Numerical estimates of the accretion rate onto intermediate-mass black holes

0.24 MB·English

by  Carolina Pepe

Mon.Not.R.Astron.Soc.000,1–9(2012) Printed14January2013 (MNLATEXstylefilev2.2)
Numerical estimates of the accretion rate onto
intermediate-mass black holes
C. Pepe1,2⋆ and L. J. Pellizza1,2
1Instituto de Astronom´ıa y F´ısica del Espacio, Casilla de Correos 67, Suc. 28, 1428, Buenos Aires, Argentina
2Consejo Nacional de InvestigacionesCient´ıficas y T´ecnicas, CONICET, Argentina
3
1
Accepted 2013January11.Received2013January11;inoriginalform2012November21
0
2

n
ABSTRACT
Ja The existence of intermediate-mass (∼ 103M⊙) black holes in the center of globular
clusters has been suggested by different observations. The X-ray sources observed in
1
NGC 6388 and in G1 in M31 could be interpreted as being powered by the accretion
1
ofmatterontosuchobjects.Inthisworkweexploreascenarioinwhichtheblackhole

accretes from the cluster interstellar medium, which is generated by the mass loss of
]
E the red giants in the cluster. We estimate the accretion rate onto the black hole and
H compare it to the values obtained via the traditional Bondi-Hoyle model. Our results
show that the accretion rate is no longer solely defined by the black hole mass and
.
h the ambient parameters but also by the host cluster itself. Furthermore, we find that
p the more massive globular clusters with large stellar velocity dispersion are the best
-
candidates in which accretion onto IMBHs could be detected.
o
r Keywords: accretion,accretiondiscs–blackholephysics–globularclusters:general
t
s
a
[

2
1 INTRODUCTION (PopulationIII)stars,whileMiller & Hamilton(2002)have
v
82 Theexistenceofintermediate-massblackholes(102–104M⊙, schluoswtenrsth(aGtCbsl)acaks haocleosnsoefcu1e0n3cMe⊙ofcothueldmfoerrmgerinofglsotbelulalarr-
IMBHs) was suggested, among other observations, by the
1 mass black holes. A similar mechanism was proposed by
detection of ultraluminous X-ray sources in nearby galax-
5 Portegies Zwart et al. (2004), who have shown that a run-
ies (see Feng & Soria 2011, for a review). The X-ray fluxes
. awaycollisionofmassivestarsinaGCresultsinanIMBHat
1 and the distances measured for these sources imply lumi-
itscentre.Hence,GCshavebecomethemain candidatesto
1 nosities (assuming isotropic emission) L > 1039 erg s−1, in
2 hosttheseobjects.Theextrapolationoftherelationbetween
excess of the Eddington luminosity for an accreting stellar-
:1 mass compact object (M ∼ 10M⊙). The simplest explana- t(he.eg.c,eMntargaolrbrliaacnkehtoalle.m19a9s8s)aanldsothpeoinbtuslgteomthaessexoifstgeanlacxeieosf
v tion for these high luminosities is the presence of accreting
IMBHs in GCs.
i objects with masses in the range of those of IMBHs, but
X
otherhypotheseshavebeenproposed.K¨ordinget al.(2002) Following thepredictions of theoretical models, several
ar suggested that the emission of ULXs could be beamed, attemptstodetectIMBHsatthecentresofGCsweredone.
hence implying lower luminosities for these sources, while Stellar density profiles and stellar dynamics measurements
King et al. (2001) proposed that ULXs emit indeed in a in the central regions of some GCs suggest the existence of
super-Eddington regime with mild geometrical collimation centralobjectswithmasses∼103M⊙.Miocchi(2007)devel-
of their photon emission (a factor . 10) due to outflows. oped a model for the stellar distribution in GCs, including
Although in these alternative models accreting stellar-mass theeffects of an IMBH. Those models containing an IMBH
black holes could explain the observed luminosities, they yield results consistent with the surface brightness and ve-
fail to account for sources with L & 1041 erg s−1, which locity dispersion profiles obtained by Noyola et al. (2006)
still require the accreting compact object to be an IMBH for the Galactic GCs NGC 2808, NGC 6388, M80, M13,
(Feng& Soria 2011). M62 and M54, and also G1 in M31. Van den Bosch et al.
On theother hand, different theoretical models for the (2006) constructed dynamical models of M15 and esti-
origin of IMBHs have been proposed. According to Fryer mated the central mass in 3400 M⊙. However, the na-
(2001), IMBHs would be the dead fossils of primordial ture of this concentrated mass can not be distinguished:
it could be either an IMBH or a large number of stellar-
mass compact objects, or a combination of both. On the
⋆ E-mail:[email protected] other hand, McLaughlin et al. (2006) fitted the proper mo-
(cid:13)c 2012RAS
2 C. Pepe et al.
tion dispersion profile of 47 Tuc, obtaining an estimate of we studied the accretion of dark matter onto IMBHs. We
its central point mass of 1000 − 1500 M⊙ at 68% confi- took into account the gravitational pull of the cluster on
dencelevel.Noyola et al.(2010)analyzedthesurfacebright- the accreted matter and found that the accretion rate de-
ness and velocity dispersion profiles of ω Cen. Both pro- pends on the cluster mass, unlike the Bondi-Hoyle accre-
files show a central cusp, a 4000M⊙ IMBH being the best tion rate that scales as the square of the IMBH mass. The
explanation for these observations. Gebhardt et al. (2002) Bondi-Hoyle accretion rate is retrieved only for ultrarela-
reported a 2 × 104M⊙ point mass at the center of G1, tivistic fluids, which are not influenced by the presence of
based on photometric and kinematical data in the central the cluster. As the ICM is a non-relativistic fluid, we ex-
areas. However, other authors arrived at different conclu- pect that its behaviour departs from that predicted by the
sions. For example, van derMarel & Anderson (2010) and standardBondi-Hoyletheory.Moreover,theICMofaGCis
Baumgardt et al. (2003) constructed dynamical models for not a homogeneous, static medium, because it is fed by the
ω Cen and G1, respectively, claiming that the presence of red giant stars and cleansed by the passages of the cluster
an IMBHisnot neededtofittheavailable data.Because of through the Galactic plane (Roberts 1988). The influence
this lack of consensus, the study of IMBHs is still an open of these effects on the accretion rate should be quantified
issueandeffortsneedtobemadeinordertoclarifythesub- tomakeapropercomparison between models and X-rayor
ject. Moreover, although kinematical and photometric data radio observations.
are suitable to show the presence of central mass concen- In this work we develop a simple numerical model for
trations in GCs, at present they are unable to determine theaccretionflowoftheICMontoanIMBHatthecentreof
thenatureof theseconcentrations, namely if theycomprise a GC, including the cluster gravitational pull and the ICM
a single massive object or a collection of stellar-mass rem- sources. We estimate the accretion rate and its dependence
nants.Complementarydataareneededtoproveanyofthese on both the IMBH mass, and the GC and ICM properties,
hypotheses. and compare them with those predicted by the standard
Asinothersystemscontainingcompactobjects,thede- Bondi-Hoylemodel.InSect.2wedescribethehydrodynami-
tectionofongoingaccretionmayhelptoplaceconstraintson calequationsoftheflowanddiscussitsmaincharacteristics,
the existence and properties of IMBHs. Hence, searches of whileinSect.3wesolvetheseequationsforMilkyWayGCs
central X-ray sources have been performed in several GCs. withdifferentmassesoftheIMBHsandtemperaturesofthe
The detection of central sources with properties consistent ICM,stressingthedifferencesbetweenourresultsandthose
with those expected from an IMBH has been reported only obtained with the Bondi-Hoyle model. Finally, in section 4
for two clusters: NGC 6388 and G1 (Nucitaet al. 2008; wediscuss theconsequences ofour resultson thesearch for
Kong et al. 2010). In other clusters, only upper limits for IMBHs, and present ourconclusions.
the X-ray luminosity of a central source have been ob-
tained, constraining it to be lower than ∼ 1031−32 erg s−1
(Maccarone 2004, and refereces therein). Assuming that
2 THE MODEL
the IMBH accretes from the intracluster medium (ICM),
that the ICM has a density similar to that measured by 2.1 Hydrodynamical equations
Freire (2001) in NGC 104, and that the radiative efficiency
is similar to that observed in other black hole systems, In order to understand the accretion process by an IMBH
Maccarone (2004) concludes that the accretion rate must in a GC, we study the dynamics of the ICM in the pres-
be far lower than that predicted by the standard Bondi- ence of the cluster plus IMBH gravitational potential. We
Hoyle model (Bondi & Hoyle 1944) in order to match the aimatcomputingtheaccretionrate,tocompareitwiththe
non-detections of central sources. Based on the correlation standard Bondi-Hoyletheory.The ICMis generated by the
betweenX-rayandradioluminosityofaccretingstellar-mass masslossoftheredgiantsofthecluster.Assumingthatthe
black holes (Gallo et al. 2003), this author also argues that distributionofthesestarsfollowsthestellarmassdensityof
the radio emission of the system would be more easily de- thecluster, and that theaverage mass loss rate is thesame
tectable than the X-ray emission. However, no central ra- for all red giants, the rate at which the density of the ICM
dio source has been detected in Galactic GCs which, ac- increases due to the injection of matter by these stars, at
cordingto Maccarone (2004),imposes strongconstraints to any position within the cluster can bewritten as
themassesofthehypotheticalIMBHs.OnlyUlvestad et al.
(2007) reported the detection of a radio source at about ρ˙=αρ∗ (1)
1arcsecfromthecentreofG1inM31.Theyconcludedthat
this radio emission is consistent with that expected for a whereρ∗ isthestellarmassdensity.Therighthandsidede-
2×104 M⊙ IMBH accreting at the center of the cluster. scribes the gas injection by the stars at a fractional rate α,
Nevertheless, this result has been challenged by recent ob- forwhichtheoreticalandobservationalworksobtainestima-
servations (Miller-Jones et al. 2012). tions in the range 10−14–10−11 yr−1 (Fusi Pecci & Renzini
Itisclearthatthedetectionof(orthefailuretodetect) 1975; Scott & Rose 1975; Dupreeet al. 1994; Mauas et al.
accretion-powered sources in the centres of GCs is crucial 2006, and references therein). The cluster plus IMBH grav-
for the investigation of the existence and nature of IMBHs. itational field can be described by the model developed by
In this context, the interpretation of both X-ray and ra- Miocchi(2007).ThismodelisbasicallyaKing(1966)model
dio data have been done in the past using simple models. with the presence of a black hole at the centre of the clus-
In particular, Bondi-Hoyle accretion has been used, which ter, which modifies the dynamics of the stars in the inner
assumesapointmass(theIMBH)accretingfromahomoge- region. Following Scott & Rose (1975), we assume that the
neous,staticmedium.Inapreviouswork(Pepe et al.2012) ICM is an ideal gas, and that its flow can be considered
(cid:13)c 2012RAS,MNRAS000,1–9
Accretion rate onto IMBHs 3
steady, spherically symmetric and isothermal. Under these integrationmethodforthissetofequationsaredescribedin
hypotheses, the flow is governed by continuity and Euler’s thefollowing section.
equations. The former is
2.2 Boundary conditions
1 d (ρr2u)=αρ∗, (2)
r2dr In order to integrate equation 8, boundary conditions must
be set. Far away from the acretting black hole, the velocity
where r is the radial coordinate and u and ρ the velocity
of the flow must be positive as there is no matter source
and density of theflow, respectively. Euler’s equation is
outside the cluster. On the other hand, on the black hole
surface, mattercanonly fall inwardsas thereisnopressure
ρudu =−kBT dρ − GM(r)ρ −αuρ∗, (3) gradient that supports the gravitational pull of the black
dr µ dr r2 holeandthecluster.Asadirectconsecuence, thereexistsa
where G is the gravitational constant, kB is Boltzmann’s stagnation radius ξst at which u=0. Evaluating Eqn. 7 at
thestagnation radius gives
constant, µ is the mean molecular mass of the ICM, and
M(r)thesumofthestellarmassM∗(r)insideradiusr and
the central IMBH mass MBH. It is assumed here that the Ω∗(ξst)=−ω0, (9)
material is injected with null velocity in theflow.
whichmeansthatallthematterejectedbytheredgiantsin-
These equationscan besimplified introducingthevari-
able q˜=q/α, where q=ρur2 is the bulk flow, giving sideξstisacrettedbytheIMBH.Thisisadirectconsequence
ofthehypothesisofstationarity.Thus,thestagnationradius
separatestwodistinctregionsinspace:aregimeofaccretion
dq˜=ρ∗r2, (4) develops in the inner region, and a wind solution exists in
dr
the outer one. It is important to highlight that, so far, the
stagnationradiuscannotbeuniquelydefinedbytheseequa-
du = u 2c2s − GM(r) − (u2+c2s)r2ρ∗ , (5) tions. For every stagnation radius there exist a solution to
dr u2−c2s (cid:18) r r2 q˜ (cid:19) the equations. However, they are not all physically accept-
able,astheRankine-Hugoniotconditionsforthecontinuity
where cs = kBTµ−1 is the sound speed. It is important to oftheflowatthestagnationradiusrequirethatdensitiesat
note that, with this definition, we can solve the equations both sides of ξst must beequal.
independentlyofα.However,αmustbeknowntocalculate AsintheBondi-Hoyleproblem(e.g.,Frank et al.2002),
thedensity and theaccretion rate. itcanbeseenfromEqn.8thatthereexistsingularvaluesξs
Equation 4 can beintegrated, giving (called sonic radii) for which either the velocity equals the
sound speed in the medium or the aceleration of the flow
q˜= rρ∗r′2dr′+q˜0 = M∗(r) +q˜0, (6) iEsqnnu.l8l.cTahnicseliss,the case when the expression in brackets in
Z0 4π
where the integration constant q˜0 is proportional to the
accretion rate of the black hole. Although the integration 2ψs2(cid:18)1ξ − ω1 ddωξ(cid:19)− 9Ωξ(2ξ) =0. (10)
shouldbedonefromtheSchwarzschildradius,thisradiusis
negligible with respect to all the scales in our model, hence For the same reasons of the classical analysis of the Bondi-
we takeit as zero. Hoyleproblem,onlythetransoniccurvesareconsistentwith
Toperformtheintegration,wedefineadimensionalvari- theproperties of an accretion flow.
ables ξ = rr0−1, ψ = uσ−1 (and, therefore, ψs = usσ−1 Given that ξst is not known a priori, the problem was
), ω = q˜(ρ0r03)−1, Ω∗(ξ) = M∗(r)(4πρ0r03)−1, ΩBH = solved numerically for a grid of values of ξst between 0 and
MBH(4πρ0r03)−1, and Ω(ξ) = Ω∗(ξ)+ΩBH, where r0 is the the cluster tidal radius. For each one of these values, sonic
King radius, ρ0 is the cluster central density, and σ2 = radiiintheinnerandouterregionsweresearchedfor,using
4πGρ0r02/9isthevelocitydispersionparameter.Withthese abisectionschemetofindtherootsofEqn.10.Then,Eqn.8
definitions, Eqns. 5 and 6 can berewritten as wasintegrated(viaafourthorderRunge-Kuttascheme)in-
wards from the outer sonic point and outwards from the
inner one, up to the stagnation radius. The difference be-
ω=Ω∗(ξ)+ω0, (7) tweenthedensitiesateachsideofthestagnationpointwere
calculated for each ξst, and interpolated to zero to find the
true stagnation radius. Once this value was found, we inte-
dψ = ψ 2ψs2 − dω(ψs2+ψ2) − 9Ω(ξ) . (8) gratedEqns.7and8toobtainthetruedensityandvelocity
dξ ψ2−ψs2 (cid:18) ξ dξ ω ξ2 (cid:19) profiles.
These equations describe the flow dynamics in our model.
TheyrepresentanextensionoftheBondi-Hoyleexpressions
tothecaseinwhichthefluidhassources,andagravitational 3 RESULTS
fieldotherthanthatoftheaccretoractsonit.Theseeffects
3.1 Individual clusters
arerepresentedbythesecond andthethirdtermsinbrack-
ets in the right-hand side of Eqn. 8, and the first term in Wefirstexplorethebehaviouroftheaccetionflowinagiven
theright-handsideofEqn.7.Theboundaryconditionsand GC. The cluster is represented by the value of the concen-
(cid:13)c 2012RAS,MNRAS000,1–9
4 C. Pepe et al.
100 100
M 15 M 15
Liller 1 Liller 1
ω Cen ω Cen
M 28 M 28
10 10
r(r) st 0 Ω∗(r) st
1 1
0.1 0.1
0 1 2 3 4 5 6 0 1 2 3 4 5 6
c2/σ2 c2 / σ2
s s
Figure 1. Stagnation radius and accretion rate vs gas temperature for different clusters (squares for M 15, triangles for Liller 1, plus
signs for ω Cen and stars for M 28). The adimensional black hole mass ΩBH is ∼0.007 inall cases. A decreasing profile arises and the
curvesareplacedinthegraphorderedwithincreasingcentralclusterpotential (frombottom totop).
100 100
10
10
r(r) st0 Ω∗(r) st 1
M 15
Liller 1
1 ω Cen M 15
M 28 Liller 1
0.1 ω Cen
M 28
0.01
0.0001 0.001 0.01 0.1 0.0001 0.001 0.01 0.1
Ω Ω
BH BH
Figure 2. Stagnation radius and accretion rate vs black hole mass for different clusters (squares for M 15, triangles for Liller 1, plus
signsforω CenandstarsforM28). Theadimensionaltemperaturec2/σ2 is∼0.34foralltheclustersintheHARregime(upper setof
s
curves). For those in a LAR regime (lower set of curves) the adimensional temperature is around 1, except for Liller 1, for which it is
∼3.Thecurves areplaced inthegraphorderedwithincreasingcentral cluster potential (frombottom totop). Itcanbeseenthatthe
dependence oftheaccretionrateontheblackholemasscanbeneglectedcomparedtothetemperaturedependence.
trationcGC(definedastheratiooftheclustertidalradiusto structeddifferentflowmodels,performingascanofthetwo
r0),andtwoofthethreeparametersr0,ρ0,σ.Theseparam- free parameters. Although NGC 7078 and Liller 1 have al-
eters, together with the IMBH mass allow us to construct mostthesamevaluefortheconcentration,theydifferinthe
the Miocchi (2007) model for the stellar mass M(r) of the scaling parameters, which allows us to extend the range of
cluster uniquely. The concentration and IMBH mass define thenon-dimensionalfreeparameters. Thetemperaturecov-
the shape of M(r), while the other parameters merely set erstherange5000−15000K,whichistherangeexpectedfor
its physicalscale. Note that thesame is truefor our hydro- the ICM (Scott & Rose 1975; Knapp 1996; Priestley et al.
dynamical model, as the flow can be fully expressed using 2011). The IMBH mass ranges from 102M⊙ up to 104M⊙.
non-dimensional variables defined in terms of those scaling Heaviermasses would produce astrong effect on thestellar
parameters. Thus, for a given GC there are only two free dynamicsandstructureofthecluster,whichisnotobserved,
parameters in our model: the non-dimensional temperature while for lower masses the assumption of the IMBH at rest
(or sound speed cs/σ) and IMBH mass ΩBH. at thecenterof thecluster would not hold.
With the above considerations in mind, we took four In Figs. 1 and 2 we show the stagnation radius ξst
sample GCs: NGC 7078 (M15), Liller 1, NGC 6626 (M28) and the (non-dimensional) accretion rate Ω(ξst) as a func-
andNGC5139(ωCen),whichspantheconcentrationrange tion of the free parameters cs/σ and ΩBH, respectively,
of Milky Way GCs (see Table 1), and for each one we con- with the other parameter kept fixed. It is worth pointing
(cid:13)c 2012RAS,MNRAS000,1–9
Accretion rate onto IMBHs 5
Cluster σ r0 ρ0 cGC
(kms−1) (pc) (M⊙ pc−3) 10000
NGC7078(M15) 10.8 0.42 1.12×105 2.29 M 15
Liller1 5.05 0.15 1.90×105 2.3 Lωi lCleern 1
NGC5139(ω Cen) 10.2 3.58 1.41×103 1.31 1000 M 28
NGC6626(M28) 7.8 0.38 7.24×104 1.67
TTahbelceen1t.raGllodbeunlsaitryclwuastseresptiamraamteedtearssstuamkeinngfraommaHsas-rlruism(i1n9o9s6it)y. r/rstacc 100
relationM⊙∼L⊙.
10
out that the true accretion rate M˙ can be recovered via 1100 1000 10000
M˙ = αΩ(ξst)ρ0r03, with a suitable value for α (see Sect. 4 MBH
foradiscussion).ItcanbeseenfromtheleftpanelofFig.1
Figure3.Ratiobetweenthestagnationradiusandtheaccretion
that the stagnation radius decreases with the ratio cs/σ. radius of black hole. In the LAR regime the stagnation radius
This can be easily understood in terms of the energetics of behaves liketheaccretionradius.
thegas:forhighertemperaturesthegashasmoreenergyand
canescapefrominnerregionsofthecluster,wherethegrav-
itational wellisdeeper,hencemovingthestagnation radius
3.2 Milky Way globular clusters
inwards.Thisresultsinaloweraccretionrate(rightpanel).
It is interesting to note that the stagnation radius curve In the previous section, we have shown that in some cases
steepens at rst ∼r0, where cs/σ ∼1. The rapid increase of there is a dependence of the accretion rate on the cluster
the stellar mass of the cluster at this radius results then in properties. Here we explore this dependence by applying
an abruptchange in theaccretion rate. This abruptchange the scheme detailed in Sect. 2 to the Milky Way GCs with
suggeststheexistenceoftwoaccretionregimes:athightem- welldeterminedparameters(listedinthecatalogueofHarris
peratures(cs/σ>1),thestagnationradiusislocatedinthe 1996), to construct flow models for different temperatures
central region of the cluster (rst . r0) resulting in a low and IMBH masses. It is worth pointing out that theresults
accretion rate(LAR),whileatlowtemperatures(cs/σ<1) from Sect. 2 extend to all the clusters in the catalogue and
the stagnation radius is far from the centre (rst ≫ r0) and thesamereasoningcanbeappliedtothem.Foreachmodel,
the accretion rate is high (HAR). Note that the accretion the accretion rate was computed using the same value of α
rate differs by almost three orders of magnitude between as in the previous section. In some cases, the models were
the LARand the HAR.Fig. 1 shows another characteristic discardedbecausetheIMBHmasswasnotmuchlowerthan
differentiating the HAR from the LAR. In the former, the the cluster mass, while in others no stagnation radius was
stagnation radius and the accretion rate increase with the found for the hottest temperatures (the ICM escapes com-
cluster concentration cGC. pletelyasawind).Wesearchedforcorrelationsbetweenthe
Fig. 2 shows the dependence of the stagnation radius accretion rate and the properties of the cluster, defined by
and accretion rate on the non-dimensional IMBH mass, at theirscalingparametersandmasses. Wefoundnotrendfor
fixednon-dimensionaltemperature.Twotemperatureswere the accretion rate M˙ with r0 or ρ0, but a clear correlation
explored, one resulting in the LAR and the other in the withtheclustermassMGC andvelocitydispersion parame-
HAR.Boththestagnationradiusandtheaccretionrateare terσ.
almost independent of ΩBH in the HAR regime. This hap- We show in Fig. 4 the dependence between M˙ and σ
pensbecausefarfromthecentretheIMBHhasnoinfluence foraconservativevalueoftheIMBHmass,MBH =1000M⊙
on the stellar distribution, and the stellar mass increases and three different temperatures T =5000, 10000, 12600 K
very slowly. On theother hand,in theLAR regime thereis (left panel), and for a fixed gas temperature (T = 5000K)
a strong dependence on the IMBH mass, because the stag- and different IMBH masses MBH = 1000, 4000, 10000M⊙
nationradiusisnearthesphereofinfluencewherethestellar (right panel). The clear trend observed in the left panel is
distributionbecomesdominatedbytheIMBH.Notethatin understood in terms of the results of the previous section.
this regime, the accretion rate varies also with the cluster For a fixed value of T, an increase in σ implies a decrease
concentration, decreasing as the latter increases. In Fig. 3 in cs/σ, changing the flow from the LAR upwards to the
we show the ratio between the stagnation radius and the HAR in the curves defined in Figure 1. This explains why
accretion radiusdefined as racc =GMBH/c2s. It can beseen theGCswithhigherσaremorelikelytobefoundinaHAR
thatintheHARregime,sincethestagnationradiusremains regime. Thechangein thelocation ofthecurveas thetem-
independent of the black hole mass, the ratio rst/racc de- peratureincreasesisalsoconsistentwiththisinterpretation.
creases with increasing MBH. IntheLARregime, thisratio Thedispersionofthiscorrelationisinpartduetothediffer-
remains almost constant implying that M˙ scales as ∼MB2H ent concentrations of the GCs, and in part to the fact that
like in the traditional Bondi-Hoyle model. However, thera- M˙ scales as ρ0r03, which is different for each cluster. The
tiois1–2ordersofmagnitudegreaterthanunity,andhence rightpanelofFig.4showsthattheeffectoftheIMBHmass
the accretion rate at a fixed IMBH mass is much higher in for a fixed temperature is small.
our models than in theBondi-Hoyle case. A trend of increasing accretion rate with the cluster
(cid:13)c 2012RAS,MNRAS000,1–9
6 C. Pepe et al.
mass can also be seen in Fig. 5 for high accretion rates.
This is explained by the fact that these clusters are accret-
0.0001
ing in the HAR regime, for which the stagnation radius is
well outside the cluster core and hence encloses almost the
wholeclusterstellarmass.AsM˙ isproportionaltothestel-
larmassinsiderst,thecorrelationwithMGCarises.Inother -1yr)
words, most of the mass ejected by the stars in the cluster M Sun
imsoarcecrmetaesds bisyatvhaeilaIbMleBHto. bAesatchcerectleuds,tewrhiischmoexreplmaiansssitvhee, ate ( 1e-08
clear trend with MGC in the HAR regime. Note that this on r
t(raesnfdariassinthdeepaecncrdeetniotnopfrtohceeetdesmipnetrhaetuHreARanrdegIiMmBe)H,amndasist Accreti MLωi lC1lee5rn 1
M 28
isverytight becausebothM˙ andMGC scale with thesame Eddington limit
combination of parameters of the cluster (ρ0r03), reducing Bondi-Hoyle
the dispersion due to the scaling from non-dimensional to 1e-12 100 10000
physical variables. For clusters in the LAR regime there is Mbh(MSun)
noclear trend,asthesmall stagnation radiusdecouplesthe
Figure 6. Comparison between our accretion rate and Bondi-
accretion rate from the cluster mass. According to the re-
Hoyleaccretionrate.WealsoshowtheEddingtonaccretionrate
sults of this section, the clusters with large values of σ and
forareference.
high MGC are most likely to be in a HAR regime, and are
therefore the best candidates to perform detections of the
accretion ontoan IMBH. sult seen in this figure in that in the HAR regime, IMBHs
reach accretion rates as high as the Eddington values (as-
suming a 100% efficient conversion of gravitational energy
3.3 Differences with the Bondi-Hoyle model
into radiation). This result is due to the development of a
The model presented in this paper differs from that of cluster-wide flow in the HAR regime, which carries matter
Bondi & Hoyle (1944) in some very fundamental aspects. from the outer regions onto the IMBH to produce a high
Our model takes into account the gravitational potential accretion rate. Itcannot bereproducedbytheBondi-Hoyle
of the cluster, which differs from cluster to cluster, and model unless a very high density or very low temperature
also includes the constant injection of gas into the ICM are assumed. This result is important for the explanation
by the red giants of the cluster, hence the black hole is no of the emission of ULXs, as will be discused in the next
longer accreting from a static and infinite medium. These section.Itisworthpointingoutthattheseflowswithsuper-
two features of our model produce a very important conse- Eddingtonaccretionratesdonotimplysuper-Eddingtonlu-
quence, which is its main difference with the Bondi-Hoyle minosities as the radiation efficiency is usually much lower
result: the accretion rate not only depends on the black that 1. Hence, these flows can still proceed without being
hole mass and gas temperature (or sound speed), but also stopped by radiation pressure.
on the cluster properties. Figure 6 shows the comparison
between our model (for different clusters; all of them at
T = 10000 K), the Bondi-Hoyle accretion rate, and the
4 DISCUSSION
Eddington rate M˙Edd = LEdd/c2, where c is the speed of
light and LEdd =1.26×1038(MBH/M⊙) erg s−1 is the Ed- In this work, we developed a simple model for the accre-
dingtonluminosityoftheIMBH.TheBondi-Hoyleaccretion tion of the ICM of a globular cluster onto an IMBH at its
ratewascalculated asM˙BH =4πG2MB2Hρac−s3,whereρa is center.Bearinginmindthatthedetectionofongoingaccre-
the ambient gas density. We used the value ρa = 0.2 cm−3 tion would be a strong piece of evidence for the existence
(Freire2001),whichisthesamevalueusedbyotherauthors of these elusive black holes, our aim was to refine the pre-
(e.g., Maccarone 2004) to compute IMBH X-ray luminosi- dictions of the expected accretion rate, improving on the
ties.Tomaketheresultsofourmodelscomparabletothose (usually assumed) classical Bondi-Hoyle accretion. In par-
obtainedwiththeBondi-Hoylescenario,foreachofourmod- ticular,weexploredtheconsequencesofincludingtheeffect
els we havechosen thefractional mass loss rate of the stars ofthegravitational fieldof theclusterandthefluidsources
α so that the density at the stagnation radius matches ρa. on the flow. Our models aims at assessing the influence of
Thischoiceisjustifiedbecauseitmakesbothmodelstohave theclusterasawholeontheaccretion flow,whichwassug-
thesamedensityatthepointwherethefluidisatrest.The gestedtobeimportantinthecaseofcosmologicalfluidssuch
valuesofαobtainedareintherange10−14−10−11 yr−1,in as dark matter by Pepe et al. (2012).
rough agreement with (but somewhat lower than) the few Wemadeseveralassumptionsinourmodel,namelythat
observational estimations of this parameter (Scott & Rose theflowisstationary,sphericallysymmetric,andisothermal.
1975;Priestley et al. 2011). While the first two can be regarded as working hypotheses
It can be seen from Fig. 6 that the accretion rate in madetosimplifythecomputations,thelattercanbeagood
the HAR regime (such as that of NGC 7078 in this plot) approximation for the whole cluster (Scott & Rose 1975).
no longer scales as MB2H, and that it is cluster-dependent. However,itmaybreakdownveryneartheblackhole,where
However,intheLARregimethebehaviourissimilartothat the transformation of gravitational into thermal energy is
of Bondi-Hoyle models, although the absolute value of M˙ higher and may heat the gas in timescales shorter than the
is one order of magnitude higher. Another interesting re- coolingone.Althoughourmodelcouldnotdescribethedy-
(cid:13)c 2012RAS,MNRAS000,1–9
Accretion rate onto IMBHs 7
1e-05 1e-05
-1n rate (M yr)Sun1e-06 Liller 1 M 28 ω CenM 15 NGC 6388 n rate (Myr-1)Sun 1e-06 Liller 1 ω CenM 15 NGC 6388
cretio 1e-07 cretio 1e-07 M 28
c c M = 1000 M
A A Sun
M = 4000 M
T = 5000 K Sun
T = 10000 K M = 10000 MSun
1e-08 T = 12600 K 1e-08
10000 10000
σ(m/s) σ (m/s)
Figure4.Accretionratevsglobularclustervelocitydispersion(σ)forthreedifferenttemperatures(leftpanel):5000K(triangles),10000K
(plus signs) and 12300 K (squares). Accretion rate vs globular cluster velocity dispersion for three different black hole masses (right
panel): 1000M⊙ (triangles) , 4000M⊙ (plus signs) and 10000M⊙ (squares). There is a nearly constant constant value of M˙ for those
clusters with low σ. The clusters located in the right end of the curve are those clusters accreting at a HAR regime. The dotted lines
indicatetheσ valuescorrespondingtotheclustersfromSect.2andtheparticularcaseofNGC6388discussedinSect.4.
M = 1000 M
Sun
TT == 51000000 0K 1e-05 M = 4000 MSun
1e-05 T = 12300 K M = 10000 MSun
-1Accretion rate (Myr)Sun11ee--0067 Liller 1 M 28 NGC 6388 M 15 ω Cen -1Accretion rate (M yr)Sun11ee--0067 Liller 1 M 28 NGC 6388 M 15 ω Cen
1e-08 1e-08
0 5e+05 1e+06 1.5e+06 0 5e+05 1e+06 1.5e+06
M (M ) M (M )
GC Sun GC Sun
Figure 5. Accretion rate vs globular cluster mass for three different temperatures (left panel): 5000K (triangles), 10000K (plus signs)
and12300 K(squares). Accretionratevs globularcluster massforthreedifferente blackholemasses(rightpanel): 1000M⊙ (triangles)
, 4000M⊙ (plus signs)and 10000M⊙ (squares). A soft trend can be observed, although there is awide spread of the data. The dotted
linesindicatetheMGC valuescorrespondingtotheclustersfromSect.2andtheparticularcaseofNGC6388discussedinSect.4.
namics of theflowin these spatial scales, it is still useful to models are stationary, the accretion rate onto the IMBH is
assess the existence of cluster-wide flows, whose dynamics determined bythestagnation radius and themass-loss rate
is governed by phenomena occurring at much larger spatial of thestars.
scales, and estimates the order of magnitude of the accre-
The accretion rate onto the IMBH in a given cluster
tion rate produced by these flows. To analyse the details of
dependsmainly ontheratio ofthesound speed intheICM
theflow verynear theblack hole, we are currentlydevelop-
to the stellar velocity dispersion parameter. This is a con-
ingmorecomplexmodelswhichincludeseveralheatingand
sequence of the energetics of the flow, and the same effect
cooling mechanisms.
observed by Pepe et al. (2012) for dark matter. The sound
Our main results are the following. First, large scale speed(relatedtothetemperatureoftheflow) measuresthe
flows can form in globular clusters with IMBHs, dueto the internalenergyoftheICM,whilethestellarvelocitydisper-
influence of the gravitational field of the cluster and the sionparameterindicatesthestrengthoftheclustergravita-
masslossoftheclusterstars.Theseflowsshowastagnation tional field. As the ratio of these two magnitudes increases,
radiusatwhich thevelocityisnull.Outsidethisradius,the the ratio of the internal to gravitational energy of the flow
ICMescapesfromtheclusterasawind,astheenergyofthe increases as well, makingeasier for theICM toescape from
flow allows it to overcome the cluster potential well. Inside the cluster. Therefore the stagnation radius decreases, the
the stagnation radius the ICM is retained by the potential wind becomes stronger and the accretion flow weaker, di-
well,andanaccretionflowontotheIMBHdevelops.Asour minishing theaccretion rate.
(cid:13)c 2012RAS,MNRAS000,1–9
8 C. Pepe et al.
The stellar mass distribution of globular clusters has thermodynamical state are still rare (e.g., Freire 2001). As
also important effects on the flow. These clusters show a pointedoutbyScott & Rose(1975),thetemperatureofthe
large concentration of their mass within a few core radii of flowdependsmainlyonthatoftheradiationfieldtowhichit
theircentres,while thepotentialwell of theclusterextends isexposed,henceaproxyfortheformerwouldbethenum-
to far beyond (typically to tidal radii one or two order of berofUVsourceswithinthecluster,suchasbluehorizontal-
magnitudeslargerthanthecoreradius).Thistranslatesinto branch stars or blue stragglers. These sources would heat
asteepincrease oftheaccretion ratewhenthetemperature thegas,preventingtheaccretionflowtodevelopouttolarge
is low enough for the stagnation radius to reach the core scales,andhencedecreasingtheaccretionrate.Interestingly,
radius, as most of the cluster stellar mass (and hence most Miocchi (2007)argues thatextremehorizontal-branchstars
of the stellar mass loss) is enclosed within the latter. This with strong UV fluxes might be the result of the stripping
steepincreaseseparatestwoaccretionregimeswithdifferent ofnormalstarspassingneartheIMBH.Ifthisisindeedthe
properties, one with a high accretion rate at low tempera- case, the presence of the IMBH would help in heating the
tures (HAR), and the other with a low accretion rate at flow, decreasing theaccretion rate.
higher temperatures (LAR). FromtheresultsofourmodelwecanestimatetheX-ray
In the HAR regime, for a fixed temperature the ac- luminosityLX duetotheaccretion processontotheIMBH.
cretion rate is high, proportional to the cluster mass, and However, in our model the luminosity depends linearly on
almost independentof theIMBH mass, as far as theIMBH twopoorly constrained parameters, thefractional mass loss
is not massive enough to severely change the whole cluster rate of the stars α and the accretion efficiency of the flow,
massdistribution(however,ifthiswerethecase,strongsig- ǫ = LX/M˙c2. If we adopt the standard values, ǫ = 0.1 for
naturesofthepresenceoftheblackholeshouldbefoundin accreting stellar-mass black holes and α = 10−11 yr−1, the
the stellar distribution and dynamics). The dependence of luminosities produced by the accretion flows of our models
theaccretion rateontheclustermassinstead ontheIMBH would be in the range 1037 −1041 erg s−1. The high end
massisduetothefactthatintheHARregime,thestagna- of this range is in good agreement with the luminosities of
tion radius is in the outerregion of theGC, and theIMBH ULXs.If,asclaimedinafewcases,thereareULXsposition-
accretesamajorfractionofthemasslostbythestars,which allycoincidentwithextragalacticGCs(Angeliniet al.2001;
isproportionaltotheclustermass.Notethedifferencewith Maccarone et al. 2011, and references therein), our models
theclassicalBondi-Hoylemodel,inwhichtheaccretionrate suggest that these systems may contain IMBHs accreting
scales as the squareof theblack hole mass. in the HAR regime at the centres of GCs, with standard
The LAR regime is qualitatively different, showing ac- accretion efficiencies.
cretion rates several orders of magnitude lower, which de- In the case of NGC 6388 our models predict (based on
pend strongly on the IMBH mass. The accretion rates in the estimation of the accretion rate and assuming the ef-
the LAR regime are still one order of magnitude greater ficiencies discussed above) X-ray luminosities in the range
thattheBondi-Hoylepredictionforsimilarboundarycondi- 1038−40 erg s−1, at least 5 orders of magnitude higher than
tions,anddependontheIMBHmassbecausethestagnation the observed value of LX, NGC6388 = 2.7 × 1033 erg s−1.
radius is near or inside the IMBH sphere of influence. This However, the prediction can be reconciled with obser-
dependence varies from cluster to cluster due to the differ- vations if either α or ǫ are lower. Nucita et al. (2008)
ent stellar-mass profiles of the clusters, but on average it reaches the same conclusion about the efficiency. It is
mimicks theMB2H dependenceof theBondi-Hoylemodel. worth mentioning that different theoretical models have
Another strong assumption of our models is that the been developed for accretion flows with very low efficien-
IMBH is at rest at the centre of the cluster. This approxi- cies, such as advection-dominated accretion flows (ADAFs,
mation holds because the mass of the IMBH is far greater Narayan & Yi1994),jet-dominatedaccretionflows(JDAFs,
thatthemassofthestarsinthecluster.Assumingthatthe Fender,Gallo & Jonker 2003), and low-radiative efficiency
stars in the cluster have a mean mass of ∼ 0.5M⊙ and a accretion flows (LRAFs, Quataert 2001). For example, in
mean velocity of the order of σ, energy equipartition im- thecase of ADAFstheefficiency scales as ǫ=M˙/M˙Edd for
plies that the IMBH would have a velocity of the order of M˙ <0.1M˙Edd,givinginourcaseestimationsoftheluminos-
σ/2 MBH/M⊙.Atthesevelocities,andtakingintoaccount ity 2–3 orders of magnitude lower, approaching the obser-
thatpcs/σ ∼ 1 in our models, the typical correction to the vational limits. On the other hand, the value of α is highly
accretion rate due to the motion of the IMBH with respect uncertain, relying in theoretical models and with few ob-
totheflow(Hoyle & Lyttleton1939)wouldbelessthan1%. servationalconstraints(Scott & Rose1975;McDonald et al.
Hence, neglecting this effect is justified for these models. 2011; Priestley et al. 2011). Given that the low luminos-
TheapplicationofourmodeltotheMilkyWayglobular ity limits of our predicted range correspond to low IMBH
clusters has shown that the higher accretion rates are pre- masses, we conclude that such an IMBH accreting in the
dictedforthoseclusterwiththelargestvaluesofthevelocity LAR is a possible explanation for the central engine of the
dispersion parameter σ. However, the higher the tempera- centralX-raysourceinNGC6388.ThecaseofG1isslightly
tureofthegas, thehighertheminimum valueof σ forGCs different,asthisclusterissomassivethattheluminosityin
accreting at high rates. This result suggests that the sig- the HAR is LX ∼ 1042 erg s−1, and the development of a
natures of the accretion onto IMBHs must be searched for LARrequirestemperaturesofseveraltimes10000K.Asthe
inglobularclusterswithhighvelocitydispersionparameters observedluminosityisonlyLX =2×1036 erg s−1,itisdiffi-
andlowgastemperatures.Alltheclustersthatwereobserva- cult toreconcile it with thepredictions, unlesstheG1 ICM
tionallyprovensofarfortheexistenceofIMBHssatisfythe isstrongly heated,theaccretion efficiencyis extremelylow,
first criterion. The satisfaction of the criterion on gas tem- or its stars lose mass at very low rates. The Galactic GCs
perature is difficult to assess, as measurements of the ICM withundetectedcentralX-raysources,withmeasuredupper
(cid:13)c 2012RAS,MNRAS000,1–9
Accretion rate onto IMBHs 9
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ACKNOWLEDGMENTS
Roberts M. S., Proceedings of theIAU,126, 411
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