Constraints on black hole spins with a general relativistic accretion disk corona model

0.37 MB·

by  Bei You

ResearchinAstron.Astrophys.Vol.0(200x)No.0,000–000 R
esearchin
http://www.raa-journal.org http://www.iop.org/journals/raa A
stronomyand
A
strophysics
Constraints on black hole spins with a general relativistic accretion
5
disk corona model
1
0
2
BeiYou1,3,XinwuCao1,andYe-FeiYuan2

v
o
1 KeyLaboratoryforResearchinGalaxiesandCosmology,ShanghaiAstronomicalObservatory,
N
ChineseAcademyofSciences,80NandanRoad,Shanghai,200030,China;[email protected],

9 [email protected]
1 2 DepartmentofAstronomy,UniversityofSciencesandTechnologyofChina,Hefei,Anhui230026,
China;[email protected]

] 3 UniversityofChineseAcademyofSciences,19AYuquanlu,Beijing100049,China
E
H
.
h Abstract Thepeaksofthespectraoftheaccretiondiskssurroundingmassiveblackholes
p inquasarsareinthefar-UVorsoftX-rayband,whichareusuallynotobserved.However,
- inthediskcoronamodel,thesoftphotonsfromthediskareComptonizedtohighenergy
o
r inthehotcorona,andthehardX-rayspectra(luminosityandspectralshape)containthe
t informationoftheincidentspectrafromthedisk.Thevaluesofblackholespinparameter
s
a a∗ are inferred from the spectral fitting, which spread over a large range, ∼ −0.94 to
[ 0.998. We find that the inclination angles and mass accretion rates are well determined

2 by the spectralfitting, while the resultsare sensitiveto theaccuracyofblack holemass
v estimates.Notightconstraintsontheblackholespinsareachieved,iftheuncertaintiesof
9 blackholemassmeasurementsareafactoroffour,whicharetypicalforthesingle-epoch
5 reverberationmappingmethod.Recently,theaccuracyofblackholemassmeasurement
6 has been significantly improved to 0.2 − 0.4 dex with velocity resolved reverberation
5
mappingmethod(Pancoastetal.,2014).Theblackholespincanbewellconstrainedifthe
.
1 massmeasurementaccuracyis<∼ 50%.Intheaccretiondiskcoronascenario,a fraction
0
ofpowerdissipatedinthediskistransportedintothecorona,andthereforetheaccretion
4
disk is thinner than a bare disk for the same mass accretion rate, because the radiation
1
pressureinthediskisreduced.Wefindthatthethindiskapproximation,H/R <∼ 0.1,is
:
v still valid if 0.3 < m˙ < 0.5, provideda half of the dissipated power is radiated in the
i
X coronaabovethedisk.
r
a Keywords: Quasars:accretiondisk;X-ray:corona;Blackholephysics;Galaxies:active
1 INTRODUCTION
It is believed that black holes reside in X-ray binaries and active galactic nuclei (AGN) with typical
masses∼ 10M⊙ and106−109M⊙ respectively(Kormendy&Richstone,1995;Ferrarese&Merritt,
2000;Remillard&McClintock,2006;Hopkinsetal.,2008;Alexander&Hickox,2012).Massiveblack
holesmaygrowupthroughgasaccretionor/andmergers,whichisalsorelatedwiththegalaxyevolution
(e.g.,Moderski& Sikora,1996;Marconiet al., 2004;Volonterietal., 2005;Berti & Volonteri,2008;
Dotti et al., 2013). The structure and radiation of accretion disks surrounding black holes have been
extensivelystudiedinthelastseveraldecades(Shakura&Sunyaev,1973;Franketal.,1992;Merloni,
2004).Relativistic jets havebeenobservedin some X-raybinariesand AGNs, whichis thoughtto be
2 B.Youetal.
relatedtoaccretiondisksor/andspinningblackholes(Blandford&Znajek,1977;Blandford&Payne,
1982).
Ablackholecanbedescribedbytwoquantities:themassM andthedimensionlessspinparameter
a∗ =cJ/GM2,whereJ istheangularmomentumoftheblackhole.Inthelastdecade,greatprogresses
havebeenachievedonthemassmeasurementofblackholesineitheractiveornonactivegalaxies.The
blackholemassinnearbygalaxiescanbeestimatedwiththecorrelationsbetweenblackholemassand
the stellar velocity dispersion or bulge luminosity of the host galaxy (Kormendy & Richstone, 1995;
Ferrarese&Merritt,2000;McConnelletal.,2011).Foractivegalaxies,thesizeofthebroad-lineregion
(BLR)ismeasuredwithreverberationmappingmethod,andtheblackholemasscanbeestimatedwith
thebroad-linewidthassumingthemotionofBLRcloudstobevirialized(Kaspietal.,2000).Thereis
atightcorrelationbetweentheBLRsizeandtheopticalcontinuumluminosity,whichiswidelyusedto
estimatemassesofblackholesinAGNs,orcalledassingleepochreverberationmappingmethod(e.g.,
Kaspietal.,2000;Woo&Urry,2002;Shenetal.,2011;Tangetal.,2012).
Compared with the mass estimates, the measurements of the black hole spins are still far from
satisfactory,thoughgreateffortshavebeendevotedonthisissue.TheobservedbroadFeKαemission
lines providea way to estimate the black holespin parameters,which is applicableforblack holesin
both galactic X-ray binariesand AGNs. The observedbroad Fe Kα emission lines are believed to be
emitted fromthe inner regionsof the accretiondisks. The radiiof the innermoststable circularorbits
(ISCO) of the gas in the accretion disks can be inferred from the observed asymmetric Fe kα line
profiles (Martocchia et al., 2002; Miller et al., 2002; Miniutti et al., 2004; Brenneman & Reynolds,
2006;Lohfinketal.,2012),andthedimensionlessspina∗canbedeterminedwiththeradiioftheISCO
(Barretal.,1985;Reynolds&Nowak,2003;Remillard&McClintock,2006).Theapplicationofthis
method requires high quality X-ray spectroscopic data, and a few black hole spins in nearby Seyfert
galaxieshavebeenmeasuredinthisway(Goosmannetal.,2006;Brenneman&Reynolds,2006;deLa
CallePe´rezetal.,2010;Patricketal.,2011;Lohfinketal.,2012).
TheblackholespinparametersinsomeX-raybinariesweremeasuredbyfittingtheobservedcon-
tinuum spectra with general relativistic accretion disk models (so-called “continuum-fitting method”,
Zhangetal.,1997;Shafeeetal.,2006;Steineretal.,2009a;Gouetal.,2011;McClintocketal.,2011;
Youetal.,2015).Inprinciple,thismethodcanbeusedtoconstraintheblackholespinsforAGNs.In
someearlyworks,theaccretiondiskmodelswereusedtofittheobservedmulti-bandspectraofquasars
(e.g., Bechtold et al., 1987;Sun & Malkan, 1989),in which the generalrelativistic effects have been
takenintoaccount(Cunningham,1975;Laor,1991).Czernyetal.(2011)estimatedtheblackholespin
to be a∗ = 0.3 by fitting the broad band (IR/optical/UV) continuum spectrum of the quasar SDSS
J094533.99+100950.1.Itwas foundthatthemodelfitting suffersfromthe degeneracyof thedisk pa-
rameters,namely,theblackholemassM,thespina∗,andthemassaccretionrateM˙ .Atthetimebefore
theinventionoftheblackholemassmeasurementsusednowadays,thederiveddiskparameterswiththe
spectralfittingsarequiteuncertain.Itispossibletoconstrainthethespinsofthemassiveblackholesin
AGNs,whentheblackholemassesaremeasured.Thedegeneracyoftheparametersinaccretiondisk
spectralfittings can be eliminated if the peak frequencyand luminosityof the accretion disk are well
observed.However,the applicationof this methodin AGNs meetsa majordifficulty,i.e., the spectral
peaksofaccretiondiskssurroundingmassiveblackholeswith∼108M⊙areinthefar-UVorsoftX-ray
band,whichareunobservableformostAGNs.
Theobservedultraviolet(UV)/opticalemissionofAGNsisthoughttobeathermalemissionfrom
the optically thick accretion flows (Shields, 1978; Malkan & Sargent, 1982; Sun & Malkan, 1989),
while they are too cold to reproduce the observed power-law hard X-ray spectra of AGNs. The hard
X-rayspectraofAGNsmostlikelyoriginatefromtheinverseComptonprocessinwhichsoftphotons
fromthedisksarescatteredbythehotelectronsinthecoronas(Galeevetal.,1979;Haardt&Maraschi,
1991;DiMatteo,1998).Accordingtothisdisk-coronascenario,mostgravitationalenergyisgenerated
inthecolddisk,whileafractionofitiscarriedintothecoronalikelybythemagneticloopsinthedisk
(DiMatteo,1998;Liuetal.,2002;Cao,2009).Cao (2009)calculatedthestructureandspectrumofa
geometricallythinaccretiondiskwithacoronainAGNs,inwhichthecoronaisassumedtobepowered
by the reconnection of the magnetic loops originating from the cold accretion disk due to buoyancy
Constraintsonblackholespinswithageneralrelativisticaccretiondiskcoronamodel 3
instability. It is foundthat both the hard X-ray bolometriccorrectionfactor Lbol/LX,2−10keV and the
hardX-rayspectralindexincreasewiththeEddingtonratio,whicharequalitativelyconsistentwiththe
observations(e.g.,Wangetal.,2004;Shemmeretal.,2006;Vasudevan&Fabian,2007;Zhou&Zhao
2010;Fanalietal.,2013).ThismodelwasexpandedinthegeneralrelativisticframeforaspinningKerr
black hole by You et al. (2012),in which the spectra of such a accretiondisk-coronaare recalculated
takingintoaccountgeneralrelativisticeffectwiththerelativisticray-tracingmethod.
Asdiscussedabove,thepeaksofthespectraoftheaccretiondiskssurroundingmassiveblackholes
inAGNsareusuallynotobserved(butalsoseeYuanetal.,2010,forthecaseofarelativesmallblack
holeinanarrow-linequasar).However,aportionofthesoftphotonsoriginatingfromthediskundergo-
ingComptonizationinthehotcorona,andthescatteredphotonsaredominantlyinthehardX-rayband.
ThisimpliesthattheobservedhardX-rayspectra(luminosityandspectralshape)containtheinforma-
tion ofthe incidentspectra fromthe cold disk.Itseems possible to constrainthe valuesof blackhole
spinparametera∗ inAGNswith ageneralrelativisticaccretiondisk coronamodel,asa variantofthe
continuum-fittingmethod,evenifthespectralpeaksoftheaccretiondisksarenotobserved.
Inthispaper,weusethegeneralrelativisticmodelofanaccretiondisk-coronasurroundingaspin-
ningblackholetofittheobservedmulti-bandspectraofAGNswithmeasuredblackholemasses.The
blackholespinningparametera∗ aretentativelyderivedforasmallsampleofAGNs.Webrieflysum-
marize the accretion disk-corona model used for AGN spectral fitting in Section 2. The results and
discussionareinSections3and4.Throughoutthispaper,acosmologywithH = 70kms−1Mpc−1,
0
Ω =0.3andΩ =0.7isadopted.
m Λ
2 MODEL
Inthiswork,weadopttheKerrblackholeaccretiondisk-coronamodeldevelopedbyYouetal.(2012).
Allthegeneralrelativistic effectsareproperlyconsideredin boththecalculationsofthestructureand
emergentspectra oftheaccretiondisks.We brieflysummarizethemodelin thenexttwo sub-sections
(seeYouetal.,2012,forthedetails).
2.1 Structureofanaccretiondiskwithcorona
Intheaccretiondiskcoronamodel,mostgravitationalenergyofthegasisdissipatedin thecolddisk,
andaportionofitiscarriedintothecoronabythemagneticloops.Theparticlesinthecoronaareheated
bythere-connectionoftheloopsinthecorona(DiMatteo,1998).Theelectronsinthecoronaarehot,
andtheobservedhardX-rayspectramostlyoriginatefromtheinverseComptonprocessinwhichsoft
photons from the thermal radiation of the cold disk are scattered to high energy by the hot electrons
(e.g.,Galeevetal.,1979;Haardt&Maraschi,1991;Svensson&Zdziarski,1994).
Thegravitationalpowerofthegasgeneratedinthecoldaccretiondisksurroundingaspiningblack
holeis
3GMM˙ L
Q+ = , (1)
dissi 8πR3 BC1/2
whereL,B,C arethegeneralrelativisticcorrectionfactors(Novikov&Thorne,1973;Page&Thorne,
1974),andM˙ isthemassaccretionrateofthedisk.
Thefractionofthedissipatedpowertransportedtothecoronacanbeestimatedprovidedthestrength
ofthemagneticfieldisknown.Themagneticpressureofthediskisusuallyassumedtoscalewiththegas
or/andradiationpressureofthediskdependingonthedifferentmagneticstressesadopted(Sakimoto&
Coroniti,1981;Stella&Rosner,1984;Taam&Lin,1984;Cao,2009;Hiroseetal.,2009).Thedetailed
physics of the field generation is still quite unclear, though the different magnetic stressed are tested
against the observations in AGNs in the previous works (Wang et al., 2004; Cao, 2009; You et al.,
2012).Forsimplicity,theratiof ofthedissipatedpowerinthecoronatothatinthediskistakenasa
parameterinourmodelcalculations,
Q+
f = cor , (2)
Q+
dissi
4 B.Youetal.
whichwillbedeterminedbythecomparisonwiththeobservedspectra.
AportionofthesoftphotonsfromthediskundergoinverseComptonizationtohighenergyinthe
corona, which correspondinglycools the electrons in the corona. Abouthalf of the scattered photons
are intercepted by the disk, some of which are reflected and the rest heats the disk and re-radiate as
blackbodyradiation(Zdziarskietal.,1999).Thereforetheenergyequationforthecolddiskis
1 4σT4
Q+ −Q+ + (1−ε)Q+ = d =σT4, (3)
dissi cor 2 cor 3τ s
where the reflection albedo ε = 0.15 is adopted in all our calculations (Zdziarski et al., 1999), τ =
τ +τ istheopticaldepthintheverticaldirectionofthedisk,andT andT arethetemperaturein
es ff d s
themid-planeofthediskandthetemperatureofthegasatthedisksurfacerespectively.
2.2 Emergentspectrumoftheaccretiondiskwithcorona
Itiswellknownthatmostoftheradiationofthediskcomesfromtheinnerregionofthedisk(Shakura
&Sunyaev,1973).Thespectrumofanaccretiondisk-coronaforaspinningblackholeisinfluencedby
therelativisticeffects,i.e.,theframedragging,gravitationalredshift,andphotontrajectorybending,due
tothestronggravityfieldoftheblackhole(Zhangetal.,1985;Laor&Netzer,1989;Laor,1991;Liet
al.,2005,2009;Youetal.,2012),especiallyfortheemissionfromtheregionneartheblackhole.The
ray-tracingmethodiswidelyadoptedinthiskindofspectralcalculations(Laor&Netzer,1989;Laor,
1991;Zhangetal.,1997;Lietal.,2005;Youetal.,2012).Consideringthecomplexityoftheradiation
transfer of Comptonized photons in the geometrically thick corona, You et al. (2012) calculated the
emergentComptonizedspectrumfromthecoronabydividinglayersinthecoronawiththeray-tracing
method.Inthis work,we followthe same routineof Youetal. (2012)to calculatethe spectrumof an
accretiondisk-coronaforarotatingblackhole,inwhichallthegeneralrelativisticeffectsaretakeninto
account(seeSection3.4inYouetal.,2012,forthedetails)
Theradiationfromgeometricallythin,opticallythickaccretiondiskisanisotropic,andthespecific
intensity of the disk emission, I ∝ (1+2µ), in the co-movingframe of the disk, where µ = cosθ
µ
(θ is the angle between the line of sight andthe normalof the disk Chandrasekhar,1960;Sunyaev&
Titarchuk,1985;Liu&Zhang,2011).
UV/softX-rayphotonsareemittedfromtheinnerregionoftheaccretiondisk,whereahotplasma
layerispresentabovethedisk surfacecausedbytheilluminationofthe externalhardX-rayemission
from the corona. The electron scattering opacity dominates over absorption in this hot layer, and the
emergentspectrumofthediskdeparturesfromtheblackbodyemissionduetoradiativetransfer.Hubeny
etal.(2001)calculatedtheverticalstructureandemergentspectraofthediskswithahotplasmalayer
by including a self-consistent treatmentof Compton scattering. Chiang (2002)found that the derived
emergentspectrumcouldbe wellreproducedbyblackbodyemission witha highercolortemperature,
T =f T .Theempiricalcolorcorrectionfactorf isgivenby
rad col d col
fcol(Td)=f∞− (f∞1+−e1x)[p1[(+νex−pν(−)ν/b∆/∆ν]ν)] (4)
p b
whereνp ≡2.82kBTd/h,f∞ =2.3andνb =∆ν =5×1015Hz.Thelocalspectrumemittedfromthe
diskcanbecalculatedwith
1
F ∝ πI (f T ), (5)
ν f4 ν col d
col
whereI isthelocalblackbodyemission.
ν
AsdescribedinSection2.1,thestructureoftheaccretiondisk-coronacanbecalculatedwhenthe
valuesof the black hole mass M, the dimensionlessspin a∗, the mass accretionrate m˙ , and the ratio
f ofthedissipatedpowerinthe coronato thatin thedisk, arespecified.With thederivedstructureof
theaccretiondisk,weusetheraytracingmethodtocalculatetheemergentspectrumofthediskcorona
systemobservedininfinityataviewingangleθ.Weplottherepresentiveemergentspectrawithdifferent
Constraintsonblackholespinswithageneralrelativisticaccretiondiskcoronamodel 5
Fig.1 Therepresenttivespectraofthedisk-coronamodelwithdifferentvaluesoftheratiofofthedissi-
patedpowerinthecoronatothatinthedisk.Thedottedlinesrepresentthespectraofthediskblackbody
radiation,whilethedashedlinesrepresenttheComptonradiationfromthecorona.Thesynchrotronand
bremsstrahlungradiationwhichdominateinradiobandsarenotplotted.ItisfoundthatthehardX-ray
spectrabecomeharderwiththeratiof.
valuesoftheratiof inFig.1toshowtheeffectoff ontheoverallshapeofthespectra.Atagivenradius,
boththeenergydissipatedinthediskandthedisktemperaturedecreasewithenergyratiof according
toEq.(2).Therefore,thespecificintensityofthesoftphotonsfromthediskwillalsodecreasewiththe
ratio f. In this case, the electron temperature will increase with f, since the cooling of the corona is
dominatedbytheinverseComptonscatteringofthesoftphotonsfromthediskbythehotelectronsin
thecorona(Cao,2009;Youetal.,2012).ThismeansthattheresultingspectrainthehardX-rayband
becomemuchharderandbrighterwiththeincreaseintheratiof,giventhatmoresoftphotonsfromthe
diskwillbeComptonscatteredtohigherenergybythehotelectronsinthecorona(seeFig.1).
3 RESULTS
Usingtheaccretiondiskmodeldescribedin Section2,wefittheobservedmulti-bandspectralenergy
distributions(SEDs)offivequasarswhichexhibitobviousfeatureofso-called“BigBlueBump”(BBB)
withmeasuredblackholemassesandgoodX-raymeasurements.ThepeakofBBBat∼1015−1016Hz
and a power-law hard X-ray spectrum are the typical features of the spectrum of an accretion disk
with corona(e.g.,Cao, 2009;You etal., 2012).Asdiscussed in Section 1,the peaksofthe BBBs are
beyond the waveband coverage of the instruments for most AGNs due to their massive black holes.
However, the left tail of the BBBs can be observed in infared/optical/UV wavebands, which provide
usefulinformationoftheBBBs(e.g.,Calderoneetal.,2013).TheSEDdataoffivequasarsinthiswork
istakenfromShangetal.(2011)andsummarizedinTables1and2.
TheblackholemassesM ofthesefivequasarsaregiveninTangetal.(2012),whichareestimated
with the broad-linewidth andthe scaling relationships(Vestergaard& Peterson,2006;Vestergaard&
Osmer,2009).Theinfraredemissionfromthese quasarsmayprobablybedominatedbythe radiation
fromthedusttori,whiletheoriginofthesoftX-rayexcessobservedinAGNsisstillcontroversial(see,
6 B.Youetal.
Table1 TheSEDdatainoptical/UVwavebands
Object Othername R.A.(J2000) Decl.(J2000) z E(B−V)a λLλ(3000A˚)b SampleID
PG1322+659 13:23:49.54 +65:41:48.0 0.1684 0.019 44.61 RQ
PG1115+407 11:18:30.20 +40:25:53.0 0.1541 0.016 46.53 RQ
4C10.06 PKS0214+10 02:17:07.66 +11:04:10.1 0.4075 0.109 45.54 RL
4C39.25 B20923+39 09:27:03.01 +39:02:20.9 0.6946 0.014 45.70 RL
OS562 16:38:13.45 +57:20:24.0 0.7506 0.013 43.51 RL
Notes: †(a)FromNED(http://nedwww.ipac.caltech.edu/)basedonSchlegeletal.(1998).(b)therest-frame
luminosityat3000A˚.
Table2 TheX-RaySpectraldata
Object E1 E2 E3 fν =f0Eα Reference
f0 α f0 α
(1) (2) (3) (4) (5) (6) (7) (8) (9)
PG1322+659 0.3 1.62 10.0 6.51E-4 −2.01+0.24 4.36E-4 −1.18+0.14 X,Po04
−0.11 −0.11
PG1115+407 0.3 2.04 10.0 6.48E-4 −1.85+0.06 4.05E-4 −1.19+0.10 X,Po04
−0.02 −0.10
4C10.06 0.1 2.4 9.08E-4 −1.13+0.52 R,Br97
−0.56
4C39.25 0.1 2.4 5.83E-4 −1.25+0.06 R,Br97
−0.06
OS562 0.1 2.4 1.36E-4 −1.38+0.04 R,Br97
−0.04
Notes: †ForthesourcesPG1322+659andPG1115+407,theirX-rayspectraarefittedbyabrokenpower
lawmodelintherangesofE1−E2andE2−E3(keV)intheobserver’sframe.Theremaindersarefittedby
asinglepowerlawmodelbetweenE1andE3.Thecolumns(5)and(7)arethefluxdensitiesat1keVinunits
ofmJy(10−26ergs−1cm−2Hz−1);EinkeV.
References: R,C,Xindicatedatasources,correspondingtoROSAT,Chandra,andXMM,respectively.Br97:
Brinkmannetal.(1997);Po04:Porquetetal.(2004)
Table3 Thefittingparametersofthesources
Object logM f a∗ m˙ θ Te(keV) χ2/dof
PG1322+659 8.29 0.47+0.10 −0.94+0.14 0.388+0.019 25.7+3.5 265.3 2.8/9
−0.12 −0.06 −0.020 −4.7
PG1115+407 8.18 0.41+0.06 −0.55+0.15 0.420+0.020 10.0+7.3 241.2 4.3/5
−0.03 −0.15 −0.005 −3.1
4C10.06 9.00 0.60+0.15 0.7+0.05 0.415+0.020 39.4+2.2 194.7 4.9/7
−0.10 −0.05 −0.017 −2.0
4C39.25 9.33 0.47+0.03 0.998+0.000 0.344+0.008 42.5+0.9 158.3 11.5/11
−0.02 −0.004 −0.011 −1.4
OS562 8.92 0.33 0.96+0.03 0.505+0.027 13.0+7.2 142.5 4.3/7
−0.04 −0.020 −9.6
Notes: TheblackholemassM istakenfromTangetal.(2012).Teisthemaximumvalueoftheelectron
temperatureofthecorona,whichiscalculatedwiththebest-fittingvaluesoffreeparametersa∗andm˙.
e.g., Doneetal.,2012,andthereferencestherein).Therefore,theinfraredandsoftX-rayspectraldata
arenotusedinourmodelfittingontheSEDs.
We use the generalrelativisticaccretiondisk coronamodelto fit the observedoptical/UVspectra
and the hard X-ray continuum spectra in 2 − 10 keV. There are four free parameters in our model
calculations,i.e., the mass accretionrate m˙ , the spin parametera∗, the powerfractionf dissipated in
the corona, and the viewing angle θ with respect to disk axis. In the spectral fitting, we find that the
fractionf ismainlydeterminedbytheX-rayspectralindex,whichisalmostindependentofthevalues
of otherparameters.Forthe broad-linetype I quasars, the valuesof the inclinationangleare tuned in
therangeofθ <∼45◦.
WeplotthespectralfittingresultsofthefivequasarsinFigures2-6.Intheoptical/UVwavebands,
thedata points(markedasbluefilled circles)whichareusedin ourspectralfittingare extractedfrom
theobservedSEDintheline-freecontinuumwindowsinordertoavoidthecontaminationbyemission
lines,asdoneinZhengetal.(1997)andVandenBerketal.(2001).
Inthis work,the measuredhardX-rayluminosityand thespectralindexareused forourspectral
fitting. In the spectralfitting, the errorsof the fitting in the hard X-ray band(both the luminosityand
Constraintsonblackholespinswithageneralrelativisticaccretiondiskcoronamodel 7
Fig.2 ThespectralfittingresultsforPG1322+659. Theradioandnearinfrareddataarenotusedin
ouraccretiondiskcoronaspectralfittings.Thephotometricpointsmarkedasbluefilledcirclesareused
forfitting.ThesolidblacklinerepresentsthepowerlawX-rayspectrumin2−10keV.Theredline
representsthebestfittingresultwiththeblackholemasslogM=8.29.Thegreendottedlinerepresents
thespectrumofthecolddisk,whilethebluedashedlineisfortheComptonradiation.Thesynchrotron
andbremsstrahlungradiationwhichdominateinradiobandsarenotplotted.
Fig.3 ThebestfittingresultforPG1115+407.
8 B.Youetal.
Fig.4 ThesameasFigure2,butfor4C10.06.
Fig.5 ThesameasFigure2,butfor4C39.25.
spectral index) are minimized by tuning the model parameters, meanwhile the observed optical/UV
spectrumistakenasupperlimits.
Thebest-fittingresultsofthediskcoronaparametersarelistedinTable3.Theerroroftheparameter
f isdeterminedaccordingtotheerrorofthehardX-rayspectralindexαinTable2,giventhatf strongly
Constraintsonblackholespinswithageneralrelativisticaccretiondiskcoronamodel 9
Fig.6 ThesameasFigure2,butforOS562.
dependsonα.Thefittingerrorχ2iscalculatedintheparameterplane,andthedistributionsoftheerror
contour for the spin a∗, the mass accretion rate m˙ and the inclination angle θ are plotted in Fig. 7.
The uncertainties on the three parameters correspond to 90% confidence level. . The 2D confidence
contours of a∗ and m˙ for PG 1322+659 with the measured black hole mass M and fraction f are
plottedinFig.8.Wefindthat,thevalueofblackholespinparameterisconstrainedinanarrowrange
−1.0 < a∗ < −0.8with best-fittingvaluea∗ = −0.94.The best-fittingaccretionrate m˙ ≃ 0.4,and
theinclinationangleofthediskθ ≃ 25◦.Weknowthatuncertaintiesoftheblackholespinconstraints
arecloselyrelatedtotheerrorsoftheblackholemassestimates.InFig.9,weplotthefittingresultsfor
PG1322+659withthedifferentvaluesofM reportedintheliterature,andfindthattheconstraintson
theblackholespinparametera∗ aresensitivetotheblackholemassestimate(seeTable4).Withinthe
uncertaintiesofafactoroffouronthemassmeasurements(Vestergaard&Peterson,2006),usingsingle
epoch reverberationmappingmethod, from 2×108M⊙ to 8×108M⊙ with the step of 108M⊙, the
best-fittingvaluesofblackholespincanvaryfrom-0.94to0.998(seeFig.10).InFig.11,weplotthe
fittingresultsforPG1322+659withasmallshift∆logL ontheobservedX-rayluminositytoassess
X
howtheblackholespina∗ isaffectedbyX-rayvariability(seeTable4).
WefittheemissionlinepropertiesofPG1322+659bysubtractingthebestspectralfittingcontinuum
in Fig. 12, to compare with the typical broad line region spectrum. The derived EW and FWHM for
MgII, Hβ, andHγ areconsistentwith thoseofthe typicalBLRspectrumgiveninShenetal. (2011).
InFig.13,weplottherelativediskthicknessH /Rvaryingwiththemassaccretionratem˙ .Itisfound
d
that the thin disk approximationH /R <∼ 0.1 can be achieved even if m˙ > 0.3. Note that when the
d
massaccretionratem˙ isashighas∼ 0.5,H /Risonlyslightlylargerthan0.1foratypicalvalueof
d
f ∼0.5.
4 DISCUSSION
ThemainfeaturesofquasarSEDwithabigbluebumpinoptical/UVwavebandsandapower-lawhard
X-raycontinuumspectrumcanbereproducedbytheaccretiondiskcoronascenario(Galeevetal.,1979;
Qiao&Liu2015).Therearevariousmodelswhicharedevelopedintheexploringtheradiationfromthe
10 B.Youetal.
Fig.7 Error contour for the spin parameter a∗, the mass accretion rate m˙ and the inclination angle
θ. Thethreehorizontal lines, frombottomtotop, arecorresponding tothe68.3%, 90.0%and 95.4%
confidence,respectively.
Table4 ThespectralfittingresultsofPG1322+659
ID logM f a∗ m˙ θ
1 8.29 0.47 -0.94 0.388 25.7
2 8.36 0.47 -0.75 0.316 29.8
3 8.16 0.47 -0.94 0.520 0.0
4 8.59 0.47 0.90 0.06 30.0
5 8.89 0.47 0.94 0.03 43.0
6 8.29 0.47 -0.7 0.5 37.0
7 8.29 0.47 -1.0 0.3 0.0
Notes: Thespectralfittingresultswithdifferentblackholemassesintheliterature(ID2-5)andthevariation
oftheX-rayluminosity(ID6,7).ThefittingresultsareplottedinFig.9and11respectively.
accretiondiskandcorona.Forinstance,KERRBBisarelativisticmulti-temperatureblackbodymodel
forthethinaccretiondiskaroundaKerrblackhole,whichisusuallyusedtofitthediskemissionfrom
disk-dominatedX-raybinariesandquasars(Lietal.,2005),whileSIMPLisaComptonizationmodel
for modeling the power-law hard X-ray spectra of X-ray binaries and AGNs (Steiner et al., 2009b).
The convolutionbetween KERRBB and SIMPL in XSPEC (Arnaud, 1996) could simulate the broad
band SEDs of quasars, namely,the opticalBBB and power-lawX-ray spectra, for the purposeof our
paper here. In the resulting spectra of SIMPL*KERRBB, the hard X-ray power-law index depends
on the parameter Γ, while the flux is determined by the parameters f which is the fraction of the
SC
soft photons from the disk being scattered to high energy (see Fig. 1 in Steiner et al., 2009b). As an
alternativeComptonizationmodel,ageneralrelativisticmodelforanaccretiondiskcoronasurrounding
aKerrblackholewasdevelopedbyYouetal.(2012),inwhichtheenergyfractiondissipatedintothe
corona was assumed in order to solve the pattern of the disk corona system. The emergent spectrum
of the disk corona observed at infinity can be calculated when the values of the disk parameters are
specified.