Grand unification of AGN activity in the CDM cosmology

22 Pages·2010·4.12 MB·English

Mon.Not.R.Astron.Soc.410,53–74(2011) doi:10.1111/j.1365-2966.2010.17427.x
(cid:2)
Grand unification of AGN activity in the CDM cosmology
(cid:2)
N. Fanidakis,1 C. M. Baugh,1 A. J. Benson,2 R. G. Bower,1 S. Cole,1 C. Done1
and C. S. Frenk1
1InstituteforComputationalCosmology,DepartmentofPhysics,UniversityofDurham,ScienceLaboratories,SouthRoad,DurhamDH13LE
2MailCode130-33,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
Accepted2010July26.Received2010July18;inoriginalform2010January13
D
o
w
n
lo
ABSTRACT a
d
e
Wetrackthecoevolutionofsupermassiveblackholes(SMBHs)andtheirhostgalaxiesthrough d
cosmictime.ThecalculationisembeddedintheGALFORMsemi-analyticmodelwhichsimulates from
theformationandevolutionofgalaxiesinacolddarkmatter(CDM)universe.Theblackhole h
(BH)andgalaxyformationmodelsarecoupled:duringtheevolutionofthehostgalaxy,hot ttps
andcoldgasareaddedtotheSMBHbyflowstriggeredbyhalogascooling,discinstabilities ://a
c
a
andgalaxymergers.ThisbuildsupthemassandspinoftheBH,andtheresultingaccretion d
e
m
power regulates gas cooling and subsequent star formation. The accretion flow is assumed
ic
to form a geometrically thin cool disc when the accretion rate exceeds 0.01M˙ , and a .o
Edd u
p
geometrically thick, radiatively inefficient hot flow when the accretion rate falls below this .c
o
value. The resulting quasar optical luminosity function matches observations well, and the m
/m
massoftheSMBHcorrelateswiththemassofthegalaxybulgeasintheobservedMbh–Mbulge nra
relation.TheBHspindistributiondependsstronglyonwhetherweassumethatthegasinany s
/a
given accretion episode remains in the same plane or it fragments into multiple, randomly rtic
aligned accretion episodes due to its self-gravity. We refer to these cases as the ‘prolonged’ le-a
and ‘chaotic’ accretion modes, respectively. In the chaotic accretion model there is a clear bs
correlationofspinwithSMBHmass(andhencehostgalaxybulgemass).MassiveBHs(M> trac
5 × 108M(cid:2)) are hosted by giant elliptical galaxies and are rapidly spinning, while lower t/4
1
0
mass BHs are hosted in spiral galaxies and have much lower spin. Using the Blandford– /1
/5
Znajek mechanism for jet production to calculate the jet power, our model reproduces the 3
/1
radio loudness of radio galaxies, low ionization emission regions (LINERS) and Seyferts, 0
3
1
suggestingthatthejetpropertiesofactivegalaxynuclei(AGN)areanaturalconsequenceof 5
0
3
boththeaccretionrateontoand thespinofthecentralSMBH.Thisisthefirstconfirmation b
y
that a CDM galaxy formation model can reproduce the observed radio phenomenology of g
u
AGN. es
t o
Keywords: methods:numerical–galaxies:active–galaxies:jets–galaxies:nuclei–quasars: n 1
0
general. A
p
ril 2
0
1
9
haveabimodaldistributionontheradio-opticalplane,withradio-
1 INTRODUCTION
loudobjectsbeingabout103timesbrighterinradiothanradio-quiet
Activegalaxiescanbeclassifiedaccordingtotheimportanceofthe objects(seealsoKellermannetal.1989;Xu,Livio&Baum1999).
radioemissionfromtheirnucleus.Objectsshowingstrongemission The origin of this dichotomy remains unknown. However, it has
atradiowavelengthsbelongtothe‘radio-loud’class,whereasthose beenproposedinmanystudiesthatafirststeptowardsexplaining
withnegligibleemissionbelongtothe‘radio-quiet’class.Radio- why some AGN launch prominent jets while others do not is to
loud active galactic nuclei (AGN) are associated with large-scale understandthenatureofthecentralengine,theaccretingblackhole
radio-emittingjetswhileradio-quietAGNshowverylittleornegli- (BH).
giblejetactivity.Sikora,Stawarz&Lasota(2007)foundthatAGN AGNjetsarebelievedtoextractrotationalenergyfromtheBH
andaccretiondiscthroughmagneticfields.Analyticalstudieshave
contributedtoanunderstandingofthenatureofthejets(Blandford
(cid:2)
E-mail:[email protected] &Znajek1977;MacDonald&Thorne1982;Blandford&Payne
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS
54 N.Fanidakisetal.
1982;Begelman,Blandford&Rees1984),butabreakthroughhas Someoftheseideashaverecentlybeenexploredusingsimplified
comefrommagnetohydrodynamical(MHD)simulationsoftheac- modelsofgalaxyformationbyBerti&Volonteri(2008)andLagos,
cretion flow. These self-consistently calculate the turbulent mag- Padilla & Cora (2009). In this paper, we calculate the growth of
neticfielddynamowhichisthephysicaloriginofthestresseswhich BHsbyaccretionandmergers,theiracquisitionofspinandtheir
transportangularmomentumoutwards,allowingmaterialtoaccrete accretion rates within a specific theory of galaxy formation in a
ontotheBH(Balbus&Hawley1998).Embeddingsuchcalcula- colddarkmatter(CDM)modelwhichhasbeenextensivelytested
tionsinafullygeneralrelativisticframeworkproducesrelativistic againstalargerangeofobservationsofthegalaxypopulation.Not
jetsfromtheaccretionflow,andthecollimationandjetpowerde- onlyisourcalculationofthejointevolutionofBHsandgalaxies
pendontheBHspin(McKinney&Gammie2004;DeVilliersetal. self-consistent, but we also incorporate several accretion and jet-
2005;Hawley&Krolik2006). launchingmodelsfromtheliteraturewhichallowsustoperforma
Thus,itnowseemsclearthatthejetpowerdependsonthevertical quantitativecomparisonwithobservationsofAGN.
(poloidal)magneticfieldstrengthclosetotheBH,butthesimula- Thepaperisorganizedasfollows.InSection2,weexplainhow
tionsarestillnotyetatthelevelwheretheycanpredictthisabinitio wecalculatethegrowthofthemassandspinofsupermassiveblack
D
(e.g.Beckwith,Hawley&Krolik2008),soanalyticmodelsarestill holes(SMBHs)throughtheaccretionofhotgasfromthehalo(radio o
w
required. The most popular of these, the Blandford–Znajek (BZ) mode) and cold gas from the galactic disc and mergers (quasar n
lo
mechanism,usesthemagneticfieldasameanstotapthespinofthe mode).MostBHsgrowprimarilybygasaccretionexceptthemost a
d
BH.Thisgivesastrongspindependenceonthejetpower.Indeed, massiveoneswhichformlateandincreasetheirmasssubstantially ed
many authors have proposed spin as the physical parameter that bymergingwithotherSMBHs.ABHisspunuptoapproximately fro
m
determines the radio loudness of an AGN (see Wilson & Colbert the maximal value when it accretes its own mass from a flow at h
1995;Hughes&Blandford2003).Thespinparadigm,asthisideais constantangularmomentum. ttp
s
called,offersaplausibletheoreticalexplanationforthewiderange InSection3,wepresenttheresultingdistributionsofBHmass, ://a
ofjetluminositiesinAGNandcouldbethebasisforunderstanding gas accretion rate and spin. We consider two possible modes of c
a
theobserveddichotomybetweenradio-loudandradio-quietAGN. accretion.Inthe‘prolongedaccretion’case,thegasisaccretedina de
m
Alternatively,theradio-loud,radio-quietswitchmayjustbede- singleepisode.Evenminormergerstriggergasflowsontothenu- ic
termined by the physical state of the accretion flow. Stellar-mass cleusthatoftendepositamassgreaterthanthemassoftherecipient .o
u
BH binary systems in our galaxy show a clear spectral transition BH.Thus,mostBHsaretypicallyspunuptothemaximalvalue. p.c
at about 1 per cent of the Eddington luminosity, L ∼ 0.01LEdd, Thismodelpredictsradiopropertiesthatdonotreproducetheob- om
fromahot,opticallythinaccretionflowatlowluminosities(e.g.an servedluminosityfunction.Wethenconsidera‘chaoticaccretion’ /m
n
advection-dominatedaccretionfloworADAF;Narayan&Yi1994) caseinwhichtheaccretionepisodesarelimitedbytheself-gravity ra
s
toacool,geometricallythindisc(TD;Shakura&Sunyaev1973)at ofthedisc.Gasflowsontothenucleusgiverisetoaseriesofac- /a
higherluminosities(Esin,McClintock&Narayan1997;seee.g.the cretionepisodeseachtypicallyaugmentingtheBHmassbyonlya rtic
review by Done, Gierlin´ski & Kubota 2007). The collapse of the smallfactor.Successiveaccretioneventsareuncorrelated,resulting le-a
radiojetobservedacrossthistransitionclearlyrelatestothelarge inlowspinsfortherelativelylowmassBHswhichgrowprimarily bs
drop in pressure (and hence scaleheight, H) between the hot and byaccretion.Ontheotherhand,themostmassiveholeswhichbuild tra
c
cool flow (e.g. Fender, Belloni & Gallo 2004). Even without BH upsubstantialmassthroughBH–BHmergersarespunuptohigh t/4
1
spin,thisproducesacleardichotomyofradioproperties(e.g.Jester (butnotmaximal)spinvalues.InSection4,weshowthatadopt- 0
/1
2005). ingthisaccretionmodel,theopticalluminosityfromdisc-accreting /5
3
Itseemsmostplausiblethatboththesemechanisms,alongwith objectsmatchesthequasarluminosityfunctionreasonablywell. /1
0
the mass accretion rate, affect jet power. The BZ mechanism has In Section 5, we incorporate an explicit BZ model for the jet 3
1
a‘hidden’dependenceonmassaccretionrateandthescaleheight power (Meier 2002) and use this to predict the radio luminosity 50
3
of the flow because the magnetic field strength close to the BH functionwhich,inthechaoticaccretioncase,agreeswellwithob- b
y
saturates, due to the dynamo, into rough equipartition with the servations.Onthebasisofourderivedopticalandradioluminosity g
u
pressure in the flow. This depends on the mass accretion rate for distributions,wepropose,inSection6,a‘grandunificationofAGN e
s
either the disc or the hot flow, but the pressure in the hot flow is activity’,inwhichtheaccretionflowandjetluminosityarerelated t o
n
muchlargerthanthatinthecooldiscsothefieldstrengthandhence toAGNtypethroughtheirmass,spinandmassaccretionrate.Fun- 1
0
jetpowerabruptlydropatthistransition(Meier2001,2002). damentally,ourcalculationshows,forthefirsttime,howthecoeval A
radSioikograalaextieasl.a(2n0d0c7o)nucsleuddethdetsheaitdtehaestdoatiantceorpurledtboebseexrvpalatiionnesdoiff gprroowpetrhtioesftBhaHtsexapnldaitnhethirehAoGstNgaaclatixviietsysreeseunltininthoeploticcaallUanndiverarsdeio. pril 2
0
thejetdependsonmassaccretionrate,accretionmodelandspin. 19
However,theyalsorequiredamass-spincorrelation,suchthatthe
2 EVOLUTION OF SPIN IN HIERARCHICAL
mostmassiveBHs,which,accordingtotheM –M relationship
bh bulge GALAXY FORMATION MODELS
(Magorrianetal.1998;McLure&Dunlop2002;Marconi&Hunt
2003; Ha¨ring & Rix 2004), reside in massive elliptical galaxies, Inthissectionwedescribethecosmologicalprocessesthatinitiate
havehigherspinthanlowermassBHs,whichresideinspirals.They and regulate the growth of SMBHs and outline the modelling of
speculatedthatthiscouldoccurthroughthehierarchicalgrowthof thespinevolutionduetogasaccretionandmergersinhierarchical
structure,wherethemajormergerswhichgiverisetogiantelliptical cosmologies. Our basic modelling tool for our predictions is the
galaxiestriggerlargeamountsofgasaccretionwithagivenangular GALFORMcode(Coleetal.2000),andweuseanupdateoftheBower
momentumdirectionontotheBH,spinningituptothemaximal et al. (2006) version for modelling the formation and evolution
value. By contrast, spiral galaxies have not experienced a recent ofgalaxiesinthe(cid:3)CDMcosmology.Thechangesrelativetothe
majormerger,andgrowmainlythroughsmoothaccretionandmul- Boweretal.modelarethefollowing.First,thefraction,(cid:4) ,of
SMBH
tipleminormergerswithrandomangularmomentum,resultingin theEddingtonluminosityofanaccretingSMBHthatisavailable
lowspinandhenceweakjetluminosities. forheatingthehaloduringanepisodeofAGNfeedbackissetto
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74
Grandunification ofAGNactivity 55
0.01(Boweretal.2006use(cid:4) =0.041).Secondly,instarbursts the SMBHs sink to the centre by dynamical friction from distant
SMBH
triggeredbygalaxymergersordiscinstabilities,weassumethatthe stars or by viscous effects from the surrounding gas. The transi-
fraction,F ,ofthecoldgasturnedintostarsthatisaccretedon tion to a bound binary state after the galaxy merger is an open
SMBH
totheBHis0.01(Boweretal.2006useF =0.017).These issue(Milosavljevic´ &Merritt2001).However,itisbelievedthat
SMBH
changesareintroducedtoimprovethemodellinginthispaperand oncetheseparationofthetwoBHsbecomessmallenough,grav-
donotaffectthefundamentalpredictionsoftheBoweretal.model. itationalradiationcarriesawaytheremainingangularmomentum
BHsareassumedtoevolveinmassinaccordancewiththemodel of the binary. The removal of energy from a SMBH binary due
developed in Malbon et al. (2007) and Bower et al. (2006). We togravitationalwaveemissionleadstoagradualshrinkageofthe
studythecosmologicalspinevolutionofSMBHseedsofatotalof relativeseparationofthetwomembers.Thereisapointwherethe
∼4.2×106galaxiesidentifiedintheMillenniumN-bodysimulation eccentricityreacheszeroandtheorbitcircularizes.Atthattime,the
(comovingvolumeof1.25×108h−3 Mpc3)fromredshift127to twoBHsareveryclosetoeachother,andgravitationalwavesare
redshiftzero(Springeletal.2005).HerehisdefinedbyH =h× emittedcopiously.TheradiatedenergyissolargethattwoSMBHs,
0
100km s−1 Mpc−1, where H is the Hubble constant at redshift which are a few au apart, lose all their potential energy within a
zero.Thecosmologyadopted0inthesimulationsish=0.72,(cid:5)m= coupleofminutesandinevitablycoalesce.Afterthecoalescenceis Dow
0.25,(cid:5)b=0.045,(cid:5)(cid:3)=0.75andσ8=0.9.2 complete,thebinaryenterstheringdownphase,wherethemerged nlo
memberssettleintoaquiescentremnanthole. a
d
A third channel for SMBH growth in our model is provided ed
2.1 ThegrowthofSMBHs bydiffusegasindarkmatterhaloesundergoingquasi-hydrostatic fro
m
The evolution of BH mass fits naturally in the CDM model of cooling.Whenamassivehalocollapsesgasisshockheatedouttoa h
galaxy formation (Kauffmann & Haehnelt 2000; Malbon et al. radiuscomparabletothevirialradiusofthedarkmatterhalo.These ttp
s
2007),wherestructuresgrowhierarchically.Smallstructuresform haloes have a cooling time that is longer than the free-fall time ://a
first,thenevolvethroughmergersintolargeones.Intheeventof ofthegasand,thus,thegassettlesintoaquasi-staticatmosphere c
a
a galaxy merger, the less massive galaxy (satellite) sinks into the surroundingthegalaxyratherthansimplyfallingtowardsthecentre de
m
gravitationalpotentialofthemassivecentralgalaxyasaresultof (White&Frenk1991).Thisatmosphere–the‘hothalo’regime–is ic
dynamical friction (Binney & Tremaine 1987). In our model we inpressure-supportedhydrostaticequilibriumandextendsbeyond .o
u
assumethatifthemassofthesatellitegalaxyiscomparabletothat thevirialradiusofthedarkmatterhalo.Thegalaxyisthensupplied p.c
ofthecentralgalaxy,themergerdisruptsthegalaxiesandresultin withcoldgasbycoolingflowsatthedisccentre,whichalsofeed om
theformationofanellipticalgalaxy(majormerger).Thiseventis thecentralSMBH. /m
n
accompaniedbyaburstofstarformationastheavailablecoldgas The formation of massive hot haloes would lead to the growth ra
s
frombothprogenitorsistransferredtothecentreandtransformed ofverymassivegalaxiesunlessaheatingmechanismregulatesthe /a
intostars.SomeofthiscoldgasreservoirfeedsthecentralSMBH. coolingflow.Boweretal.(2006)andCrotonetal.(2006)invoke rtic
Anadditionalprocessofcoldgasaccretionthatcontributestothe energyinjectionfromthecentralSMBHduringtheso-called‘radio le-a
SMBHmassisthecollapseofgalaxydiscstriggeredbydynami- mode’ feedback (see also De Lucia et al. 2006; Cattaneo et al. bs
calinstabilities(Efstathiou,Lake&Negroponte1982).Whenthe 2007;Lagos,Cora&Padilla2008).Theradiomodeisassumedto tra
c
self-gravityofthegalacticdiscbecomessufficientlylarge,thedisc occurduringthequiescentaccretionofgasfromthehydrostatically t/4
1
equilibriumisdisruptedresultingintheformationofabarwhich supportedhothaloontotheSMBH,duringwhichenergyfromthe 0
/1
enablesgastobetransferredtowardsthecentreasthedisc.Afrac- BHaccretionisinjecteddirectlyintothehothalosuppressingthe /5
3
tion of that gas is directly fed into the BH through an accretion cooling flow. In the Bower et al. (2006) model, the cooling flow /1
0
disc and powers an AGN (Lynden-Bell 1969), while the rest un- stopswhenthepowerfromtheSMBHissufficienttooffsettherate 3
1
dergoes star formation. For highly efficient accretion activity, the atwhichenergyisbeingradiatedaway. 50
3
disc/BH system dominates the energetics of the nucleus, and the Boweretal.(2006)adoptaBHgrowthmodelinwhichduringa b
y
galaxybecomesvisibleasaquasar.SMBHsatthecentresofgalax- discinstabilityorgalaxymerger,theBHaccretesafixedfractionof g
u
iesobservedinthelocalUniverseareregardedastheremnantsof thegasthatturnsintostarsinthebursttakingintoaccountprocesses e
s
quasarphasesatearlierepochs. suchasfeedbackandrecyclinginthegalaxy(Malbonetal.2007). t o
n
The consequence of a minor merger, namely the accretion of TheamountofgasdepositedontotheBHissetbyanefficiency 1
0
satellites of low mass compared to the central galaxy, is usually factor,whichdeterminesthefractionoftheavailablegasreservoir A
lsetesslladrracmonatteicn.tDofutrhinegsaatemlliinteosramreeragdedreidntooutrhemcoednetlr,aclogladlagxays,aanndd fpoarrasmtaertefrorimscahtioosnenthtoatfiitsthacecnroetremdalbiyzatthioenhoofleth.eTlhoecavlaMlue–oMf that pril 2
bh bulge 0
amerger-drivenstarburstmaysupplyfreshgastothecentralBH. relation.Hereafter,werefertotheaccretionofcoldgastriggered 19
SMBHsacquirepartoftheirmassmergingwithotherSMBHs. bydiscinstabilitiesandgalaxymergersas‘quasarmode’accretion
TheformationofaBH–BHbinaryandthesubsequentcoalescence andtotheaccretionfromquasi-hydrostatichaloesas‘radiomode’
oftheBHsisanaturalevolutionarystageforaSMBHifthehost accretion, following the terminology introduced by Croton et al.
galaxyexperiencesamerger.Asthegalaxiesmerge,weassumethat (2006).
1NotethatduetoanerrorinBoweretal.(2006),coolingluminositieswere 2.2 AstrophysicalprocessesaffectingBHspinevolution
overestimatedbyafactorof4π.Thus,whilethepaperquotestheefficiency SMBHs are expected to possess angular momentum J =
parameter(cid:4)SMBH as0.5,thisshouldhavebeen0.5/4π=0.04.Withthis |a|GM2 /c, where a is the spinparameter, 0 ≤ |a| ≤1. Thebhspin
correction,therestoftheparametersandresultsareunchanged. bh
2(cid:5)m,(cid:5)band(cid:5)(cid:3)expressthepresentdensityofthebaryonic,totalmatter hasasignificantimpactintheclosevicinityoftheBH.Forexam-
anddarkenergycomponentsoftheUniverserelativetothecriticaldensity ple,itdeterminestheefficiencyforconvertingmatterintoradiation
(ρcrit =3H2/8πG).σ8 measuresthermsmassfluctuationsinspheresof in an accretion disc (Novikov & Thorne 1973) and it is believed
radius8h−1Mpclinearlyextrapolatedtothepresentday. toinfluencetheformationanddirectionoftheradiojetsinAGN
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74
56 N.Fanidakisetal.
(Blandford&Znajek1977;MacDonald&Thorne1982;Begelman changeinthehole’stotalmass,M ,andangularmomentum,J ,
bh bh
etal.1984).Inaddition,itisofspecialinterestinthemodellingof equalto
gravitationalwavesfromBH–BHbinaries(Merritt&Milosavljevic´
dM =(e˜ /c2)dM , dJ =˜l dM . (4)
2005;Bakeretal.2006b,2007;Campanellietal.2007;Buonanno, bh lso 0 bh lso 0
Cook&Pretorius2007). Thechangeinthehole’sspininducedbytheaccretionofdM is
0
Theevolutionofspiniscloselyrelatedtothechannelsthatcon- governedbythedifferentialequation
tributetothegrowthoftheBH.EachmechanismforBHgrowthis
da 1 c3 ˜l
associatedwithdifferentspinevolution.Forexample,accretionof = lso −2a. (5)
gasthatco-rotateswiththeBHshouldspinupthehole(Bardeen dlnMbh Mbh Ge˜lso
1970), whereas the merger of two equal-mass non-spinning BHs This was integrated by Bardeen (1970) using the explicit expres-
resultsinafinalspinof0.69(Bakeretal.2006a;Bertietal.2007; sionsfore˜and˜l,resultinginthefollowingsolution:
Hiniflnudeenrceettahle.2sp0i0n8)o.fWaeBeHxparlaeininbcleulodwedhionwouthremseodmele.chanismsthat af = 1rˆ1/2Mbh ⎡⎣1−(cid:11)3rˆ (cid:12)Mbh(cid:13)2−2(cid:14)1/2⎤⎦ (6) D
3 lso Mf lso Mf o
bh bh w
n
2.2.1 Gasaccretion lo
After the SMBH seed forms at the centre of a galaxy, accretion swphinenpaMrabfmh/eMtebrhof≤thrˆel1s/Bo2H. H.IefreM,fM/fbMh and>arˆf1/a2r,ethtehnetfihneafilnmalassspiannids aded
usuallyinitiatesthegrowthera.Weassumethatanaccretiondiscis alwaysequaltounity.Theexpbrhessiobnh inelsqouation(6)governsthe fro
m
formedintheequatorialplaneofthehole.AsproposedbyLynden- evolutionofaduringaccretionfromaninitialstateofaco-rotating h
Bell(1969),gasparcelsinthediscgraduallyloseangularmomen- orcounter-rotatingdisc.Accordingtothis,anon-rotatingBHwill ttp
s
tum due to viscous torques exerted by magnetic fields and drift bespunuptoamaximumrotation(a=1,rˆlso=1)afterincreasing ://a
radially inwards until they reach the inner edge of the accretion itsmassby1.44Mbh (seeFig.1).ForamaximallyrotatingBHin cad
disc.Theinneredgeofthediscisusuallytakenasthelocationof acounter-rotatingaccretiondisc(a =−1,rˆ =9),anincreasein e
lso m
tohfethlaesthostlaeb’sleanogrbuilta(rLmSoOm)eanrotuumndatnhdecBaHn.bTehwerLitSteOnaissa(Bfuanrdcetieonn, m0aasnsdosfu2bMsebqhuisenretlqyusirpeudninuportdoermfoaxritmheuBmHrototabtieosnp.uAnsdiomwpnliteodab=y ic.ou
p
Press&Teukolsky1972) equation(6),furtheraccretionontotheBHkeepsaequaltounity. .c
(cid:2) (cid:3) o
rˆlso≡rlso/Rg= 3+Z2±[(3−Z1)(3+Z1+2Z2)]1/2 , (1) Bardeen’scalculationsarelimitedtotheprocessofangularmo- m/m
mentumtransportbetweentheaccretedmatterandtheBH,without n
where the gravitational radius, Rg, is defined as half of the taking into account any other processes. In fact, Thorne (1974) ras
ZSc1,hZw2arazrsecdheilfidnreaddiinuste(cid:4)ormf tshoefBthHe,sRpiSnch,wa,=as2R(cid:5)g = 2GMbh/c2, and atirognuewdiltlhbaettahcecraectcerdetbioynthdeishcorlaed.iCataepstuarnedotfhapthsootomnesowfitthhisanrgaudliaar- /article
ZZ12 ≡≡1(cid:6)3+a2(1+−Za12(cid:7)2)11/2/3. (1+a)1/3+(1−a)1/3 , (2) omtaeurot=aflmcotei0wnn.gt9su9omt8of(rmrqoˆlupsaoepttoe=thsriait1nte.pt2thro3ee,vtfh(cid:4)eoanrt(cid:9)tmsosfo0pf.ti3hnj2eeut)sB.pmHIbnaewyyadoihldnlaidvttihetohesnnet,rlopainmrxogiiadtiliumncrgpeellvaiacatialcvutoiieosutnonicsf- -abstract/4
Note that when the spin parameter has a negative sign the BH is 1
0
counter-rotating with respect to the orbit of a particle around it. for the upper spin limit, since these outflows are accelerated by /1
Then,forcounter-rotatingorbits(−1≤a<0)9 ≤ rˆ < 6and, /53
forco-rotatingorbits(0<a≤1),6<rˆ ≤1. lso /10
lso 3
AnimportantpropertyisthebindingenergyofthegasattheLSO, 15
0
definedasthedifferencebetweentherestenergyofagasparcelat 3
b
infinity and its energy at the LSO as measured by an observer at y
infinity.Ife˜expressestheenergyperunitrestmass,wecandefine gu
e
thebindingenergyas1−e˜/c2.Thisprovidesasimplerelationfor st o
theaccretionefficiency, n
(cid:8) 1
0
2 1 A
(cid:4)≡1−e˜lso/c2=1− 1− 3rˆlso (3) pril 2
0
(Novikov&Thorne1973)whichcorrespondstothefractionofthe 19
energyreleasedbymatterspirallingintowardstheBHthrougha
successionofalmostcircularorbits.Equation(3)showsthataccre-
tionofmatterontoslowlyrotatingBHshasmoderat√eefficiency.
Fornon-rotatingBHs(a=0,rˆ =6)thisisjust1− 8/9or5.7
lso
per cent. However, for co-rotating matter around rapidly ro√tating
BHs, the efficiency increases significantly reaching 1−1/ 3 or
42.3percentasa→1.Thissetsanupperlimittotheefficiencyof
theaccretionontoBHs,aprocessobviouslymuchmoreefficient
thanthermonuclearburning.
Once the gas reaches the LSO, we assume that it falls directly
intotheBH.Inthisway,theaccretioncarriesintotheBHtheenergy Figure1. ThefinalBHspinaftertheaccretionofgasinacounter-and
perunitmass,e˜,andangularmomentumperunitmass,˜l,thatthe co-rotatingconfiguration.Thedottedlinesindicatethefinalmassneededto
gashasattheLSO.Thus,accretionofarestmassdM leadstoa spinupanon-rotatingandmaximallycounter-rotatingBHtoaf=0.998.
0
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74
Grandunification ofAGNactivity 57
magneticfieldsthatarepoweredbytheextractionoftherotational whereRisthedistancefromtheBH(Pringle1992).Theprecession
energyoftheBH(Hawley&Krolik2006;Benson&Babul2009). time-scaleisthereforedefinedas
2π
t ≡ , (11)
prec (cid:5) (R)
p
2.2.2 Gasaccretionthroughamisaligneddisc whichisproportionaltoR3 andthusismuchshorterclosertothe
BH.Othertime-scalesrelevanttothisproblemaretheviscoustime-
In the general case, where the accretion disc does not lie on the
equatorial plane of the BH but has a random orientation relative scalesofaccretion,tν1,andwarppropagation,tν2,
totheangularmomentumoftheBH,theevolutionofthespinisa R2
complicatedprocess.FollowingthediscussioninKingetal.(2005), tν1,2 ≡ ν1,2(R). (12)
weassumethataTDisinclinedatsomerandomangleθ relative Here, ν and ν are the kinematic viscosities acting on velocity
totheorientationofthehole’sangularmomentumvector, J .We 1 2
bh gradientsparallelandnormaltotheplaneofthedisc,respectively.
2de0n0o5teantdheVaonlognutlearri,mSoikmoernat&umLoasfotthae2d0i0s7c faosrJadd(issceuesKsiionngoefttahle. Thebalancebetweenthetime-scalestprecandtν2determineswhether Do
theLense–Thirringtorqueisabletoaligntheinnerdiscwiththe w
natureof Jd),anddefinethetotalangularmomentumvector Jtotas spinaxis.Theconditionforalignmentistprec (cid:2) tν2 (Natarajan& nloa
TJhtoet=angJlbeh+θ bJedt.ween J and J is defined such that 0 ≤ θ(7≤) Astipmrimne-iatstacrgaaeldei1io9wf9rh9ae)dr.ieaIntlhdeoitfphfrueesrcieowsnosiorodfnst,htitemhwee-adsricpsa.clTewhisielmlcbhueacrhaalcsihtgeonrrietsdetircwthriaathdnittuhhsee, ded fro
bh d m
πoanr;iteit-hnaetlaitvgiaonlnmueeasnstθwre=eslpl0eacsatnivmdealθyg.n=iJtutπodtecisofroarrecsaopnogsnitvadennttovafecuccltrloeratilioi.genn.mehvaeesnntfit.axnIetdds RtpwreTacrph=,eoetνfv2o.thluetiaolnigonfedJbphaarntdofJdthdeurdiinsgctfhoellporwecsefsrsoiomnitshedectoenrmdiitnioend https://a
magnitudeisgivenby bytheconditions(Kingetal.2005) ca
d
e
Jt2ot=Jb2h+Jd2+2JbhJdcosθ. (8) ddtJb2h=0, ddtJd2≤0 and ddt cosθt ≥0. (13) mic.o
WeWfuhretnhetrheinvtreocdtourcseJthbehaanngdleJθdtaarsethmeisaanlgiglenebde,twtheeentilJtebdhaonrbditJstootf. AJsbhcraenmabiensincfoenrrsetadnbtywhthileefithrestantwgloecθotnddeictrioeansse,sthweitmhtaigmneit,uwdheicohf up.com
thegasparcelsintheaccretiondiscexperienceatorqueduetothe impliesthat Jbh alwaysalignswith Jtot.Incontrast, Jd decreases /m
Lense–Thirringeffect,whichcausestheplaneoftheaccretiondisc inmagnitudeas Jbhaligns,whichistobeexpected,sincethetotal nra
to precess about the rotational axis of the BH (Lense & Thirring angularmomentumhastoremainconstant. s/a
1918;Wilkins1972;Bardeen&Petterson1975;Scheuer&Feiler TheendresultoftheLense–ThirringprecessionisaBHwhich rtic
1996).Ifviscosityisstrongenoughthiscanforcetheinnerpartsof isalignedoranti-alignedwiththesurroundingaccretiondisc.The le
-a
thedisctorotateintotheequatorialplaneoftheholeresultingina expressionforthemagnitudeof J allowsustoexaminethefinal b
tot s
‘wTarhpeedLednissce’–T(sheierrFinigg.t2o)r.quecanbeexpressedas cooccnufirgsu,rwathiiocnhorfeqthueirseysstem.Obviously,ifJ2bh>J2totanti-alignment tract/4
1
∂∂Lt =(cid:3)p×L, (9) cosθ <−2JJd . (14) 0/1/5
bh 3
iwshthereepLrecisesthsieoannrgautelaarnmdoismgeinvteunmbpyerunitareaofthediscand(cid:3)p wHeitnhceth,eaaBcHcrewtioitnhdθis>c.πT/h2isamndea2nJsbhth>atJifdJedve>ntu2aJlblhytahnetni-aalliiggnns- /10315
ment always happens as the disc angular momentum completely 03
(cid:3) = 2GJbh, (10) overwhelmsthatoftheBH. by
p c2R3 Further investigation of the warped discs requires the knowl- gu
e
edge of the accretion flow properties. We use the TD solution s
of Shakura & Sunyaev (1973) for our analysis. In the standard t on
J Accretion Disk Shakura–Sunyaevdiscmodel,theanalyticexpressionforthewarp 10
tot radiusdependsonthevaluesofJ ,M andtheaccretionrateM˙ A
bh bh p
oftheBH.IntermsoftheSchwarzschildradius,Rwarpiswrittenas ril 2
J (Volonterietal.2007) 01
θ t θ d R (cid:12) M (cid:13)1/8 9
Jbh Rwarp RSwcahrwp =3.6(cid:12)×1(cid:13)03a5/8 108Mbh(cid:2) λ−1/4
ν −5/8
× 2 α−1/2. (15)
ν
Black Hole 1
HereαistheShakura–Sunyaevviscosityparameterandλ=L/L
Edd
istheEddingtonratio.Theaccretionluminosity,L,andEddington
luminosity,L ,aredefinedas
Edd
L=(cid:4)M˙c2, (16)
Figure 2. Schematic illustration of a warped accretion disc. Jbh is the with(cid:4)denotingtheaccretionefficiency,and
manogmuleanrtummomofenthtuemdisocfgtihveenBbHye(|qJubahti|o=n(2|a0|)GaMnd2bhJ/cto)t,reJpdreissenthtsethanegtuoltaarl L = 4πGMbhc =1.4×1046(cid:12) Mbh (cid:13) ergs−1, (17)
angularmomentumofthesystem, Jbh+Jd. Edd κ 108M(cid:2)
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74
58 N.Fanidakisetal.
whereκ∼0.3cm2g−1istheelectronscatteringopacity.Associated Themuchsmallerangularmomentumassociatedwitheachac-
with the Eddington luminosity is an accretion rate expressed as cretionepisodemeansthat,ingeneral,J <2J ,sointhechaotic
d bh
M˙ = L /(cid:4)c2. This is the accretion rate for which the BH accretionmodelcounter-alignmentoccursinafraction(Kingetal.
Edd Edd
radiatesattheEddingtonluminosity.Itisthenconvenienttoexpress 2005)
(cid:12) (cid:13)
thephysicalaccretionrateinunitsoftheEddingtonaccretionrate, 1 J
m˙ ≡M˙/M˙ .WenotethattheaccretionefficiencyinM˙ isspin f = 1− d (cid:9)1/2 (22)
Edd Edd 2 2J
dependent; however, we chose to keep (cid:4) = 0.1 when calculating bh
M˙ forsimplicity. oftheaccretionepisodes.Therefore,counter-andco-alignmentare
Edd
Giventhattheaccretionischaracterizedbyatime-scaletν ,the equallylikelyoutcomesoftheLense–Thirringeffect.However,ac-
1
massofthediscinsidetheradiusR is cretionofgasinacounter-rotatingdiscismoreefficientinspinning
warp
uptheBHsincethegasisbeingdumpedontotheBHfromalarger
Md(Rwarp)=M˙tν1(Rwarp), (18) distance and, thus, carries more angular momentum into the BH.
wheretheaccretiontime-scaleisgivenby Hence, a succession of counter-aligned and co-aligned accretion
R2 (cid:12) M (cid:13)11/8 episodeswithequalfrequencyshouldsystematicallyspindownthe Dow
tν1 = νwarp =3×106a7/8 108Mbh(cid:2) BH,resultinginaglobalspindistributionoscillatingaroundzero. nlo
1 (cid:12) (cid:13) The physics of self-gravitating accretion discs is described by ad
ν −7/8 Pringle(1981).Briefly,thecriterionthattheself-gravityofadisc e
×λ−3/4 ν21 α−3/2yr. (19) bzˆednireegcltiigoinblbeeisdtohmeirneaqtueidrebmyetnhtethcaetntthraelgBraHv.itTathioisncaalnfobrceeeaxlpornegssthede d from
WaJsde(Rcawnarpw)r(cid:2)iteMthde(Rtowtaarlpa)(nGguMlabrhRmwoamrpe)1n/t2u.m,Jd,passingthroughR(w2a0rp) taMhset(hq<euasRnut)rift(cid:11)aycMeHdbheMHns/iRt.y3,oofrthinetdeirsmc,s(cid:14)of,btheeindgisncemglaigsisblecompare(d23to) https://acad
Intermsoftheanti-alignmentcriterionthisgives d R bh em
2JJdbh = aMMdbh (cid:12)(cid:12)RRtSwcahrw(cid:13)p(cid:13)(cid:12)1/R2 (cid:13)1/2 T(HRh/eR)sMelfb-hg,rwavhiitcyhdbeeficnoems(cid:12)etsheMmsealrfg-ginr(cid:13)aalv−liy2t6y/2ir7madpiourst,aRntsg,wghiveennbMyd (cid:9) ic.oup.com
=10−9λ 1νy1r RSwcahrwp a−1. (21) RScshgw =1.5×103(cid:4)8/27 108Mbh(cid:2) λ−8/27α14/27. (24) /mnra
Evaluation of this quantity determines whether or not the anti- Thesemi-thicknessofthediscintheShakura–Sunyaevmodelis s/a
alignmentconditioninamisaligneddiscissatisfied. givenby rtic
le
H (cid:12) M (cid:13)−1/10 -ab
2.2.3 Thecaseofself-gravitylimiteddiscs R =1.36×10−3(cid:4)−1/5 108Mbh(cid:2) λ1/5 stra
The BH growth process in AGN environments involves vast (cid:12) R (cid:13)1/20 ct/4
amounts of accreted gas, often comparable to the initial mass of × α−1/10. (25) 10
theaccretinghole.Thisamountofmasssettlesontotheaccretion RSchw /1/5
3
discaroundthehole,whichoftenextendstoseveralthousandsof Hence, by replacing R with Rsg in expression (25), we obtain an /1
sgMuraapcvpci,ltyaistoiofcncooanlnsrusatmadnieitd.aInitngisuaulassruinmagllolyemaaescnscutrumemteiod(nVthoealpotintshtoeedraieve,attihallua.bs2l0pe0rmo7va).isdIsnifnutgheilas, Mansagly=ticHRexMprbehs(cid:17)(cid:17)(cid:17)(cid:17)sionfo=rt2h.e1d3i×sc1m0a5s(cid:4)s−w5/i2t7hinRsg, 031503 b
case,theamountofmassconsumedisenoughtospinupthehole (cid:12) R=Rsg(cid:13) y g
uptoa=0.998,eveniftheBHinitiallywasmaximallyspinningin M 23/27 ue
acounter-rotatingdirection.Inevitably,repeatedaccretionepisodes × 108Mbh(cid:2) λ5/27α−2/17M(cid:2). (26) st o
n
during major galaxy mergers act to spin up the BH to maximum 1
AswedescribedinSection2.2.2,oncethediscformswithanon- 0
rotation. A
theReencdenretlsyu,lhtoowfethveera,cKcrinetgi,oPnrginrogwlet&hcHhaonfmnealnmni(g2h0t0b8e)caorgmupeldettehlayt zbeerosumbjiescatlitgonmtheenLteannsgel–eTahbiroruintgthperescpeisnsiaoxni.sIfofthteherahdoiules,, Ritsgw,iilsl pril 2
0
different if we take into account the fact that an accretion disc greaterthanRwarp,then 19
becomesself-gravitatingatsomeradius,Rsg,whereitsmassexceeds Jd=Jd(<Rwarp). (27)
M ∼(H/R)M .Asaresult,themassofthediscislimitedbyits
selsfg-gravity to (cid:13)bhM (cid:11) M , which gives rise to a series of If,however,Rsg<Rwarp,thentheentirediscwillbesubjecttoLense–
episode acc
N (cid:9)M /(cid:13)M well-separatedaccretionepisodes.Kingetal. Thirringprecession,thusaligningitselfintheequatorialplaneof
acc episode
theBH.
suggestthattheseaccretionepisodesarerandomlyorientedaround
the BH, an assumption supported by observations indicating that
there is no apparent relation between the accretion disc (or radio
2.2.4 BH–BHbinarycoalescence
jet) orientation and the host galaxy disc (Nagar & Wilson 1999;
Kinneyetal.2000).Theyfurtherarguethatthiscouldbetheresult We now examine the evolution of spin during BH binary coales-
of intense star formation outside R randomizing the input gas cence.Duringthemergerofthetwobinarymembers,thesmaller
sg
direction,whichcouldprovideaqualitativeexplanationforthering BHaddsitsspinandorbitalangularmomentumattheLSOtothe
ofstarsseeninthenearvicinityofthecentralBHintheMilkyWay spinofthelargerone.Eveniftheprogenitorholesdonotpossess
(Genzeletal.2003). anyorbitalangularmomentum,thefinalremnantwillpreservethe
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74
Grandunification ofAGNactivity 59
residualorbitalmomentumofthebinary(i.e.theangularmomen- Here μ expresses the symmetric mass ratio, μ = q/(1 + q)2 =
tumthathasnotbeenradiatedaway).Thus,theremnantwillalways M M /(M + M )2, and the coefficients take the values s =
1 2 1 2 4
beaKerrBH. −0.129, s = −0.384, t = −2.686, t = −3.454, t = 2.353.
5 0 2 3
Recentbreakthroughsinnumericalrelativityhaveprovidedro- Thefinalspinasgivenbyequation(31)isingoodagreementwith
bust simulations of BH mergers by directly solving the Einstein numericaldata,withresidualsoflessthan3percent.
equations in a fully relativistic frame (Campanelli et al. 2007;
Herrmann et al. 2007; Marronetti et al. 2007). These simulations
have explored the parameter space for different configurations of 2.3 Simulatingthespinevolution
initial masses and spins, allowing accurate measurements of the
WefinallydescribeourmethodforcalculatingtheevolutionofBH
final spin. For example, a merger of two equal mass BHs both
spin.DuringtheformationhistoryoftheBHswetakeintoaccount
witha=0inacircularorbitresultsinaKerrBHwithaf (cid:9)0.69
spinchangesduetotheaccretionofgasandmergers,asdescribed
(Bakeretal.2006a).Thisisarobustpredictionandisalsovalidfor
inSection2.2.Theinitialpopulationofseedsisassumedtobenon-
eccentric orbitswitheccentricities smallerthan0.4(Hinderetal.
rotating. These seeds grow through accretion of gas during disc D
2008). instabilities,galaxymergers,andradiomodeaccretionandthrough ow
Analyticfitsextendthepredictionsforaf totheentirespaceof n
BH–BHmergers. lo
parametersandreproducecloselyalltheavailablenumericaldata. a
d
Inthecasewherethemassesareunequalbutthespinsarezero,the ed
finalspincanbeestimatedbytheanalyticexpression 2.3.1 Thindiscs from
√ q q2 h
wafh(cid:9)ere2q=3(M12+/Mq)12≤−12i.s0t2h9e(m1a+ssqra)4ti,ooftheprogenitorBHs(B(2er8t)i Tmp˙hoy≥msico0sd.0eisl1dtthehesecprgihabysesdficobsrymotsfheathnsetaaanccdccraerretditoeSndhadgkiasuscr,aaw–rSoeuuannsydsautemhveedBitshHcatmwwohdhoeeslne. ttps://aca
etal.2007). WeassumethatthegasavailabletobefedintotheBHafteragalaxy de
m
A more general analytic fitting formula for predicting the fi- merger or the collapse of a dynamically unstable disc is accreted ic
nal spin of any given binary configuration has been provided by over a time-scale proportional to the dynamical time-scale of the .o
u
Rezzolla et al. (2008a,b). In their analysis, they assume that the galacticbulgethathoststheBH.Afixedproportionalityfactorof4 p.c
finalspinparametervectorcanbeexpressedas isusedinthisanalysisresultingintypicalaccretiontime-scalesof om
theorderof107–108yr.Notethatthisfactoris∼10longerthanthe /m
af = (1+1q)2(a1+a2q2+(cid:4)q), (29) onechosenbyMalbonetal.(2007).Thenewvalueisintroduced nras
in order to match the local quasar luminosity function and does /a
wthheedreiffae1r,e2n=cecbJet1bw,h2e/e(GnMth21e,2o)rabnitdal(cid:4)an=gu(cid:4)l(cid:12)a/r(Mm1oMm2e)n,twumith,l(cid:4),(cid:12)wdheefinnitnhge ndoutrianfgfetchtetrhaedgioalmaxoydefoirsmcoatmiopnumtedoddeilr.eTcthlyebtiymteh-escgaalleaxfoyrfaocrmcreattiioonn rticle-a
binary is widely separated, and the angular momentum radiated modelanddependsonthecoolingtime-scaleofthegasinthehot bs
awayingravitationalwavesbeforethemerger, j ,namely halo. tra
rad c
(cid:4)(cid:12)=l− jrad. (30) toTthheesapcicnreatxioisnodfistcheishaoslseu.mIfedθ t(cid:13)=ob0e◦,ra1n8d0o◦,mwlyeoarsiseunmteedtrhealatttihvee t/410
/1
Thevector(cid:4)(cid:12)istakentobeparalleltotheorbitalmomentumvector discprecessesaroundtheBHasdescribedinSection2.2.2.Inbrief, /5
3
throughouttheevolutionofthebinary,anassumptionwhichisnot foragivenBHmass,Mbh,wedeterminethenumericalvaluesof /1
0
strictlyvalidsincethesystemcouldradiateawayangularmomen- the warp parameters, Rwarp, Md(Rwarp), and the angular momenta 31
tuminanon-symmetricway.However,theerrorintroducedbythis Jd(Rwarp)andJbh.TheratioJd/2Jbhallowsus,throughthecriterion 50
3
assumptionisrelativelysmallforthebinaryconfigurationsstudied inequation(14),tocheckifanti-alignmentoralignmentoccurs.We b
y
in the simulations of Rezzolla et al. A further assumption is that thenallowtheBHtoaccreteanamountofgasequaltoMd(Rwarp)and g
u
the mass radiated away in gravitational waves is negligible, as it evolvethespinusingequation(6).Finally,weupdatethemassof e
s
accountsforonlyasmallfraction(5–7percent)ofthetotalmass theBHandestimatethehole’snewspinandthediscprecessionby t o
energyofthebinaryconfigurationsanalysed.Undertheseassump- calculatingthenewanglesθ andθt.Thisprocessisrepeateduntil: n 1
tions,Rezzollaetal.(2008a,b)proposedananalyticexpressionfor (a)thediscalignsorcounteraligns(θ =0◦or180◦)withthehole’s 0 A
themagnitudeo(cid:18)fthefinalspin,givenby sepvionl,vientwhehiscphincaascecworedijnugsttcooneqsuumateiotnhe(6re),stoorf(bth)ethaveaailcacbrleetigoansdainsdc pril 2
af|= (1+1q)2 |a1|2+|a2|2q4+2a2a1q2cosϕ is entirely consumed, without being able to align or counteralign 019
+2(cid:6)a cosϑ+a q2cosξ(cid:7)(cid:4)q+(cid:4)2q2(cid:19)1/2, (31) idtisseclfuwltiimthattehleysapliing.nsIn(oarlmanotsit-aallilgncsa)seitssewlfeofinntdhetheaqtutahteoraicaclrpeltaionne
1 2
oftheBH.
withthecosineanglesϕ,ϑ andξ definedas Inadditiontothecasewheretheentiregasreservoirisconsumed
withinoneaccretionepisodewithconstantangularmomentumori-
cosϕ≡aˆ ·aˆ , cosϑ ≡aˆ ·(cid:4)ˆ, cosξ ≡aˆ ·(cid:4)ˆ. (32)
1 2 1 2 entation(prolongedaccretion),wetestthemodelproposedbyKing
Thenormof(cid:4)isgivenby etal.(2008)inwhichthemassoftheaccretiondiscislimitedbyits
s (cid:6) (cid:7) self-gravity(chaoticaccretion).Werepeatthesamestepsasbefore.
|(cid:4)|= (1+4q2)2 |a1|2+|a2|2q4+2|a1||a2|q2cosϕ However,inthiscaseMd=Msg.OnceMsgisconsumed,weupdate
(cid:12) (cid:13) themassoftheholeanddeterminethemassofthenewdisc,M(cid:12) ,
s μ+t +2 (cid:6) (cid:7) sg
+ 5 0 |a |cosϑ+|a |q2cosξ computed from the updated BH mass. At this point we make the
1+q2 1 2
√ assumptionthatthediscretainsnomemoryoftheangularmomen-
+2 3+t μ+t μ2. (33) tumoftheinitialflowandthatitsettlesintoarandomorientation
2 3
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74
60 N.Fanidakisetal.
aroundtheBHwithM =M(cid:12) .WethenaddthenextM untilthe duringtheformationofthebinarydonotinfluencetheorientation
d sg sg
discisconsumed. oftheBHspinspriortothemerger.WethereforetaketheBHspin
Itisimportanttoclarifyatthispointthatthetwoaccretionmodels vectors in equation (31) to be randomly oriented relative to each
consideredhereserveasameansforstudyingtheeffectofaccretion other and the orbital angular momentum. In addition, we assume
onBHspin.Theforthcominganalysisthatfollowsandresultsdo that allbinaries willultimately mergeon averyshorttime-scale,
notrelyonanyspecificmodelforthephysicalprocessesthatdrive effectivelyrightafterthehostgalaxiesmergerandinducedaccretion
thegasflowstowardsthevicinityoftheBH.Severalmechanisms havebeencompleted.
forfeedingtheSMBHwithgascouldresultinsimilarspindistribu- During the merger, the smaller BH plunges into the larger one
tionstothechaoticaccretioncaseconsideredhere(seee.g.Heller, carryingalongitsangularmomentumattheLSO.Atthispointwe
Shlosman & Englmaier 2001 for the properties of non-self- neglectanymasslossduetogravitationalwaveemission,asitcor-
gravitating gaseous bars in barred disc galaxies and Hobbs et al. respondstoasmallfractionofthetotalmassenergyofthesystem.
2010foraninvestigationoftheeffectofsupernova-driventurbu- Inaddition,wedonottakeintoaccounttheeffectsofgravitational
lenceonthefuellingofSMBHsingalacticbulges).Here,ourmain recoilduetoasymmetricemissionofgravitationalwavesinbinary
D
goalistoexploretheeffectofdifferentSMBHspindistributionson mergers. In our model, mergers with recoil velocities as high as o
w
propertiesofthehostgalaxy. υ ∼ 500 km s−1 (Libeskind et al. 2006) could displace BH n
recoil lo
remnants. However, we assume that dynamical friction relocates a
d
them in the galactic centre. Therefore, we do not expect gravita- ed
2.3.2 ADAFs tionalrecoiltohaveasignificantimpactonthespindistributionof fro
m
Manyoftheaccretioneventsinoursimulationsarecharacterized theremnantBHs.Thespinofthefinalremnantisfinallyestimated h
by low, sub-Eddington, accretion rates, with m˙ ≤ 0.01. For such usingthefittingformulaeinequations(31)and(33),asdescribed ttps
lowaccretionrates,thegasflowhaslowdensityandisunableto inSection2.2.4. ://a
c
coolefficientlysinceradiativecoolingdoesnotbalancetheenergy a
d
e
generatedbyviscosity.Thus,theviscousenergyistrappedinthe 3 EVOLUTION OF BH MASS AND SPIN m
gasasentropyandultimatelyadvectedintothehole.Thistypeof ic.o
accretionisknownasanADAF(Rees1982;Narayan&Yi1994; 3.1 EvolutionofBHmass up
Abramowiczetal.1995). .co
ADAFshaveanumberofdistinctpropertiesthatsomeofthem WenowbrieflypresentourmainpredictionsforBHgrowth,using m/m
willbeessentialfortheanalysisinlatersections(seeSections5and ourupdates(seeSection2)tothemodeldevelopedbyBoweretal. n
6).Forexample,foranADAFaroundaBH,onlyafractionofthe (2006)andMalbonetal.(2007).InFig.3weshowthefractionof ras
standardaccretionluminosity,L = (cid:4)M˙c2,isemittedasradiation. allBHsthataccreteatagivenrateinthequasarandradiomodes, /artic
The remainder of the viscously dissipated energy is stored in the le
gas as entropy, resulting in hot flows with almost virial tempera- -a
b
s
tures.Wenotethat,asshownbyIchimaru(1977),theionsandthe tra
electrons in an ADAF are not thermally coupled and, thus, reach c
t/4
differenttemperatures.Thistwo-temperaturevirializedplasmaflow 1
0
isopticallythinand,forhigh-viscosityparameters(α ∼0.2–0.3), /1
it acquires a quasi-spherical geometry around the BH (H ∼ R), /53
which resembles spherical Bondi accretion. However, despite the /10
3
geometricalsimilarity,thedynamicsoftheADAFarefundamen- 1
5
tallydifferent,sincetheaccretionisentirelyduetodissipationvia 03
viscousforcesratherthangravity. by
Ingeneral,theaccretionflowcanhavearathercomplicatedstruc- gu
e
ture.Forexample,itispossiblethataTDdominatesthegeometryof s
theouterparts,thenswitchestoanADAFastheflowapproachesthe t on
BH(Esinetal.1997).Thetransitiontothethickflowdependsonthe 10
accretionrateandusuallyoccursclosertotheBHforhigheraccre- Ap
tionrates.Suchacomplexdiscmodelfordescribingtheaccretion ril 2
flowsduringtheradiomodeis,however,beyondthescopeofthis 01
9
paper.Here,weassumeasimpleconfigurationofaquasi-spherical
corona around the hole with any transition to a TD occurring far
fromtheBH.Precessioneffectsduetomisalignmentbetweenthe
discandtheBHaremodelledasintheTDcase.
Figure3. ThedistributionofaccretionratesforselectedBHmassranges
2.3.3 Binarymergers at various redshifts. Line styles represent two different accretion modes:
solidlinesforthequasarmodeanddashedlinesfortheradiomode.The
Finally, we consider the spin changes due to binary coalescence.
shadedarearepresentstheregimewheretheaccretionflowisdescribedby
FollowingthemergeroftwogalaxiesinGALFORM,eachharbouring anADAF.Theaccretionratesintheradiomodearetruncatedatm˙ =0.01,
aSMBH,theonehostedbythesatellitesinkstowardstheSMBH
inaccordancewithourmodelofAGNfeedback.InthepresentUniverse,the
ofthecentralgalaxyandeventuallyformsabinary.Inourmodel, quasarmodeaccretionpeaksatm˙ (cid:9)0.5.However,thepeakshiftstohigher
BHmergerstendtooccuringas-poorenvironments,andthuswe valuesathigherredshifts,indicatingthataccretionwasmoreefficientinthe
assume that torques from accreting flows that might be present past.
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74
Grandunification ofAGNactivity 61
atdifferentredshifts.AllBHsthataccreteintheradiomodeduring
theredshiftbinareincluded,butthequasarmodeisepisodicsowe
only include objects which experience a starburst within a given
redshift bin. Our updated parameters give a longer duration for
the accretion episode in the quasar mode (four times rather than
0.5timesthedynamicaltime-scaleofthebulge)thanassumedin
Malbonetal.(2007).ThisresultsinthemajorityofBHswithmass
106– 108M(cid:2) which accrete in the quasar mode at z = 0 having
m˙ (cid:9)0.5,asobserved(Heckmanetal.2004).Athigherredshiftthe
distributionsshifttohighermassaccretionrates,withameanm˙ (cid:9)3
atz=6.Thisimpliesthatinahierarchicaluniverse,accretionof
gaswasmoreefficientatearlyepochs,sotherateofgrowthofBHs
isfasterinthepast.Bycontrast,accretionactivityduringtheradio
modepeaksatverylowaccretionrates(m˙ ≈10−4)atz=0,witha Do
w
longtailtolowervaluestogetherwithsomespreadtothemaximum n
lo
allowedvalueof0.01(imposedbytheAGNfeedbackmodel).This a
d
upperlimittothemassaccretionratelimitstheshifttohigherm˙ at ed
higherz,sothemainevolutionarytrendisthattherearefewervery fro
lowm˙ radiomodeobjectsathighz. m h
Thismodelresultsinabimodalmassaccretionratedistribution ttp
s
for all BH masses and redshifts. Activity triggered by accretion ://a
of cold gas from the galaxy disc, replenished by galaxy mergers, c
a
hasameanmassaccretionratewhichis∼1000timeshigherthan Figure5. ContributiontothefinalBHmassatdifferentredshiftsfromgas de
m
thatofactivityfedbyradiomodeaccretion.However,therelative accretioninquasarmode(solidgreenline),radiomode(dashedblueline) ic
importanceofthesetwomodesvarieswithBHmass.Thisisshown and BH–BH mergers (dash–dotted red line) in the updated Bower et al. .ou
inmoredetailinFig.4whichshowsthe20–80and5–95percentile (2006)model.Thelinesrepresentthemediansandtheshadedregionthe p.c
massaccretionratesasafunctionofBHmassatz=0.Morethan5 20–80percentilesofthedistributions.ThedominantchannelofBHgrowth om
percentofthelowermassAGNhaveaccretionviaaTD(m˙ >0.01). forBHswithMbh(cid:2)108M(cid:2)isthequasarmode.Abovethatmass,mergers /m
However,thisfractiondropswithmassatBHmassesof∼108M(cid:2). andradiomodebecomethedominantgrowthchannels. nra
s
Moremassivegalaxiesarehostedbylargeellipticalswhicharegas /a
poorandaredominatedbyradiomodeaccretion. rtic
Fig.5showsthecontributionfromthedifferentgrowthchannels le-a
tothefinalBHmassatdifferentredshifts.Athighredshifts,SMBHs bs
with masses up to ∼108M(cid:2) build their mass almost exclusively tra
c
throughtheaccretionofcoldgas(quasarmode).Thisformseithera t/4
TDoranADAFdependingonm˙ (seealsoFig.3).Accretionofgas 10
/1
fromthehothaloofmassivegalaxiesduringtheradiomodealways /5
occurswithm˙ ≤0.01byconstruction,thusgivingrisetoanADAF. 3/1
0
Thislowupperlimittothemassaccretionrateinthismodemeans 3
1
thatitmakescomparativelylittleimpactonBHgrowth,exceptfor 50
massesgreaterthan108M(cid:2).NotethatintheoriginalBoweretal. 3 b
y
(2006)model,theradiomodeisresponsibleforbuildingmostof g
theBHmassabove108M(cid:2),whileintheupdatedversionpresented ue
s
here,bothBHmergersandradiomodecontributesignificantlyfor t o
n
high-massBHs. 1
0
InFig.6,wecomparetherelationshipbetweentheBHmassand A
hTohsetmbuoldgeelmparessdipcrtesdaicnteadlmboysotulirnmeaordreellatotiothneaonbdserrevpartoiodnuaclesdatthae. pril 2
0
observationsreasonablywell.ThescatterinBHmassis∼1dexfor 19
bulge masses below 1011M(cid:2) and becomes gradually smaller for
higherbulgemasses.TheresultingBHmassfunctionsareshown
inFig.7.Theseshowthatthepresent-dayUniverseisdominated
byBHswithmassesof∼107–108M(cid:2)whichisingoodagreement
withtherecentresultsfromtheSloanDigitalSkySurvey(SDSS)
ofHeckmanetal.(2004).Theshapeofthemassfunctionsisfairly
Figure4. ThecorrelationofaccretionratewithBHmassatredshiftzero.
similaratallredshifts.Themaindifferenceoccursatthehigh-mass
Themedian(red–yellowpoints),20–80percentiles(darkgrey)and5–95
endwheretheBHmassfunctionextendstohighermassesatlow
percentiles (light grey) are shown. The shaded background indicates the
redshifts.ForM (cid:2)108M(cid:2)themassfunctionisalmostconstant
regionwheretheaccretiondiscismodelledasanADAF.Onlylowermass bh
BHs(<108M(cid:2)),i.e.thosehostedbylowermassgalaxies(typicallyspirals), and flat up to z = 2. This indicates that these BH are already in
have enough gas left to trigger accretion via a TD (m˙ > 0.01). More place at z = 2. Above that mass, there is a sharp decrease in the
massiveBHshavelittlecoldgasandundergoradiomodeaccretionwhich spacedensityofBHatz=2and4.Atz=0,themassfunctionhas
(byconstruction)hasm˙ <0.01. ashallowerslope.TheestimatedBHmassdensityatz=0isρ =
bh
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74
62 N.Fanidakisetal.
accretionmodels,respectively.TheBHspindistributionspredicted
bythetwoaccretionmodelsarealreadyestablishedatz=6and
do not change appreciably with time. The major channel for BH
growthatz=6isthequasarmodeandthisgenerallytakesplace
viaaTD.Theaccretedmassineachepisodeisgenerallylargerthan
themassoftheBH,sothisspinsituptomaximalintheprolonged
accretionmodel.Thebreakupoftheaccretionintomuchsmaller
individualeventsinthechaoticmodelleadstoarandomwalkspin
distribution around a low spin value. None the less, the plots in
Fig. 5 show that BH–BHmergers contribute to the most massive
BHsatlateredshifts,asdoesaccretionwithm˙ <0.01inboththe
radioandquasarmode.Inordertogainfurtherinsightintotheeffect
ofthesegrowthchannelsweshowthedifferentspindistributions
asafunctionofBHmassatz=0intheright-handpanels. Do
w
Asalreadymentioned,low-massBHsarebuiltviathin-discac- n
cretion in the quasar mode (Fig. 5) with mean m˙ ∼ 0.5. For a loa
d
BHofmass106M(cid:2) atypicalaccretioneventinthequasarmode ed
contains106M(cid:2) ofgas,sointheprolongedaccretionmodelthis fro
m
willspintheBHuptomaximal.However,inthechaoticaccretion h
model in which the size of the disc is limited by its self-gravity, ttp
s
t∼he(Hm/Ras)s×of1t0h6eMg(cid:2)as ∼tha1t0s4eMttle(cid:2)s.oTnhuthse, adi1sc06cMan(cid:2)noatcbceremtioonreetvheannt ://ac
a
consistof∼100accretionepisodeswith(cid:13)Mepisode (cid:11)Mbh,allran- de
Figure6. TheMbh−MbulgerelationpredictedbytheupdatedBoweretal. BdoHmslyexopreireinetnecdeaaronuentdsptihne-dBowHn. Atosmaocdoensstevqauluenescec,e1n0tr6e–d10ar8oMun(cid:2)d mic.o
(2006)model(solidline).Theshadedareasindicatethe10–90(light)and u
20–80(dark)percentilespreadofthetheoreticalpredictions.Theobserva- af ∼0.15. p.c
tionaldataaretakenfromHa¨ring&Rix(2004). The growth channel for more massive BHs (Mbh > 108M(cid:2)) om
includesalargefractionofobjectswhichaccreteintheradiomode /m
orquasarmodewithm˙ <0.01,i.e.viaageometricallythickADAF. nra
s
However,wedonotincludefullmodellingofthisprocessasour /a
spinevolutioncalculationsalwaysassumeaShakura–SunyaevTD rtic
ratherthantheappropriateADAFequations.Wedonotexpectthis le-a
tohavealargeeffectonthespindistributionsasmostoftheBH bs
massandspinarebuiltupfromhigh-massaccretionrates. tra
c
ThemergersbetweenmassiveBHshaveasignificantimpacton t/4
1
thefinalspindistributions.Accordingtotheanalyticfittingformula, 0
/1
equation(31)fromRezzolla(2008,2009),thepost-mergerspinof /5
3
thefinalremnantdependsstronglyonthemassesoftheprogenitors. /1
0
For example, BHs acquiring a final spin greater than 0.69 after a 3
1
merger are the end product of binaries of comparable mass (q (cid:9) 50
3
1)inwhichthemembersalreadyhadsignificantspins(recallthat b
af =0.69isthefinalspinofabinaryofequal-massnon-spinning y g
u
BHs).MergersbetweenBHsofcomparablemassarecommononly e
s
for the most massive BHs (Mbh (cid:3) 108M(cid:2)) in our simulation.3 t o
n
ThisisbecausetheseBHsarehostedbymassiveellipticalgalaxies 1
0
thathaveexperiencedatleastonemajormerger(mergerbetween A
gHaelnaxceie,sthoefireBquHasl,mwahsisc)hinarethoefirspimasitlahrimstoarsys,(wPailrlryaceqtuiarle. 2a0fi0n9a)l. pril 2
0
spincloseto0.69orhigher(dependingonthevalueoftheinitial 19
spins)aftertheycoalesce.IncontrasttothemostmassiveBHs,lower
Figure7. Thepredictedmassfunctionsatz=0,2,4and6fortheBH massBHs(Mbh(cid:2)108M(cid:2))whichexperienceamergerareinvolved
populationinoursimulation. mostlyinminormergers(q(cid:11)1).Thesedonothaveasignificant
3.84×105M(cid:2)Mpc−3,whichisingoodagreementwiththevalue
impliedbytheX-raybackground(Fabian&Iwasawa1999).
3Atredshiftzeronearly19percentoftheBHswithMbh≥106M(cid:2)have
3.2 GlobalBHspindistributions experiencedatleastonemerger.ThelowmergerrateofBHsmerelyreflects
the relative minor effect that mergers have in the formation of galactic
ThehistogramsinFig.8showthedistributionsofBHspinsatdiffer-
spheroids (except in very massive ellipticals) in the Bower et al. (2006)
entredshifts(left-handpanel)andfordifferentmassranges(right- model,asfoundbyParry,Eke&Frenk(2009).Forexample,atredshiftzero
hand panel) for all galaxies that host BHs with Mbh ≥ 106M(cid:2). about6percentoftheBHswithMbh ≥106M(cid:2) haveexperiencedmore
Thetopandbottompanelscorrespondtothechaoticandprolonged thanonemergerandonly∼8percenthavehadarecentmajormerger.
(cid:3)C 2010TheAuthors.Journalcompilation(cid:3)C 2010RAS,MNRAS410,53–74