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- ingKERRBBresults,theequationsrelatedtothepeakfrequency REFERENCES andluminosityofthespectrum,foundbyCalderoneetal.,canbe Abramowicz,M.A.&Kato,S.1989, ApJ,336,304. adaptedtorotatingblackholes:newequations9and10andFig.4 Afshordi,N.&Paczynski,B.2003, ApJ,592,354. showedthatthesamespectrumcanbereproducedbyafamilyof Arnaud,K.A.1996, AstronomicalDataAnalysisSoftwareand solutions,withdifferentvaluefortheBHmass,theaccretionrate SystemsV.ASPConferenceSeries,Vol.101,eds.G.H.Jacoby andthespin. andJ.Barnes,p.17. 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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