The condensation of the corona for the correlation between the hard X-ray photon index $\Gamma$ and the reflection scaling factor $\Re$ in active galactic nuclei PDF
Preview The condensation of the corona for the correlation between the hard X-ray photon index $\Gamma$ and the reflection scaling factor $\Re$ in active galactic nuclei
MNRAS000,1–8(2016) Preprint17January2017 CompiledusingMNRASLATEXstylefilev3.0 The condensation of the corona for the correlation between the hard X-ray photon index Γ and the reflection scaling factor in active ℜ galactic nuclei Erlin Qiao 1,2⋆ and B.F. Liu 1,2 1KeyLaboratoryofSpaceAstronomyandTechnology,NationalAstronomicalObservatories,ChineseAcademyofSciences,Beijing100012,China 7 2SchoolofAstronomyandSpaceSciences,UniversityofChineseAcademyofSciences,19AYuquanRoad,Beijing100049,China 1 0 2 n AcceptedXXX.ReceivedYYY;inoriginalformZZZ a J 6 ABSTRACT 1 Observationally,it isfoundthatthereisa strongcorrelationbetweenthe hardX-rayphoton indexΓ andthe Comptonreflectionscaling factor in active galacticnuclei.Inthispaper, ] ℜ E weproposethattheΓ correlationcanbeexplainedwithintheframeworkoftheconden- −ℜ H sationofthehotcoronaontothecoldaccretiondiscaroundasupermassiveblackhole.Inthe model,itispresumedthat,initially,averticallyextendedhotgas(corona)issuppliedtothe . h centralsupermassiveblackholebycapturingtheinterstellarmediumandstellarwind.Inthis p scenario,whentheinitialmassaccretionrate M˙/M˙ & 0.01,atacriticalradiusr ,partof Edd d - o the hot gas beginsto condense onto the equatorialdisc plane of the black hole, formingan r innercoldaccretiondisc.Thenthematterisaccretedintheformofthedisc-coronastructure t extendingdownto the innermoststable circularorbitsof the blackhole. Thesize of the in- s a nerdiscisdeterminedbytheinitialmassaccretionrate.Withtheincreaseoftheinitialmass [ accretion rate, the size of the inner disc increases, which results in both the increase of the Comptonreflectionscaling factor and theincrease ofthe hardX-rayphotonindexΓ. By 1 ℜ v comparingwith a sample of Seyfertgalaxies with well-fitted X-ray spectra, it is found that 1 ourmodelcanroughlyexplaintheobservations.Finally,we discuss thepossibility to apply 1 ourmodeltohighmassX-raybinaries,whicharebelievedtobefueledbythehotwindfrom 2 thecompanionstar. 4 0 Keywords: accretion,accretiondiscs–blackholephysics–galaxies:active . 1 0 7 1 1 INTRODUCTION factor ,definedas =Ω/2π(withΩbeingthesolidanglecov- v: eredbyℜtheaccretionℜdiscasseenfromthehotcorona).Originally, The hard X-ray spectra of bright Seyfert galaxies can phe- i itwasfoundthatthereispositivecorrelationbetweenΓand by X nomenologically be described by a power law with a high en- ℜ analyzingtheobservationaldataofGingaforblackholeX-raytran- ergy exponential cut off at a few hundred keV, i.e., F(E) r ∝ sientGX339-4inthelow/hardstate(Uedaetal.1994),whichwas a E−Γ+1exp(−E/Ec)(withΓbeingthehardX-rayphotonindex,and confirmedbyanalyzingtheRXTE/PCAdatainthelow/hardstate E beingthee-foldingcut-offenergy),plusareflectioncomponent c of GX 339-4 during 1996-1997 (Revnivtsevetal. 2001). A simi- (e.g. Nandra&Pounds 1994). It is believed that the hard power- larcorrelationbetweenΓand wasalsofoundinblackholehigh lawX-rayspectrumisproducedbytheinverseComptonscattering ℜ massX-raybinaryCygX-1(Gilfanovetal.1999;Ibragimovetal. ofthesoftphotonsfromacoldaccretiondiscinanopticallythin, 2005; Gilfanov 2010, for review). The first report on the corre- hotcorona(Sunyaev&Titarchuk1980).Whilethereflectionspec- lation between Γ and in active galactic nuclei (AGNs) was trumisgenerallythoughttobeproducedbytheilluminationofthe ℜ by Magdziarzetal. (1998), in the type 1 Seyfert galaxy NGC cold disc by the hot corona (Whiteetal. 1988; George&Fabian 5548. Later, this correlation was confirmed by a comprehensive 1991; Laor1991;Doneetal. 1992;Magdziarz&Zdziarski 1995; study of a sample, including Seyfert galaxies, radio galaxies and Ross&Fabian2005;Dauseretal.2010;Garc´ıaetal.2011,2013). black hole X-ray binaries and weakly magnetized neutron stars The relative strength of the reflection component depends on the (Zdziarskietal. 1999, 2003). A recent study on a sample of 28 fractionoftheemissionofthehotcoronainterceptedbythecold Seyfertgalaxiesfurtherconfirmedthiscorrelation(Lubin´skietal. accretion disc, which is often normalized by a reflection scaling 2016). ThecorrelationbetweenΓand issuggested tobeusedto ℜ ⋆ E-mail:[email protected] explorethegeometryoftheaccretionflowaroundasupermassive c 2016TheAuthors (cid:13) 2 ErlinQiaoandB.F.Liu blackholeinAGNs(e.g.Zdziarskietal.1999).Generally,itisbe- clarity.Westudytheinteractionbetweenthehotcoronaandapre- lievedthat,forluminousAGNs,mainlyincludingthebrightSeyfert existinggeometrically thin, optically thick, cold accretion discin galaxiesandradioquietquasars,theaccretionmodeisacoldac- theverticaldirection.Theinteractionbetweenthediscandcorona cretion disc sandwiched by a hot corona extending down to the leadstoeitherthematterintheaccretiondisctobeevaporatedto innermost stable circular orbits (ISCO) of the central black hole thecoronaorthematterinthecoronatobecondensedtothedisc (Shakura&Sunyaev1973;Shields1978;Malkan&Sargent1982; untiladynamicequilibriumisestablishedbetweenthediscandthe Elvisetal.1994;Kishimotoetal.2005;Shangetal.2005).Insuch corona.Wetaketheverticallystratifiedmethodtotreattheinterac- adisc-coronamodel,thecoldaccretiondiscmainlycontributesto tionbetweenthediscandthecorona,inwhichtheaccretionflowis theoptical/UVemission,andthehotcoronamainlycontributesto dividedintothreepartsintheverticaldirection,i.e.,ahotcorona,a theX-rayemission.Meanwhile,ifthehotcoronaisassumedtobe thindisc,andatransitionlayerbetweenthem. movingawayfromortowardsthecoldaccretiondiscwithamildly relativisticbulkvelocity,themodelcanwellexplaintheobserved correlation between the hard X-ray photon index Γ and the re- 2.1 Thecorona flectionscalingfactor (Beloborodov1999;Malzacetal.2001). The corona is treated as a two-temperature hot accretion flow, ℜ However,theoretically,inthedisc-coronamodel,theformationof which is described by the self-similar solution of the advection thecoronaisabigproblem.Historically,itwasonlyassumedthata dominatedaccretionflow(ADAF)(Narayan&Yi1994,1995).The fractionofthematterinthediscistransferedtothecoronalregion pressure p, the electron number density n , the viscous heating e toexplaintheobservedstrongX-rayemissionsinluminousAGNs rate q+ and the isothermal sound speed c as functions of black s (e.g.Haardt&Maraschi1991,1993;Svensson&Zdziarski1994; holemassm,massaccretionratem˙ ,viscosityparameterαandthe c Sternetal.1995).Althoughsomeimportantprogresseshavebeen magneticparameterβ(definedas p = B2/8π = (1 β)p,where achievedviamagnetohydrodynamic(MHD)simulationsinthepast p = p + p ,with p beingthemgaspressureand−p beingthe gas m gas m few decades, it is still in debate currently (e.g. Miller&Stone magneticpressure)areexpressedas(Narayan&Yi1995), 2000;Hiroseetal.2006;Bai&Stone2013;Fromangetal.2013; Uzdensky 2013; Takahashietal. 2016). Jiangetal. (2014) per- p=1.71×1016α−1c−11c13/2m−1m˙cr−5/2 gcm−1s−2, formedathreedimensionalradiationMHDsimulation,theyfound ne=2.00×1019α−1c−11c3−1/2m−1m˙cr−3/2 cm−3, (1) that the fraction of the energy dissipated in the corona increases q+=1.84×1021ε′c13/2m−2m˙cr−4 ergscm−3s−1, withdecreasingthesurfacedensityoftheaccretiondisc.However, c2 =4.50 1020c r 1 cm2s 2, s × 3 − − with a lowest surface density in their simulation, the maximum wheremistheblackholemassscaledwiththesolarmassM ,m˙ fractionoftheenergydissipatedinthecoronaisonly3.4%,which isthecoronal/ADAFmassaccretionratescaledwiththeEddin⊙gtonc is inconsistent with the strong X-ray emission often observed in accretionrate M˙ ,ristheradiusscaledwiththeSchwarzschild luminousAGNs(Vasudevan&Fabian2007,2009). Edd radiusR (withR =2.95 105mcm),and As suggested by Liuetal. (2015), the physical properties of S S × tAhGeNinsit.iaInlgtahsefupealpiesr,imwpeoirntatenrtpfroerttthheeXΓ-rayemciosrsrieolnatfioornluinmilnuomuis- c1 = (53+α22ε′)g(α,ε′), −ℜ nous AGNs within the framework of the condensation of the hot corona onto the cold accretion disc around a supermassive black c3 = 2ε(95+α22ε′)g(α,ε′), hole.Inthismodel, itispresumedthat,initially,thematterisac- crfeorgertieoM˙dn/ibMn˙eEytdhodend&fothr0me.0B1oof(nwvdeiitrhrtiacMda˙iluEldsydoe=fxtt1eh.ne3d9bel×dac1hk0oh1t8ogMlaes/.MI(tc⊙iosrgofson−ua1)n),dawtthhtahetne, ε′= εf = 1f5γ/3−−1γ, 1/2 (2) thegasflowstowardsthecentralblackhole,atacriticalradiusr d g(α,ε′)= 1+ 18α2 1, , a fraction of the hot gasbegins to condense onto the equatorial (5+2ε′)2 − discplaneoftheblackhole,forminganinnercoldaccretiondisc. Thenthegaswillbeaccretedintheformofthedisc-coronastruc- γ= 32−24β−3β2, ture extending down to the ISCO of the black hole. The size of 24−21β theinnerdiscisdeterminedbytheinitialmassaccretionrate.With with f beingtheadvectedfractionoftheviscouslydissipateden- theincreaseoftheinitialmassaccretionrate,thesizeoftheinner ergy. The energy balance of theelectrons inthe corona/ADAF is discincreases,whichresultsinboththeincreaseof theCompton determinedbythefollowingequation, reflectionscalingfactor andtheincreaseofthehardX-raypho- tonindexΓ.BycompariℜngwithasampleofSeyfertgalaxieswith ∆Fc/H=qie−qrad, (3) well-fitted X-ray spectra, it is found that our model can roughly where∆F referstothefluxtransferedfromtheupperboundaryof c explaintheobservations.OurmodelisbrieflydescribedinSection thecorona/ADAF totheinterfaceofthetransitionlayer. H refers 2.Thenumericalresultsandsomecomparisonswithobservations totheheightofthecorona/ADAF,givenbyH =(2.5c )1/2rR .q 3 S ie areshowninSection3.SomediscussionsareinSection4,andthe refers to the energy transfer rate from ions to electrons (Stepney conclusionsareinSection5. 1983), which is re-expressed for a two-temperature hot accretion flowas(Liuetal.2002), 2 THEMODEL qie=(3.59×10−32gcm5s−3K−1)neniTi mkTce2!−3/2. (4) e Initially,thematterispresumedtobeaccretedintheformofaver- q (n ,T )=q +q +q +q referstothecoolingrateof rad e e brem syn cmp excmp ticallyextended,opticallythin,hotgas(corona)beyondtheBondi theelectronsinthecorona/ADAF, withq ,q andq being brem syn cmp radiusofthecentralblackhole.OnecanseethepanelaofFig.1for thebremsstrahlung coolingrate,synchrotron cooling rateandthe MNRAS000,1–8(2016) Γ- correlationinAGNs 3 ℜ corresponding self-Compton cooling rate respectively. q , q thiscriticalradiusoutwards,m˙ >0,indicatesthatthematterevap- brem syn z and q are all the functions of electron number density n and oratesfromthedisctothecorona/ADAF. cmp e electrontemperatureT (Narayan&Yi1995).q istheComp- Currently,wedon’tconsiderthepotentialcoolingmechanism e excmp toncoolingrateoftheunderlingdiscphotonstotheelectronsinthe suchastheComptoncoolinginthetransitionlayer.Althoughthe corona/ADAF,whichisgivenby Comptoncoolinginthetransitionlayershouldbenotveryimpor- tant, as analyzed by Meyeretal. (2007), it will also increase the 4kT q = en σ cu, (5) radiationofthetransitionlayertosomeextent.Consequently, the excmp m c2 e T e energytransferedfromthecorona/ADAFtothetransitionlayerwill withubeingtheseedphotonenergydensityoftheunderlyingdisc. bemoreeasilytoberadiatedout,anincreasedamountofthematter ucanbeexpressedasafunctionofthedistancefromtheblackhole, willmovetowardsthediscplane,leadingtoaslightincreaseofthe i.e.,u = 12aTe4ff(r) = 12a{2.05Teff,max(3r)3/4[1−(3r)1/2]1/4}4 (witha condensationrate. beingtheradiationconstant,T beingthemaximumtempera- eff,max tureoftheaccretiondisc).AccordingtotheSpitzer’sformulafor thethermalconduction(Spitzer1962),i.e.,F (z) = k T5/2dT /dz, 2.3 Thedisc c 0 e e wesimplyexpresstheconductive fluxtransferredfromtheupper Combiningequations(8)and(9),theintegratedcondensationrate boundaryofthecorona/ADAFtothesurfaceofthetransitionlayer inunitsofEddingtonratefromR toanyradiusRofthediscinward d as, reads, ∆Fc = k0Te5m/2(Tem−Tcpl)/H m˙ (R)=m˙ (R)= Rd 4πRm˙ dR. (10) = k0Te7m/2(1−Tcpl/Tem)/H cnd disc ZR M˙Edd z ⋍ k0Te7m/2/H, (6) According tomassconservation, if theinitialmassaccretion rate inthecorona/ADAFism˙,themassaccretionrateinthecoronaisa where T refers to the electron temperature at the upper bound- em functionofdistance, aryofthecorona/ADAF,whichisthemaximumtemperatureofthe electronsatafixedradius.T referstothecouplingtemperatureof m˙ (R)=m˙ m˙ (R). (11) cpl c − cnd thetransitionlayer,whichismuchlessthetemperatureoftheelec- Thetotalluminosityofthecorona/ADAFisderivedbyintegrating tronsinthecorona/ADAF(Meyeretal.2007;Liuetal.2007).As thecorona/ADAFregion,i.e., demonstratedbyLiuetal.(2002),theelectrontemperatureinthe mainbodyofcoronaforafixedradiusisapproximatelyaconstant, Rd L = q H4πRdR. (12) wesimplyreplaceT bythelocaltemperatureT . c,t Z rad em e 3RS Wesolvetheequations(1),(3),(4),(5)and(6)fortheelec- trontemperatureT inthecoronabyspecifyingm,m˙ ,α,βandthe e c 2.4 Theeffectoftheirradiationofthediscbythecorona maximumeffectivetemperatureoftheaccretiondiscT .Then eff,max wecancalculatethefluxtransferredfromthecorona/ADAFtothe We consider the irradiation of the accretion disc by the transitionlayer∆F viaequation(6).Meanwhile,ifitispresumed corona/ADAF.Forsimplicity,weassumethattheradiationofthe c thatthefluxattheupperboundaryofthecorona/ADAFisapprox- corona/ADAF can be regarded as a point source lying above the imatelyzero,thefluxarrivedattheinterfaceofthetransitionlayer disccenterataheightofH.Inthiscase,thereflectionscalingfac- s FADAFequalsto∆F . tor (equivalenttothecoveringfactor)canbedefinedasfollows, c c ℜ = Ω = 1 Rd Hs 2πRdR (13) 2.2 Thetransitionlayer ℜ 2π 2πZRin (R2+Hs2)3/2 2 1/2 2 1/2 FollowingtheworkofLiuetal.(2007),theenergybalanceinthe = 1+ Rin − 1+ Rd − , ftrraonmsittihoencloaryoenra/iAsDdeAteFr,mthineebdrebmystshtreahinlucnogmirnagdiaetnioerng,yantrdanthsfeereend- where R and R are theHinsner bou−ndaryaHndsthe outer bound- in d thalpycarriedby themasscondensation, whichcanbeexpressed aryoftheaccretiondiscrespectively.Theeffectivetemperatureof as, the accretion disc can be expressed as follows by including both d γ 1 ˜T the accretion fed by the condensation and the irradiation of the dz"m˙zγ 1βℜµ +Fc#=−neniΛ(T), (7) corona/ADAF, − 3/4 1/2 1/4 Lwtioiiutnh/eetvℜa˜alp.o2trh0ae0ti7og)n,asratceonpsetranutn.itFarroema iseqguivaetinonas((7M),eythereetcoaln.d2e0n0sa7-; Teff(r)=2.05Te′ff,m1ax+3r6L (11−a)3rH 1/4 c,t − s m˙z= γ−γ1β−ℜFTcAiD/AµFi(1− √C), (8) =2.05T × 3M˙3/c4nd1c2 331R/2s1/4, (14) with eff,maxr −r 0.25β2p2 T 2 whereaisalbedo,whichisdefinedastheenergyratioofreflected C κ b 0 cpl . (9) ≡ 0 k2 ! FADAF! radiationfromthesurfaceoftheaccretiondisctoincidentradiation c uponitfromthecorona/ADAF,T isexpressedas, eff,max From equation (9), setting C = 1, a critical radius r is deter- d 1/4 mined. From this critical radius inwards, m˙ < 0, indicates that 1+6L (1 a) H thecorona/ADAF mattercondensesontothezdisc.However,from Teff,max =Te′ff,max M˙ccn,tdc2− 3Rss . (15) MNRAS000,1–8(2016) 4 ErlinQiaoandB.F.Liu T referstothemaximumeffectivetemperaturefromdiscac- 2009;Cao2009;Veledinaetal.2011;Liuetal.2012;Qiao&Liu e′ff,max cretion which is reached at r = (49/12). The expression of 2013;Liuetal.2015). tmax T isgivenas(Liuetal.2007), Inourmodel,inordertosimplifythecalculationoftheirradi- e′ff,max ationoftheaccretiondiscbythecorona/ADAF,weassumethatthe 1/4 1/4 Te′ff,max =0.20461m0− m˙c0nd.0(rt1max) keV. (16) choerigohnta/oAfDHA.FAicstluoaclalyteidnaosuarpmooindtels,oiunrictiealalby,otvheethmeadttiesrcicseanctcerreatteda Wecalculatethecondensationrateofthecorona/ADAFfrom intheformws ithanextendedgeometry,weshouldcheckthatthelu- minosityofthecorona/ADAFisindeedcompact.InFig.3,weplot equation(10)andtheluminosityofthecorona/ADAFfromequa- the ratio between the integrated luminosity of the corona/ADAF tion(12)byspecifyingtheblackholemassm,theinitialmassac- fromR = 3R tosome radiusR and the total luminosity of the cretionratem˙,theviscosityparameterα,magneticparameterβ,the in S corona/ADAF. It can be seen that most of the luminosity of the albedoa,theheightofthecorona/ADAFH andamaximumeffec- s coronaisreleasedinaverynarrowregion.Forexample,R=10R , tivetemperatureof the accretiondiscT .Then anew T S eff,max eff,max L (R/R )/L 60%, and for R = 20R , L (R/R )/L 80%, is derived by combining the equations (10), (12), (15) and (16). c S c,t ≈ S c S c,t ≈ It is clear that, although the corona/ADAF is geometrically dis- We make iterations by changing the value of T until a self- eff,max tributed within a wider range in the radial direction, most of the consistent solution of the discand the corona isfound, including luminosity of the corona/ADAF is still dissipated in a very com- thesizeoftheinnerdiscr ,thecondensationratem˙ (r),theelec- d cnd pactregion,whichisroughlyconsistentwithourassumptionthat trontemperatureoftheelectronsinthecorona/ADAFT (r),andthe e thecorona/ADAFisassumedtobelocatedasapointsourceabove scatteringopticaldepthoftheelectronsinthecorona/ADAFτ (r). es thedisccenterattheheightofH =10R . Withthederivedstructureofthediscandthecoronainradialdi- s S Inordertocomparewithobservationsforthecorrelationbe- rection, we calculated the emergent spectrum of the disc-corona tweenΓand ,wesearchforobservationaldatafromliteratures. systemwithMonteCarlosimulations,onecanrefertoQiao&Liu ℜ Wecompiledasamplewith118Seyfertgalaxiesbycombiningtwo (2012,2013)fordetails. samplesfromLubin´skietal.(2016)andZdziarskietal.(2003).In thesampleofLubin´skietal.(2016),thereare28Seyfertgalaxies, including8type1,8intermediateand12type2Seyfertgalaxies.In 3 RESULTS thesampleofZdziarskietal.(2003),thereare90Seyfertgalaxies, Wesolveequations (1), (3), (4),(5), (6), (10) (12), (15) and (16) including11radioloudAGNs,and79radioquietAGNs.Wefitall numericallybyspecifyingtheblackholemassm,theinitialmass theobservationaldataofthe118sourcesfortheΓ correlation −ℜ accretionratem˙,theviscosityparameterα,themagneticparameter withasecond-orderpolynomial.Thebestfittingresultisexpressed β,andthereflectionalbedoa.Inthispaper,wesetm=108.Weset asfollows, α = 0.3asusual(Kingetal.2007),andβ = 0.95assuggestedby =3.2 5.3Γ+2.1Γ2, (17) MHDnumericalsimulations(Hawley&Krolik2001).Thereflec- ℜ − tionalbedoisrelativelow,whichissuggestedtobea 0.1 0.2 onecanseethedottedlineinthepanelaofFig.4.Weplotourthe- ∼ − (e.g.Magdziarz&Zdziarski1995).Wefixa=0.15.Theheightof oreticalresultsfortherelationbetweenΓand asacomparison. ℜ thecorona/ADAF isdifficult todetermine, asan example, we fix Onecanseethesolidredlinewithfivelargerredfilledcirclesin Hs=10R . thepanelaofFig.4.Thefivelargerredfilledcirclesfromleftto S In the panel a of Fig. 2, we plot the mass accretion rate in rightcorrespondtoourtheoreticalresultsform˙ =0.015,0.02,0.03, thecoronaandmassaccretionrateinthediscasafunctionofra- 0.05and0.1respectively.Itcanbeseenthatourmodelresultcan diusfromtheblack holefordifferent initialmassaccretion rates. roughlymatchthecasefor . 1.Weshouldnotethat,sincewe ℜ It is found that the inner disc can be formed when the mass ac- donotconsidertheeffectssuchasthelightbendinginthevicinity cretionratem˙ & 0.01.Form˙ = 0.015,thecondensation radiusis of theblack holeand therelativisticmotion of thecorona, tothe r = 3.75. Withthe increase of the mass accretion rate, the con- strengthofthereflection,thecurrent modelintrinsicallycanonly d densation radius increases, i.e., as examples, for m˙ = 0.02, 0.03, beappliedtothecasefor .1. ℜ 0.05and0.1thecondensationsradiiarer = 23,70,175and444 Wetesttheeffectoftheheightofthecorona/ADAFH tothe d s respectively. One can see the panel b of Fig. 1 for the schematic theoreticalrelationbetweenΓand .Onecanrefertothepanelb ℜ description for the change of the geometry of the accretion flow ofFig.4.Thesolidgreylinewithlargergreycirclepointsrefers withincreasing the initialmass accretion rate. Assuming that the to the case for m = 108, α = 0.3, β = 0.95, a = 0.15, and blackholeisanon-rotatingSchwarzschildblackhole,sotheinner H =3R ,andthefourpointsfromlefttorightcorrespondtom˙ = s S boundaryoftheaccretiondiscR ,i.e.,theISCOis3R .Wethen 0.015,0.02,0.05,0.1respectively.Thecorrespondingcondensation in S calculate the reflection scaling factor with equation (13). The radiiofthefourpointsfromlefttorightarer =3.7,6.3,108,343 ℜ d correspondingreflectionfactorsare =0.02,0.56,0.82,0.90,and respectively. The solid black line with larger black circle points ℜ 0.94 for m˙ = 0.015,0.02, 0.03, 0.05 and 0.1 respectively. In the referstothecaseform = 108,α = 0.3,β = 0.95,a = 0.15,and panel b of Fig. 2, we plot the corresponding emergent spectra of H = 20R , and the three points from left to right correspond to s S themodel (notincluding thereflectionspectra).Fromthebottom m˙ = 0.014,0.015,0.02respectively.Thecorrespondingcondensa- up, the mass accretion rates are m˙ = 0.015, 0.02, 0.03, 0.05 and tionradiiofthethreepointsfromlefttorightarer = 3.4,26,63 d 0.1 respectively. Fromthe emergent spectra, the hard X-ray pho- respectively. It is clearly that the assumption for height of the ton indices between 2-30 keV are Γ = 1.58,1.88,2.01,2.17, and corona/ADAF can affect the relation between Γ and . Theoret- ℜ 2.29 for m˙ = 0.015,0.02, 0.03, 0.05 and 0.1 respectively. It is ically,based on thegeometry of theaccretion flowinour model, clear that the hard X-ray photon index Γ increases with increas- we can more accurately determine the relative location between ing m˙, which is consistent with the trend observed in luminous thediscandcoronaandthenthecorrespondingstrengthofthere- AGNs(e.g.Lu&Yu1999;Porquetetal.2004;Wangetal.2004; flection,whichisacomplicatedtasktobedoneinthefuture,and Shemmeretal. 2006; Saezetal. 2008; Sobolewska&Papadakis beyond the current study. On the other hand, observationally, the MNRAS000,1–8(2016) Γ- correlationinAGNs 5 ℜ Figure1. Panela:Schematicdescriptionofthemassandenergyflowinthediscandcoronafortheinitialmasssupplywithaverticallyextendeddistribution.Panelb: Changeofthegeometryoftheaccretionflowwithincreasingtheinitialmassaccretionratefromthebottomup. Figure2. Panela:Massaccretionrateinthecorona(blackline)andmassaccretionrateintheaccretiondisc(redline)asafunctionofradius.Thetriple-dotted-dashed line,dotted-dashedline,dashedline,dottedlineandthesolidlineareform˙ =0.015,0.02,0.03,0.05and0.1respectively.Inthecalculation,wefixm=108, α=0.3,β=0.95,a=0.15andHs=10RSrespectively.Panelb:Thecorrespondingemergentspectra.Fromthebottomup,themassaccretionratesare m˙ =0.015,0.02,0.03,0.05and0.1respectively. heightofthecoronacanberoughlyestimatedbysomemethod,like geometryofaccretionflowaroundasupermassiveblackhole.His- thetimelagbetweentheemissionindifferentX-raybands,which torically,severalmodelshavebeenproposedtoexplainsuchacor- isalsoroughlyconsistentwiththevalueweadoptedinthecalcu- relation.Zdziarskietal.(1999)proposedaso-calleddisc-spheroid lation(e.g.McHardyetal.2007;DeMarcoetal.2013;Karaetal. model to explain the correlation between Γ and . In the disc- ℜ 2016). spheroid model, an optically thin, hot sphere is surrounded by a flatopticallythickaccretiondisc,whichcanextendfromatrunca- tionradius,d = 0tod = ,totheblackhole.Inthismodel,the ∞ seedphotons fromthecoldaccretiondiscarescatteredinthehot 4 DISCUSSIONS sphere, producing the power-law X-ray emission. In turn, a frac- tionofthepower-lawX-rayemissionfromthehotsphereisinter- Inthepresentpaper,weproposedamodelofthecondensationof cepted by the disc. The intercepted X-ray photons will be partly thecoronatoexplaintheobservedcorrelationbetweenthehardX- ray photon index Γ and the reflection scaling factor in active reflected by the accretion disc, forming the reflection spectrum, ℜ and partlybe absorbed by theaccretion disc, then reprocessed in galactic nuclei, which is believed to be employed to explore the MNRAS000,1–8(2016) 6 ErlinQiaoandB.F.Liu Figure3. Ratiobetweentheintegratedluminosityofthecorona/ADAFfromRin=3RStosomeradiusR,Lc(R/RS),andthetotalluminosityofthecorona/ADAFLc,t asafunctionofradiusfromtheblackhole. Figure4. Panela:CorrelationbetweenthehardX-rayphotonindexΓandthereflectionscalingfactor .Theblackdiamondisasamplecomposedof28Seyfert ℜ galaxies,including8type1,8intermediateand12type2Seyfertgalaxies,onecanrefertoLubin´skietal.(2016)fordetails.Thecoloredsmallpointsare fromasamplecomposedof90SeyfertgalaxiessummarizedbyZdziarskietal.(2003),including11radioloudAGNs(filledsymbols),and79radioquiet AGNs(Opensymbols).Thebluecircles,greentrianglesandredsquarescorrespondtotheobservationsbyGinga,RXTEandBeppoSAXrespectively.The dottedlinereferstothebest-fittingresultwithasecond-orderpolynomial.Theredsolidlinewithfivelargerredfilledcirclesisourtheoreticalresults,andthe fivelargerredfilledcirclescorrespondtothemodelresultsform˙ =0.015,0.02,0.03,0.05and0.1respectively.Inthecalculation,wetakem=108,α=0.3, β=0.95,a=0.15andHs=10RSrespectively.Panelb:Alltheobservationaldataaresamewiththepanela.Weaddtwosetsoftheoreticalresultsby assumingdifferentheightofthecorona.Thesolidgreylinereferstothecaseform=108,α=0.3,β=0.95,a=0.15andHs=3RS.Thesolidblackline referstothecaseform=108,α=0.3,β=0.95,a=0.15andHs=20RS. thediscastheseedphotons tobescatteredinthehot sphere. Fi- (1999),theauthordidnotconsidertheself-Comptonscatteringof nally, a equilibrium feedback between the cold disc and the hot theseedphotons(synchrotronradiationandbremsstrahlung)inthe sphereisestablished.Withthedecreaseofthetruncationradiusof hot sphere, which may have obvious effects to the X-ray spectra theaccretiondiscd,thesolidanglecoveredbytheaccretiondisc forthetruncationradiusoftheaccretiondiscgreaterthan 30R ∼ S asseenfromthehotsphereincreases,leadingtoastrongerreflec- (Poutanenetal.1997;Esinetal.1997;Poutanen&Veledina2014, tion.Meanwhile,thecoolingofthedisctothehotspherealsoin- forreview). creases,leadingtoasofterX-rayspectrum.Bychangingdarbitrar- Theoretically,thephysicalmechanismofthetruncationofthe ilyfromd=0tod= ,themodelcanwellreproducetheobserved accretiondischasbeenstudiedformanyyears,e.g.,thediscevapo- ∞ Γ correlationfor .1.Weshouldnotethat,inZdziarskietal. rationmodelproposedbyMeyeretal.(2000a,b),Liuetal.(2002), −ℜ ℜ MNRAS000,1–8(2016) Γ- correlationinAGNs 7 ℜ Qiao&Liu (2009, 2010), Qiaoetal. (2013), and Taametal. ISCO being 3R , and we don’t consider the effect of the spin of S (2012), or the strong ADAF principle by Narayan&Yi (1995), thecentralblackholetoourresults.ForanextremelyrotatingKerr Abramowiczetal. (1995) and Mahadevan (1997) etc. There are blackhole,theaccretiondisccanextenddowntotheISCObeing two main findings of the previous studies, (1) the truncation ra- 1/2R ,whichtheoreticallycanincreasethestrengthofthereflec- S diusoftheaccretiondiscdependsontheinitialmassaccretionrate tion.Meanwhile,inthevicinityoftheblackhole,theeffectoflight m˙,i.e.,thetruncationradiusdecreaseswithincreasingm˙;(2)there bendingcanalsoincreasethestrengthofthereflection,whichprob- exists a critical mass accretion rate m˙ α2. When m˙ & m˙ , ablycanhelptoexplainthecasefor &1(e.g.Fabian&Vaughan crit ∼ crit ℜ theaccretiondiscdoesnottruncate,andextendsdowntotheISCO 2003; Reynolds&Nowak 2003; Dauseretal. 2014; Garc´ıaetal. of the black hole. In this case, due to the strong Compton cool- 2013).Inthiswork,asthefirststep,wesimplysuggestedthatthe ingoftheseedphotonsfromtheaccretiondisctothecorona,the condensationofthecoronacanincreasethestrengthofthereflec- viscous heating of the corona itself can not sustain the existence tion, then can explain the observed Γ- correlation for . 1. ℜ ℜ of the corona. The corona collapses very quickly, and the accre- Thestudyoftheeffectssuchasthespinoftheblackhole,thelight tion will be dominated by the cold accretion disc. Consequently bendingtothereflectionisverycomplicated,whichisbeyondthe the corona above the disc is too weak to explain the observed scopeofthecurrentwork. power-lawhardX-rayemission,nottomentionthereflection,un- We should notice that, in the AGNs case, the scattering of lessanotherheatingmechanism(probablythemagneticreconnec- the X-ray emission by the outer torus can also contribute to the tion heating) for the corona isconsidered, which is still not very reflection. Currently, it is difficult to distinguish the contribution clear (e.g. Meyer-Hofmeisteretal. 2012). Withthese two restric- ofthediscandtorustothereflectionexceptforsomeX-rayvery tions, it is unclear whether the correlation between Γ and can bright sources, such as NGC 4151 (Lubin´skietal. 2010). Mean- ℜ stillbereproducedwell. while,sincemost ofthesourcesinour samplearetype1Seyfert Based on the disc evaporation model, Taametal. (2012) in- galaxies,duetotherelativesmallerviewingangle,thecontribution vestigated thetruncation radius of the accretion discasfunctions ofthetorustothereflectionshouldbenotveryimportant. ofα,βandm˙.Theauthorformulizedtheexpressionofthetrunca- We suggest that our model can also be applied to the high tionradiusas, massX-raybinaries(HMXBs).InHMXBs,thecompanionstaris a massive, bright O or B star. A fraction of the matter from the rtr≈17.3m˙−0.886α0.07β4.61. (18) companion star is captured by the compact object via the stellar Itisclearthatthestrengthofthemagneticfield,i.e.,β,isimportant windratherthantheRochelobeoverflow(RLO)asinthelowmass tothetruncationradiusoftheaccretiondisc.Meanwhile,thecriti- X-raybinaries(LMXBs)(alsocalledX-raytransients).Soitisrea- calmassaccretionrateandthecorrespondingtruncationradiuscan sonablethat,initially,theaccretedmattershouldbeintheformof beexpressedas, avertically,extendedhotgasinHMXBsratherthanbeconstrained inaverynarrowequatorialplaneinLMXBs.Sincethehotgasis m˙crit≈0.38α2.34β−0.41 (19) fully ionized, it can not trigger the ionization instability (viscous instability)oftenobservedinLMXBs,whichcanhelpustounder- rmin≈18.80α−2.00β4.97. (20) standwhyHMXBsareoftenrelativelystableandtheLMXBsare Bychanging α,β andm˙,especiallyβ,themodelcanfitthetrun- oftenunstable(e.g.Lasota2001;Shakuraetal.2015). cation of the accretion disc for most of low-luminosity AGNs and theblack holeX-ray binaries intherelow/hard spectral state verywell(Gilfanovetal.2000;Taametal.2012;Plantetal.2015; DeMarcoetal.2015;Basak&Zdziarski2016).However,weno- ticethat,forexample,takingα = 0.3andβ = 0.95,thepredicted minimum truncation radius is r = 162; taking α = 0.3 and min 5 CONCLUSIONS β = 0.8, the predicted minimum truncation radius is r = 69. min The maximum reflection scaling factors predicted by the above In the work, we proposed a model within the framework of the twosetsofparametersare = 0.05and = 0.13respec- condensationofthecorona/ADAFtointerprettheobservedcorre- ℜmax ℜmax tively,whicharetooweaktoexplainthestrengthofthereflection lation between the hard X-ray photon index Γ and the reflection in a wider range of & . If an extreme value of β = 0.5 scalingfactor inAGNs.Inthemodel,beyondtheBondiradius ℜ ℜmax ℜ is adopted, the predicted minimum truncation radius r = 6.7, the matter is assumed to be accreted in the form of a vertically min then = 0.8, which indeed can extend the reflection in a extended,hotgasbycapturingtheinterstellarmediumandstellar widerℜramnaxge. However, as the author stressed, the effect of such wind.Withinthisframework, when M˙/M˙ & 0.01,atacritical Edd a strong magnetic field to the structure of the corona is still un- radiusr ,afractionoftheaccretedhotgascancondenseontothe d clear, which probably makes some assumptions in the model not equatorialdiscplane,formingainnerdisk.Thentheaccretionwill work.Forexample,inthemodelSpitzer’sformulaisusedforthe occurintheformadisk-coronasystemflowingtowardstheblack description of the electron thermal conduction in the corona. If hole. We found that the size of the inner disk increases with in- a strong magnetic field is considered, Spitzer’s formula may not creasing theinitialmass accretion rate. Withthegeometry of the work,whichmakestheresultspredictedbythemodelveryuncer- accretionflow,wecalculatethereflectionscalingfactor .Mean- ℜ tain(Meyer&Meyer-Hofmeister2002).Morestudiesbyconsider- while,wecalculatetheemergentspectrumwiththestructureofthe ingthedetailedmorphologyofthemagneticfieldorotherunknown discandthecoronaforthehardX-rayphotonindexΓ.Bycompar- physical mechanism arestillneeded toreproduce smaller trunca- ingwiththeobservationsofasamplewith118Seyfertgalaxies,it tionradiiinthefuturetoextendthestrengthofthereflectionina isfoundthatourtheoreticalrelationbetweenΓand canroughly ℜ widerrange. matchtheobservationsfor . 1.Finally,wesuggestedthatthe ℜ Inthiswork,throughout thecalculation,weassumethecen- currentmodelcanbeappliedtoHMXBs,whicharebelievedtobe tralblackholeasanon-rotatingSchwarzschildblackholewiththe fueledbythehotwindfromthebrightOorBcompanionstar. MNRAS000,1–8(2016) 8 ErlinQiaoandB.F.Liu ACKNOWLEDGMENTS MahadevanR.,1997,ApJ,477,585 MalkanM.A.,SargentW.L.W.,1982,ApJ,254,22 We thank professor W.M. Yuan from NAOC and Professor R.E. MalzacJ.,BeloborodovA.M.,PoutanenJ.,2001,MNRAS,326,417 Taam from ASIAA for very useful discussions and suggestions. McHardy I. M., Are´valo P., Uttley P., Papadakis I. E., Summons D. 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