The Evolution of Black Hole Mass and Spin in Active Galactic Nuclei

0.28 MB·English

by  A. R. King

Mon.Not.R.Astron.Soc.000,1–8(2006) Printed2February2008 (MNLATEXstylefilev2.2)
The Evolution of Black Hole Mass and Spin in Active
Galactic Nuclei
A.R. King1, J.E. Pringle1,2 & J.A. Hofmann3
8 1Theoretical Astrophysics Group, Universityof Leicester,LeicesterLE1 7RH
0 2Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA
0 3Institut fu¨r Theoretische Physik und Astrophysik, Leibnizstrasse 15, 24118 Kiel, Germany
2

n
a 2February2008
J

0
1 ABSTRACT
Observations show that the central black hole in galaxies has a mass M only ∼10−3

]
of the stellar bulge mass. Thus whatever process grows the black hole also promotes
h
p star formation with far higher efficiency. We interpret this in terms of the generic
- tendencyofAGNaccretiondiscstobecomeself–gravitatingoutsidesomesmallradius
o R ∼ 0.01−0.1 pc from the black hole. We argue that mergers consist of sequences
sg
tr of such episodes, each limited by self–gravity to a mass ∆Mepisode ∼ 10−3M, with
s
angularmomentumcharacteristicofthesmallpartoftheaccretionflowwhichformed
a
[ it. In this picture a major merger with ∆Mmerger ∼ M gives rise to a long series of
low–massaccretiondisc episodes,allchaoticallyorientedwith respectto oneanother.

1 Thus the angularmomentum vector oscillatesrandomly during the accretionprocess,
v on mass scales ∼ 103 times smaller than the total mass accreted in a major merger
4
event.
6
WeshowthatforessentiallyallAGNparameters,thediscproducedbyanyaccre-
5
tion episode of this type has lower angular momentum than the hole, allowing stable
1
. co–andcounter–alignmentofthe discsthroughthe Lense–Thirringeffect.Asequence
1
of randomly–oriented accretion episodes as envisaged above then produces accretion
0
discs stably co– or counter–aligned with the black hole spin with almost equal fre-
8
quency. Accretion from these discs very rapidly adjusts the hole’s spin parameter to
0
averagevalues a¯∼0.1−0.3 (the precise range depending slightly on the disc vertical
:
v viscosity coefficient α ) from any initial conditions, but with significant fluctuations
2
Xi (∆a ∼ ±0.2) about these. We conclude (a) AGN black holes should on average spin
moderately,withthemeanvaluea¯ decreasingslowlyasthemassincreases;(b)SMBH
r
a coalescencesleave little long–termeffect on a¯; (c) SNBH coalescence products in gen-
eralhavemodestrecoilvelocities,sothatthereislittlelikelihoodoftheirbeingejected
fromthehostgalaxy;(d)blackholescangrowevenfromstellarmassesto∼5×109M⊙
athighredshiftz ∼6;(e)jetsproducedinsuccessiveaccretionepisodescanhavesim-
ilar directions, but after several episodes the jet direction deviates significantly. Rare
examples of massive holes with larger spin parameters could result from prograde
coalescences with SMBH of similar mass, and are most likely to be found in giant
ellipticals. We compare these results with observation.
Key words: accretion, accretion discs – black holes, galaxies – active
1 INTRODUCTION and references therein). This is reasonable, as growth rates
of this order are needed in order to reach the hole masses
observedinthenearbyuniverse.Ifoneassumesfurtherthat
Astronomersgenerallyagreethatthenucleiofmostgalaxies
contain supermassive black holes (SMBH), but as yet have all the mass gain ∆Mmerger carries the same sense of spe-
cificangularmomentum,theholealwaysspinsrapidlyupto
little clear idea of how they grow. Cosmological large–scale
Kerrparametersaclosetounity(e.g.Volonterietal.,2005;
structure simulations strongly suggest that galaxy mergers
Volonteri & Rees, 2005). The resulting high radiative effi-
arethebasicmotorofgrowth,andpredictthatablackhole
ciencyofaccretion meansthattheEddingtonlimit severely
typicallyacquiresamass∆Mmerger oforderitsownmassM
restricts the rate of black hole mass growth, creating diffi-
in such mergers (e.g. di Matteo et al., 2005, Li et al., 2007,
2 A.R. King, J.E. Pringle, J.A. Hofmann
cultiesinunderstandingobservationsofhighmassesatearly Thispaperdealsnotwithsuchlow–luminosityAGNbut
cosmic times. instead with high–luminosity events causing major black–
However it is sigmificant that ∆Mmerger is only a very hole mass growth. In these cases our discussion suggests a
small part of the total mass involved in a major galaxy picture in which, within an event, the flow on to the hole
merger,inlinewiththeobservationthattheblackholemass is episodic, via a rapid succession of accretion discs limited
in nearby galaxies is typically only ∼ 10−3 of the galaxy by self–gravity (as in the well–separated episodes in low–
bulge mass. Moreover, current cosmological simulations are luminosity AGN), and thus with masses ∆Mepisode ∼ Msg.
entirely unable to resolve the hydrodynamics of matter ac- Thenumberofsuchepisodesisevidently∼∆Mmerger/Msg.
cretion on to the central black hole. The mass inflow time Given that the flow is constantly stirred up by energy and
throughanaccretion discof1pcradiusisalready >∼109 yr momentum input,and that weare concerned only with the
(Shlosman,Begelman &Frank,1990),afactor >∼10greater verysmall fraction of thegas with almost zero angular mo-
than the timescale on which rapid hole growth occurs. The mentum, it is reasonable to expect the disc orientations to
best cosmological simulations resolve only lengthscales at beessentially random.
least 100 times larger. Inevitably the simulations have to We thus arrive at a picture of AGN accretion close to
resort to sub–resolution recipes in trying to model the ac- that suggested by Sanders (1984) (see also Heckman et al.,
cretion process. Forexampleit isusualtoassumeaccretion 2004, Greene & Ho, 2007, and KP07). This picture repro-
attheBondiratefromsomelargeradius,whereasinreality ducesseveralcharacteristicfeaturesofnearbyAGN.Inpar-
theinfallinggasmustpossessangularmomentumwhichwill ticular the random orientations of the accretion episodes
complicate things. meanthattheradiojetsshownocorrelationwiththegrand–
Inthispicturetheverysmallmass∆Mmerger whichac- design structure of the host, as observed for low–redshift
cretesontothecentralblackholemusthavealmostzeroan- AGN (Kinney et al., 2000), and inferred at higher redshift
gularmomentumaboutit.Moreover,sincetheholeisgrow- by Sajina et al (2007). Moreover the picture implies a lu-
ingitsmassasrapidlyaspossible,itmustbeclosetoitsEd- minosity function in broad agreement with that observed
dington limit andthusfeeding backenergy and momentum for moderate mass central black holes (KP07). Finally this
intoitssurroundings.Indeedsimplemodels oftheresulting picture provides a plausible explanation for the ring(s) of
momentum–drivenfeedbackon tothehost galaxy correctly young stars seen around the black hole in the centre of our
reproduce theM −σ relation between black hole mass and own Galaxy (Genzel et al., 2003; Lu et al., 2006).
velocitydispersion withoutanyfreeparameter(King,2003; Since observational properties such as the luminosity
2005, Murray et al., 2005). Along with this, there must be function of AGN and quasars are sensitive only to the to-
copious star formation from material close to the hole, i.e. tal mass increase ∆Mmerger, and not to its internal angular
at distances ∼ 0.01−0.1 pc (see below) where the angular momentumbudget,ourpicturepredictsthesamevaluesfor
momentumneglectedinestimatingtheBondiaccretionrate these observables as found in previous studies (e.g. Wyithe
becomes significant in supporting the captured gas against &Loeb,2003) andisthereforeinbroad agreement withex-
gravity.Thisstarformationalsoinjectsenergyandmomen- pected galaxy merger rates.
tum into the gas near the hole. Hence although it is very Clearlytheproperhydrodynamictreatmentofthekind
difficult to predict thegas flow neartheblack hole it seems of complex event we envisage must await advances in com-
more likely to be chaotic than well–ordered, and rather in- puting power. However we can gain some insight by simple
efficient in growing the hole compared with forming stars, arguments.
thus keeping the latter a fairly constant fraction ∼10−3 of
thestellar bulge mass.
An obvious reason for the relative inefficiency of hole 2 SMBH SPINUP
growth versus star formation is that an accretion disc be-
comes self–gravitating at radii R where its mass exceeds To understand how a supermassive black hole grows we
Msg ∼ (H/R)M, where H is the disc scaleheight and M have to know its spin rate. This governs the accretion ef-
thecentral(hereblackhole)mass(e.g.Pringle,1981,Frank ficiency and thus, through the Eddington limit, the maxi-
et al. 2002). For AGN this means that discs must be self– mumgrowthrateforthemass.Therangehereisverylarge.
gravitating outside some radius Rsg ∼0.01−0.1 pc (Shlos- Given an adequate mass supply,a low Kerr spin parameter
man et al., 1990; Collin–Souffrin & Dumont, 1990; Hur´e et aandconsequentlowaccretionefficiencyǫallowsverymuch
al., 1994). Cooling times in these regions are so short that faster mass growth than a rapidly spinning hole with a∼1
self–gravity is likely to result in star formation (Shlosman (cf. King & Pringle, 2006).
& Begelman, 1989; Collin & Zahn, 1999). As discussed by Accretionontoablackholeaffectsitsspinintwoways.
King&Pringle,2007(hereafterKP07)weexpectthatmost First,ittorquestheholebyaddingmattercarryingthespe-
of the gas initially at radii R > Rsg is either turned into cificangularmomentumoftheinnermoststablecircularor-
stars, or expelled by those stars which do form, on a rapid bit(ISCO),andsecond,theLense–Thirring (LT)effectcre-
(almostdynamical)timescale,whilethegaswhichisinitially ates a viscously mediated torque between the hole and any
at radii R < Rsg forms a standard accretion disc, which accretion disc not completely co– or counter–aligned with
slowly drains on to the black hole and powers the AGN. the hole spin. Gas within the disc follows precessing orbits
KP07 show that for local, low–luminosity AGN,fuelling by because of the LT effect, and the resulting viscous dissipa-
well separated episodes of this type explains observational tiontriestoproduceanaxisymmetricsituationbyadjusting
features suchas theluminosity function formoderate–mass thedirection of theangular momentum vectors of both the
blackholes,andthepresenceandlocationofaringofyoung hole spin and thedisc.
stars observed about theGalactic Centre. The LT effect is very important in AGN, as the ac-
The Evolution of Black Hole Mass and Spin in AGN 3
cretion episodes all initially have random directions with holedowntolowspinparameters,andwefindameanvalue
respect to the hole spin. As it has a large lever arm, it op- a¯∼0.2−0.3.
erates more quickly than the accretion torque, which can-
not greatly change the hole spin until the hole has signif-
icantly increased its mass. Accordingly, the LT effect de- 3 SMBH ACCRETION EPISODES
termines whether the disc and hole angular momenta are
Wefirstrequireasimpledescriptionofthepropertiesofthe
co– or counter–aligned long before the accretion torque
accretiondiscduringafeedingepisodeofthetypedescribed
has had a real effect. Early studies of the LT effect (e.g.
above.Onecanregardanevolvingdiscaspassingthrougha
Scheuer & Feiler, 1996) concluded that it always produced
sequenceofsteadystates,sowefollowKP07inadoptingthe
co–alignment. Accordingly studies of SMBH growth (e.g.
discpropertiesderivedbyCollin–Souffrin&Dumont(1990)
Volonteri et al., 2005; Volonteri & Rees, 2005) argued that
in the context of AGN. These are essentially the same as
the main effect of the accretion torque was always to pro-
thosederivedbyShakura&Sunyaev(1973)forsteadydiscs
duce spinup. However King et al. (2005; hereafter KLOP)
in close binaries. The disc surface density is
showed on verygeneral groundsthatan initially retrograde
accretion disc with total angular momentum J would end
d
upstablycounter–alignedwithaholeofangularmomentum Σ=7.5×106 α1 −4/5 ǫ −3/5
Jh provided that Jd < 2Jh and the initial angle θ between (cid:16)0.03(cid:17) (cid:16)0.1(cid:17)
theangular momenta satisfied L 3/5 R −3/5
× M1/5 gcm−2. (3)
0.1L 8 R
J (cid:16) E(cid:17) (cid:16) s(cid:17)
cosθ<− d . (1)
2Jh Here α1 is the Shakura & Sunyaev (1973) viscosity param-
eter,ǫistheaccretion efficiency,with theluminosity L and
(Throughout this paper Jd,Jh and the Kerr parameter a accetion rate M˙ related by
denotethe absolute values of these quantities.)
Thus Scheuer & Feiler’s (1996) conclusion that co– L=ǫM˙c2. (4)
alignmentalwaysoccursdependedontheirimplicitassump-
Further
tionthattheouterdiscwasfixed,i.e.J ≫J .Bycontrast,
d h
KLOP’s result implies that counter–alignment occurs in a LE =1.4×1046M8ergs−1, (5)
fraction
is theEddington luminosity, M8 is theblack hole mass, M,
1 J in units of 108 M⊙, R is theradius and
f = 1− d (2)
2(cid:18) 2Jh(cid:19) Rs=2.96×1013M8cm (6)
of accretion episodes. is theSchwarzschild radius of thecentral black hole.
Intheiranalysis,KLOPconsideredadiscoffixedtotal The mass of thedisc M(<R) inside radius R is then
aannygumlairsamligonmeedntduismc mJdusmtibsaeliwgnaerpdewdiatht stohmeehorlaed.iuIns rRewaalirtpy, M(<R)=2.94×1034 0α.013 −4/5 0ǫ.1 −3/5
(cid:16) (cid:17) (cid:16) (cid:17)
in general rather smaller thanits outeredge. Only thepart
L 3/5 R 7/5
of the disc around Rwarp exerts an LT torque on the hole, × 0.1L M811/5 R g. (7)
so the correct interpretation of J in such cases is the total (cid:16) E(cid:17) (cid:16) s(cid:17)
d
angularmomentumpassingthroughRwarp duringthealign- The disc semi–thickness H obeys
muleantitopnrsoocefstsh.eThaliisginnmteernptreptaroticoenssaginreesuscwhitchanseusm(eLroicdaaltsoim&- H =1.94×10−3 α1 −1/10 ǫ −1/5
R 0.03 0.1
(cid:16) (cid:17) (cid:16) (cid:17)
Pringle, 2006). Of course if the whole disc is warped J is
simply thetotal angular momentum of thedisc as beforde. × L 1/5M−1/10 R 1/20. (8)
0.1L 8 R
Equation(2)showsthatarandomsequenceofepisodes (cid:16) E(cid:17) (cid:16) s(cid:17)
with J ≪J must havef ≃1/2, so that co– and counter– Thediscbecomesself–gravitatingwhenitsmassreaches
d h
alignment are almost equally common. Since the ISCO for H
retrograde accretion carries higher specific angular momen- Md =Msg ≃ RM, (9)
tum than for prograde accretion if the Kerr a parameter is
significant (in theratio 11:3 for maximal spins) we see that (e.g. Pringle, 1981) which occurs at radii R≥Rsg, where
spindown is more effective than spinup, ultimately produc- Rsg =1.13×103 α1 14/27 ǫ 8/27
ingaslowly–spinninghole(Hughes&Blandford,2003reach R 0.03 0.1
s (cid:16) (cid:17) (cid:16) (cid:17)
aofssimpiinlanrincgonhcolluessi,ofnorfotrhethseameffeecrteaosfonre;pseeaeteadlsocoCaloelsbceerntce&s ×. L −8/27M−26/27 (10)
0.1L 8
Wilson, 1995). King & Pringle (2006) used this argument (cid:16) E(cid:17)
to show how suitably random accretion episodes could al- In line with our discussion in the Introduction we iden-
lowthegrowth ofverylarge SMBH(∼5×109M⊙) athigh tify Msg as the mass ∆Mepisode in an individual accretion
redshift z∼6 from even stellar–mass seed black holes. episode. The evolution timescale of the disc, and thus the
In this paper we show that the fact that self–gravity characteristic duration of an episode, is given by τsg =
cutsoff theaccretion discsinAGNfeedingepisodesimplies Msg/M˙, i.e.
oJfdt<hi2sJthyipnealwmitohstraalnldcaosmes.oAriesnetqauteionncesotfhreerpeefoarteedspepinissodthees τsg =1.12×106 α −2/27 ǫ 22/27
0.03 0.1
(cid:16) (cid:17) (cid:16) (cid:17)
4 A.R. King, J.E. Pringle, J.A. Hofmann
×. L −22/27M−4/27yr (11) 1 10
0.1L 8
(cid:16) E(cid:17)
Inlinewith ourdiscussion aboveweexpectamajor merger
eventtoconsistofachaoticsequenceofsuchepisodes.These 0.8 8
may overlap, or be slightly separated in time. Our conclu-
sionsareindependentofthethetimeprofileoftheaccretion,
0.6 6
anddependonly on thetotal mass andangular momentum
aa log MBH
accreted. We differ from other treatments such as those of M
Volonteri & Rees (2005) and Volonteri et al. (2007) in not 0.4 4
following their assumption that all the mass in a given ma-
jor merger accretes at the same angle, even at the scale of
0.2 2
theinner accretion disc.
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6
4 THE EVOLUTION OF BLACK HOLE MASS accretion time t [Gyr]
AND SPIN
Figure 1. Numerical simulation of black hole mass M (circles)
Wecannowseehowasequenceofaccretionepisodeslimited andspinparametera(solidcurve)evolutionversusaccretiontime
bydiscself–gravity willaffecttheblackholemassandspin. (i.e. neglecting epochs when no mass accretes). The hole is as-
We compare the angular momentum J of the warped disc sumedtoaccretealwaysatitscurrentEddingtonrate,througha
d
sequenceofdiscaccretionepisodeswithdiscangularmomentum
withtheangularmomentumJ ofthehole.Theinterpreta-
tion of J as the total angularhmomentum passing through Jd randomly oriented with respect to that of the hole, Jh. (In
Rwarp shodws that irsealliimtyittehdesbeyeepqisnod(1e2s)c,aonrbbeyseeqpnar(a1t4ed) iifntthime ew.a)rTphreadvaiuluseRowfaJrpd
Jd<∼Msg(GMRwarp)1/2 (12) exceeds theself–gravityradiusRsg.Thedashedcurveshowsthe
expected means spin parameter a¯ (see eqns 18, 19), and bends
(i.e. the total accreted disc mass ∆Mepisode = Msg mul- sharplyatlowerright,reflectingthechangefromeqn(12)toeqn
tiplied by the disc specific angular momentum at Rwarp ), (14).InthissimulationtheholehasinitialmassM0=10M⊙and
providedthatRwarp islessthantheouterdiscradiusRsg.In spin parameter a0 = 0.2, but a very rapidly converges towards
the mean value a¯ (cf eqn 19) whatever its initial value. We take
thislattercase wereplaceRwarp withRsg.Thewarp radius
is given by(cf KLOP) viscosity parameters α1 = α2 = 0.03. Note that the hole mass
grows more slowly at epochs when a is large, and more rapidly
Rwarp =990 α1 1/8 α2 −5/8a5/8 whenitissmall.
R 0.03 0.03
s (cid:16) (cid:17) (cid:16) (cid:17)
ǫ 1/4 L −1/4
× M1/8. (13) 1 10
0.1 0.1L 8
(cid:16) (cid:17) (cid:16) E(cid:17)
HereR =2GM/c2istheSchwarzschildradiusandathedi-
s
0.8 8
mensionlessKerrspinparameter,withα2 thedimensionless
viscositycoefficientrelevanttoverticaldiscmotions,andwe
notethatLodato&Pringle(2007)haverecentlyarguedthat 0.6 6
α2 neverexceeds a value ∼1 in a warped disc. a log MBH
If thecondition Rwarp <Rsg fails we instead take M
0.4 4
Jd =Msg(GMRsg)1/2, (14)
whichimpliesthatJ canneverbelargerthangivenby(12).
d 0.2 2
Given thehole angular momentum
J =2−1/2Ma(GMR )1/2, (15)
h s 0 0
0 0.1 0.2 0.3 0.4 0.5 0.6
we find from (12) accretion time t [Gyr]
Jd ≤0.061a−11/16 α1 −5/432 α2 −5/16 Figure 2. As for Figure 1, but with α2 = 1 The same random
2J 0.03 0.03 sequenceofaccretionepisodeorientationsisusedinordertohigh-
h (cid:16) (cid:17) (cid:16) (cid:17)
ǫ −13/216 L 13/216 lighttheeffectofchangingα2.
× M8−37/432 . (16)
0.1 0.1L
(cid:16) (cid:17) (cid:16) E(cid:17)
thegeneraleffectofasequenceofrandomly–oriented accre-
WeseethattheratioJ /2J determiningthetendency tionepisodesasenvisagedintheIntroductionistodecrease
d h
ofthedisctoco–orcounter–alignunderLTtorquesisalways a.Thusonemightimaginethattheendresultisavalueofa
<1unlessthespinparameteraissmallitself,effectivelyin- oscillatingaroundzero,andbothKing&Pringle(2006)and
dependentlyofallotherparametersexceptforaweakdepen- Volonterietal (2007) assume this.Howeveras J decreases
h
denceonα2.WithJd/2Jh <1retrogradeaccretionepisodes for a given Jd, the probability p(spindown) of accreting in
stablycounter–alignwiththeholeangularmomentum,spin- a stably retrograde fashion (given by f in (2)) approaches
ning it down. Since spindown is more efficient than spinup, zero. Evidently J must on average decrease only to the
h
The Evolution of Black Hole Mass and Spin in AGN 5
point where the expected spinup in prograde accretion is
equaltotheexpectedspindown in retrograde accretion, i.e. 0.3 0.3
MBH = 1e6 MBH = 1e7
p(spindown) da =p(spinup) da (17) 0.2 0.2
dMdown dMup
0.1 0.1
or equivalently
p(spindown)= 1 1− Jd = ddMadown . (18) 00 0.2 0.4 a 0.6 0.8 1.0 00 0.2 0.4 a 0.6 0.8 1.0
2(cid:18) 2Jh(cid:19) ddMa down+ ddMaup
0.3 0.3
Onehas tousetheequations ofBardeen (1970) toevaluate MBH = 1e8 MBH = 1e9
thederivativesinthisequation,whichthendefinesthemean 0.2 0.2
valuea¯.
0.1 0.1
0 0
0 0.2 0.4 a 0.6 0.8 1.0 0 0.2 0.4 a 0.6 0.8 1.0
4.1 Numerical Simulations
Figure3.Distributionofaaroundthemeana¯(arrow)atmasses
Wesimulatetheeffect of repeated accretion episodes of the M/M⊙=106,107,108,109,forthecaseα2=0.03.
typedicussed here as follows. Weassume that each episode
hasmass Msg and accretes at theEddington rate appropri-
ate to the current black hole mass M. The total angular
0.3 0.3
momentum Jd of each episode is given by the recipe (12), MBH = 1e6 MBH = 1e7
replacing Rwarp by Rsg if Rwarp >Rsg. The direction of Jd 0.2 0.2
ischosenatrandomfromanisotropicdistribution.Thedisc
and hole are assumed to co– or counter–align according to
0.1 0.1
thecriterion(1).Incontrast toVolonterietal(2007) wedo
not assume that all the mass accreting in a major merger
0 0
has the same orientation of angular momentum, but only 0 0.2 0.4 a 0.6 0.8 1.0 0 0.2 0.4 a 0.6 0.8 1.0
the mass Msg contained within each self–gravitating disc
episode, allowing each episode to berandomly oriented. 0.3 0.3
MBH = 1e8 MBH = 1e9
Figures 1, 2 show the results of two such simulations.
Thefigures omit theepochswhen thehole is not accreting, 0.2 0.2
so the horizontal axes measure the accretion time rather
thanthetotalelapsedtime.Sincetheaccretionisalwaysat 0.1 0.1
thecurrentEddington rate, theaccretion timeis theshort-
est possible time for the hole to acquire its mass through 00 0.2 0.4 a 0.6 0.8 1.0 00 0.2 0.4 a 0.6 0.8 1.0
accretion.
Thetwosimulationsdifferonlyinthevalueofthe‘ver- Figure 4.AsFigure3,butforα2=1.0.
tical’viscositycoefficientα2,whichis0.03inFig.1and1in
Fig 2. The two simulations use the same random sequence 5 SMBH COALESCENCES
of accretion episode orientations in order to highlight the
effect of changing α2. The main difference is that themean We have so far discussed the evolution of SMBH mass and
valuea¯ is somewhat lower in thesecond case (see below). spin via accretion. Figures 1, 2 show that the Kerr param-
As one can see, the main features inferred above do eter a always has a strong tendency to move towards the
appear. Although thevalue of a fluctuateswidely, its mean mean value a¯ from any initial value as the hole mass dou-
a¯ does indeed quickly settle to the value predicted by (18) bles through accretion. Thus although a coalescence of two
in each case. We can fit this as SMBH with comparable masses would presumably cause
a discontinuous departure from the mean accretion–driven
a¯≃AM8−0.048, Rwarp <Rsg trend of Figs. 1, 2, accretion would drive it back towards
≃0.3M8−0.3, Rwarp >Rsg (19) themeantrendasitdoublesthemassofthecoalescedhole.
In other words coalescences have little long–term effect on
where A= 0.246 (for α2 = 0.03), 0.13 (for α2 =1) is fixed a¯,although theycanstrongly affectthespinforarelatively
bycontinuityatthechangeoverbetweenthetwopower–law short time (see theend of the Discussion).
regimesineachcase.InFig.1,thisoccursatM8 ≃2,which
in the case with higher vertical viscosity the changeover is
only at a mass of almost 1010M⊙. We see that for holes 6 SMBH RECOIL VELOCITIES
of masses ∼ 106 −109M⊙ we expect a¯ ∼ 0.3−0.2 in the
first case and a¯ ∼ 0.2−0.1 in the second, with excursions The relatively modest spin rates we find mean that SMBH
∆a ∼ ±0.2 about these means. Figures 4.1, 4.1 show the coalescences even with similar masses (i.e. in major merg-
normalized distributions of a as a function of mass in the ers) are likely to produce fairly small gravitational radia-
case α2 =1. tion recoil velocities <∼200 kms−1 (Gonz´alez et al., 2006).
6 A.R. King, J.E. Pringle, J.A. Hofmann
These are well below the escape velocity from the merged III] emission in a sample of nearby Seyfert galaxies is well
host.Thisagreeswiththeobservationalevidencethatmost aligned with the radio, and it is usual to assume that the
massivegalaxiesdohavenuclearSMBH.WenotethatBog- orientation of the[O III]emission isgoverned bythegeom-
danovi´c et al. (2007) reach similar conclusions, but for en- etry of the inner torus, of typical radius 0.1 – 1.0 pc (e.g.
tirely orthogonal reasons. Specifically, they argue that the Antonucci, 1993). Similarly, Verdoes–Kleijn and de Zeeuw
effect of gas accretion is to cause any SMBH to align with (2005) andSparksetal(2000) findevidencethatdustdiscs
the angular momentum of the accretion flow. They further tend to be perpendicular to radio and optical jets respec-
argue that this flow is of such large scale and well–defined tively.
angularmomentumthatinanyincipientSMBHcoalescence As the torques between the disc and the hole are all
both holes have their spins aligned with the flow and thus internal, the outcome of an accretion event is that the hole
with each other. It is known that co–aligned spins reduce spin vector aligns along the total a.m. vector J =J +J
t h d
the recoil velocity to modest values <∼200 kms−1, similar (cf KLOP).We can therefore track thedirection of Jh over
to those for non–spinning holes as we found above. time.Fromeqn(16)weseethatJ ≪J fora∼1,andeven
d h
Ourargumentsdifferfromtheseinthreeways.First,we for a near the mean value (19) we have Jd/Jh<∼0.26,0.16
have argued above that self–gravity severely limits the an- forthetwocasesα2 =0.03,1.Bysimplevectoradditionwe
gular momentumof anyaccretion flowso that J <2J ,so havethatinsuchanaccretionepisodethevectorJ deviates
d h h
thatco–andcounter–alignment occuralmost equallyoften, from its original direction by an angle ψ given by
ratherthantheholeanddiscalwaysco–aligningasasserted
J
by Bogdanovi´c et al. Second, we have also argued that the sinψ= d sinθ<∼0.26,16 (20)
J
accretion flow on to an SMBH cannot have as large a scale h
as claimed by Bogdanovi´c et al., who suggest a linear size for any original misalignment angle θ, so that ψ<∼15◦,9◦
of 105Rs ∼ 1 pc, because the viscous timescale from such for α2 = 0.03,1. Thus successive accretion episodes have a
radiiexceedsaHubbletime.FinallyBogdanovi´cetal.’spic- tendency to produce jets in similar directions, even more
turepredictsthattheSMBHspinandtheaccretionfloware soifahappenstobelarger thanaverage.Thismayexplain
both aligned with the large–scale structure of the galaxy. whysome‘double–double’radiosourcesareseenwithappar-
One would thus expect any jet structure to show a similar entlynear–common projectedaxesforsuccessivejet events.
alignment othogonal to the galaxy disc apparently feeding Kharb et al. (2006, Figs. 1, 5) give a beautiful example of
it. However this disagrees with the observational evidence this.Ultimately theeffectof evenafewepisodes isthat the
discussed by Kinney et al (2000; see also Nagar & Wilson spin axis loses all memory of its original direction. There
1999): in a sample of nearby (z ≤ 0.031) Seyfert galaxies is clearly no correlation at all with structures in the host
the direction of the jet from the central black hole is unre- galaxy. Coalescences havesimilar disorienting effects.
lated to the orientation of the disc of the host galaxy. As
mentioned above Sajina et al. (2007) haverecently inferred
a similar result at higher redshift. 9 DISCUSSION
We have argued that accretion on to supermassive black
holes in AGN occurs through events with total mass
7 MAXIMUM MASS GROWTH RATE FOR
SMBH ∆Mmerger ∼ M, consisting of long sequences of randomly
oriented disc accretion episodes with masses ∆M ∼
episode
Inanearlierpaper(King&Pringle2006)weconsideredthe 10−3M. The mass of these disc episodes is strongly con-
problemofgrowingthelargest SMBHmasses∼5×109M⊙ strained by the generic tendency of such discs to become
at cosmological redshift z ≃ 6, i.e. only ∼ 109 yr after the self–gravitating and form stars outside quite small radii
Big Bang. We concluded that growth to these masses from ∼ 0.01−0.1 pc. The self–gravity constraint also limits the
evenstellarinitialvalueswaspossibleprovidedthatthespin disc angular momentum. This is always less than the max-
rate,andthustheradiationefficiencyofmassaccretion,was imum that the hole’s mass would allow it to have. A hole
kept sufficiently low (a<∼0.5). From the discussion of the withsignificantspinthereforestablyco–andcounter–aligns
last section we see that this condition is always satisfied by initially progradeandretrogradediscsrespectively,produc-
accretion from randomly–oriented accretion episodes. The ing a net spindown until its angular momentum is reduced
main limit on black hole mass growth is therefore themass to a relatively modest value, with fluctuations ∆a ∼ ±0.2
supplyratherthantheEddingtonlimit.Figure1showsthis about the average a¯ given by (19). We note from (19) that
explicitly, giving M as a function of the accretion time. a¯ itself decreases slowly as theblack hole mass decreases.
We can compare these results with those for coales-
cences of SMBH as a consequence of the mergers of their
host galaxies. Hughes and Blandford (2003) show that the
8 SMBH SPIN DIRECTIONS
average secular evolution of black hole spin under coales-
We have so far considered the magnitude J of the black– cences is
h
holeangularmomentumvectorJ ratherthanitsdirection. 2.4
The latter may be important, ashthe central part of theac- a=a0 M0 (21)
(cid:18)M (cid:19)
cretion disc always has its axis parallel or antiparallel to
it, and this is probably the direction of any jet produced wherea0 istheoriginal spinparameter ofaholewithin ini-
by the system. Observations of various types support this tialmassM0.Thuscoalescencesdoublingthemassofahole
conclusion. Schmitt et al (2003) show that the extended[O onaveragereduceabyafactor∼5.3.Thisisanevenfaster
The Evolution of Black Hole Mass and Spin in AGN 7
0.45 rotating and rotating black holes, and argue that the outer
radius of the emission region for the non–rotating hole is
0.4
unreasonably small in the case of the Type I AGN in their
0.35 sample, although the statistical fit is marginally preferred
over a rotating hole. The best–fitting rotating hole for this
0.3 group has an inner emission radius of 3GM/c2, which re-
quires a > 0.78. Since holes with such values of a would
0.25
ε automatically have higher efficiencies and thus presumably
0.2 stronger iron lines, it is difficult to know what fraction of
theSMBHinthesampleactuallyhavesuchfairlylargespin
0.15
parameters (note that, as discussed below, our arguments
0.1 only refer to the statistical mean a¯, and individual SMBH
can differsignificantly from this).
0.05 We remark as a general caveat that the search for any
effect depending on a, or the interpretation of a particular
0
-1 -0.5 0 0.5 1 observational effect as being due to a, is likely to produce
a
a bias towards a ∼ 1. This effect is particularly marked
becauseoftheveryhigh efficiencyofprograde accretion for
Figure 5. Accretion efficiency ǫ versus spin parameter a (solid
curve). Negative values of a denote retrograde accretion. The a ≃ 1 (see Fig 5). We note that Brenneman & Reynolds
dashed curve shows the effective ǫ=ǫ± given by assuming that (2006) estimate a = 0.989+−00..000092 in the specific case of the
retrograde and prograde accretion are equally probable. This is broad iron line in MCG-06-30-15, remarkably close to the
theefficiencyrelevanttotheSoltanargument. maximum allowed value.
Of course since our treatment is statistical it does not
ruleoutlargerspinvaluesinindividualcases.Themostim-
decline of spin than given by assuming fixed angular mo-
portantwayofproducinglargevaluesofainindividualcases
mentumbutgrowingmass,whichwould producea∝M−2,
isincoalescencesoftwoSMBH.Hughes&Blandford(2003)
reflecting the fact that retrograde coalescences are more ef-
showthatthiscanproducesignificantaforprogradecoales-
fective in reducing spin than prograde ones are in growing
cences of two black holes with similar mass. This suggests
it.Evidentlycoalescencescanreduceaquitequicklytozero.
that giant ellipticals offer a promising site for significant
However we have seen that accretion very rapidly pushes a
values of a. Another conceivable way of producing excep-
back towards the mean value a¯ (cf eqns 18, 19) from any
tions to our general conclusions is that we have assumed
startingpoint.Henceonaveragecoalescenceshavelittlenet
accretion from thin discs, in which cooling is efficient, so
effectonthemeanvalueofa.Weconcludethatrepeatedac-
that the total disc mass is limited to ∼ H/R times that of
cretionepisodesandcoalescencesonaveragedrivetheblack
the hole. In some situations it may happen that cooling is
holesinAGNtowardsfairly lowvaluesofa,withstatistical
inefficient and our conclusions do not hold. More work is
fluctuationsaround them of order ∆a=±0.2.
needed here, although such cases are likely to be rare. Fi-
Given this result, we should consider the observational
nallyonemightarguethatinsomecasesanaccretionevent
evidenceconstrainingvaluesofaforSMBH.Muchofthisis
derivedfromtheSoltan(1982)argumentthatthetotalityof with mass ≫ Msg may contrive to retain some memory of
anoverallangularmomentum,despitethefactthatit must
backgroundlight inthelow–redshift Universeiscompatible
have a near–radial orbit within the host galaxy in order to
withblack–holegrowthwithradiativeefficiencyǫ≃0.1(e.g.
accrete at all (cf. Kendall et al., 2003: recall that accretion
Wang et al., 2006; Treister & Urry,2006; Hopkins, Richard
&Hernquist,2007),whichformally requiresa≃0.67. How- withinaHubbletimerequiresdiscradii <∼1pc).Inthiscase
individualepisodesnolongerhaveuncorrelatedangularmo-
ever ǫ is such a slow function of a except near a = 1 (see
mentaasweassumed.Thisiseffectivelywhatisassumedby
Figure 5) that this conclusion is vulnerable to the system-
Volonteri & Rees (2005), and Volonteri et al (2007) for all
aticeffectsinherentintryingtoestimatetheefficiencyfrom
episodes.
observation. Evidently we have to find ǫ ≃ 0.1 from obser-
vation to an accuracy of a factor <∼2 in order to exclude
anyvalueofaotherthanthoseverynearunity.Sinceretro-
10 ACKNOWLEDGMENTS
grade and prograde accretion appear equally probable, the
effective efficiency for the Soltan argument is actually the ARK acknowledges a Royal Society Wolfson Research
symmetrized curve Merit Award. JH acknowledges a scholarship from the
1 Deutscher Akademischer Austausch Dienst (DAAD) and
ǫ± = 2[ǫ(a)+ǫ(−a)] (22) partial support from the Deutsche Forschungsgesellschaft
(DFG) through grant SFB 439.
in Figure (5), whose vertical range is now compressed to
0.057−0.230. In this case only a very accurate estimate of
ǫ can give real information about a at all.
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