Apparent quasar disc sizes in the "bird's nest" paradigm

0.31 MB·

by  Pavel Abolmasov

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 sin-
gle 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 accre-
tion 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 lu-
minosity 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