sun
Fig. 1.— Central carbon abundance at the end of He-core Fig. 2.— Comparison of the iron core masses at the time
burning for “clothed” (single) stars as function of ZAMS mass. of iron core implosion for a finely spaced grid of stellar masses
The rapid drop in the central carbon abundance at ZAMS mass (Hegeretal. 2001). The circular black dots were calculated with
MZAMS ∼ 20M⊙ signals the disappearance of convective carbon theWoosley&Weaver(1995)code,whereasthecrossesemploythe
burning. TheresultingironcoremassesaresummarizedinFig.2. vastlyimprovedLanganke &Martinez-Pinedo(2000)ratesforelec-
troncaptureandbetadecay. Iftheassembledcoremassisgreater
entropy can go is into an increase in the number of nu- thanMPC =1.8M⊙,whereMPC istheproto-compact star mass
as defined by Brown&Bethe (1994), there is no stability and no
cleons in the Fe core, once the burning is confined to bounce;thecorecollapsesintoahighmassBH.MNS =1.5M⊙de-
the low-temperature (T ∼ 20keV) region. Thus at the notesthemaximummassofNS(Brown&Bethe1994). Themass
ZAMS mass at which the two-body process takes over oftheheaviestknownwell-measuredpulsar,PSR1913+16,isalso
completely from the three-body process, central abun- indicated with dashed horizontal line (Thorsett&Chakrabarty
1999).
dance of 12C has decreased below 15% and the increase
in entropy comes from the Fe cores increasing rapidly linesneednotbezero,butsomeweakonescancomefrom
with M . carbon shell burning.) Therefore, the explosion was I
ZAMS bcd
Giventhe Woosley value for the 12C+α→16O process in nature. This occurs only in one venue in astronomy;
of170keVbarns(atE =100keV),tobecorrect,thebest namely,inablackholeformedatthelowestZAMSmass
experiments (Kunz et al. 2001) obtaining 165 ± 50keV possible,∼18M⊙. Thus,clearlythecentralenginewasa
barns, the 12C abundance drops below 15% just around rotating black hole. The hypernova had a low energy of
18M⊙,themassoftheprogenitorofSN1987a,Sanduleak 2 bethes (Mazzali et al. 2006) and the GRB was highly
69◦202. WereproduceFig.1ofBrown et al.(2001)asour subluminous. However, from equipartition of energy we
Fig 1. Thus with Case C mass transfer the threshold for expectthekineticenergytoberoughlyequaltothether-
blackholeproductionis ZAMS18M⊙. We showthe cal- malenergy,sothe kinetic energymusthavebeenmostly
culatedcompactobjectmassesfromBrown et al.(2001) used up by the ram pressure which cleared the way for
as Fig 2. This is independent of metallicity, depending theGRB(ForGRB980425,MacFadyen(2000)estimates
only on the low-energy burning. In fact, the metallicity this to be ∼ 1052ergs). Clearly the “unwinding” of the
is essentially zero at the threshold in ZAMS masses for lowest possible rotational energy will continue until the
black hole formation. GRB falls below observational threshold. The T90 of
Fryer (2004) and Fryer (2006) independently ob- GRB 060218 is 2100±100s (Campana et al. 2006), ex-
taineda ZAMSmass of∼20M⊙ to givethe lowestmass tremelylong. Itisclearthatifalloftherotationalenergy
black hole. Our use of the Woosley Ansatz is not only a canbe accepted,whichshouldcertainly be true for ato-
simple, elegant argument, based on the behavior of en- talenergyofonlyE ∼4×1051ergs,thenthe time will
rot
tropy, but also a practical one. For example, it settles be very long because the kinetic energy is mostly used
thequestionsofcentralengineinGRB060218/SN2006aj, up in clearing the way through the star for the GRB,
whichwasdebatedbythe 119observerswhowrote5pa- which has power ∝ B2, where B is the magnetic field,
pers: Young (2006), Campana et al. (2006), Pian et al. which would be expected to be low for the lowest mass
(2006),Soderberg et al.(2006)andMazzali et al.(2006), black hole. We understand in this way that the sublu-
mostly speculating that the centralengine wasa magne- minous bursts from low rotational energy have a long
tar, althoughthey gaveno mechanism by which it could lifetime. Kaneko et al. (2007) find the energies, kinetic
unwinditsmagneticfieldtopowertheGRB.Ifonelooks and thermal, to be the smallest in GRB 060218 among
at the additional information of Mazzali et al. (2006), the subluminous bursts, GRBs 980425, 030329, 031203
oneseesthattherewerenocarbonlines. (Actually,these and 060218, they investigate. It should be noted that
4 Brown, Lee, Moreno M´endez
theverylowGRBenergycomesfromanearcancellation Fruchter et al. (2006) is correct in that they do keep
of the kinetic energy by the ram pressure work. Note the high-luminosity long GRBs away. The idea that
that with such an energy this would be a nearly “dark” the black hole progenitor does not tidally lock with the
explosion like Cyg X−1 (Brown, et al. 2007), which was donor, until the explosion, found in the literature of the
estimated to have ∼ 6 bethes of rotational energy, ex- soft X-ray transient sources is incorrect, as we now dis-
ceptthatthe blackholemustbe &1.5M⊙ atthetime of cuss.
explosion rather than the ∼ 7M⊙ black hole calculated The tidal locking between the donor and the He
byBrown, et al.(2007)forCygX−1. Inthe caseofCyg star has been demonstrated by Van den Heuvel & Yoon
X−1,extensivemasstransferfromthedonortotheblack (2007). Given this, when the center of the He star falls
holeandmasslossfromthedonorcomplicatedthesimple intothepreformingblackhole,themetalstowhichithas
interpretationusing Kepler’slaw. Nonethe less,the∼6 burned before Case C mass transfer undergo a perfect
bethesobtainedfromCygX−1shouldnotbesodifferent magnetohydrodynamics. The magnetic field is frozen in
fromthe∼4bethesforGRS060218becausethedonorin thematerial,andtransferstheangularmomentumofthe
the lattermustbe afactorof∼3 lowerinmassthanthe Hestartotheblackholeasitforms. Thehigh,∼1015G,
estimated ∼30M⊙ inCyg X−1, in orderto be less mas- B-field threading the disk of the black hole transfers the
sive than the black hole progenitor. This would increase tidallockingofthe He star withthe donorto that ofthe
therotationalenergybyafactorof∼3whereastheblack blackholeandthedonor. Thenthedonordecouplesand
hole in GRS 060218 is &1.5M⊙, a factor of ∼ 14/3 less takes the role of an observer to the explosion. As noted
than that of the 7M⊙ black hole in Cyg X−1 at the earlier, by the material the donor accretes the nature of
time of common envelope evolution. Scaling 6 bethes by the explosion can be reconstructed.
3/(14/3) gives ∼ 4 bethes. Mirabel & Rodrigues (2003) Aside from the help in tidal locking by the metals
found the mass loss in the explosion of Cyg X−1 to be threading the disk, the Case C mass transfer accom-
very small, the space velocity of Cyg X−1 relative to plishes two other mechanisms. Firstly, the remaining
the Cyg OB3 association to be typical of the velocities heliumisoriginallyontheouterpartoftheHestar,later
of stars in expanding O-star associations and gave other supported by the angular momentum that cannot be
arguments supporting a “dark” explosion. The truly re- transferredintotheblackhole,sothatitcanleaveinthe
markable property of GRB 060218 is the long T time Blaauw-Boersma explosion without interaction, thereby
90
of the GRB of 2000±100 seconds. This is, however,the leaving a type I explosion. There is then no helium en-
c
typical time of the Blandford-Znajek engine (eq.(8) of velope to tie the magnetic fields to, which wouldtend to
Lee et al. (2000)) reducetheangularmomentumofblackhole(Heger et al.
2003). As can be seen from Lee et al. (2002), Case C
τ = MBHc2 ∼2.7×103 1015G 2 M⊙ sec.(1) mass transfer also makes the initial binary separationai
BZ B2R2c2 (cid:18) B (cid:19) (cid:18) M (cid:19) insensitivetodonormass,makingouruseofKepler’slaw
easy.
The low magnetic field, appropriate for low-mass black Thislatterargumentresultsfromthe natureofthe He
holes and the low MBH tend to make τBZ longer, such supergiantatthe timeofmasstransfer(Lee et al.2002).
thatthemeasuredT90 =2000±100sec. isjustthe right In order for the mass transfer to be delayed sufficiently
time for the GRB in 060218. That of GRB 060218 is for Case C mass transfer to take place, winds must be
certainly no more than “dusky”. sufficientlylowthatthemetalsobtainedattheendofthe
We feel confident that the subluminous GRBs can be burning do not blow away, but extend in space beyond
described by our binary scenario. As noted, we do not the helium. Otherwise Case B mass transfer would take
claimthatthecosmologicalonescanbe;theymayrequire place. For a ZAMS 20M⊙ giant, this means that the
the original Woosley model. However, our model is not He supergiant must expand to ∼ 1000R⊙. Adding the
really different, in the sense that we use the Woosley ∼ (200− 300)R⊙ Roche lobe of the donor gives mass
collapsar model after the donor has spun up the black transfer beginning at (1200−1300)R⊙.
holeprogenitorto thenecessaryamountofrotation,and Ineither case,the masstransfer is welllocalizedin ra-
we feel that understanding the dynamics of LMC X−3 dius because the donors will have sizes which are small
willhelpus makeaconnectionbetweenthe subluminous compared with the radii of the supergiants, so that
and cosmologicalGRBs. their Roche lobes are typically about 0.2 of that ra-
dius. In Lee et al. (2002) we found all supergiants to
3. WHYCASECMASSTRANSFER? have ∼ 30M⊙. This was because in the Galactic evo-
ItisclearthatCaseC,masstransferfollowingHeburn- lution, binaries with lesser mass black holes would lose
ing, is useful in the binary evolution of black holes, es- more than half their mass in the explosion and not be
pecially in a Galaxy with solar metallicity. In this case stable (Moreno M´endez et al. 2007), but with the lower
the large winds off the stellar surfaces, referred to by rotational energies of the low metallicity binaries, the
Fruchter et al. (2006), do not blow awaythe He because lower-massgiants of ∼20M⊙ are likely to be more copi-
it is covered by hydrogen -just like in single stars, and, ous. Thesewouldallowanai ∼(1200−1300)R⊙forCase
therefore,protectedfromthestrongHe-starwinds. Were Cmasstransfer,withoutstrongvariation,soitisreason-
this not so, the winds in our solar-metallicity Galaxy, able to forget the binary dependence of ai. If we can do
would blow off the helium, lowering the stellar masses this we can easily use eq. (4) of Moreno M´endez et al.
sufficiently that they would go into neutron stars, the (2007)
fate described by Fruchter et al. (2006). The idea that
3/2 1/2 0.83
life in the Milky Way is protected by the metals keep- days 4.2R⊙/ai Md+MHe Mgiant
= (2,)
ing the GRBs away (Stanek et al. 2006) obtained from Pb (cid:18) Md/M⊙ (cid:19) (cid:18) M⊙ (cid:19) (cid:18) M⊙ (cid:19)
LMC X-3 as a relic of GRBs 5
beaboutequaltothisbecauseofequipartitionofkinetic
andthermalenergies. TheGRB980425isa“smothered”
1 one (MacFadyen 2000), but, in general, the visible GRB
isonly∼1bethe,soinmostcasestheenergyusedupby
the ram pressure in order to clear out material from the
0.8 path of the jet must be cancelled in first order of mag-
a) nitude by the kinetic energy. We believe this near can-
er (0.6 cellation between kinetic energy and energy needed for
met the ram pressure work to produce the various observed
ara low-energy GRBs (see our later discussion).
Kerr p0.4 toWtheeb3e0liebveetthheastmaedadsiunrgedthhey∼pe3r0nobveatheexspkloinsieotniceenneerrggyy,
assumingequipartitionbetweenthetwoenergies,givesus
0.2 7 Msun an energy of ∼ 60 bethes, possibly that of cosmological
11 Msun GRBs. At least, this is the largest energy that we have
0 found recently in the literature. We argue that GRB
0.1 1 10
980425 is subluminous because of the near cancellation
Preexplosion Orbital Periods [days]
betweenworkoftherampressureandthekineticenergy.
(Butitcouldbesubluminousbecauseofthehigh,nearly
Fig. 3.—TheKerrparameteroftheblackholeresultingfromthe
solar metallicity in the background galaxy.)
collapseofaheliumstarsynchronouswiththeorbit,asafunction
of orbital period (LBW). Note that the result depends very little The present rotational energy of LMC X−3 is ∼ 54
on the mass of the helium star, or on whether we use a simple bethes, obtained from the 1.7 day period and Fig. 3,
polytropeoramoresophisticatedmodel. roughly half of the natal rotational energy if we assume
theexplosiontouseup∼60bethes. GoingbacktoFig.3
where it is seen that the approximation
again,thiswouldimplyapre-explosionperiodof∼1day
days ∝ 4.2R⊙/ai 3/2 Md+MHe 1/2 (3) forItLiMs CcleXar−t3h.at the black-hole binaries cannot deliver
Pb (cid:18) Md/M⊙ (cid:19) (cid:18) M⊙ (cid:19) all of their rotational energy into the explosion because
gives the main dependence as their power goes to zero as they slow down. From di-
mensional analysis it would seem reasonable that their
3/2
Pb ∝ Md (4) pre-explosion energy would be about double the post-
days (cid:18)M⊙(cid:19) explosion energy, assuming that all of this latter energy
could be accepted. In any case, our assumption can be
for M ≪M and
d He tested in that the system velocity, at least the radial
P M component of it, will be measured by the Smithsonian-
b ∝ d (5) Harvard collaboration. In our above model we have a
days M⊙
mass loss of ∆M =3.25M⊙, which implies a system ve-
for higher M . These can be normalized to Galactic bi- locityof43km/s,definitelynota“dark”explosion. This
d
naries, say Nova Sco or Il Lupi, where the theoretical is a little less than 1/3 the system velocity of Nova Sco,
and observationalKerr parameters agree. We reproduce where the peculiar velocity in the direction of the local
Fig.1 of Brown, et al. (2007) as our Fig. 3 to show how standardof restis (114±19)km/s (Brandt et al.1995),
it is possible to obtain the Kerr parameter from P . where only one component of Nova Sco’s space velocity
b
has been measured, so that its space velocity may be
4. LMCX−3 higher. In any case, it is clear that LMC X−3 is an in-
The poster child subluminous explosion GRB teresting black-hole binary, and better measurement of
980425/SN1998bw came from an LMC-size star- its properties is underway.
forming Galaxy but with metallicity nearly solar We note that LMC X−1 has a donor mass more mas-
(Sollerman et al. 2005). sive than its black hole (Orosz 2002), MBH/Mdonor ∼
The LMC has ∼ 1/3 solar metallicity, so it should go (0.3−0.7) with black hole mass ∼ (7−13)M⊙. Thus
a long way towards having low metallicity stars. The LMC X−1 is similar to Cyg X−1, and would have been
donor of LMC X−1 is more massive than the black hole expected to undergo a “dark” explosion.
and it seems to be somewhere between Cyg X−1 and
M33 X−7 in mass and their relative sizes. In the case of 5. CONCLUSIONS
LMC X-3 Davis et al. (2006) find a ≃ 0.26 now; using We suggest that LMC X-3 may be similar to relics
⋆
the Cowley et al. (1983) 7M⊙ for the black hole would of cosmological GRBs, to the extent that some or most
imply using 4M⊙ for the donor. The present period is of these latter result from binaries. The number of the
1.7days,with present rotationalenergyof 54 bethes, ob- latter is certainly sufficient to produce the GRBs and
tained from the period from Fig. 3. the binary nature takes care of the necessary angular
Note that the much lower energy than those of most momentum,whichcanbeachievedbychoosingthedonor
Galacticbinariesresultfromthe2−4timeslargerdonor mass.
mass here. It would be natural for the natal rotational Fruchter et al. (2006) have made a case that high-
energy to be about double this, with energy ∼ 108 luminosity, long GRBs came from irregular low-metal
bethes. The hypernova energy of SN1998bw is ∼ 30 galaxies. We suggest LMC X-3 as the closest nearby
bethes, and, as noted earlier, the kinetic energy should binary from a region of 1/3 solar metallicity in an irreg-
6 Brown, Lee, Moreno M´endez
ulargalaxy. Thelowermetallicityenvironmenthasmore Ourpresentestimateof∼60bethesforthetotalcoso-
massive stars, and the donor in LMC X-3 is probably at logical GRB energy is the highest recent estimate that
least twice as massive as the donors of the relics of the we have seen. For this estimate we invoke equipartition
Galactic GRBs, which should slow the binary down to a of kinetic and thermal energies.
rotational energy that can be accepted. We believe that Finally, we wish to point out that our ability to cal-
LMCX-3bringsusinthemetallicitytowardstheirregu- culate Kerrparametersand the confirmationof these by
lar low-metallicity binaries considered by Fruchter et al. the Smithsonian-Harvard measurements makes it possi-
(2006). ble to calculate in a reliable way the amount of angular
Working in this region of energies we feel that we can momentum carried by the black-hole binary. This an-
makepredictions,becauseourcalculationofKerrparam- gular momentum is conserved, but the questions to be
etersmakesitpossibletomakequantitativecalculations. answeredconcern how it is to be distributed: how much
Our predictions are, however, at best, order of mag- remainsinthebinary,howmuchenergygoesintotheex-
nitude, because the necessary properties of the binary plosion and what is the fraction of the latter which goes
LMC X-3 have not been accurately measured. As they intokinetic andinto heatenergy? Progressinanswering
aremeasuredwewillprobablyhavetoreadjustournum- thesequestionsisnecessarytomakeaquantitativestudy
bers. In particular, it would be most valuable to obtain out of the GRBs, and to put order into their classifica-
a reliable value for the explosion energy. Whereas the tion.
hypernova energy is probably measured with reasonable
accuracy, the kinetic energy required by the work per-
formedbytherampressuretoclearthewayfortheGRB
is mostlyhidden, andthe netGRB energyis asmalldif- ACKNOWLEDGMENTS
ference between large kinetic energy and a large amount We would like to thank Jeff McClintock for many use-
ofworkdonebythe rampressure. We believethatthere fuldiscussions. G.E.B.wassupportedbytheUSDepart-
will be great variations in GRBs depending upon these ment of Energy under Grant No. DE-FG02-88ER40388.
smalldifferenceswhichmust be sensitive to stellarprop- CHL was supported by Creative Research Initiatives
erties. (MEMS Space Telescope) of MOST/KOSEF.
APPENDIX
APPENDIX A: ENERGIES
The energies of cosmologicalGRBs and hypernovae are generally taken to be ∼1 bethe and a few times 10 bethes,
respectively. However, the GRB involves, in order that it can begin, the clearing of the matter in the way of the
jet in order that it can leave the star. This energy is roughly the mass still contained within the beaming angle of
the jet times the square of the velocity with which it is displaced. This is about 1M⊙ (including both poles) times
(1%−10%)c2 (MacFadyen 2000). MacFadyen estimates this to be a few ×1051 ergs. However, the(1%−10%)M⊙c2
is ∼(2×1052−2×1053) ergs. We compare this with the thermalenergy of ∼3×1053 ergsin the hypernova 1998bw.
The mechanism for heat production is viscous heating, from the field lines frozen in the rotating disk, different from
ordinary supernovae. In the latter, the kinetic and heat energy are roughly equal, from energy equipartition. We
assumethistobetruealsointheGRB-Hypernovacase. Otherwisewecannotmakesenseoutofourenergies. Inother
words,we believe thatGRB 980425/SN1998bwhadessentiallythe samerotationalenergyascosmologicalGRBs, the
GRB, however,being nearly smothered by the work from the ram pressure.
This means that the kinetic energy is roughly equal to that of the hypernova, ∼ 30 bethes in the case of GRB
980425. The GRB is, however, a “smothered” one (MacFadyen 2000), which means that the ram pressure to free the
jet uses up nearly all of the kinetic energy.
In cosmological GRBs the typical GRB energy is ∼ 1 bethe, resulting from a near cancellation of work from ram
pressure work and other kinetic energy.
Ifsuchanearcancellationis to be engineered,the helium starsinwhichthe jet is producedcannotbe verydifferent
in properties. However,the cosmologicalGRBs are only a tiny population compared with subluminal ones, which are
probablyformedwith helium starsthatdo not showthis nearcancellationofkinetic energieswith rampressurework.
To summarize,we aresayingthatthere is a large“invisible energy”whichgoesinto workdone bythe rampressure,
so that our energies calculated in the Blandford-Znajek mechanism look much larger than those found empirically in
the observations,althoughitis generallyrealizedthat extraenergy mustbe furnishedto getphotons outin the GRB,
by saying that there is an efficiency, often taken to be ∼10%.
APPENDIX B: ANEW CONSTRAINT FORGRBPROGENITOR MASS
We wish to point out that there is observational support for our choice of the LMC as a site for the long gamma-
ray bursts. Larssonet al. (2007) provide a new constraint for gamma-ray burst progenitor mass. They show that
long-duration gamma-ray bursts (L-GRBs) are much more concentrated on their host galaxy light than core collapse
supernova explosions. From this they way “Assuming core collapse supernova arise from stars with main-sequence
masses > 8M⊙, GRBs are likely to arise from stars with initial masses > 20M⊙. This difference can naturally be
explained by the requirement that stars which create a L-GRB must also create a black hole.”
As a template and close by natural analogue of starburst galaxies they use NGC 4038/39. They say, however,
the luminosity function and surface density of clusters on NGC 4038/39 is comparable to that seen in other local
LMC X-3 as a relic of GRBs 7
star-forming galaxies of varying morphology in which they include the LMC.
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