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Astronomy&Astrophysicsmanuscriptno.28842_am cESO2017 (cid:13) February10,2017 Apparent quasar disc sizes in the “bird’s nest” paradigm P.Abolmasov 1 TuorlaObservatory,UniversityofTurku,Väisäläntie20,FI-21500Piikkiö,Finland 2 SternbergAstronomicalInstitute,MoscowStateUniversity,Universitetskypr.,13,Moscow119992,Russia, e-mail:[email protected] ABSTRACT 7 1 Context.Quasarmicrolensingeffectsmakeitpossibletomeasuretheaccretiondiscsizesarounddistantsupermassiveblackholes 0 that are still well beyond the spatial resolution of contemporary instrumentation. The sizes measured with this technique appear 2 inconsistentwiththestandardaccretiondiscmodel.Notonlyarethemeasuredaccretiondiscsizeslarger,buttheirdependence on b wavelengthisinmostcasescompletelydifferentfromthepredictionsofthestandardmodel. e Aims.Wesuggest thatthesediscrepanciesmayarisenotfromnon-standard accretiondiscstructureorsystematicerrors,asitwas F proposed before, but rather from scatteringand reprocession of the radiation of the disc. In particular, the matter falling fromthe gaseoustorusandpresumablyfeedingtheaccretiondiscmayatcertaindistancesbecomeionizedandproduceanextendedhalothat 9 isfreefromcolourgradients. Methods.Asimpleanalyticalmodel isproposed assumingthatageometricallythicktranslucent inflowactsasascatteringmirror ] changingtheapparentspatialpropertiesofthedisc.Thisinflowmaybealsoidentifiedwiththebroadlineregionoritsinnerparts. E Results.Suchamodelisabletoexplainthebasicpropertiesoftheapparentdiscsizes,primarilytheirlargevaluesandtheirshallow H dependenceonwavelength.Theonlyconditionrequiredistoscatterasignificantportionoftheluminosityofthedisc.Thiscaneasily . befulfilledifthescatteringinflowhasalargegeometricalthicknessandclumpystructure. h p Keywords. accretion,accretiondiscs–quasars:general–scattering–gravitationallensing:micro - o r st1. Introduction tingregionisclosetotheseparationofthebinarythatdoesnot a dependonwavelength. [Whilethemechanismsresponsibleforthespectralenergydistri- It seems that this large-radius problem is typical for lower butions of bright active galactic nuclei are more or less under- mass black holes (M . 109M ), while the discs of the most 2 vstood in the frameworkof standard disc accretiontheory,there massive supermassive black ho⊙les better conform to the stan- 7arecertainobservationalresultsthatlackunambiguousinterpre- dard disc theory (see Fig. 4 in Abolmasov&Shakura (2012)). 5tation.Oneofthemostpuzzlingisthebehaviouroftheaccretion Becausethesampleoflensedquasarsisbiasedtowardsbrighter 9disc sizes measured through microlensing effects. Pooleyetal. objects, one can suggest that smaller black holes should have 8(2007) have found the observed disc sizes to exceed the stan- higherEddingtonratios, i.e. highermass accretion rates in Ed- 0dardtheorypredictionsbyone-twoordersof magnitude.Later, dingtonunits.Accretionuponsmallerblackholesshouldproba- 1.in Morganetal. (2010), thisresultwasqualitativelyconfirmed, blybeevensuper-Eddington(Shakura&Sunyaev1973).Super- 0buttherealvalueofdiscrepancyappearedtobesmaller–about Eddington mass accretion is expected to lead to the formation 7a factorofseveral.Evenmoreimportantis thatthe wavelength of outflows that are able to cover the inner parts of the visi- 1dependenceoftheradiusseemstodeviategenerallyfromtheex- bledisc.InAbolmasov&Shakura(2012),weconsiderthespa- :pectedstandardtheoryscaling.Foramulti-temperaturediscwith tialpropertiesofquasaraccretiondiscsaffectedbyaspherically v Xipaaproewnterr-aldaiwustesmhopuelrdatsucraeledeapseRndenλce1/opn.Irnadpiaurst,icTul∝arR,f−opr,tahsetaanp-- ssuypmemr-eEtdridcinscgatottneraicncgreetnivoenl.oHpeowfoervmere,dinbysuthchewaimndodperlo,dthuecemdabsys rdarddisc, T R−3/4 (Shakura&∝Sunyaev1973) and R λ4/3. accretionratesrequiredseemtobeunexpectedlylarge,m˙ >103, aBlackburneet∝al.(2011)findthatforthelargerpartofth∝eir12- wherem˙ ismassaccretionrateindimensionlessunits, object sample, the apparent disc sizes in the UV/optical range M˙c2 M˙κc (0.1 1µm) show much weaker dependenceon wavelength. An m˙ = = , (1) evid−ent solution is to assume steeper temperature profile, but L 4πGM Edd thereseemto benophysicallymotivatedmodelsabletorepro- 4πGMc duce the very small temperature slopes p > 1 (see also dis- where LEdd = κ is the Eddington luminosity and κ is cussion in Floydetal. (2009)). Another solution is to assume the (presumably, Thomson) opacity of the medium. The black thatthemeasuredspatialscalesarenotrepresentativeoftheac- hole masses inferred are inconsistent with the virial mass esti- cretiondiscitself. Forinstance,Yanetal. (2014) suggestthata mates (Abolmasov&Shakura 2013). There are no indications sub-parsecblackholebinarymodelmayreproducesomeofthe for mass accretion rates this high even in the brightest AGN observationalresults.Inthiscase,theobservedsizeoftheemit- while some estimates around m˙ 100 exist (see for example ∼ Collinetal.2002).Fornear-Eddingtonmassaccretionrates,the Send offprint requests to: Pavel Abolmasov, e-mail: super-Eddington region is small and the outflows are too fast [email protected] (v m˙ 1/2)and,hence,theopticaldepthsaretoosmall. − ∼ Articlenumber,page1of13 A&Aproofs:manuscriptno.28842_am This actually means that if one wants to reproduce the ob- torus servedquasarsizeswith the outflow-basedmodel,theoutflows eithershouldbebound(andthusbedisqualifiedasoutflows)or possessnearlyzerototalmechanicalenergy(parabolicmotion). BLR Thelatterseemsunrealisticbecausesomeprocessisrequiredto θ ensuretheoutflowhasthekineticenergythatisexactlyequalto 0 its potentialenergy.This problemis eliminated if one assumes that an inflow rather than an outflow is responsible for the for- disc mationofthescatteringenvelope.Inthiscase,nearlyparabolic motionisanaturaloutcomeofenergyconservationiftheenergy is lost at dynamical or longer timescales. The infalling matter may scatter or reprocess the radiation of the disc, thus making theeffectivesizeofthedisclarger. There is another line of evidence for the existence of scat- teringmaterialaroundsupermassiveblackholeaccretiondiscs, Fig.1. SketchillustratingtheunifiedschemeofAGNaccretioncom- which comes from the broad emission-line regions (BLRs). bining the dusty torus, BLR and accretion disc. Radiation from the Broad components of emission lines in quasars and other ac- accretion disc ionizes and destroys the dusty clouds of the torus and tivegalacticnuclei(AGNs)arecurrentlybelievedtobeformed makesthemradiateinbroademissionlines.Thematerialfromthetorus in a clumpy partiallyionized gasin a geometricallythick disc- andBLRfinallyprecipitatesintothedisc.AcoupleofBLRcloudsare likestructure(Dultzin-Hacyanetal.2000;Bonetal.2009)sur- shownschematicallywithonesidehighlightedtoindicatethephotoion- roundingtheaccretiondiscitself.ThematterinBLRsmovesat izationoftheilluminatedsideofthecloud. near-virialvelocitiespossessingatthesametimelargenetorbital momentum that is comparable to Keplerian orbital momentum InSect.2,weconstructamodeloftheionizedinflow.Inten- Untilrecently,theexistenceanddirection(inflowsoroutflows) sitydistributionsfordifferentparametersetsandhalf-lightradii of net radial motions in BLR were debated. However, recent arecalculatedinSect.3. InSect.4,ourresultsarecomparedto velocity-resolvedreverberationmappingresultssupporttheex- observationaldata. WediscusstheresultsinSect.5. istenceofaninflowinBLRsofmanyAGNs(Doroshenkoetal. 2012; Grieretal. 2013). Besides, the long-lasting problem of blueshiftedhigh-ionizationbroademissionlines(Gaskell1982) 2. Accretionflowstructure iselegantlyexplainedbyscatteringinsideaconvergingtranslu- We assume thatthegasaccretesatthe totalrateof M˙ fromthe centflow(Gaskell2009;Gaskell&Goosmann2013)existingin gaseoustorusintheformofindividualopticallythickcloudsthat theinnerpartsofBLRs. aresubsequentlydestroyedbydiscradiationandmutualinterac- This new observational evidence allows us to unify tions.Weassumethattheradialvelocity(directedinward,taken the dusty torus surrounding most of the well-studied AGN byabsolutevalue)scaleswiththelocalvirialvelocity (Tristrametal. 2012), BLR, and the accretion disc into a single accretion flow, as shown in Fig. 1. Transition from the GM torus towards the partially ionized gas of the BLR is thought vr =β , (2) R tobeconnectedtodustsublimationatthetemperatureofabout r T (1 2) 103K , which is weakly dependent on den- where β < 1 is a dimensionless coefficient. The exist- dust ≃ − × sity (Draine 2003; Nenkovaetal. 2008). The source of heating ing net radial velocity estimates for BLR are sub-virial istheradiationoftheaccretiondisc.Thebroadlinesthemselves (Gaskell&Goosmann2013)byafactorofseveral,closetobut areemittedinacertainrangeofradii,generallyinbetweenthe smallerthanthechaoticvelocitiesobservedinBLRsandrespon- sizeofthedustytorusandtheaccretiondisc.Togetherwiththe sibleforthe virialfactorof f 5.5.Virialfactorisdefinedas v ∼ evidenceforinflowthismeansthatinsidecertainradiusthema- theratio ofthevirialvelocitysquaredtothe measuredvelocity terial emitting in the BLR either enters the accretion disc or at GM dispersion, f = (seeforexampleParketal.2012andref- leastbecomestoohottomanifestitselfviaultravioletlines.Both v R v2 processes probablywork and BLR cloudsare graduallyheated erencestherein). RhanidommotionvelocitiesofBLRcloudsare and destroyed and then precipitate into a thin disc because of sub-virialbya factor of 1/ f 0.4and the radialinflow ve- v energylossmechanisms(collisionsandsubsequentcooling).At locitiesmeasuredfor Seyfertgal∼axies(Doroshenkoetal. 2012) the same time, thegeometryof the BLR inheritsthe oblatege- are a factor of several smalpler, which implies β 0.1. As we ometry of the torus and thus forms a structure sometimes re- see below in Sect. 4, the likely value of the relati∼ve velocity is ferredtoasa“bird’snest”(Mannuccietal.1992). smaller,oftheorder0.01,butinSect.3weadoptβ=0.1. Itfollowsthatthereshouldbeionizedgasscatteringnotonly Weexpressthemassaccretionrateinthediscas fd(r)M˙.All thephotonsofthebroademissionlinesbutalsotheradiationof therest(1 fd)M˙ isassumedto existin theformofageomet- − thedisc.Thedistancerangeforthismaterialisdictated,first,by ricallythickhighlyinhomogeneousflowsubtendingsomesolid heating by the disc radiation, and, second, by the cooling pro- angle Ω = 4πcosθ0, where θ0 is the half-openingangle of the cessesthatleadtoformationofthethinaccretiondisc. flow(seeFig.1),whichwehereafterconsiderfixed.Thedensity ofthegeometricallythickflowiswrittenas In this paper we show that a geometrically thick scattering inflow with sub-virial radial velocity, which gradually deposits (1 f )M˙ itsmaterialintotheaccretiondisc,infactcanexplainthebasic ρ= − d . (3) Ωv R2 featuresofthequasarspatialstructureresolvedbymicrolensing, r suchaslargersizesandweakerdependenceonwavelengththan Thisestimatedoesnotincludeclumpingeffects,whichareoflit- predictedbythestandarddiscmodel. tleimportance,aslongasscatteringisalinearprocessindensity. Articlenumber,page2of13 P.Abolmasov: Apparentquasardiscsizesinthe“bird’snest”paradigm β 1 1 InappendixA,itisshownthatscatteringopacitybecomessen- wherea= θ = ,R istheoutermostradiuswhere sitivetoclumpingonlyiftheopticaldepthsofindividualclumps β cosθ cosθ out 0 0 becomeconsiderablylarge. accretiondiscexistsor,alternatively,theoutermostradiuswhere Normally, BLR clouds are assumed to have hydrogen col- the geometrically thick inflow starts to deplete. Outside Rout, umndensitiesof1022 1024cm−2,whichmakesthemoptically fd = 0. For simplicity, we assume here the outer radius of the thin to Thomson scatt−ering. These values are relevant for ion- discequalstheouterradiusoftheionizedflowaswell.Thera- izedgas;theamountofneutralgasismoredifficulttoconstrain. dialopticaldepthofthescatteringflowisthen Totallineluminositiestogetherwithsomelimitonthetotalcov- r 4π m˙ ra 1/2 eringfactorallowustoestimatethethicknessofthecloudtobe τ = κρ(r)dR − . (10) closeto1023cm−2 (seeforinstancePeterson1997,2006).How- r Zrin ≃ (a−1/2)Ω β roaut ever, these estimates are biased by the presence of neutral gas Here,r istheinnerradiusofthedisc.Inourassumptions,a 1 and the real column densities may be larger,although there no in ≥ andthusthelowerintegrationlimitisrelativelyunimportant,es- indicationsthatthesedensitiesshouldbelarger.Thereareindi- pecially if r r . The outer radius R should be a physi- cations (Maiolinoetal. 2010) for the existence of a population out in out ≫ cally motivatedquantity that is connectedto the processes that of cometary-shaped clumps that are visible through X-ray ab- lead to destruction of the torus clouds and are responsible for sorptionandprobablyidenticaltothepopulationofBLRclouds radial drift. The actual value of R is determined by the radi- oratleastalargepartofit.Thecolumndensitiesoftheseobjects out areestimatedas1023 1024cm 2. ation fields that destroy dust and ionize the gas composing the − − clouds. Ionizationis the key process because the most relevant In case of optically thick clouds,the effective opticaldepth processis scattering by freeelectrons.1 Largedensitycontrasts is reduced by about the characteristic optical depth of a cloud. andcomplexgeometrymakeitimpossibletoestimatetheouter Here,clumpingwastakenintoaccountasacorrectionmultiplier foropticaldepthsξ =(1−e−τ1)/τ1,whereτ1istheopticaldepth rsaimdipulseacsritthereiosniztehaotfctahnebSetruösmedgirnenthriesgciaosne.iTshtheeornaldyiactlieoanrdaennd- ofanindividualcloud.Dependenceoftheobservedintensitydis- sitytemperatureoftheradiationofthedisc.Weassumethatthe tributionsuponclumpingis also addressedin Sect. 3.1. Below, outerradiusoftheionizedinflowisdeterminedbysomeradia- thelimitofτ 0isusedeverywherebydefault. 1 → tionenergydensitytemperatureT , Itisinstructivetointroducedimensionlessquantities out 4σ L Rc2 SBT4 = , (11) r= , (4) c out 4πR2 GM l0aan.r3d4mcme˙mta2lalgci−cc1iot,yrwd).ihnTigchhetnoisthteh(e1d)eT,nhsaointmydsboethnceosmcoaeptstaecriitnyg iospasceittyto(foκr so≃- rout = sGMcκ5σηSm˙BTo4ut ≃103 1T0o4uKt !2 rηm˙ 10M9M⊙. (12) Here, η is radiative efficiency of the accretion disc, which we 4π(1 f )m˙ c2 ρ= Ω −βd GMκr−3/2. (5) later assume is equal to 0.057 in all numerical estimates, as fora Schwarzschildblackhole.Thisconditionissimilar to the Theopticaldepthtoscatteringinverticaldirectionis dust evaporation condition defined by energy density tempera- tureT = T (1 2) 103K(Draine2003).Theproposed ED dust τ (r) Rcosθ0κρdz= 1− fd m˙ . (6) limitTout should≃beno−tlow×erthanTdustbecausedustefficiently v ≃ β √r absorbsEUVradiationandshieldsthegasfromthesource. Z0 Totalradialopticaldepthiscalculatedusingexpression(10) Thereis one unknownquantityleftin our problem, f = f (r). evaluatedatr=r , d d out Weconsiderthatradialandverticalmotionsbothscalewiththe 4π m˙ 1 virialvelocityandthescatteringcloudsaredestroyedwhilepass- τ (r )= ingthroughtheequatorialplane.AsweshowlaterinSect.5.3, r out (a 1/2)Ω β √rout the column density of the disc is many orders of magnitude −0.03 m˙3/4 M 1/4 T (13) out . largerthan thatof a single cloud,thereforeBLR cloudsshould ≃ (a 1/2)cosθ βη1/4 109M 104K bestronglyaffectedbypassagethroughtheaccretiondiscifthe − 0 ⊙! latterhasalreadyformedattheradiusunderconsideration.The Afractionofaboutτ Ω/4πofthediscluminosityiscapturedby r geometricallythickflowisthendepletedas the inflow and may contribute to the observed radiation of the disc. This quantity may be considerably high even for m˙ . 1 d (1 f )M˙ =4πRρv , (7) dependingonβandtheradialextentofthescatteringregion. d θ dR − (cid:16) (cid:17) where the vertical velocity is also considered virial v = θ 3. Observedintensitydistribution β √GM/R with β = β (for simplicity). The partial mass ac- θ θ cretionrateitself,accordingto (5),equals(1 f )M˙ =ΩR2ρv , We consider the observed intensity distribution composed of d r − hence, two parts: directly visible accretion disc radiation and accretion disc radiation scattered by the inflow. Accretion disc ra- d(1 f ) 4πβ 1 f − d = θ − d. (8) diation conforms more or less to the standard disc model dR Ω β R 1 RayleighscatteringisalsoexpectedtobeimportantinBLRclouds; Thisimpliesapower-lawdependenceontheradius seeGaskell&Goosmann(2013).Scatteringbydustmaybeofimpor- tance at larger distances because dust albedo is still high, about 0.5, R a inthefarultravioletaccordingtoobservational dataandmostpopular f (r)=1 , (9) d − R interstellardustmodels(seeDraine(2003)andreferencestherein). out! Articlenumber,page3of13 A&Aproofs:manuscriptno.28842_am (Shakura&Sunyaev1973),butthemassaccretionratebecomes smalleratlargeradii,M˙ = f M˙.Besides,atlargeradiitheradi- d d 103 ationofthediscisseenthroughthescatteringmaterial.Thedisc isassumedtobeobservedattheinclinationofi,whiletheeffects ofinclinationareneglectedforthescatteredradiation.Belowwe 102 assumecosi=1(discseenfaceon)ifnotstatedotherwise.Prop- agation of the photonsscattered by the outer scattering flow is 101 difficulttotreatexactly,hence,weconsiderthatsomefractionof 2 thediscmonochromaticluminosity(primarilytheradiationini- ×r tiallygoingatlargeinclinations,whichhavealargeprobability ν 100 I tobescatteredintheinflow)isscrambledandredistributedover theradialcoordinates.Finally,monochromaticintensitymaybe 10−1 expressedintheform, fL Iν = Iνdisce−τvξcosi+(1−e−τvξ)4π2Rν2, (14) 10−2 100 101 102 103 104 105 106 whereτvistheverticalopticaldepthcalculatedaccordingto(6), r,GM/c2units ξ = (1−e−τ1)/τ1 isclumpingcorrectionfactor(τ1 istheoptical Fig. 2. Intensity (multiplied by r2, in relative units) as a function of depthof a single cloud;see AppendixA), f cosθ is the luminosity fractionscattered inside the inflow (≃f = co0sθ0 for an ira=diu0s(fdoirscθ0is=se4e5n◦,fma˙ce=-o1n0)0.,TMhrBeHe=di1ff0e8rMen⊙t,eTnoeurtg=ies10w3Ker,eacnodnβsid=er0e.d1,, isotropicsourceand f =cos2θ foraflatdiscwithoutlimbdark- 0 0.1,0.3,and1Ry(fromthethinnesttothethickestcurves).Solidlines ening;relativistic effects also distortthe angulardistributionof showobserved(face-on)intensitydistributions,dottedlinescorrespond theradiationoftheinnerpartsofthedisc),Lν isthecumulative tostandarddiscintensitydistribution,anddashedlinestothecontribu- luminosityofthediscreleasedinsidecurrentradius tionofthescatteredradiation. R L =4π2 IdiscRdR, (15) ν ν ′ ′ Z0 an additional π multiplier originating from the transition be- 101 tween flux and intensity, and the local disc intensity is defined bythestandarddisctheoryasthelocalPlanckradiationwiththe 100 temperaturedeterminedbytheenergyreleaseinthedisc,given hereaccordingtoShakura&Sunyaev(1973)asfollows: 2 10−1 r 2hν3 1 × Idisc = , (16) ν ν c2 exp(hν/kT(R)) 1 I − 10−2 3 c5 1/4 r 1/4 T(R)≃ 2σ κGMm˙ 1− rin ×r−3/4fd1/4. (17) 10−3 SB ! r ! Thisexpressionisforastandard,non-relativisticaccretiondisc 10−4 withamassaccretionratemodifiedbythefactor f .Ifthedimen- 100 101 102 103 104 105 d sionless mass accretion rate becomes more than about several r,GM/c2units tens, the accretion disc may become super-Eddingtonin its in- Fig.3. Sameaspreviousfigure,butform˙ =1. nerparts(Poutanenetal.2007).Inthiscase,theinnermostparts of the disc are covered by another scattering envelope that we found,inAbolmasov&Shakura(2013),tobetoosmall,forrea- BHmasses,theinnerandouteredgesofthediscseta limitfor sonablylowaccretionrates,toexplaintheobservationaldata. discsizevariationsthatisindependentofthescatteringcompo- The photon flux from the disc as seen by the scattering nent. At higher dimensionless mass accretion rates, the optical mediumshouldalsobeattenuatedbyafactorofe−τrξ.However, depth of the outer flow becomes large and the contribution of the scattered radiation has a large (of the order of f) probabil- thescatteredradiationbecomesimportant.Inthislattercase,the ity to be scattered once more. Multiple scatterings effectively effectivesizeofthediscincreasesifthescatteredluminositybe- maketheradiationisotropic,eventuallyreproducingtheinverse- comescomparabletothevisibleluminosityofthedisc. square scaling with distance, hence we neglect this multiplier and assume that the emissivity of the scattered radiation de- creases simply R−2. The real intensity distribution should be 3.1.Apparentdiscradius ∝ ofcoursemorecomplicated. SampleintensitiesareshowninFigs.2and3fortwodiffer- The quantity primarily responsible for microlensing properties entmassaccretionrates(m˙ =1andm˙ =100)andthreedifferent of a source is its half-lightradiusR1/2 inside which half of the photon frequencies. Depending on the model parameters, spa- observedfluxisemitted(Mortonsonetal.2005), tialpropertiesmaybedifferent,butthetwo-componentstructure holds for a very broad range of mass accretion rates and outer 1 R1/2IRdR temperatures.Afterintroducingthescatteringcomponent,radius = 0 . (18) dependenceonwavelengthalwaysbecomesshallower.Atlarger 2 R +∞IRdR 0 Articlenumber,page4of13 R P.Abolmasov: Apparentquasardiscsizesinthe“bird’snest”paradigm This parameter may be contrasted to another quantity that is Clumping effects make the scattering medium more trans- more convenient for analytical estimates and much more sen- parentthusdecreasingthescatteredfractionofradiationbutin- sitive to scattered light contribution, namely to the intensity- creasing the mean radius of the halo. Sometimes it leads to a weightedmeanradius, non-monotonicdependenceofradiuscorrectionfactoruponthe wavelength.Effectsoftheopticaldepthofasinglecloudτ are + 1 ∞IR2dR showninFig.6fortwodifferentoutertemperatures.Generally, R = 0 . (19) h i R +∞IRdR increase in τ1 decreasesthe contributionof the scattered radia- 0 tion,buttheeffectbecomesprofoundonlyforτ 1andscales 1 ≫ Two quRantities may differ severely if the intensity distribution inverselyproportionaltoτ1forlargeopticaldepths. hasextendedwings.Half-lightradiusisactuallyanon-linearme- dianestimate,whichpracticallyignoreswingsinintensitydistri- butioniftheircontributiontothefluxissmall.Atthesametime, 4. Numericalestimatesandcomparisonto the mean radiusmay be affected even by a very faint extended observations halo. This is well illustrated by Fig. 4 where the mean radius Weuseobservationalaccretiondiscsizesandstructureparame- is strongly increased by the existence of faint scattering wings terstocheckthevalidityofthemodelandconstrainitsparame- while the half-light radius remains practically unaffected if the ters. mass accretion rate is small (m˙ . 1). In Fig. 5, identicalquan- tities are plotted for inclination i = 60 . In this case, accretion ◦ disccontributionistwotimessmallerandtheeffectiveradiiare 4.1.Dependenceonwavelength largerforlargemassaccretionrates. Wedonotconsiderherethe contractionoftheapparentdiscsizeduetoprojectioneffects. We parameterize the dependence of the half-light radius on Fortheintensitydistributionintroducedabove(Eq.14),one wavelengthasR1/2 λζ andcalculatethevaluesofζ intherel- ∝ canestimatetheeffectivemeanradiusinassumptionofτ 1 evant(“bigbluebump”)wavelengthrange0.1 1µm.Outside andL independentofdistance, v ≫ this wavelength range, the accretion disc is lik−ely not the pri- ν maryradiationsource.Shorterwavelengthsmaybealsoaffected R = hRi0Lν+ f (Rout−Rin)×Lν R + fR , (20) bytherelativisticeffectsandsuper-Eddingtonaccretionregime h i (1+ f)L ≃h i0 out intheinnerdiscparts.Foraninfinitestandarddisc(withoutany ν outerorinnerlimits), ζ = 4/3,butseveralprocesses,including where R isthehalf-lightradiusofanaccretiondiscwithouta h i0 thoseconsideredin the currentwork,makethisquantitylower. scatteringhalo. In Fig. 7, the estimated values of ζ are comparedto the values At the same time, it may be shown that wheneverthe half- obtained by fitting the observational data of Blackburneetal. light radius considerably exceeds the half-light radius of the (2011).Wesetβ = 0.01becausethisvalueallowsustoexplain accretion disc R , it should scale approximately as √R R . d out d the apparentaccretion disc sizes (see Sect. 4.2). In each panel, Indeed, approximating the expression for intensity as Idisc + ν thetemperatureisfixedandeachofthecurvescorrespondstoa fL /R2, one can re-write the definition of the half-light radius ν fixedphysicalmassaccretionrate. (18)as Several observational radius estimates for different wave- lengthsbetween0.1and1µmwereusedtomakethepower-law R1/2IdiscRdR+ f Lν ln R1/2 = 1 Lν 1+ f ln Rout . fitsshownbyblackcrosses.Thepower-lawslopesbestfittedto Z0 ν 4π2 Rd ! 24π2 · Rd !! theobservationaldatawerepresentedinAbolmasov&Shakura (21) (2012).TherealshapesoftheR(λ)dependencesaremorecom- plexthanatruepowerlaw,usuallyshowingflatteningsathigher TheR limitherearisesfromthecumulativenatureofthelumi- d and lower wavelengths similar to those expected for an accre- nosity L , which makes the scattering term vanish at the radii ν tiondisc modelwith finite outerandinneredges(see Figs. 6-8 smallerthanR .Finally,onecanre-writethisconditionas d in Blackburneetal. 2011). The average slopes of these curves areoftenclosetozerobutstillpositiveand,inacoupleofcases, R1/2IdiscRdR= 1 Lν 1+ f ln RoutRd . (22) theyareveryclosetotheslopeofastandarddisc. ν 24π2 · R2 Z0   1/2  Themassaccretionrateintheouterpartsofthediscisprob- Because the expressionon the left should be larger than aobnlythdeemtearsmsionfedthbeybelaxctkerhnoallep.rIofctehsesemsaasnsdacthcurestdioonesranteotindeppheynsd- RdIdiscRdR = Lν , the radius R can not exceed the geo- ical units M˙ does not depend on the black hole mass M, di- 0 ν 8π2 1/2 mensionlessmassaccretionratescalesinverselyproportionalto mR etric mean √RdRout. If the contribution of the halo becomes the mass m˙ M 1, and the effects of scattering and distorted importantandintensitybehavesasIν ∝R−2,thehalf-lightradius intensity dist∝ributi−on become more importantfor smaller black approachesthegeometricalmeanofsomeeffectiveinnerradius holes (since the inflow optical depth for a smaller black hole andtheouterradiusR , out massishigher;seeequation(13)).InFig.7,weattempttorepro- ducethe M ζ plotwith a populationof blackholesaccreting R R R . (23) 1/2 ≃ d out at fixed mas−s accretion rate (from 1 to 100M yr 1) for differ- − There apre two importantoutcomesof this estimate. Firstly, the entoutertemperatures.Usinganoutertemper⊙aturevalueabout radius scales with wavelength as R λ2/3 in this limit. De- (1 3) 103Kanddimensionalmassaccretionratesintherange 1/2 pendence on wavelength becomes eve∝n shallower as the disc 3 −10×M yr 1allowsustoexplaintheoverallbehaviourofthe − radius becomes comparable to R . Second, R exceeds R ζ − M pl⊙ot. Lower mass supermassive black holes appear in a out 1/2 d only if f and R are largeenough.The scattered flux fraction m−oderately super-Eddingtonregime (m˙ & 100, at least several out cosθ /cosidependsonthegeometricalthicknessoftheflow timeslargerthanthecriticalmassaccretionrate),andforthese 0 ∼ andontheinclinationofthedisc. black holes outflows can also affect the properties of the disc. Articlenumber,page5of13 A&Aproofs:manuscriptno.28842_am 103 103 s s nit nit u u 2 2 c/ 102 c/ 102 M M G G ,2 ,2 / / 1 1 R R M=107M ,m˙ =1 ⊙ M=109M ,m˙ =1 ⊙ 101 101 10−2 10−1 100 101 10−2 10−1 100 101 λ,µm λ,µm 104 104 s 103 s 103 nit nit u u 2 2 c c / / M M G G ,2/ 102 ,2/ 102 1 1 R R M=107M ,m˙ =100 ⊙ M=109M ,m˙ =100 ⊙ 101 101 10−2 10−1 100 101 10−2 10−1 100 101 λ,µm λ,µm Fig.4. Half-lightradii(opencircles)andintensity-weightedmeanradii(crosses)forθ = 45 ,T = 5 103K,andβ = 0.1.BHmassesare 107M (leftpanels) and109M (rightpanels). Dimensionlessmassaccretion ratesarem0˙ = 1◦(uppouetr pane×ls) and100(lower panels). Standard accret⊙iondiscpredictionsarep⊙lottedwithred solidlines,dottedlinesshowapproximationR = √R R ,whereR isthehalf-lightradiusofa 1/2 d out d standardaccretiondisc. However,thesize ofthe envelopeformedbythe outflowisex- applied our model to single spectral band accretion disc size pectedtobesmallerthanthescatteringregion(seeSect.5.4for measurementsmadebyMorganetal.(2010).Theadvantageof amoredetaileddiscussion). these data set is the existence of de-lensed fluxes that allow us Thedimensionlessmassaccretionrateentersalltheoptical tomakeindependentmassaccretionrateestimates.Weconsider depths(equations6and13)incombinationm˙/β.Hence,forin- thesamplefromMorganetal.(2010)with thedataforoneob- stance, apparentlylargemass accretionratesmaybe mimicked ject(QJ0158-4325)replacedbynewerdatafromMorganetal. bysmallradialvelocities. (2012), following our work in Abolmasov&Shakura (2012). Mass accretion rate was estimated using monochromatic de- Strongdeviationsfromthestandarddiscpower-lawasymp- lensed fluxes from Morganetal. (2010). Standardmulti-colour toticoccurwhenthefluxfromthehalobecomescomparableto accretion disc approximation allows us to link monochromatic thevisiblefluxfromthedisc.Thisrequiresasignificantpercent- ageofthediscluminositytobescattered(f & 0.5,m˙/β & 10). flux (at observer frame λobs = 0.79µm, which is close to the observable I-band flux) with the mass accretion rate assum- Fortheheaviestobjectsofthesample(seeFig.7),observational ing the black hole mass is known. Using expression (9) from ζ seems to marginally exceed the standard disc value. There areseveraleffectsthatcanreproducelargestructuralparameters Abolmasov&Shakura(2012),wecanestimatethemassaccre- tionrate asa functionoftheobservedmagnitudeofthediscas ζ 2,oneofwhichisirradiation.Iftheradiationofthediscis ≃ thermalizedinsteadofscattered,thetemperatureofthermalized radiationscaleswithradiusasT ∝r−1/2,whichimpliesr∝λ2. m˙ 7.4 10−4 109M⊙ 2 DA 310−0.4(I−19)(1+z)4cos−3/2i, ≃ × M Gpc ! ! 4.2.Apparentdiscsizesforobjectswithknownfluxes (24) Additionalconstraintsmaybe madebyconsideringthe objects whereIistheobservedI-bandmagnitude,zisredshift,andD is A wheremassaccretionratesmaybeestimatedindependently.We theangular-sizedistance(about1.5 1.7Gpcforalltheobjects). − Articlenumber,page6of13 P.Abolmasov: Apparentquasardiscsizesinthe“bird’snest”paradigm 103 103 s s nit nit u u 2 2 c/ 102 c/ 102 M M G G ,2 ,2 / / 1 1 R R M=107M ,m˙ =1 ⊙ M=109M ,m˙ =1 ⊙ 101 101 10−2 10−1 100 101 10−2 10−1 100 101 λ,µm λ,µm 104 104 s 103 s 103 nit nit u u 2 2 c c / / M M G G ,2/ 102 ,2/ 102 1 1 R R M=107M ,m˙ =100 ⊙ M=109M ,m˙ =100 ⊙ 101 101 10−2 10−1 100 101 10−2 10−1 100 101 λ,µm λ,µm Fig.5. Sameaspreviousfigure,butforanaccretiondiscinclinedbyi=60 . ◦ 1.35 3.0 T =104K T =103K 1.30 2.5 1.25 D 1.20 D 2.0 S S 2, 2, R1/ 1.15 R1/ / / 12/ 1.10 12/ 1.5 R R 1.05 1.0 1.00 0.9150−1 100 101 102 0.150−1 100 101 102 τ τ 1 1 Fig.6. Half-lightradiusdependenceupontheopticaldepthofasinglecloudfordifferentwavelengthsof0.1(solidline),0.3(dashed),and1µm (dotted). Radius is expressed inthe units of the standard disc half-light radius at the same wavelength. Black hole mass M = 108M mass BH accretionratem˙ =10,temperaturelimits103(left),and104K(right).Thediscisseenface-on,θ =45 . ⊙ 0 ◦ Articlenumber,page7of13 A&Aproofs:manuscriptno.28842_am 3.5 3.5 T =103K T =3×103K 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 ζ ζ 1.0 1.0 0.5 0.5 0.0 0.0 −0.5 −0.5 10−2 10−1 100 101 10−2 10−1 100 101 M,109M units M,109M units ⊙ ⊙ 3.5 3.5 3.0 T =5×103K 3.0 T =104K 2.5 2.5 2.0 2.0 ζ 1.5 ζ 1.5 1.0 1.0 0.5 0.5 0.0 0.0 −0.150−2 10−1 100 101 −0.150−2 10−1 100 101 M,109M⊙units M,109M⊙units Fig.7. Comparisonofstructureparameterζestimatedfromobservationaldata(blackcrosses)andcalculatednumerically. Virialmassisshown alongthex-axes. Fromlefttoright,theouter temperaturevaluesare103,3 103 (upperpanels), 5 103,and104K(lowerpanels).Thethree curvesineverycasecorrespondtodifferentabsolutemassaccretionrates(soli×dblue,dashedred,andd×ottedgreencurvescorrespondto1,10,and 100M yr 1,respectively). Thehorizontaldottedlinecorrespondstothestandarddisclimitζ=4/3.Everywheretheinflowhalf-openingangleis − θ =4⊙5 . 0 ◦ ObjectsandtheirbasicpropertiesarelistedinTable1.Half- prehensivestatistical analysis is requiredin future. However,it lightradiicalculatedusingourmodelwerecomparedtotheob- also means that observed accretion disc sizes are easily repro- servationalresultsusingacrudeχ2criterion, ducedbyamodellikeours. lgR lgR 2 Thevaluesofβfavouredbyourfittingaremuchsmallerthan χ2 = 1/2− 1/2,model , (25) thecharacteristicvelocitydispersionsinvirialunits( 1/ fv ∆lgR1/2 ! 0.4)knownfrombroadline profiles. Smallradialve∼locity ma≃y X result from the inefficient angular momentum transportpin the wheretheuncertainty∆lgR is half-widthof70%intervalin 1/2 inflowingmatter.To moveinward,thescatteringcloudsshould lgR measured by microlensing effects. Summation was per- 1/2 have some means of angular momentum loss or redistribution. formedforthewholesetof11objects.Weassumedi=60 and ◦ For the case of accretiondiscs, this is providedby turbulentor θ = 60 .InFig.8,contoursofconstantχ2 areoverplottedwith 0 ◦ magnetic viscosity. We consider several processes possibly re- theshadesrepresentingthepredictedvaluesofζ inthespectral sponsibleforangularmomentumtransferintheouter,geometri- range0.1 1µm. − callythick,partoftheflowinSect.5. The fluxes allow us to estimate the mass accretion rates of m˙ 0.1 10forthe sampleof objects,whichmakesan upper LowertemperaturesT . 3 103Kmakeitpossibletore- out ∼ − × limit for β. As it is shown in Fig. 8, observational data favour producedifferentζfordifferentobjects.AscanbeseeninFig.7, β . 0.1andsetan upperlimitforT . 2 104K.Theminimal hightemperaturesrestricttherangeofpossiblestructureparame- valueofχ2isabout1.5for9degreesoffre×edomthatmeansthat ters,hencetheexistenceofnearlystandarddiscsinsomequasars the modelis over-definedfor the existing data, and more com- isnaturallyreproducedonlyforβ 0.01,T (1 3) 103K. out ∼ ∼ − × Articlenumber,page8of13 P.Abolmasov: Apparentquasardiscsizesinthe“bird’snest”paradigm Table1. ObjectsetusedforfittinginSect.4.2.Redshiftsz,virialmassesM ,half-lightradiiR ,de-lensedI-bandmagnitudesI ,dimensionless vir 1/2 corr massaccretionrates,andobservedstructureparametersζ aregiven. obs ObjectID z M ,109M R ,1015cm I m˙ ζ vir 1/2 corr obs QJ01584325 1.29 0.16 ⊙ 4.9 19.4 19.09 0.12 2.3 50.5 – − ± − HE04351223 1.689 0.5 2.4 38.7 20.76 0.25 0.0 1.2 0.1 1.1 − ± − − SDSS0924+0219 1.52297 0.11 1.0 4.9 21.24 0.25 0.3 10.1 0.15 0.8 − ± − − FBQ0951+2635 1.24603 0.89 12.2 77.2 17.16 0.11 1.0 21.1 – − ± − SDSS1004+4112 1.73995 0.39 1.0 3.9 20.97 0.22 0.1 1.6 – − ± − HE11041805 2.3192 2.37 9.7 30.7 18.17 0.31 0.1 4.5 1.65 0.5 − ± − ± PG1115+080 1.73547 1.23 38.7 193.8 19.52 0.27 0.0 1.2 0.4 0.5 − ± − ± RXJ11311231 0.654 0.06 3.1 7.7 20.73 0.4 0.2 8.3 0.4 0.5 − ± − ± SDSS1138+0314 2.44375 0.04 0.5 7.7 21.97 0.19 2.9 79.0 0.4 0.5 − ± − ± SBS1520+530 1.855 0.88 7.7 19.4 18.92 0.13 0.2 5.3 – − ± − Q2237+030 1.695 0.9 4.9 19.4 17.9 0.44 0.5 25.7 1.15 0.2 − ± − ± Notes.ExceptforQJ01584325,alltheradiiwereestimatedinMorganetal.(2010),dataforQJ01584325weretakenfromMorganetal.(2012). Massaccretionrateswereestimatedusingthede-lensedfluxvaluesofMorganetal.(2010)inassumptionofisotropicemissionusingformula24. Structureparameterestimatesζ weremadebyus(Abolmasov&Shakura2012)usingthedatabyBlackburneetal.(2011). obs 5. Discussion 1.4 1.2 5.1.Openingangleandcoveringfactor 101 1.0 K For our model, it is importantthat the scattered flux is compa- 30 0.8 rable or exceedsthe primary flux from the accreΩtion d1isc. This 1 requiresalargecoveringfactorofthescatterer, & . ,ut 0.6 4π 2 o T The geometry of dusty tori was studied for different kinds 0.4 of AGN, mainly obscured Seyfert galaxies, in several papers (Ibar&Lira 2007; Ichikawaetal. 2015). The estimated half- 0.2 openinganglesareabout60 .Ontheotherhand,about50 60% ◦ − 10100−3 10−2 10−1 100 101 0.0 θofAG5N0 ar6e0ob.scured(Daviesetal.2015),whichagainresultsin 0 ◦ ∼ − β The vertical structure of a BLR, however, does not show any single characteristic opening angle. For instance, Kollatschny&Zetzl(2013)studythegeometryoftheregionin Fig.8. Structureparameterζ foraM =109M ,m˙ =1discasafunc- tion of β and outer temperature. Overplotted a⊙re lines of constant χ2 differentlines for several nearby well-studied Seyfert galaxies. (minimal value +2.3and +4.6, corresponding to70% and 90% prob- TherelativethicknessoftheregionH/Rvariesfrom 0.1 0.3 abilityiftwoparametersarevaried)forthefittingofdiscsizesofthe for Balmer lines to 1 for hotter lines such as Civ∼λ1549−. Al- ∼ Morganetal.(2010)sample. thoughtheuncertaintiesarestillveryhighandtheeffectisless studiedathigherluminosities,itsuggeststhesolidangleofthe regionislarge,possiblylargerthanthesolidangleofthedusty torus.ThepossibleinwardincreaseofthesolidangleoftheBLR doesnotnecessarilycontradicttheproposedsettlingofthema- Theexistenceoftwopopulationsofquasars,withsmallandlarge terialintothedisc.Iftheformationofthediscisdrivenbycloud (about unity) ζ , is then easily explained by roughly universal collisions,as proposedin Sect. 5.3, some of the matter maybe physical mass accretion rates about M˙ 3 30M yr 1 (see − easilybroughtto orbitswith largerinclinations.Besides, radia- Fig.7). ∼ − ⊙ tionandwindfromtheaccretiondisccanaffectthemotionofthe cloudsthroughdragandoutwardpressure.Detailsofthephysics All these results were obtained for θ = 60 . Smaller solid 0 ◦ andtheinterpretationoftheobservationalresultsareuncertain. anglesrequiresmallerβ, approximatelyasβ Ω.Itisdifficult ∝ toputanyconstraintswith thedataonhand,butΩ 1would If the radiation of the disc is scattered efficiently, it is ge- ≪ make it impossible for the scattered component to provide an ometrically channelled in a solid angle 4π Ω, which means important contribution. Probability of secondary scatterings is that the observed scattered flux is enhance−d by the factor of likelytogotozerointhiscasehenceexpression(14)isnolonger 4π −1 valid, and the secondary (scattered) intensity is limited by the 1 . This becomes importantfor θ . π/3, when the Ω ∼ Ω − value Idisc.Therefore,wearguethatthesolidangleshould am plificatio!nfactorisabouttwo.Wedidnotconsiderthiseffect ∼ 4π ν Ω 1 as it relies on the unknownbeamingpattern of the radiationof belarge,probably & ,forourmodeltowork. thediscandexactgeometryofthescatteringregion. 4π 2 Articlenumber,page9of13 A&Aproofs:manuscriptno.28842_am 5.2.Massaccretionrates disctheory(Shakura&Sunyaev1973)as Aswehavealreadyshown,thehalf-lightradiusstartsdeviating M 1/5 from that of the disc if the scattered flux becomes comparable 7.2×1030α−4/5m˙3/5 109M r−3/5cm−2, r.1200m˙2/3, wtohtehreeflτuxisfrgoimvetnhebydis(1c.3T),haensdcabtetecroemdefsraacrtoiounndisurnoiutyghwlyhe4Ωnπdτir-, Ndisc ≃ 2.6×1031α−4/5m˙7/10 10M9M⊙!1/5r−3/4cm−2, r&1200m˙2/3. mensionrlessmassaccretionratereachesthecriticalvalue  ⊙! (28) Thetwo cases correspondto the so-calledzonesb andc of the m˙ 100 a 1 4/3 β104K 4/3η1/3 M −1/3. (26) standarddisc,differingindominatingopacitysources.Theinner cr ≃ − 2! Tout ! 109M⊙! radiation-pressure dominated zone a is too compact compared tothe size of theBLR. Thedimensionlessradiusr hereshould Forrealisticvaluesofβ 0.01,T 103K,a 1 1,η 0.06, beclose totheouterradiusthatweconsiderinourmodelorto ∼ ≃ − 2 ≃ ≃ the empirical size of the BLR that is known to scale approxi- and M 109M thecriticalmassaccretionratem˙cr 2.More matelyasR √L(Kaspietal.2007).Therelationgivenby detailed∼calcula⊙tions presentedin Sect. 4 supportthis≃estimate. BLR ∝ Kaspietal.(2007)maybewrittenas Adjusting the free parametersβ, T , and a (or the solid angle out oftheflow,asa cos−1θ0)allowsustoshiftthecriticalvalue M Hinobwreovaedr,lilmarigtse,m˙w≃hcaenrenothtebemeoxsctlupdroebdaebiltehevra.lue is aroundunity. RBLR ≃0.2sηm˙ 109M pc, (29) ⊙ Evidently,m˙ &100producesaluminosityof&100η 10in ifoneidentifiestheUVluminositywiththebolometricluminos- ∼ Eddingtonunits,whichmakestheaccretiondiscsupercriticalin ity λL (1350Å) L = ηm˙L . Here, m˙ is the dimensionless λ Edd itsinnerparts.TheregionofthediscwherethelocalEddington massaccretionra≃te(introducedabove,seeEq.(1))andηisthe 3 L limitisviolatedisrestrictedbythespherizationradiusrsph ≃ 2m˙ overallaccretionefficiencyη= M˙c2.Theproductηm˙ = L/LEdd (Shakura&Sunyaev 1973; Poutanenetal. 2007). On the other isalsoknownastheEddingtonfactor. hand,themonochromatichalf-lightradiusofthestandardaccre- Intermsofdimensionlessradius, tiondiscis,accordingtoMorganetal.(2010),about2.44radial scalelengths,or 109M rBLR 4000 ηm˙ ⊙, (30) ≃ M r1/2 ≃170 1λµ 4/3m˙1/3 10M9M −1/3, (27) c2l0o0s0eKto. the router radius given by expression (12) for Tout ≃ ! ⊙! At this radius, the hydrogen column density in the disc whichiseitherclosetoorlargerthanthespherizationradiusfor N 1028 1030cm 2, greatly exceeding the column densi- disc − theparametersweareinterestedin.Therefore,theeffectsofdisc ∼ − ties estimated for BLR clouds. Unless all the column densities thickness,advection,andoutflowsareimportantmainlyforthe in BLR are vastly underestimated,the accretion disc easily be- EUV part of the spectrum λ . 0.1µm that is still poorly cov- comesanimpassiblebarrierforBLRcloudsandthusefficiently ered by microlensingstudies. However,if the supercriticalpart absorbstheirmaterial. ofthediscformsanoutflow,thepseudo-photosphereoftheout- Formationoftheseedaccretiondiscthatsubsequentlydrains flowmaybecomecomparableinsize totheaccretiondisc. We mass from the BLR may be connected to cloud-cloud interac- considered this pseudo-photosphere in Abolmasov&Shakura tion.Indeed,foragivensphericalcloudofdensitynandcolumn (2012, 2013) and conclude that its contribution can affect the densityN,thenumberofcollisionspersecondisabout apparentdiscsizeatshorterwavelengthsλ 0.1µm.However, asweshowbelowinSect. 5.4,theeffectof∼aninflowisgener- π N 2 ν vn , (31) allystrongerthanthatofasupercriticalwind. cc ≃ 4 n c (cid:18) (cid:19) 2GM 5.3.AccretiondiscformationandinteractionwiththeBLR wherenc isthevolumedensityoftheclouds,andv clouds ∼ s fvR is the mean relative collision velocity. The number density of Inourmodel,thematerialissupposedtoentertheaccretiondisc thecloudsmaybeexpressedthroughmassaccretionrate,radial throughdestructionof BLRclouds.Thisstatementneedssome velocity,andcloudmassas justification.Forinstance,ifthesurfacemassdensityofacloud M˙ is much larger than that of the accretion disc, it easily passes n , (32) throughthe disc losing only a small portion of its verticalmo- c ≃ 4πR2vrMc mentum. On the other hand, collision with a high surface den- where M N3n 2m isthemassofacloud.Neglectingmulti- sitydiscleadstosignificantlossinverticalmomentum.Besides, c ≃ − p pliersoftheorderunity, such a collision is highly supersonic and probablyleads to de- structionofthecloud. GMm˙ ν . (33) Column number densities of BLR clouds are usually esti- cc ≃ βσ cR2N T mated as N 1023 1024cm 2 although most of the esti- BLR ∼ − − AttheempiricalBLRradiusgivenbytheestimate(29),thisfre- matesonlyconstraintheamountofionizedgasandarethusonly quencybecomes lowerlimits(Korista&Goad2000;Goad&Korista2015).The surfacecolumndensityoftheaccretiondisc,ontheotherhand, 1 M 1023cm 2 ν 10 9 − s 1, (34) may be estimated usingthe equationsof the standardaccretion cc ∼ − β109M N − ⊙ Articlenumber,page10of13