The origin of intergalactic thermonuclear supernovae

0.2 MB·English

by  A. G. Kuranov

THE ORIGIN OF INTERGALACTIC THERMONUCLEAR SUPERNOVAE
A. G. Kuranov* and K. A. Postnov**
SternbergAstronomicalInstitute,Universitetskiipr.13,Moscow,119992Russia
Abstract—Thepopulationsynthesismethodisusedtostudythepossibilityofexplainingtheappreciable
fractionof the intergalactic type-Ia supernovae (SN Ia), 20+12%, observedin galaxy clusters(Gal-Yam
−15
4 etal.2003)whenclosewhitedwarfbinariesmergeinthecoresofglobularclusters.Inatypicalglobular
0 cluster,thenumberofmergingdoublewhitedwarfsdoesnotexceed∼10−13peryearperaveragecluster
0 starintheentireevolutiontimeofthecluster,whichisafactorof∼3higherthanthatinaMilky-Way-type
2 spiralgalaxy.From5to30%ofthemergingwhitedwarfsaredynamicallyexpelledfromtheclusterwith
n barycentervelocities up to 150 km s−1. SN Ia explosionsduring the mergersof double white dwarfs in
a densestarclustersmayaccountfor∼1%ofthetotalrateofthermonuclearsupernovaeinthecentralparts
J ofgalaxyclustersifthebaryonmassfractioninsuchstarclustersis∼0.3%.

2
2

INTRODUCTION Atpresent,itisnotcompletelyclearwhichofthese
2
scenarios (or both) make the largest contribution
5
SNIa(thermonuclearsupernovae)areamongthe to the SN Ia rate, and a considerable observational
4
most important objects in modern astrophysics, be- effort is being made to find the Galactic population
1
0 causethemaximumoftheiropticallightcurveshasa of double WDs with a total mass close to or larger
4 smallspread.Therefore,theycanbeused1 asastan- than the Chandrasekhar limit (e.g., the SPY project
0 dard candle in modern cosmology, and fundamental by Napiwotzki et al. 2003). We will adhere to the
h/ conclusions about the expansion kinematics of the scenarioofmergingdoubleWDsastheprogenitorsof
p Universe can be reached (Riess et al. 1998; Perl- type-Iasupernovae.Therecentdiscoveryofhydrogen
- mutter et al. 1999). The physical cause of an SN Ia emissionlinesinthespectrumofSN2002ic(Hamyu
o explosion is the thermonuclear explosion of a white et al. 2003) cannot rule out this formation channel
r
of thermonuclear supernovae, because such SN Ia
t dwarf(WD)withamassclosetotheChandrasekhar
s areatypical;theiroriginisthesubjectofdebate(e.g.,
a limit(HoyleandFauler1960).Thisideaisfullycon-
: firmed by detailed numerical calculations ofthe evo- LivioandRiess2003;ChugaiandYungelson2003).
v
lution of a white dwarf in a binary system (see, e.g., A purposeful search for supernovae in Abell
i
X Dunina-Barkovskataetal.2001).TheWDmasscan clusters of galaxies (the WOOTS project; Gal-Yam
r increase during mass transfer between the compo- et al. 2003) has revealed that two of the seven
a nents in a close binary, which determines the astro- SN Ia found (SN 1998fc in Abell 403 at z =0.1
physical sources of such supernovae—double WDs and SN 2001al in Abell 2122/4 at z =0.066)) have
thatmerge throughgravitationalradiation(Ibenand no obvious host galaxies and can be associated
Tutukov 1984; Webbink 1984) or accreting WDs in only with faint dwarf galaxies with M >−12m.
R
semidetached or symbiotic binaries with a second In reality, in both cases, the supernovae are pro-
nondegenerate component (Whelan and Iben 1973; jected onto the halos of the central cD galaxies in
Nomoto 1982). SN Ia are observed in galaxies of all the corresponding clusters. However, the authors
morphologicaltypes(van denBergetal.2003),and rejected the possibility that they belong to these
theirrate(inSNUunits)dependsweaklyonthetype galaxiesbecauseofthelargeradial-velocitydifference
ofgalaxy(seeCapellaroetal.(1997)andmorerecent (750−2000kms−1)betweenthesupernovaeandthe
papersofCapellaro’sgroup). correspondingcDgalaxies.
Assuming a modified Schechter luminosityfunc-
*E-mail:[email protected]
tion for the galaxies in the Virgo cluster (Trentham
**E-mail:[email protected]
and Tully 2002), the luminosity fraction of the dwarf
1However,withcertainreservations,upto40%ofSNIamay
be peculiar (Lee et al. 2001), plus a significant spread of galaxiesintheclustersunderstudyshouldhavebeen
theoreticallightcurves(Sorokinaetal.2000). less than 0.3%. If the supernova rate is proportional
THE ORIGIN OF INTERGALACTIC SUPERNOVAE 141
to luminosity, and if the fraction of the supernovae then ∼1% of SN Ia in the cluster centers can be
withoutobvioushostgalaxiesisontheorderof10%, naturallyexplainedwithoutadditionallyassumingthe
thisleadstotheir∼30-foldexcessamongsuchdwarf presence of 10% of the intergalactic stellar popula-
galaxiescomparedtohigh-massgalaxies.Therefore, tion.
these supernovae are believed to have exploded out-
sidegalaxies(intergalacticsupernovae)asaresultof
theevolutionoftheintergalacticstellarpopulation.It
isconcludedfromtheseobservationsthatthefraction THE MODEL
of the intergalactic SN Ia is 20+12%, in accordance
−15
with the theoretically expected fraction of the stars
TheStructureandEvolutionofaGlobularCluster
outsidegalaxiesduetotidalinteractionbetweenclus-
ter galaxies. The authors themselves (Gal-Yam et
al. 2003) point out a low statistical significance of Inourcalculations,weusedaMichie–Kingmodel
theresultobtained,andsuchahighpercentageofthe (Michie1963)todescribethestructureoftheglobular
extragalactic SN Ia should be confirmed by further cluster.Thestellarpopulationwasbrokendowninto
observations. subsystems (depending on their mass); their space
There is a potential channel for the formation of densityραwasfittedbyapowerlaw:
intergalactic supernovae through the dynamical ex-
pulsionofcloseWDpairsfromthesystemofgalactic ρcα, r ≤rcα,
globularclusters (GCs).Thestellar mass fractionin ρα(r)=ρcα(r/rcα)−2, rcα <r ≤rhα, (1)
theobservedGCsis∼0.1%ofthevisiblemassofthe ρ (r/r )−4, r <r ≤r ,
galaxies. However, dynamical interactions between hα hα hα t
sotfacrlsosinedpeanirssewcliuthstwerhcitoeredswianrcfrse.aSseomtheeomfethrgeemr rcaatne where rcα and rhα are, respectively, the radii of the
coreandthespherewithinwhichhalfthemassofthe
beexpelledfromtheclustersduringtripplecollisions,
α subsystem ofstars is contained; and r is the tidal
t
and the assumption that the rate of thermonuclear
radius of the cluster. Thus, ρ =ρ (r /r )−2
supernovae is proportional to luminosity for dense hα cα hα cα
and ρ =M / 8πr2 r − 2r . Under the
clustersbecomesinvalid.Thenumericalcalculations cα totα cα hα 3 cα
of the WD evolution in open star clusters by Shara energy equipartition condition for the GC stars, we
and Hurley(2002)confirmed thesignificant increase mayassumethat (cid:5) (cid:6) (cid:7)(cid:8)
in the formation rate of close WD pairs through dy-
m¯
namicalinteractions. c
r = r ,
Thegoalofourstudyistoquantitativelyestimate cα (cid:9)mα c
the merger rate of double WDs in the dense cores
where m¯ is the mean mass of the stars in the GC
of star clusters by using the method of population c
synthesisoftheevolutionofbinarystarswithasemi- core,r = 3vm(0)2 istheradiusoftheGCcore,v
c 4πGρc m
analytical allowance made for their dynamical inter-
isthermss(cid:10)pacevelocity,andρ isthecentralnumber
c
actionswithsinglestarsintheclustercore.Weshow
densityofthestars:
that the the merger rate of double WDs in GCs per
averageclusterstarisafactorof∼3higherthanthe ρ = ρ . (2)
merger rate of double WDs in spiral galaxies with c cα
α
continuous star formation. If ∼0.3% of the baryon (cid:11)
mass in the cluster centers is concentrated in virial- The stellar velocities can be described by the trun-
izeddensestarclusterswithmassesof105−108M , catedMaxwelliandistribution
⊙
Ke−γJ2[e−βE −e−βEt], E <E , J <J (E),
g(E) = t c (3)
(cid:12)0, E >Et,
where E and J are the energy and the angular mo- energy E. Stars with E >E are assumed to have
t
mentum (per unit mass), respectively; and J (E) is
c
the value of J for a star in a circular orbit with escapedfromthecluster.
142 KURANOV, POSTNOV
Detailedcalculationsoftheevolutionofthestan- of close systems. Note also that the above authors
dard modelreveal acollapseoftheinner coreand an assumed up to 30% of the initially hard binaries in
expansionoftheouterlayers.Asfollowsfromnumer- the GC. We choose an upper limit of 1000R for
⊙
icalcalculations(see,e.g.,Kimetal.1998),thecore theinitialbinarysemiaxes,whichcorresponds tothe
contraction atlatestages asymptotically approaches criteria by Fregeau et al. (2003). Thus, the high (at
apowerlaw: firstglance)percentageoftheinitiallybinarysystems
thatweadoptedisnotoverestimatedandisconsistent
ρ (t)∼ρ (0)(1−t/t )−1.2,v (t) (4)
c c coll m with available observations. Unfortunately, we can-
∼v (0)(1−t/t )−0.12, notaccuratelyestimatethenumberofnewlyforming
m coll
binaries duetotidalcaptures. Inanycase, theirper-
where t is the current time, and t is the collapse
coll centageismuchlowerthanthefractionoftheinitially
timeofthecluster.
binarysystems.
As the GC core contracts, the central number
density of the stars rapidly increases, which causes
the formation rate and the binding energy of bi- TheEvolutionofaBinaryinaGlobularCluster
nary stars to increase. Once the initial collapse has
TheScenario Machinepopulationsynthesis code
stopped,onemightexpectanexpansionofthecluster
(Lipunovetal.1996)wasusedtomodeltheevolution
core due to the tidal formation of binary stars. For
ofbinaryandsinglestars.Weusedtheinitialdistribu-
some time, the newly forming binaries will be the
tions
energy source that maintains the expansion of the
innerclusterregions.Theexpansioncanbedescribed f(lga)=const, (6)
by the following asymptotic formulas (see Kim et
10R
al.1998): max ⊙ <a<103R ,
⊙
 
ρc(t)∼t−2,vm(t)∼t−0.32. (5) RL(M1)
WeusedseveralGCmodelsthatdifferedincentral forthesemimajoraxesofthebinariesand
densityρcandradiusrcatthetimeofthecollapse.In f(M )∝M−2.35, 0.1M <M <120M (7)
all models, the collapse time was taken to be t = 1 1 ⊙ 1 ⊙
coll
7×109yr,thecentralpotentialwasW0 =−βφ(0) = forthemassoftheprimarycomponent.Here,RL(M1)
9(β ≡3/v2 , φ(r)isthegravitationalpotential),the is the Roche lobe radius for the primary component.
m Forthecomponent massratio,weusedapower-law
totalnumberofstarswas106,andthefractionofthe
distribution:
initiallybinarystarswasassumedtobe30%.
f(q)∝qαq, q =M /M <1. (8)
Thelasttwoparametersspecifythenormalization 2 1
of our calculations. For the adopted Salpeter initial Here,α istheparameter ofthedistributionincom-
q
mass function with a minimum mass of the forming
ponent mass ratio. In our calculations, we varied it
stars of 0.1M , the mean stellar mass in the GC is
⊙ between0and1.
0.38M , and the total mass of the GC of 106 stars
⊙ Another important parameter of the evolution of
is ∼4×105M⊙. Thus, the results presented below binaries that significantly affects the results of all
pertain to modeling the evolutionof3×105 binaries such calculations is the energy transfer efficiency
inaGCwithamassof1.3×105M . at the common-envelope stage, α . Having been
⊙ CE
determined both by Lipunov et al. (1996) and in
Here,itispertinenttomakeanoteonthechosen
other studies of this group, it was assumed to be
30%fractionoftheinitiallybinarystarsintheGC.It
α =0.5. This value corresponds to the (standard)
followsfromHubbleSpaceTelescopeobservationsof CE
valueofα =1inthedeterminationusedbyHurley
GCs (Rubinstein and Bailyn 1997) that the fraction CE
et al. (2002), in which the common envelope was
of the main-sequence binary stars in the GC with a
ejected only through the transfer of the gravitational
collapsedcoreNGC6752mayrangefrom15to38%.
energy of orbital motion of the approaching compo-
In NGC 288, this fraction is estimated to be in the
nentstoit.
range from 8 to 38% (Bellazzinni et al. 2002). A
numericalanalysisoftheevolutionofbinariesinaGC When describing the dynamical interactions be-
(Fregeauetal.2003)indicatesthatthefractionofthe tweenclusterstars,wetookintoaccountthefollow-
ingprocessesinthecode:
binaries with respect to their initial number after the
corecollapseranges(dependingontheclustermodel) (1)Thepassagesthatledtochangesintheorbital
from ∼0 to 35% (the table in the paper by Fregeau parametersofbinaries(semimajoraxisandeccentric-
et al. 2003). This is attributable to the effective dy- ity).
namical disruption of wide pairs and to the mergers (2)Thepassagesthatledtostarexchanges.
THE ORIGIN OF INTERGALACTIC SUPERNOVAE 143

(3) The passages that led to the disruption of bi- 0.20
naries. 1
(4) Theinteractions between single stars that led –1 2
totheformationofbinaries. 6 yr 0.15 3
–
The corresponding cross sections σ for these 0
ij 1
processeswerenumericallycalculatedbyseveralau- e, 0.10
at
thors(seeHeggeietal.1996;Mikkola1984;Kimand r
r
Lee1999). ge
r 0.05
The rate of interaction between an individual bi- Me
nary b at distance r from the GC center with an
α subsystem of stars with a space density n (r) at
α
0 5 10 15
timetisgivenbytheformula
t, 10–9 yr
R (t)=n (r,t)v (r,t)σ .
αb α m αb
Fig. 1. Mergerrateofdouble WDsina GC(model C,
The determination of the binary position in the GC
andRisdescribedinmoredetailintheAppendix. αq =1)versustime,inunitsof10−6yr−1.Theincrease
in merger rate at t∼7×109 yr corresponds to an in-
Todraw achange inthebindingenergy∆Eb and creaseinΓduringtheGCcorecollapse:1—forsystems
orbital eccentricity e of a binary during the close mergingwithintheGC;2—forsystemsexpelledfromthe
passageofathirdstar,weusedthefollowingformulas GC;and3—thetotalmergerrateofdoubleWDsinthe
forthedifferentialinteractioncrosssection(Heggeiet GC.
al.1993;Davisetal.1992):
dσ X−∆(1+X)−4.5+∆ X-ray sources in the GC. In this case, the power-
=kπa2 , (9)
dX 0 V2 law index for X-ray sources, which are associated
withhigher-massbinaries,isclosetounity:N ∝
X-ray
dσe Γ0.74±0.36(Pooleyetal.2003).
∼e, (10)
de
The merger rate of double WDs (SN Ia candi-
where X =∆Eb/Eb, and the parameter ∆ depends dates) is plotted against time for one model (C) in
ontherelative velocityoftheapproachingstars. The
meanvalueis(cid:9)X(cid:10)=0.4.
The change in binding energy during a tripple ThenumberofmergingdoublewhitedwarfsinaGCof106
collision results in momentum transfer to the binary starswithatotalmassof4×105M⊙inanevolutiontime
barycenter. Thus, it determines the possibility of the of1.5×1010yr
binarybeingexpelledfromtheclusterthroughsucha
ccoanllisgioonbeafnodrethitesdmisetragnecr.etowhichtheescapedbinary Model ρc,pc−3 rc,pc Γ NWD2 NWGDC2 NWejD2 ν
A 105 0.1 1 108 98 10 0.09
RESULTS OF THE CALCULATIONS 228 216 12 0.05
B 105 0.3 10 120 108 12 0.10
SincetheexpulsionofmergingdoubleWDsfrom
a cluster is directly related to the dynamical inter- 252 216 36 0.14
actions between stars, one mightexpectthe number C 106 0.1 32 138 114 24 0.17
of such systems Nej in the evolution time to be 330 264 66 0.20
WD2
determinedbyΓ: D 107 0.03 90 174 132 42 0.24
ρ2r3 381 261 120 0.31
Γ∝ R n dV ∼ 0 c ∼ρ1.5r2,
WD2 WD2 v 0 c Note. The initial number of binaries is 3×105. The GC pa-
m
(cid:16) rametersaregivenatthetimeofthecorecollapse(7×109 yr).
wheretheintegrationisovertheentireGCvolume. The value of Γ was normalized in such a way that Γ=1 for
Thecalculated numbers ofmerging WDpairsfor model A. NWD2 is the total number of merging double WDs
variousGCmodelsaregiveninthetable.Weseefrom in theentireevolutiontime;NWGDC2 and NWejD2 are the numbers
thistablethatthechoiceofaGCmodel(parameterΓ) of merging double WDs within the GC and expelled from the
most strongly affects precisely the number of WDs GC, respectively;and ν is thefractionof the expelledmerging
doubleWDs.Foreachmodel,theresultsofourcalculationsare
dynamicallyexpelledfromthecluster:NWejD2 ∝Γ0.65. presentedfortwovaluesofαq:αq =0(upperrows)andαq =1
Asimilarpowerlawalsofollowsfromobservationsfor (lowerrows)(seeformula(8)).
144 KURANOV, POSTNOV

0.20 0.8
0.15 0.6
0.10 0.4
0.05 0.2
0 0
50 100 150 100 200 300
v, km s–1 R, kpc
Fig. 3.Histogram(normalizedtounity)illustratingthe
Fig.2.Histogram(normalizedtounity)illustratingthe distributionofWDpairsexpelledfromtheclusterindis-
distributionofWDpairsexpelledfromtheclusterintheir tancefromthecenterofanisolatedglobularclusteratthe
barycentervelocities(modelC). timeoftheirmerger(modelC).
Fig. 1. The increase in merger rate at time t∼7× Our Galaxy. The GC system of our Galaxy
109yrcorrespondstoasharpincreaseinΓduringthe extends to distances as large as 100 kpc above
evolutionofthecluster(GCcorecollapse). the Galactic plane and has a velocity dispersion
The ratio of the total number of mergers to the of about 150 km s−1 (see, e.g., the catalogs by
numberofevolvingbinariesmayserveasacharacter- Kukarkin (1974) and Harris (1996)). The escape
isticoftheefficiencyofthemergerrateofdoubleWDs velocity from the Galactic potential is ∼550 km s−1
thatisindependentoftheadoptednormalization(the intheGalacticplaneanddecreasesto∼100kms−1
total number of stars in the cluster and the fraction at distances of ∼100 kpc from the Galactic center
of the initially binary stars). Taking 300 mergers per (accurate estimates are difficult to obtain, because
3×105 binaries (table) as a typical value, we obtain the form of the Galactic potential is not known at
an efficiency of 10−3 per initial binary system. This suchdistances).Theescapeofthestarsexpelledfrom
value is half the value obtained by Shara and Hur- the GC is preferred at the apocenter of the cluster
ley (2002) (2002) intheir numerical simulations and orbit,nearwhichtheclusterislocatedforthelongest
canbeexplained,inparticular,bythehigherefficiency time.Clearly,notallofthebinariesexpelledfromthe
ofthecommonenvelope(αCE =3)inthesenumerical clusterwillbeabletoescapefromthehostgalaxy,but
simulations. willremaininitshalo.Thus,thetotalnumberofWD
Note also that the total number of merging WDs pairs escaping from the cluster gives an upper limit
asaresultoftheGCevolution,NWD2,changesbya on the number of intergalactic type-Ia supernovae.
factorofonly1.5fordifferentGCmodelsandis100– The rough upper limit on the probable number of
500percluster;most(70%)ofthemergersoccurafter intergalactic SN Ia, N , that emerge during the
SNIa
thecollapseoftheclustercore.Themeanmergerrate mergers of double WDs expelled from the GC in a
of double WDs in the past 5 Gyr is then ∼10−13 clusterofNgalgalaxiesis
per year per average cluster star, which is a factor
N =N ×N ×N
of∼3higherthantheestimatedmergerrateofdouble SNIa WD2 GC gal
WDsinourGalaxy(Nelemansetal.2001;Hurleyet ∼100×500×103 =5×107
al.2002).
(N isthenumberofGCsinanaveragegalaxy).In
The distributions of WD pairs expelled from the GC
a characteristic time of 1010 yr, the formation rate of
cluster in their barycenter velocities and distances
SNe Ia through this channel throughout the cluster
from the center of the host GC are shown inFigs. 2
is ∼5×10−3 per year, which is comparable to the
and3.AsweseefromFig.2,thefractionofthestars
merger rateofdoublewhitedwarfswithatotalmass
withescapevelocitiesofabout100kms−1issmall(a
exceedingtheChandrasekharlimitinasinglegalaxy
fewpercent).
(Nelemans et al. 2001). Even this upper limit is an
order of magnitude smaller than the high fraction of
DISCUSSION
theextragalacticSNeIadeducedfromobservations.
TheEscapeofMergingDoubleWDsfromthe
Dwarf elliptical galaxies. We may also assume
SystemofGalacticGlobularClusters
thatdwarfellipticaldEgalaxies(thefractionofwhich
THE ORIGIN OF INTERGALACTIC SUPERNOVAE 145
inaclustercanbesignificant)withamassuptoM ∼ cdgalaxiesaswellasthroughthetidalcaptureofthe
108M arethemainsourceofobservedintergalactic GCsystem ofthegalaxiesthatpassnear thecluster
⊙
supernovae.Thecentralnumberdensitiesofthestars center.Assumingthat0.3%oftheluminosity(andthe
insuchgalaxiesarecomparabletothoseindenseGC baryonmass)intheclustercentersisconcentratedin
cores.Theescapevelocityfromaclusterataconstant densecollapsedstellarstructuresandtakingintoac-
star space density is v ∝M1/3. Therefore, applying countthethreefoldincreaseinthemergerrateofdou-
e
ourcalculationstoasystemof108 stars,wefindthat ble WDs in such clusters, we naturally obtain∼1%
the number of escaping WD pairs from this system of the observed SN Ia rate outside the visible host
is 108/106×f(>v )×100∼100 (in this estimate, galaxies. Thus, this model predicts the existence of
e faint ((cid:9)M (cid:10)∼−10m) host clusters for SN Ia. It is
the fractionofthe binaries with barycenter velocities R
hoped that the purposeful search for type-Ia super-
higherthantheescapevelocityfromthecluster,v ∼
e
novae being conducted at present as part of various
100kms−1,wastakentobef(>v )∼1/100),which
is also insufficient to explain theetruly intergalactic projects(see,e.g.,Tonryetal.2003)willimprovethe
existingstatisticsandwillrevealthefainthostgalax-
supernovae.
ies and astrophysical sources of intergalactic SN Ia.
cD galaxies. Let us now turn to the central
The absence of host clusters with M >−8m...−
cD galaxy as the source of intergalactic supernovae. R
10m may serve as a critical argument for choosing
AlthoughSN 1998fc in Abell403 and SN 2001al in
the model of intergalactic type-Ia supernovae from
Abell2122/4areprojectedontothehalosofthecen-
intergalacticstars.
tral сD galaxies (the projected distance is 160 kpc),
we rule out the possibility that they belong to these
galaxies because of the radial-velocity difference
CONCLUSIONS
(Gal-Yam et al. 2003). The giant elliptical galaxies
in the cluster centers are known to have a huge We haveconsidered the possibleformation chan-
system ofglobularclusters (upto10000). Usingthe nelofintergalacticSNIaduringthemergersofdou-
results obtained in our calculations (∼300 mergers ble white dwarfs that evolve and are expelled from
of double WDs in cluster cores and ∼100 merging dense star clusters (GCs, the nuclei of dwarf ellip-
pairs escaped after the core collapse), we obtain an tical galaxies, etc.) when they dynamically interact
average estimate of the maximum SN Ia rate due to with a third star (tripple collisions). The population
the WD evolution in the GC in projection onto the synthesis method was used to model the formation
halo of the cD galaxy, 400×104/(5×109)∼10−3 and evolutionofclose doublewhitedwarfs incluster
peryear.OnlyafewpercentofsuchdoubleWDscan cores by taking into account the time evolution of
havebarycentervelocitieshighenoughtoescapefrom their physical parameters and to calculate the rates
thegalaxy;theycannotexplaintheobservations. ofescape ofsuchobjects fromclusters. Weobtained
their distributions in escape velocities from clusters,
inmergerrates,and indistances fromthehostclus-
TheMergersofDoubleWDsinVirializedStar
ter after escape. We showed that the mean merger
ClustersattheCentersofGalaxyClusters
rate of double WDs after the core collapse could
Is there an alternative explanation for the obser- reach ∼10−13 per year per average GC star. In a
vations by Gal-Yam et al. (2003) that does not ap- typical GC, the number of expelled WD pairs after
peal to the evolution of 10−20% of the stars out- the GC core collapse is ∼100. For a galaxy cluster
side galaxies? In our view, the most plausible expla- with several thousand members, this estimate yields
nation for intergalactic supernovae is the merger of a formation rate of intergalactic thermonuclear su-
WD pairs in a virialized system of star clusters in pernovae<0.005peryear,whichismuchlowerthan
thecentral regionsofgalaxyclusters.Itfollowsfrom the observed highrate of intergalactic supernovae of
observations of the central parts ofthe Virgo cluster this type. We hypothesize that the possible observed
and from simulations (see the recent review article high percentage of intergalactic SN Ia without any
by Lee (2002) and references therein) that a huge obvious host galaxies can be explained in part by
system of blue GCs with a large velocity dispersion the evolution of double WDs in dense star clusters.
mustbeformedduringtheformationofgiantelliptical A system of such clusters naturally arises near the
galaxies. Recent observations of intergalactic (up to centerofgalaxyclustersduringtheformationofgiant
250 kpcfrom the center) GCs inthenearby Hydra I elliptical galaxies and, subsequently, during the tidal
andCentaurusclusters(Hilker2002)leadtoasimilar interaction between the galaxies that pass near the
conclusion. An enhanced density of star clusters in clustercenter.Thishypothesiscanbeverifiedthrough
thecentralregionsofgalaxyclustersnaturallyarises observations of the faint host clusters of SN Ia in
during the merger of the galaxies that form giant nearbygalaxyclusters.
146 KURANOV, POSTNOV
ACKNOWLEDGMENTS The diffusion coefficients that contain E and J
can be expressed in terms of (cid:9)∆v (cid:10),(cid:9)∆v2(cid:10) and
WewishtothankA.S.RastorguevandO.K.Sil’chenko || ||
for helpful discussions and L.R Yungelson for valu- (cid:9)(∆v⊥)2(cid:10) are the locally calculated velocity diffusion
able remarks. This work was supported by the Rus- coefficients:
sianFoundationforBasicResearch(projectnos.03- 1 1
(cid:9)∆E(cid:10)=v(cid:9)∆v (cid:10)+ (cid:9)∆v2(cid:10)+ (cid:9)∆v2(cid:10),
02-16110аand03-02-06733mas). || 2 ⊥ 2 ||
(cid:9)∆E2(cid:10)=v2(cid:9)∆v2(cid:10),
||
APPENDIX
J r2
(cid:9)∆J(cid:10)= (cid:9)∆v (cid:10)+ (cid:9)∆v2(cid:10),
v || 4J ⊥
Thepositionofabinaryinaglobularcluster.
J2 1 J2
The relative motions of stars in a GC produce (cid:9)∆J2(cid:10)= (cid:9)∆v2(cid:10)+ (r2− )(cid:9)∆v2(cid:10),
continuous fluctuations of the gravitational field. In v2 || 2 v2 ⊥
turn,thesefluctuationscausethemagnitudeanddi- (cid:9)∆E∆J(cid:10)=J(cid:9)∆v2(cid:10)
||
rection of the velocity of each star to change. As a
result, the energy E =v2/2+φ(r) and the angular For a Maxwellian velocity distribution of the field
stars, the diffusion coefficients reduce to standard
momentumJ =r×v ofthestarwillchange.When
t form(Spitzer1987):
calculating each evolutionary track, the changes in
these parameters in a time interval ∆t due to the m
(cid:9)∆v (cid:10)=−2 1+ n Γk2G(x),
gravitational interaction with the surrounding stars || m f
canbecalculatedbyusingtheformula (cid:17) f(cid:18)
G(x)
∆E =nǫ1+n1/2y1ǫ22, (cid:9)(∆v||)2(cid:10)=2nfΓk x ,
∆J =nj1+n1/2y2j22, (cid:9)(∆v )2(cid:10)=2n ΓkΦ(x)−G(x),
⊥ f
x
∆t=nP(E,J),
whereΦ(x)istheerrorfunction
where y and y , the random numbers drawn from
1 2
x
a normal distribution with a mean (cid:9)y (cid:10)=(cid:9)y (cid:10)=0
1 2 2
and a variance (cid:9)y2(cid:10)=(cid:9)y2(cid:10)=1, are chosen from the Φ(x)= exp−y2dy
1 2 π1/2
correlationconditioninsuchawaythat (cid:16)
0
(cid:9)y y (cid:10)=ξ2/ǫ j . and
1 2 2 2
Φ(x)−xΦ′(x)
Weusedamethodwithaveragingovertheorbitsim- G(x)≡ .
ilar to the method by Shapiro and Marchant (1978) 2x2
and Marchant and Shapiro (1979) to determine the
Using E and J,we can determine the orbitalpa-
coefficientsǫ andǫ :
1 2,1 rametersofthestarwithintheGC:theorbitalperiod
ra ra
ǫ1 =2 (cid:9)∆E(cid:10)dr/vr, P(E,J)=2 dr/vr,
(cid:16)rp (cid:16)rp
ra
ǫ22 =2 (cid:9)∆E2(cid:10)dr/vr, the pericenter rp and the apocenter ra of the stellar
(cid:16)rp orbitintheclusterfromthecondition
ra v2 =2E−2φ(r)−J2/r2 =0.
j =2 (cid:9)∆J(cid:10)dr/v , r
1 r
(cid:16)rp The rate of interaction between a binary b and an
j2 =2 ra(cid:9)∆J2(cid:10)dr/v , αsubsystemofstarsattimetisgivenbytheformula
2 r
(cid:16)rp 2 ra
R (t)=σ × n (r,t)v(r,t)/v dr.
ξ2 =2 ra(cid:9)∆E∆J(cid:10)dr/vr. αb αb P(E,J) (cid:16)rp α r
(cid:16)rp
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