Radiatively efficient disks around Kerr black holes

1.4 MB·

by  S. Campitiello

Mon.Not.R.Astron.Soc.000,1–9(2016) Printed2February2017 (MNLATEXstylefilev2.2)
Radiatively efficient disks around Kerr black holes
S. Campitiello1(cid:63), G. Ghisellini2, T. Sbarrato1, G. Calderone3
1DipartimentodiFisica“G.Occhialini”,Universita`diMilano–Bicocca,piazzadellaScienza3,I–20126Milano,Italy
2INAF–OsservatorioAstronomicodiBrera,viaE.Bianchi46,I–23807Merate,Italy
3INAF–OsservatorioAstronomicodiTrieste,ViaTiepolo11,I-34131,Trieste,Italy
7
1
2February2017
0
2

n ABSTRACT
a Thepropertiesoftheradiationemittedbytheaccretiondiskaroundrotatingblackholeshave
J beenstudiedonlyrecently,duetothedifficultiesincalculatingrelativisticeffectscausedby

1 the strong gravitational field and to the large velocities of the matter in the vicinity of the
3 laststablecircularorbit.Anumericalapproachisrequired,asdonebyLiandcollaborators,
for the case of star-sized black holes. In this paper we build upon this numerical treatment
]
E trying to interpolate the numerical results with simple analytical functions, that can be used
for black holes of any size and spin. We especially emphasize the pattern of the produced
H
radiation,namelythefluxproducedasafunctionoftheangleθbetweenthelineofsightand
.
h thenormalofthedisk,andtherotationofboththeblackholeandthedisk.Lightbendingand
p relativisticbeamingconcurtoenhancethefluxobservableatlargeθ (i.e.edgeon),contrary
- tothestandardcosθpatternofdiskemissionaroundSchwarzschildblackholes.
o
r Keywords: galaxies:active—(galaxies:)quasars:general—blackholephysics—accre-
t
s tion,accretiondisks
a
[

1
v
1 INTRODUCTION 2008,Wangetal.2009,Shenetal.2011).Or,alternatively,wecan
1
fit the accretion disk spectrum that depends in the simplest case
01 Inthelasttwodecadeswelearntthatatthecoreofalmostevery only from the black mass M and the accretion rate M˙, to find
galaxythereisasupermassiveblackhole(SMBH),nowinactivein
0 both quantities through the two observables: the disk total lumi-
mostcases,butthatwasnecessarilyactiveinthepast,tohaveac-
0 nosityanditspeakfrequency(hereafter“SEDfittingmethod”;see
cretedthemassithas.TheproblemofmeasuringtheSMBHmass
. Calderoneetal.2013).Allthesemethodsbearlargeuncertainties
02 abtlatchkehcoenlete-rgoaflgaxalyaxcioerseismcarsuscicaolrrteolaatsisoenss(et.hge. pMroapgeorrtriieasnoefttahle. (∼>0.4−0.5dex)thatarebuiltintheintermediatecorrelationused
tofindthevirialmassandthesystematicsinvolvedinthecalibra-
7 1998,Gebhardtetal.2000,Ferrarese&Merrit2000,Marconi&
1 tion which will lead to significant biases of these BH mass esti-
Hunt2003,Ha¨ring&Rix2004,Gu¨ltekinetal.2009,Beifiorietal.
: mates (Peterson et al. 2004, Collin et al. 2006, Shen et al. 2008,
v 2012, Kormendy & Ho 2013, McConnell & Ma 2013, Reines &
Marconi et al. 2008, Kelly et al. 2009, Shen & Kelly 2010, Park
i Volenteri2015),andhenceuptowhatdegreeweneedafeedback
X et al. 2012), and in the model used for the SED fitting method:
mechanismtoexplainthisrelation,orifmorethanoneprocessis
standard accretion disk, geometrically thin, optically thick down
r needed(seethereviewbyFabian2012andreferencestherein).Re-
a totheinnermostradius(Shakura&Sunyaev1973),oftenwithno
semblingtheladderofdistances,thereisnowasimilarladdercon-
rotationandnorelativisticeffectsincluded.
cerning black hole masses, starting from primary measurements,
usingmaserswithintheaccretiondisk(seee.g.Greeneetal.2010, Giventheimportanceoffindingareliable(oratleastcomple-
Kuoetal.2011andthereviewbyTarchi2012),orindirectlymea- mentary)methodtoestablishthemassofSMBHsandtheirspins,in
suringthewidthofthebroademissionlinestoknowtheirvelocity thispaperwestudythepropertiesofaccretiondiskthatarestillge-
dispersion, and the reverberation mapping (e.g. Blandfors & Mc- ometricallythinandopticallythickdowntotheirinnermostcircular
Kee1982,Peterson1993,Petersonetal.2004)toknowtheirdis- orbit(R ),butconsideringKerrblackholesandaccountingfor
ISCO
tancefromtheblackhole.Ifthetwoquantitiesarelinkedbygrav- Special and General relativistic effects. Many authors tried to in-
ity,wecanusedthevirialargumentandfindtheblackholemass cluderelativisticeffectstodescribetheemissionfromthedisk(e.g.
(hereafter“virialmethod”,e.g.Vestergaard2002,McLure&Jarvis Novikov&Thorne1973,Page&Thorne1974,Riffert&Herold
2002,Greene&Ho2005,Vestergaard&Osmer2009,McGilletal. 1995),andwewillbaseourstudyonannumericalcodedeveloped
by Li et al. (2005) called KERRBB and implemented on the in-
teractiveX-rayspectral-fittingprogramXSPEC(seeArnaud1996
(cid:63) E–mail:[email protected] and references therein). This is specifically designed for Galactic
(cid:13)c 2016RAS
2 Campitielloetal.
binaries,butwewillshowastraightforwardmethodforextending G=c=h=1):
itscapabilitiesforblackholesofanymass. (cid:90) dΩ
Inthispaper,weconcentrateonaspecificaspectoftheprob- Fν,obs =2g3fc−o4lEe3m exp[E /k oTbs (R)]−1 (2)
lem:howtofindaccurateanalyticalformulaethatcanreadilyde- Ω em B col
scribetheemissionpatternoftheradiationemittedbydisksaround whereg istheredshiftofthephotonandEem thephotonenergy
Kerr holes (including the case of zero spin) accounting for rela- whenitisemittedfromthedisk.Ifweaccountforlimbdarkening,
tivisticeffectsthatarefoundbythenumericaltreatment,andhow thenweshouldaddtheextratermY(θ)(see§2.2andEq.(4)).The
topassfromstar-sizeblackholestoSMBHs. colortemperatureTcolisdefinedas:
In this work, we adopt a flat cosmology with H = 68 km
s−1Mpc−1andΩM =0.3,asfoundbyPlanckCollab0orationXIII Tcol(R)=fcolTeff(R)=fcol(cid:20)Foσut(R)(cid:21)1/4 (3)
(2015). SB
2.2 Relativisticeffects
2 KERRBBSOFTWARE The effects of Special and General relativity caused by the black
holegravitationalfield,stronglymodifytheemissionpatternofthe
2.1 Basicequations
accretiondisk.Theseeffectsaremoreimportantintheinnerregion
TheKERRBlackBody(KERRBB)accretiondiskmodelisanu- of the disk (closer to the black hole) than in the outer region (in
mericalcodethatdescribestheemissionfromathin,steadystate, KERRBB the accretion disk extends from the inner most stable
generalrelativisticaccretiondiskaroundarotatingblackhole,us- orbitRISCOto106Rg,Li05).Themainrelativisticeffectsare:
ing the ray-tracing technique. It is described and compared with
• Doppler beaming — Matter orbiting close to the black hole
othermodelsinLietal.(2005,hereafterLi05)andrepresentsan
movesatrelativisticvelocities.Thiscausesrelativisticaberration,
extensionofapreviousrelativisticmodelcalledGRAD(Hanawa
timedilationandblue/red–shifting(seebelow).
1989, Ebisawa et al. 1991), which assumes a non rotating black
• Gravitationalredshift—Electromagneticradiation,produced
hole. KERRBB is the best public code for fitting accretion disk
in a deep potential well is observed redshifted by a distant ob-
spectrabecauseittakesintoaccountalltherelativisticeffectsthat server2.Thiseffectbecomesmoreimportantasgravityincreases,
othermodelsdonotordoonlyinpart.Italsotakesintoaccount
i.e.closertotheblackhole
theeffectsrelatedtotheangularmomentumoftheblackhole,that
• Self irradiation of the disk — Photons emitted by the inner
determinestheinnerradiusofthedisk(InnermostStableCircular
diskcanirradiatetheouterregionsofthediskitselforcanbecap-
Orbit,i.e.ISCO),andthereforetheradiativeefficiency.Theaccre-
turedbytheblackholebecauseofthestronggravitationalpullthat
tion disk emission is approximated by a diluted black body: the
bendstheirtrajectories(forthisreason,itiscalledreturningradia-
specificintensityatradiusRcanbewrittenas
tion).Theeffectisstrongerforrapidlyrotatingblackholes,because
(cid:20)T (R)(cid:21)4 RISCObecomessmallerforlargerspins.Forotherdetails,seeFig.
Iν(R)= eTff Bν(Tcol) (1) 1inLi05.
col • Limbdarkening—Thisopticalphenomenonmakesthecen-
where B is the Planck function, T (R) is the effective tem- tralpartofthediskbrighterthanitsedge.Foranelectron-scattering
ν eff
peratureofthediskgivenbytheStefan-BoltzmannlawF(R) = atmosphere,itcanbedescribedbyafunctionY(θ)oftheanglebe-
σ T4 (R),F(R)isthetotaldiskfluxdependingonthedistance tweenthewavevectoroftheemittedphotonsandthenormalofthe
SB eff
Rfromthecenter1,andT isthecolortemperature.Thelatter disksurface:Y(θ)=1forisotropicemission.Y(θ)= 1+3cosθ
col 2 4
couldbehigherthantheeffectivelocaltemperatureif,forinstance, for limb-darkened emission (Chandrasekhar 1950, Cunningham
ahotcoronaislocatedaboveandbelowanotherwisestandardac- 1975, Li et al. 2005). Hence, the outgoing intensity is related to
cretiondisk.InthiscasetheComptonizationprocessoccurringin theoutgoingfluxby
thecoronacanbemimickedbyassumingTcol >Teff. 1
ItistruethatinanAGNthehotcoronaispresentandactive, Iout = πFoutY(θ) (4)
butonlyintheinnerregionsoftheaccretiondisk,whileitisunim-
ThiseffectmodifiesEq.(2)withtheextratermY(θ)intheinte-
portantatlargerradii.InthiscasethecorrespondingX-rayemis-
gral.
sioncannotbeapproximatedbyalargertemperatureofthediskbut
• Frame dragging — (also known as Lense-Thirring effect).
mustbetreatedasaseparateprocess.Therefore,inthiswork,we
Thisprocessisrelatedtoarotatingblackholethatdragsthespace-
assumedthehardeningfactorf = T /T = 1,anddidnot
col col eff
time itself: during its fall towards the black hole, a particle will
accountfortheComptonizationprocess.
gain angular momentum that forces it to a co-rotation with the
An important feature of KERRBB is the self–irradiation of
blackholespin,eveninthecaseofaradialfreefall.Whena = 0
the disk (i.e. returning radiation on the disk). The outgoing flux
(Schwarzschildblackhole),sucheffectisnull.
F isthesumofthestandardemittedfluxF calculatedignoring
out 0
• Innertorque—Inthestandardtheoryofaccretiondisks,the
returningradiation,andthecontributionoftheingoingcomponent
torqueattheinneredgeofthediskisassumedzero.Thisisvalid
F ,duetotheradiationfocusedbackontothediskbytheblack
in
fornon-magnetizedorweaklymagnetizedflowsandforthindisks
hole gravity (for other details, see Section 3 and Appendix D in
(Muchotrzeb & Paczynski 1985, Abramowicz & Kato 1989, Af-
Li05).TheobservedspecificfluxF ,canbeexpressedas(with
ν,obs
shordi & Paczynski 2003). It has been suggested that a non-zero
1 The general form of F(R), depending also on the black hole spin, is 2 Aboutthiseffect,Li05correctedthewrongformulaintheGRADcode
describedinPage&Thorne1974. fortheredshiftfactor.
(cid:13)c 2016RAS,MNRAS000,1–9
DisksaroundKerrblackholes 3
torquecanarisefromtheconnectionbetweenthediskandthecen-
tralblackholebyamagneticfield(Krolik1999,Gammie1999,Li
2000,Wangetal.2002),buttheproblemisstillunderdebate.A
torqueg (cid:62) 0attheinnerboundaryoftheaccretiondiskmodifies
thetotalpowerofthediskasfollows(withG=c=h=1):
L =g Ω +ηM˙ (5)
disk in in
where Ω is the angular velocity of the inner boundary, η is the
in
radiativeefficiency,M˙ istheblackholeaccretionrate.Bydefining
thedimensionlessparameterζ =g Ω /(ηM˙),Eq.(5)becomes
in in
L =(1+ζ)ηM˙ =ηM˙ (6)
disk eff
whereM˙ =M˙(1+ζ)istheeffectiveaccretionrate.Whenζ =
eff
0,thesolutioncorrespondstoastandardKepleriandiskwithzero
torqueattheinnerboundary.Forotherdetails,seeFig.7andFig.
8inLi05.
• Light-bending—Photontrajectoriesarebentbecauseofthe
strong gravitational pull exerted by the central black hole. Since
R decreasesforlargerspins,thiseffectbecomesmoreimpor-
ISCO
tantforlargerspins.Asaresultoflight-bending,thephotonsini-
tially emitted in a direction normal to the disk, are bent to larger
angles,increasingtheradiationobservededgeon(seeFig.1).For
otherdetails,seeFig.4andFig.9inLi05.
The effect of light-bending on the photon emission pattern
fromtheaccretiondiskissketchedinFig.1.Thetoppanelshows
a Shakura-Sunyaev accretion disk (Shakura & Sunyaev 1973).
In this model all relativistic effects are neglected and even the
photons emitted close to R = 3R are not bent by gravity.
ISCO S
The KERRBB code instead accounts for all relativistic effects
evenifthediskdoesnotrotate.Thiscaseissketchedinthecentral
panel. The dotted blue circle around the black hole contains the
regionwhererelativisticeffectsarestronger.Thephotonsproduced
by the outer regions are not affected by the gravitational field
of the black hole and are not (or only moderately) bent: for this
Figure1.Schematicviewofthephotonemissionpatternofanaccretion
reason, the Shakura–Sunyaev and KERRBB models are similar
diskaround ablackhole.A distantobserveris atthebottomof thefig-
in the low frequency part, because the flux at these frequencies
uresandsees thediskface-on.PanelA:Shakura-Sunyaevcase. Theab-
is produced by regions far from the black hole where relativistic senceofrelativisticeffects(mostnotably,light-bending)makesallthepho-
effects are weak and can be neglected. Instead, photons emitted tonsemittedfromthediskgostraighttowardsadistantobserverwithno
inside the strong influence region (i.e. inside the blue circle) are modification.Alsothereisnoreturningradiationontothediskbecausethe
stronglyaffectedbylight-bending:theseenergeticphotonsarelost light-bending effect is absent. Panel B: a/M = 0 KERRBB case. The
inonedirectionandappearinanother.Furthermore,somephotons circularregionaroundtheBHiswheretheeffects,likelight-bending,are
emittedclosetotheblackholehavetrajectoriessobentthatcould moreimportant.Photonsemittedfromtheouterpartofthediskgostraight
tothedistantobserverbecausethegravitationalfieldoftheblackholeis
bedirectedtowardsthediskitself:theseenergeticphotonscanbe
notsostrongtobendtheirtrajectories.Photonsinitiallyemittedinthedi-
identifiedasthereturningradiationontothediskthatisre-emitted
rectionnormaltothedisk(initsinnerregions),cannotreachthedistant
as an additional component of the disk flux. The bottom panel
observeranymorebecausetheirtrajectoriesarebentindifferentdirections
shows a KERRBB model with a/M = 0.998: this is the case
by the strong gravitational field. Together with the gravitational redshift
of the strongest light-bending. In fact the extreme spin moves (neglected in the Shakura-Sunyaev model), this explains why the KER-
the inner radius of the disk down to almost the event horizon: RBBspectrum(witha/M = 0)islessbrightthantheShakura-Sunyaev
correspondingly, the radiative efficiency is maximum, due to the onewhenviewedface-on.Furthermore,someclosestphotonstotheblack
enhanced emission of high energy radiation (see Fig. (8)). All holehavetrajectoriessobentthatcouldbedirectedtowardsthediskitself:
these features make the disk emission brighter than the previous theseenergeticphotonscanbeidentifiedasthereturningradiationontothe
cases. diskthatisre-emittedasanadditionalcomponentofthediskflux.Panel
C:a/M = 0.998M KERRBBcase.Photonsemittedintheouterregion
gostraighttothedistantobserver.Innerphotonshavetrajectoriesbentby
Thesketchcanbeusefulalsoforlargeviewingangles:inthis
thestronggravitationalfield.Themostenergeticphotonscouldbestrongly
case photons emitted in the outer region can intercept the strong
bentinalldirectionsandsomeofthemcanbedirectedontothedisk.The
influenceregion(bluecircle)andcanbebentinotherdirections.
strongrelativisticeffectsandthehighradiativeefficiencymaketheemis-
However the main effect is that energetic photons emitted in the sionbrighterthenpreviouscases,evenifthediskisedge-on.Thetopaxis
innerregionofthediskcanbeseenbyobserversatlargeangles: isthelogarithmicdistancefromthecentralblackholeinunitofRg:RISCO
this explains why the KERRBB model with a large spin has an andradiusRPEAK (radiusatwhichthefluxismaximized)areshownas
observedbolometricluminosityalmostconstantwhilevaryingthe well.
viewingangle(seeFig.(8)).
(cid:13)c 2016RAS,MNRAS000,1–9
4 Campitielloetal.
2.3 Mainparameters
TheKERRBBcodeisbasedon4mainparameters,listedbelow:
(i) BlackholemassM:inunitofsolarmassesM .KERRBB
(cid:12)
isdesignedforstellarblackholeshencetheblackholemassrange
isfrom0.01to100solarmasses.Fig.2showshowaccretiondisk
spectrascalewithdifferentblackholemasses.Thiswillbeuseful
toscaletheKERRBBresultsfromastellartoasupermassiveblack
hole(see§3).
(ii) AccretionrateM˙:inunitof1018gs−1.Therange,designed
forstellarblackhole,isfrom0.01to1000.Thisparametercorre-
spondstotheeffectiveaccretionratedescribedinEq.(6)andthere-
foredependsontheinnertorqueζ.Inthiswork,weassumethat
theinnertorqueisalwaysnull,sotheaccretionrateissimplythe
oneofthediskM˙ = M˙ .Fig.2andFig.3showhowaccretion
eff
diskspectrascalewithdifferentaccretionrates.
(iii) Blackholespina:inunitofM,theratioa/M isdimen-
sionless. KERRBB allows to compute the spectrum for all spins
between the values a/M = −1 and +0.9999. In this work, we Figure2. KERRBBaccretiondiskspectrawithdifferentmassesandac-
have considered this range, even if it goes beyond the canonical cretionrates.Theinitial10M(cid:12)spectrum(bluesolidline)moveshorizon-
tally(i.e.infrequency)whentheblackholemasschanges(orangearrow),
maximumspina/M = 0.9982(Thorne1974).Fig.3showshow
diagonally(i.e.inbothfrequencyandluminosity)whentheaccretionrate
theaccretiondiskspectrumchangeswiththespin.
changes(redarrow).Thepositionofthepeakinfrequencyνandluminosity
(iv) Viewingangleofthediskθ:betweenthenormaltothedisk
νLν,foragivenspinvalue,followsthesameshiftingequationsfoundby
andthelineofsight.KERRBBallowstocomputethespectrumin
Calderoneetal.(2013):correctedEq.(9)andEq.(10)(dependingalsoon
therange0◦ (cid:54)θ(cid:54)85◦.InthecaseofaclassicalShakura-Sunyaev theblackholespina)canbefoundusingKERRBBresults(inthespecific
model,theflux(andsotheluminosity)scalesas∝cosθ(Shakura caseofaface-ondisk).
&Sunyaev1973,Calderoneetal.2013).IntheKERRBBmodel
the accretion disk luminosity does not follow this simple scaling
becauseoftherelativisticeffects.Theviewingangleatwhichthe
observerseesthemaximumluminosityislargerthanθ = 0◦ and
dependsontheblackholespin.SeeTable2.
Along with these main parameters, KERRBB allows to reg-
ulate the influence of the other features described above. For our
work, we considered the self-irradiation of the disk, in order to
haveamorerealisticdescriptionofthediskemission.Ontheother
hand,weneglectedtheeffectsoflimb-darkeningandinnertorque
ζ.Regardingthehardeningfactorf ,wechosetosetitequalto1,
col
becausetheComptonizationeffectisover-simplifiediftreatedwith
thisapproximation,asalreadydiscussedin§2.1.
3 SCALINGWITHBLACKHOLEMASSAND
ACCRETIONRATE
AsshowninFig.2andFig.3,thechangesinaccretionrateM˙ and
blackholemassM shiftthespectrumpeakbyaprecisevaluere-
latedtothetwoparametersthemselves.Fortheclassicaldescription Figure3. KERRBBaccretiondiskspectrawithdifferentaccretionrates
ofageometricallythinandopticallythickaccretiondiskarounda andspins,butsameblackholemass(10M(cid:12)).WhenM˙ increases,thespec-
trumpeakincreasesalso(redarrow),butlessthantheincreaserelatedtothe
nonrotatingblackhole,givenbytheShakura-Sunyaevmodel,the
increaseofthespin(orangearrow).Thelowfrequencypartofthespectrum,
positionofthepeakfrequencyν andthepeakluminosityν L
p p νp whenachanges,remainsconstant,butthisoccursatfrequenciesnotvisible
scalewiththemassandtheaccretionrateaccordingtoCalderone
inthefigure.Thelowfrequencyfluxisproducedbytheouterregionsofthe
etal.(2013).Theseequations,adaptedfortheobserveddiskemis-
disk,unaffectedbytheblackholespin.Aboutthiseffect,seealsoFig.(8).
sion3,canbewrittenas:
ν (cid:16) η (cid:17)3/4(cid:20) M˙ (cid:21)1/4(cid:20) M (cid:21)−1/2
p =A (7) whereηistheradiativeefficiencyandtheconstantsareLogA =
[Hz] 0.1 M yr−1 109M
(cid:12) (cid:12) 15.25 and Log B = 45.66. Note that for the Shakura-Sunyaev
νpLνp =B(cid:16) η (cid:17)(cid:20) M˙ (cid:21)cosθ (8) modeTl,hηet∼wo0.e1qusoatitohnesη,/fo0r.1afi∼xe1dinvibewotihngeqaunagtlieonθs,.giveunivocally
[erg/s] 0.1 M yr−1
(cid:12) the position of the spectrum peak corresponding to a given mass
and accretion rate for a Shakura-Sunyaev model. For KERRBB,
3 In Calderone et al. 2013, these equations are related to the total disk Eq.(7)andEq.(8)havethesameform,butdependalsoonthespin,
emission. that changes the efficiency η and the radiation pattern, no longer
(cid:13)c 2016RAS,MNRAS000,1–9
DisksaroundKerrblackholes 5
Figure5.Scalingmassandaccretionratefromastellarblackholetoasu-
Figure4. WecanobtainthesameKERRBBspectrum(i.e.theemission permassiveblackhole.Theinitialmodel(thinsolidblueline)isfirstshifted
peakinthesameposition)fordifferentvaluesoftheblackholemassM, infrequencybychangingthemassvaluetheninluminosity(andfrequency)
accretionrateM˙ andspina/M.Thereisthereforeafamilyofsolutions. bychangingtheaccretiondiskvalueaccordingtoEq.(12)andEq.(13).In
ConsiderthespectrumA(thicksolidblueline),andsupposetoincrease thisway,itispossibletodescribetheaccretiondiskemissiondatarelated
M atconstantM˙ andspin.WewouldobtainthespectrumB.Nowsup- tosupermassiveblackholeswithKERRBB.Notethattheinitialandthe
posetoincreasethespin,butkeepM andM˙ constant.Sincealargerspin finalspectrahavethesameEddingtonratioL/LEdd =0.053.IntheAp-
correspondstoalargerefficiency,weobtainthespectrumC.Finally,sup- pendix,Eq.(12)andEq.(13)areexpressedasafunctiononlyoftheblack
posetodecreaseM˙:wecanobtainthespectrumD,whichisalmostexactly holemass,byconsideringthattheEddingtonratiooftheinitialandthefinal
equaltotheinitialspectrumA(theyslightlydifferinthehighfrequency, modelsmustbeequal.
exponentialpart).
trumD,whichisequaltotheinitialspectrumA.Theoverlappingof
equal to cosθ. The mass and accretion rate dependences are the twomodelscanbedonebecause,asshowninFig.3,theaccretion
samebecausetheassumptionofageometricallythinandoptically rate and the spin move the spectrum peak in different directions.
thickdiskisvalidforbothmodels.ByusingKERRBBresults,it Strictlyspeaking,modelAandDarenotexactlyequal:thereare
ispossibletofindnewequations,applicableforKerrblackholes. verysmalldifferencesinthehighfrequency,exponentialpartofthe
Theycanbewrittenas spectra:forlargerspinvalues,theexponentialtailisbrighter.
ν (cid:20) M˙ (cid:21)1/4(cid:20) M (cid:21)−1/2 SinceKERRBBspectrummovesontheLogν,LogνLν plane
p =A g (a,θ) (9) accordingtotheproportionalitiesofEq.(9)andEq.(10),wecan
[Hz] M yr−1 109M 1
(cid:12) (cid:12) scalethespectrumfromstellarblackholemasstoasupermassive
blackholemass,withthesamespinvalue.Itisimportanttonote
νpLνp =B(cid:20) M˙ (cid:21)g (a,θ) (10) thatthisprocedurecanbedoneiftheemissionprocessesfroman
[erg/s] M(cid:12)yr−1 2 accretion disk around a star-sized and a supermassive black hole
arethesame.Inourcase,inbothsystemstheemissionisrequired
The functions g and g describe the dependencies on the
1 2
tobeamulti-colorblackbody.Someauthorshavepointedoutsome
black hole spin a (and hence also on the radiative efficiency)
problemswiththemulti-colorblackbodyinterpretationforSMBHs
and the viewing angle θ. We derived a functional form for both
g (a,θ = 0◦) and g (a,θ = 0◦) by fitting the variation of the (Koratkar&Blaes1999,Davis&Laor2011),buthereweassume
1 2
spectrumpeakpositionatdifferentspinvaluesandfixedθ=0◦: thatitisagoodenoughapproximationinordertouseEq.(9)and
Eq.(10).Fig.5showsanexampleofourprocedure:startingfrom
gi(a,θ=0◦) = α+βx1+γx2+δx3+(cid:15)x4+ζx5+ιx6 a stellar black hole mass (thin solid blue line), the spectrum can
x ≡ log(n−a) i=1,2 (11) beshiftedinfrequencyandluminositybyquantitiesrelatedtothe
n
initialandthefinalmass:
whereaistheadimensionalblackholespin.Table1givesthepa-
rametervaluesofthetwofunctions.TheaccuracyofEq.11isof νfin =(cid:20)M˙fin(cid:21)1/4(cid:114)Min (12)
theorderof∼ 1%.Itisimportanttonotethatthepeakfrequency νin M˙in Mfin
andluminosity,describedbyEq.(9)andEq.(10),aredegenerate
inmass,accretionrateandspin(ifθisfixed).Inotherwords,the νL M˙
samespectrumcanbefittedbyafamilyofsolutions,bychanging νfin = fin (13)
theblackholemass,accretionrateandspininaproperway.Fig.4 νLνin M˙in
showsthisdegeneracy:startingfrommodelA,supposetoincrease Inthisway,itispossibletodescribetheaccretiondiskemis-
M atconstantM˙ andspin:wewouldobtainthespectrumB.Now siondatarelatedtosupermassiveblackholeswithKERRBB.Inthe
supposetoincreasethespin,butkeepM andM˙ constant:sincea Appendix,Eq.(12)andEq.(13)areexpressedasafunctiononlyof
largerspincorrespondstoalargerefficiency,weobtainthespec- theblackholemass,byrequiringtheEddingtonratiooftheinitial
trumC.Finally,supposetodecreaseM˙:wecanobtainthespec- andthefinalmodelsequal[Eq.(A4)andEq.(A5)].
(cid:13)c 2016RAS,MNRAS000,1–9
6 Campitielloetal.
Parameters α β γ δ (cid:15) ζ ι
g1 1001.3894 -0.061735 -381.64942 8282.077258 -40453.436 66860.08872 -34974.1536
g2 2003.6451 -0.166612 -737.3402 16310.0596 -80127.1436 132803.23932 -69584.31533
Table1.Parametersofthefunctionsg1 andg2 inEq.(9)andEq.(10),writtenasageneralfunctional(11),inthecasewiththeviewingangle0◦.Using
KERRBBresults,itispossibletofindsimilarequationsforthecaseswiththediskviewingangle>0◦.
4 EMISSIONPATTERN
Spinvalue[a/M] θmax Spinvalue[a/M] θmax
4.1 Accretiondiskbolometricluminosity
0.998 63.9◦ 0.6 0.3◦
IntheclassicalnonrelativisticShakura-Sunyaevcase,thebolomet- 0.98 46.4◦ 0.5 0.1◦
ric luminosity observed at an angle θ is related to the total disk 0.95 24.6◦ 0.4 0.1◦
luminosityL bythefollowingexpression(Calderoneetal.2013): 0.94 19.1◦ 0.3 0.028◦
d
0.92 12.5◦ 0.2 0.009◦
LoSbSs(θ)=2cosθLd =2cosθηM˙c2 (14) 0.9 8.7◦ 0.1 0.002◦
As mentioned before, this is no longer valid in the general 0.8 2.2◦ 0 0.0005◦
0.7 0.8◦ −1 0◦
relativistic case for a Kerr black hole. In this case, the emission
pattern depends not only on M˙ and θ but also on the black hole
spina.Ourtaskwastofindananalyticexpressionforit.Tothis Table2.Valuesoftheviewingangleθmax atwhichthediskbolometric
aim,wewrotetheobservedbolometricluminosityas: luminosityismaximized,fordifferentspins.Notetheextremeeffectatspin
a/M =0.998andlittlefluctuationsfrom0◦forsmallerspins.Inasense,
LoKbesrr =f(θ,a)η(a)M˙c2 (15) theseresultsshowthedeviationfromthecosθ-lawatdifferentspins.
wheretheradiativeefficiencyη(a)dependsontheblackholespin.
Werequiredthatthislastequationsatisfies:
1 (cid:90) Thevaluesofα,β,γ,δ,(cid:15),ι,κofEq.20areinTableB2inthe
Lobs (θ,a)dΩ=η(a)M˙c2 (16) Appendix.TheaccuracyofEq.20isof∼ 1%.Asanexample,in
4π Ω Kerr thecasewitha/M = 0(η = 0.057),M˙ = 1018g/s,thefunction
UsingEq.15,thisimplies: thatinterpolatestheobserveddiskluminosityfordifferentanglesθ
(cid:90) π/2 is:
f(θ,a)sinθdθ=1 (17)
0 LoKbesrr(θ) =8.475·1037cosθ[1−0.958·(sinθ)6.675]0.121
First,weconsideredthebolometricluminosityofthespectragiven [erg/s] [1−(sinθ)1.918]0.202
bytheKERRBBcode,fixingthevaluesM˙,M,aandvaryingonly (21)
the viewing angle. In this way, we derived the emission pattern FollowingtheparallelstructureofemissionpatternsinKER-
numerically. Therefore we looked for an analytic expression that RBBandShakura-Sunayev,wedidnotexpectanydependenceof
couldinterpolatethenumericalresultswithagoodaccuracy.Since thefunctionf(θ,a)ontheblackholemassM andindeedwedid
wewantedtocomparethefinalpatternwiththesimple2cosθterm notfindany.Therefore,thebolometricluminosityLobs isonlya
Kerr
inEq.(14),weassumedthatthefirsttermofthefinalanalyticex- functionofθ,aandM˙,aswritteninEq.(15).
pression (function of θ) is just cosθ. Therefore, considering Eq.
(15),weassumed:
Lobs =[cosθ·(termsfunctionsofθ)] η(a)M˙c2 (18) 4.2 Pattern
Kerr
(cid:124) (cid:123)(cid:122) (cid:125)
ThroughEq.(15),whoseexplicitformisEq.(19),itispossibleto
=f(θ,a)
studytheemissionpatternoftheKerrblackholeaccretiondisk.For
Weobtainedagoodmatchusingthefollowingfunctionalform:
theShakura-Sunyaevmodel,thepatternfollowsthecosθ-law[Eq.
LoKbesrr =Acosθ[1−(sinθ)C]B[1−E(sinθ)F]D η(a)M˙c2 (14)],themaximumluminosityisat0◦anditisnullat90◦.KER-
(cid:124) (cid:123)(cid:122) (cid:125)
RBB radically changes the θ-dependence and the angle at which
=f(θ,a)
(19) theobservedluminosityismaximized(θmax)isnotnecessarily0◦.
TheaccuracyofEq.(19)isof<1%,hencerepresentsavery Itispossibletofindthisθmaxbyputtingequalto0thederivativeof
goodapproximation.Wealsotriedapolynomialfunctionwiththe Eq.(19)withrespecttheviewingangleθ.Settingη(a)M˙c2 = 1,
samenumberofparameterbuttheaccuracywaslower(> 1%)at weobtain:
smallandlargeangles,fordifferentspinvalues.Alltheparameters ∂Lobs (θ,a)
A,B,C,D,EandF arefunctionsofthespina.Wethenrepeated Kerr = [1−(sinθ)C]·[1−E(sinθ)F]+
∂θ
thisanalysisusingdifferentspinvalues.TableB1inAppendixlists
+CB(sinθ)C−2cos2θ[1−E(sinθ)F]+
thevaluesoftheparametersfordifferentspins.Withthisapproach,
we studied how the values of A, B, C, D, E and F change by +FED(sinθ)F−2cos2θ[1−(sinθ)C]=0
changingthespin.Again,welookedforananalyticalfunctionthat (22)
interpolatesthenumericalresults(seeFig.B1andFig.B2inthe
Appendix).Wefoundthatagoodrepresentationisgivenby: Theanglesθmaxincreasesforincreasingspina.Asanexam-
ple,θ (cid:39) 64◦ fora/M = 0.998and(cid:39) 25◦ fora/M = 0.95.
H(a) = α+βx1+γx21+δx31+(cid:15)x41+ιx51+κx61 By remduacxing the spin value, θmax approaches 0◦ (see Table 2 for
x = log(1−a) (20) other spin values). Note that Eq. (22) is equal to zero also for
1
(cid:13)c 2016RAS,MNRAS000,1–9
DisksaroundKerrblackholes 7
Figure 6. Emission pattern for different models: the radial axis is the
observed efficiency, given by ηobs ≡ f(θ,a)η(a). The classical non
relativistic Shakura-Sunyaev model (dashed blue line) is compared with
a/M =−1(orangeline),a/M =0(greenline)anda/M =0.797(red Figure8.KERRBBmodelbehavioratdifferentanglesandspins.Allmod-
line)KERRBBpatterns.NotethatfortheShakura-Sunyaevmodel,theob- elshavethesameM andM˙.Whena/M = 0case(bluelines),thedisk
servedefficiencyisηobs =2cosθηSS ≈0.17cosθ.Thea/M =0.797 luminositydecreasesincreasingtheviewingangleinawaysimilartocosθ.
casedescribesthemostsimilarKERRBBmodeltoaShakura-Sunyaevwith Whenthespinismaximala/M =0.998(redlines),light-bendingeffect,
sameparameters(massandaccretionrate)atθ=0◦.Atdifferentviewing alongwithDopplerbeamingandgravitationalredshiftaresostrongthatthe
angles,theKERRBBmodelisbrighter.Notealso,howfortheShakura- luminosityisalmostthesame,evenatlargeangles.Inthiscase,thetrajecto-
Sunyaevmodel,theemissionhasacircularpatternwhileforKERRBB,the riesoftheenergeticphotons,producedclosetotheblackholehorizon,are
patternis“warped”bytherelativisticeffects.Reddotsindicatetheangleat bentinalldirectionsmakingtheradiationintensitytobealmostthesame
whichtheKERRBBluminosityismaximized(Table2andFig.7). atallviewingangles.
and0.797(redline)KERRBBpatterns.ForShakura-Sunyaev,the
emissionisgivenbyacircle:forKERRBB,thiscircleis“warped”
byrelativisticeffects.Thea/M = 0.797casedescribesthemost
similar KERRBB model to a Shakura-Sunyaev, at θ = 0◦, with
thesameblackholemassandaccretionrate4:inordertoemitthe
sameluminosity,theKERRBBmodelmusthavealargerefficiency,
hencealargerspinvalue.Inthisway,thetwomodelsareequiva-
lentaroundθ = 0◦ butatdifferentviewingangles,theKERRBB
model is brighter because of the strong relativistic effects. Fig. 7
isequivalenttoFig.6:theShakura-Sunyaeviscomparedwiththe
a/M = −1(greenline),0.95(orangeline)and0.998(redline)
KERRBBpatterns.Notehow,fortheextremeKERRBB,theemis-
sionisstronglymodifiedwithrespecttotheothercases.Reddots
indicatetheanglesatwhichtheemissionpatternismaximized.
Figure 7. Emission pattern for different models: the radial axis is the
Thestrongmodificationofthespectrumemissionduetothe
observed efficiency, given by ηobs = f(θ,a)η(a). The classical non
combinationofviewingangleandspinisalsovisualizedinFig.8:
relativistic Shakura-Sunyaev model (dashed blue line) is compared with
a/M = −1(greenline),a/M = 0.95(orangeline)anda/M = 0.998 thebluespectracorrespondtoa/M =0,theredonesarethesame
(red line) KERRBB patterns. Note how, for the extreme KERRBB, the witha/M =0.998(MandM˙ areconstant).Inthea/M =0case,
emission is strongly modified with respect to the others. The angles at relativisticeffectsareveryweakandthespectrumseemtofollow
whichtheKERRBBluminosityismaximizedareindicatedwithreddots: theclassicalcosθ -law.Instead,inthea/M = 0.998cases,the
fora/M =0.998,θmax(cid:39)64◦,andfora/M =0.95,θmax(cid:39)25◦(see emitted luminosity is almost constant, even for the largest view-
Table2forotherspinvalues). ingangles.Thisisduetothecombinationofdifferentrelativistic
effects(Dopplerbeaming,gravitationalredshiftandlight-bending)
alongwiththeblackholespin:thetrajectoriesoftheenergeticpho-
θ = 90◦: this solution must not be considered because Eq. (19)
tons coming from the innermost region of the disk (that are very
isdefinedintheinterval[0◦ :85◦](see§2.3).
closetothehorizonfora/M → 1),arebentinalldirectionsand
In order to plot the different emission patterns for different
theintensityofradiationisalmostthesameatallviewingangles.
anglesandspins,wekeptconstantthevalueoftheaccretionrate
M˙ in Eq. (15) and computed the observed efficiency defined by
η ≡f(θ,a)η(a).Fig.6showstheemissionpatternfordifferent
obs
models:theradialaxisistheobservedefficiencyplottedatdifferent 4 Forthesamemass(oraccretionrate)value,thepreciseoverlappinghas
viewingangles.FortheclassicalnonrelativisticShakura-Sunyaev asmallerspin(∼0.76)butadifferentaccretionrate(ormass)value(dif-
model (dashed blue line), ηobs = 2cosθηSS ≈ 0.17cosθ. It ferencesoftheorderof∼1%).Theoverlappingcasewitha/M =0.797
is compared with the a/M = −1 (orange line), 0 (green line) istheclosestonewiththesameMandM˙,evenifimprecise.
(cid:13)c 2016RAS,MNRAS000,1–9
8 Campitielloetal.
5 DISCUSSIONANDCONCLUSIONS KERRBB emission must have a larger efficiency, hence a larger
BHspinvalue.Inthisway,thetwomodelsareequivalentaround
In this work, we studied the radiation and emission pattern from
θ = 0◦ but, at larger viewing angles, the KERRBB model is
an accretion disk around a spinning black hole. Our aim was to
brighterbecauseofthestrongrelativisticeffects.
buildananalyticapproximationoftheemissionpattern,givenby
• Relativistic effects modifies the pattern at different viewing
thenumericalmodelKERRBB(Lietal.2005),alreadydeveloped
angles:thesimplecosθ-law(followedbyShakura-Sunyaev)isno
forX-raybinariesandaccountingforalltherelativisticeffects(i.e.
longer valid, mostly due to light-bending (see Fig. 1). Hence the
Dopplerbeaming,gravitationalredshift,self-irradiationofthedisk,
maximumobservedluminosityisnolongeratθ=0◦butatlarger
light-bending),inordertoefficientlyextenditsusetosupermassive
viewingvalues.
blackholes.
• Theobservedluminosityismaximizedatdifferentanglesde-
Wefirsttestedsomefeaturesofthemodelcomparingittothe
pendingontheblackholespin:forlargerspinvalues,θ hasa
classicalnonrelativisticdescriptionbyShakura&Sunyaev(1973): max
largervalue(seeTab.2foritsvalueatdifferentspins).
weverifiedhowtherelativisticeffectsaffecttheemissionprofileof
thedisk,mainlyinthevicinityoftheblackhole.Afterwards,we
focusedonthechangesinthespectrumimpliedbychangesinthe With our approach, we introduced a simple analytic tool to
4mainparametersofthemodel:blackholemass,accretionrate, studytheimpactofdifferentspinvaluesonobservablefeaturesof
spin and inclination angle of the disk with respect to the line of the accretion disks surrounding supermassive black holes. In up-
sight.AfirstanalysisshowedthattheKERRBBmodelisaffected comingpublications,wewilldetailhowouranalyticapproximation
bychangesinmassandaccretionrateinthesamewayasastan- canbeusedtoderiveaccretionandblackholefeatures(introduc-
dard Shakura-Sunyaev model: the assumption of a geometrically inganewmethodtoestimatetheblackholespinfromthediskSED
thinandopticallythickdiskisvalidforbothmodelshencethedisk fittingprocess),andlikelybeusedforawideactivegalacticnuclei
emissionscaleswiththesameblackholemassandaccretionrate sample.
proportionalitiesfoundbyCalderoneetal.(2013).Thisusefulin-
formationallowstoshifttheKERRBBspectrum,fromastar-sized
toasupermassiveblackhole,byusingEq.12andEq.13(orequiv-
alentlyEq.A4andEq.A5if,inthetrasformation,theEddington
ratiooftheinitialandthefinalmodeliskeptconstant).Also,us-
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∼ 0.01%,hencefunction(B1)representsaverygoodapproxima-
APPENDIXA: SHIFTINGEQUATIONS tion.
Fig.B2showstheparametersA,B,C,D,E andF fordif-
Eq. (12) and Eq. (13) describe the relations that allow to rescale
the initial spectrum with mass M and M˙ , to a new spectrum ferent values of the spin, in order to find their dependence on it.
in in
with final M and M˙ , with a fixed black hole spin. The peak Asshownin§4.1,thecommonfittingequation(blueline)canbe
fin fin
frequencyν andluminosityν L followtheserelations: writtenas:
p p νp
ν (cid:16)M˙ (cid:17)1/4(cid:114)M H(a) = α+βx1+γx21+δx31+(cid:15)x41+ιx51+κx61
p,fin = fin in (A1)
νp,in M˙in Mfin x1 = log(1−a) (B2)
Thevaluesofα,β,γ,δ,(cid:15),ι,κareinTableB2.Notehowthe
νLνp,fin = M˙fin (A2) residualsarealwaysoftheorderof1%(orless),exceptforF (of
νLνp,in M˙in theorderof∼ 10%).Theselargeuncertaintiescouldbereduced
usingmorethan6parametersforthefit,butwefounditunneces-
If, in the transformation, the Eddington ratio (cid:96) = L/L
Edd saryfortheaimofthiswork.
remainsthesame,thepreviousequationsbecomefunctionsonlyof
theblackholemass.Infact,onecanwrite:
L =ηM˙ c2 L =ηM˙ c2
in in fin fin
L =CM L =CM
Edd,in in Edd,fin fin
whereC =6.9·104ergs−1g−1.TheEddingtonratioswillbe:
ηM˙ c2 ηM˙ c2
(cid:96) = in (cid:96) = fin
in CM fin CM
in fin
If(cid:96) =(cid:96) ,oneobtains:
in fin
M˙ M
in = in (A3)
M˙fin Mfin
5 TheKERRBBcodeallowstochoseaviewinganglebetweenthesetwo
Inthisway,Eq.(12)andEq.(13)become:
limitingvalues.
νp,fin =(cid:16)Min(cid:17)1/4 (A4) 6 Inthecasewithnorelativisticeffects(i.e.Shakura-Sunyaevcase),there
ν M wouldhavebeenafactorof∼11.5,betweenthecases0◦and85◦.
p,in fin
(cid:13)c 2016RAS,MNRAS000,1–9
10 Campitielloetal.
FigureB1.KERRBBdiskbolometricluminosityasafunctionoftheviewingangleofthedisk,inthecaseswitha/M =−1,a/M =−0.6,a/M =0.6
anda/M = 0.998.Thefittingfunction(blueline)hasthegeneralformof(19)withdifferentvaluesfortheparameters,inthedifferentcases.Notethe
differentbehaviors:inthecasesa/M =−1,−0.6and0.6,thebolometricluminosity,betweenthecases0◦and85◦,decreasesbyafactorof∼7,6.6and
4.4,respectively.Inthecasewitha/M =0.998,theluminosityreachesamaximumvalueat∼64◦(seeTable2)thendropsatlargerviewingangles.
Par a=0.998M a=0.95M a=0.8M a=0.6M a=0.4M
LogA −0.04950±0.00029 0.06055±0.00029 0.13668±0.00017 0.17345±0.00011 0.19378±0.00008
B −0.59008±0.00508 −0.46635±0.00576 −0.35544±0.00471 −0.28623±0.00382 −0.24788±0.00346
C 1.90743±0.01574 1.87187±0.02087 1.87208±0.01959 1.87674±0.01806 1.89046±0.01758
D 0.16910±0.00415 0.20649±0.00473 0.18649±0.00393 0.15944±0.00311 0.14239±0.00272
E 1.00344±0.00121 0.99368±0.00146 0.98357±0.00159 0.97466±0.00203 0.96833±0.00251
F 12.00334±0.56470 10.6638±0.4250 8.75966±0.26306 7.86278±0.20588 7.28707±0.18300
Par a=0 a=−0.2M a=−0.4M a=−0.6M a=−1M
LogA 0.21802±0.00007 0.21991±0.00008 0.22547±0.00008 0.23317±0.00008 0.23836±0.00007
B −0.20169±0.00433 −0.18812±0.00527 −0.18213±0.00716 −0.17351±0.00730 −0.16644±0.00889
C 1.91789±0.02450 1.94180±0.03034 1.99089±0.03919 2.01418±0.04000 2.08771±0.04433
D 0.12095±0.00310 0.11501±0.00369 0.11359±0.00537 0.10989±0.00548 0.10898±0.00723
E 0.95794±0.00514 0.95591±0.00694 0.96097±0.00785 0.96081±0.00828 0.96786±0.00751
F 6.67478±0.23957 6.37660±0.28415 5.88852±0.32039 5.67540±0.31558 5.09754±0.29246
TableB1.Parametervaluesofthefunction(B1),fordifferentspinvalues.Byusingthesevalues,itwaspossibletodescribethemwithageneralfunctional
(B2),whoseparametersareinTableB2.Thedifferentparametersalongwiththefittingfunctionandresiduals,areshowninFig.B2.
(cid:13)c 2016RAS,MNRAS000,1–9