On the determination of the spin and disc truncation of accreting black holes using X-ray reflection

0.19 MB·

by  A. C. Fabian

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed9January2014 (MNLATEXstylefilev2.2)
On the determination of the spin and disc truncation of accreting
black holes using X-ray reflection
A.C. Fabian1∗, M.L. Parker1, D.R. Wilkins1, J.M. Miller2, E. Kara1,
C.S. Reynolds3 and T. Dauser4
4 1InstituteofAstronomy,MadingleyRoad,CambridgeCB30HA
1 2DepartmentofAstronomy,UniversityofMichigan,AnnArbor,MI48109,USA
0 3DepartmentofAstronomy,UniversityofMaryland,CollegePark,MD20742-2421,USA
2 4DrKarlRemeis-ObservatoryandErlangenCentreforAstroparticlePhysics,Sternwartstr.7,D-96049Bamberg,Germany

n
a
J
9January2014
8

]
E ABSTRACT
H We discuss the application of simple relativistically-blurred X-ray reflection models to the
determinationofthespinandtheinnerradiusofthe discin accretingblackholes.Observa-
.
h tionally,thenatureofthecoronaisuncertainapriori,butarobustdeterminationoftheinner
p diskradiuscanbemadewhenthediscemissivityindexistightlyconstrained.Whentheinner
- disc is well illuminated, the black hole spin can also be determined. Using reflection mod-
o
ellingderivedfromraytracing,weshowthatrobustdeterminationofdisctruncationrequires
r
t thatthelocationofthecoronalsourceisquasi-staticandataheightandradiuslessthanthe
s
a truncationradiusofthedisc.Robustspinmeasurementsrequirethatatleastpartofthecorona
[ lies less than about 10 gravitational radii above the black hole in order that the innermost

regions,includingtheinnermoststablecircularorbit,arewellilluminated.Thewidthofthe
1
blurringkernel(e.g.,theironline)hasastrongdependenceoncoronalheight.Theselimita-
v
5 tionsmaybeparticularlyapplicableatlow Eddingtonfractions(e.g.the low/hardstate, and
1 low-luminosityAGN)wheretheheightofthecoronamayberelativelylarge,oroutflowing,
6 andtiedtojetproduction.
1
Keywords: blackholephysics:accretiondiscs,X-rays:binaries,galaxies
.
1
0
4
1
: 1 INTRODUCTION thisisidentifiedwiththeInnermost StableCircularOrbit(ISCO)
v
then the spin of the black hole can be determined. Alternatively,
i X-rayreflectionisbecoming acommon toolinX-rayastronomy,
X thediscmaybetruncatedwellbeforetheISCOinwhichcasethe
particularlyforstudyingtheinnerregionsaroundcompactobjects.
r truncationradiusmaybelocated.
a ItoccurswhenmatterisirradiatedbyaprimaryX-raysource,giv-
ing rise to backscattering, fluorescence and secondary1 emission Considerable effort has been expended recently on tracing
(seeFabian&Ross2010forareview).Theprimarysourceishere light rays in the Kerr metric in order to determine the emissiv-
referred to as the corona. We initially assume it to be pointlike, ityprofileon thedisc plane produced by asource above the disc
staticandonaxis,butinpracticeitmaybeextendedandoutflow- (Wilkins&Fabian2012;Dauseretal2013).Theemissivityprofile
ing,perhapsthebaseofajet.Thereflectionspectrumprovidesdi- andtheradiusrangeoverwhichitisrelevantarekeyfactorsincon-
agnostics for elemental abundances and ionization parameters of structingthespectralresponseofthereflectionspectrum.Herewe
thematteraswellasitsvelocityandradiusrangethroughdoppler discusslimitationstothemeasurementofspinanddisctruncation
andgravitationalredshifts.Forsourceswithsignificantvariability, whentheprimarysourceorcoronaismuchmorethan10gravita-
reverberation of the reflectioncomponent compared withthe pri- tional radii (10GM/c2) above thedisc. Whenthat occurs, thein-
mary can reveal the source size and geometry in physical units. nermostregionsaroundtheISCOarepoorlyirradiatedsomaking
HereweareconcernedwiththeuseofX-rayreflectiontomeasure thereflectionspectrumfromthatpartofthedisc,whichiscrucial
theinnermost radiusof theaccretion discaround ablack hole. If tospindetermination,difficulttomeasure.Thisissuehasbeendis-
cussed before (e.g. Vaughan & Fabian 2004; Chiang et al 2012;
Dauser et al 2013) but is not widely recognised. The purpose of
thispaperistoexploreitfurther.
∗ E-mail:[email protected]
1 Primaryheremeansthattheenergyderives directly fromtheaccretion Spin determinations of AGN using X-ray reflection have
process, through magnetic fields for example, whereas secondary means beenongoingsinceDabrowskietal(1997)withrecentresultsin-
thattheenergysourceistheirradiatingX-rays. cluding MCG–6-30-15 (a=cJ/GM2 =0.989−+00..000029, Brenneman
(cid:13)c 0000RAS
2 A.C. Fabianet al
& Reynolds 2006), 1H0707-495 (a>0.98, Fabian et al 2009);
IRAS13224-3809 (a =0.989±0.001, Fabian et al 2013a) (see 1 Source Height
recent reviews by Reynolds 2013 and Brenneman 2013 for sum-
3 R
maries of more AGN spin results). The high statistical precision 6 RG
G
of theabove three resultsisduetoa combination of low coronal 0.8 10 RG
heightandhighironabundance,whichemphasizesfeaturesinthe 30 RG
reflectionspectrum.X-rayreflection-basedspindeterminationsin
blackholebinaries(BHB)arereviewedbyMiller(2007)following ux
Milleretal(2002,2004);morerecentresultsincludeCygX-1(a= d Fl 0.6
a0n.9d7G+−00R..00S1241,9F1a5b+ia1n05et(aal=20102.9;8a±=00.0.918,±M0il.l0e1r,eTtoamls2i0ck13e)t.aOl2b0je1c4ts) malise
0w.i2t,hMinitneirumtteideitaatel2s0p0in9,),inXcTluEdJe1S7W52I-F2T23J2(1a2=7.04.+5526±540.(1a1=,R0e.i6s±et Nor 0.4
al2011)andFairall9(a=0.52−+00..1159,Lohfinketal2012).Comple-
mentaryworkonBHBusingthethermalcontinuummethod(Mc-
Clintock,Narayan&Steiner2013)isprovidinggeneralagreement 0.2
withX-rayreflectionmethods,forobjectswherebothmethodscan
beapplied(e.g.Steineretal2012).
Bothmethodsrelyontheidentificationof r ,theinnermost
in 0
discradiusdetected,withtheISCO2.Thissystematicuncertainty 2 3 4 5 6
isbeingtackledwithnumericalsimulationsofdiscs(Reynolds&
Energy (keV)
Fabian2008;Shafeeetal2008;Nobleetal2011;Pennaetal2010;
Kulkarnietal2011).Thesimulationsindicatethatr ,theinnerra-
in Figure1.Modelironlineprofilesexpectedfromanon-axispointsource
diusofthedensedisc(whichisrelevanttoreflection),lieswithin atheightsof3,6,10and30rg aboveanaccretiondiscinclinedat30deg
apressurescale-height oftheISCO,whichcorrespondstoanun- aroundarapidlyspinningblackholewithspina=0.998.Theprofileswere
certainty of ∼(L/LEdd)rg. Different methods may give (slightly) obtainedusingRELLINE-LPofDauseretal2013.
differentvaluesforr dependingupontheirsensitivitytosurface
in
density or emission. The disc may be thicker when the thermal
continuummethodisoptimallyappliedthanforthereflectionap- theillumination,whichmeansthatthecoronalemissionsourceis
proach.Thelowionizationparameteroftenfoundwiththereflec- point-like,liesalongthecentralrotationaxisofthediscandemits
tionmethod for theinner discof AGN (e.g. x <20 for 1H0707- isotropically.Weconsidermorecomplexgeometrieslater.Weuse
495andIRAS13224-3809)indicatesthatthediscisdenseandthus RELLINE-LPofDauseretal(2013)inthespectral-modellingpack-
probablythin.Therearecurrentlynopublishedsimulationsofthe age XSPEC to model the response of a single emission line from
ISCOregionofthinirradiateddiscs. the disc. This model is based on ray-tracing calculations, so ac-
Disctruncation inwhich theoptically-thick, physically-thin, curatelyrepresentsthestronggravityeffectsof lightbendingand
discterminatesbeyondtheISCOandtheinnerregionisfilledwith gravitational redshift. The disc emissivity profile of the reflected
ahot advection-dominated flow isoften invoked as amechanism spectrumisimplicitlyincludedinRELLINE-LP.Itcanbeexplicity
forexplainingstatechangesinstellarmassblackholesources(Esin specifiedusingabroeknpower-lawwithRELLINE,itsconvolution
etal1997;Doneetal2007).Inthismodelasourceinthelow,hard kernelRELCONV,oranyothersimilarkernel.
state is interpreted as having a truncated disc. Spectral evidence Theemissionlinepredictedasthesourceisraisedfrom3to
fortruncationisconfusedintheliteraturewithsomeworkclaim- 30gravitationalradii(3−30GM/c2)isshowninFig.1.Theblack
ingthatitdoesnot occur (e.g. Milleretal2004, 2006, Reiset al holehasahighspinof0.998,andthedisccontinuesdowntothe
2008,2010)andsome(Done&DiazTrigo2010)thatitdoes.Re- ISCO,whichisat1.235r .Whenthesourceishighabovethedisc,
g
centworkbyPlantetal(2013)onthesourceGX339-4,whichhas atheighth>6r ,thelinedoesnotlookverybroad.Thisisbecause
g
featured much in this debate, appears to show disc truncation in little illumination is reaching the innermost part near the ISCO.
thelowstateusingasimplerelativisticreflectionmodel(seealso Astheheightisincreasedfurthertheilluminationonthediscap-
Petruccietal2013).AsdiscussedhereinSection2,thisinterpre- proaches the Newtonian pattern which is roughly constant out to
tationmaybetoosimplistic.Truncationhasalsobeeninferredin radius r∼h, beyond which theemissivity of thedisc drops with
some radio-loud AGN, such as 3C120 (Marscher et al 2003; Lo- radiusasapower-lawofindex3,asplottedinVaughan&Fabian
hfinketal2013). (2004).
To highlight the effect of illumination on detection of the
emissionfromtheISCOwhenthesourceisataheightof30rg,we
2 DETERMINATIONOFTHEINNERRADIUSOFTHE show inFig.2thelineprofilefor r=rISCO =1.235rg andwhen
DISCBYREFLECTION thediscistruncatedat30rg.Thesmalldifferenceinnormalization
would in reality be unmeasurable since the absolute value is un-
For the purpose of illustration, we concentrate here on a single
known;themainindicationoftruncationliesbelow5.5keV,where
emission line, but the principle is the same if the whole reflec-
the line strength is less than 10 per cent of its peak. Truncation
tionspectrumisused.Weemployasimplelamp-postgeometryfor
would be difficult to detect clearly with current instruments and
evenabrightsource.
2 In principle both rin and acan be measured, butthe deviations to the Theconclusionhereisthatatleastpartofthecoronalsource
energyshiftsproducedatagivenradiusbyspinalonearesmall,seee,g, needs to be situated at a height below at most 10rg in order that
Fig.2ofDovciaketal(2004) clearevidencefortruncationbeyondtheISCOisobtainable.This
(cid:13)c 0000RAS,MNRAS000,000–000
Reflectionon truncationandspin 3
1 1
Truncation Radius
1.25 R
G
0.8 0.8 3 RG
5 R
G
10 R
G
30 R
G
x x
u 0.6 u 0.6
Fl Fl
d d
e e
s s
ali ali
m m
or 0.4 or 0.4
N N
0.2 0.2
0 0
4 4.5 5 5.5 6 6.5 7 2 3 4 5 6
Energy (keV) Energy (keV)
-1V)]1000 Fthiegucernet3re.oMfaoddeislclitnreunpcraotfiedlesatprriend=ict1e.d25fo,3r,a5,s1o0uracnedh3e0igrhgt.h=3rgabove
e
k
-2-2m s mimickingtheappearanceofahighsource.Thiscouldwellberel-
ns c 500 evant to the situation when a jet develops in a source where the
o coronalemissionoccursfromthebaseofthatjet(Markoff,Nowak
ot
ph &Wilms2005).Thelocationofthebaseofthatjetmaybequasi-
2V ( stationary,yettheemittingmatterwithinthebasemayflowrapidly
ke upward. This rapid motion is of course what is relevant for the
F [E emissionpatternanddiscirradiation.
2E Both coronal height and motion could be relevant to jetted
200 sources, suchasthelow stateofblackholebinaries,forexample
GX339-4,andradio-loudAGNsuchas3C120.Lohfinketal(2013)
1.1
findthattheinnerradiusofthediscin3C120appearstochangefol-
atio 1 lowingaradiooutburst(seealsoMarscheretal2002).Possibleal-
R ternativeinterpretationsarethateithercoronalheighthasincreased
0.9
orthecoronalmaterialhasdevelopedanoutwardvelocityofafew
0.8
tenthsofthespeedoflight.
5 10 Whenadiscistruncatedandtheinner radiusiswellillumi-
Energy (keV) natedthentheeffectsoftruncationcanbeseeninthelineprofile
(Fig.3).Inthisextremeexamplethesourceheightisonly3rg.Note
Figure2.Upperpanel:Thesolidlineisthelineprofileforh=30rgfrom thatthepeakofallmodellinesisnormalisedtounityanddoesnot
Fig.1.Thedashedlineshowstheeffectoftruncatingthediscatr=30rg. showthedropinoverallreflectedemissionwhichwouldresultfrom
Note that the mostmeasurable change is relatively small and occurs be- thelargertruncationradiiinthisexample.
low5.5keV.Lowerpanel:Asabovebutwithreflectionspectrum(Ross&
Fabian2005)ratherthanjustaline.
2.1 Simulations
alsoappliestospinmeasurements,whichonlyyieldalowerlimit The height of the X-ray source above the disc is manifest in the
tothe spin unless thecoronal height issmall sothat theISCO is emissivityprofileoftheaccretiondiscusedinthecomputationof
well illuminated (Dauser et al 2013). Any firm recommendation the broadened line profile. If the spin of the black hole is to be
dependsonthequalityofthedata,andsourceconditionsincluding obtained by fitting to the profile of the broadened emission line,
ironabundance, inclination and the spinitself, (Fig.1 inWilkins someassumptionisrequiredastotheemissivityprofileofthedisc
&Fabian2011showsanexampleoftheenergyrangerelevantto inthemodelthatisfittothedata.
variousinnerdiscradii).Theaboverecommendation,h<10rg,is Toquantifytheeffectoftheassumedemissivityprofile,spec-
relevanttocurrentdataquality. traweresimulatedusing amodel consisting of apower-law con-
Notethat similarresultsapply ifthesource iseven amildly tinuumandrelativistically-broadened ironKa lineresultingfrom
relativisticoutflow, as suggested by Beloborodov (1999; see also illuminationbyanX-raysourceatdifferentheights(3,5and10rg)
Malzac,Beloborodov&Poutanen2001).Duetospecialrelativis- above thedisc. (Thisextends thesimulations shown inDauser et
ticaberrationtheemissionwouldbebeamedawayfromthedisc, al 2013.) The emission line was modelled with the RELLINE-LP
(cid:13)c 0000RAS,MNRAS000,000–000
4 A.C. Fabianet al
100 h = 3r , ξ = 2000 bothofwhichassumethecorrectemissivityprofilefortherelevant
g
h = 5r , ξ = 2000 illuminationcase.
g
h = 10rg, ξ = 2000 In Fig. 5 we show results from simulations using RELLINE,
h = 3r , Line
g whichusespower-lawapproximationsoftheemissivityprofile.In
80 h = 5r , Line
g
h = 10r , Line this case, when the source is at a greater height above the disc,
g
overestimation of the emissivity index parameter (the power law
slope) causes the measured spin to be systematically underesti-
of 60 mated(Fig.4).Whenthesourceishigh,thereexistsasignificant
d
4 flattenedportion of the emissivityprofile at radii between thein-
4
2Δχ / 8 40 nNeerwmtoosntiapneaekmainsdsivrihty(Windilekxinqs&=F3acbaiaunse2s0t1h2e).reEdvsehnifatessduemminisgsitohne
fromtheinnerpartofthedisctobeoverestimated(sincetheover-
allnormalisationof theemissionlineisdrivenbytheblueshifted
peak).Thebestfittotheobservedspectrumthereforerequiresthat
20 thediscbeartificiallytruncatedatlargerradiustocompensatefor
this extra emission created from the inner part of the disc. This
artificialtruncation of the accretion disc isinterpreted as a lower
measurementoftheblackhole’sspin.
0
0 0.2 0.4 0.6 0.8 1 Thusanassumedemissivityprofileindexof3canmeanthat
a thespinisunderestimatedwhentheilluminatingX-raysourceisas
100 littleas5rg abovetheplaneoftheaccretiondisc,thoughitisim-
h = 3rg, ξ = 2000 portanttoemphasisethatthiseffectsystematicallyunderestimates
h = 5r , ξ = 2000
g theblackholespin,meaningthatconclusionsdrawnaboutrapidly
h = 10r , ξ = 2000
g
h = 3r , Line spinningblackholesremainvalid.
g
80 h = 5r , Line
g
h = 10r , Line
g
3 MORECOMPLEXGEOMETRIES
of 60
d The situation becomes more complicated if the coronal source is
4
4 extended, as it surely must be. Observations including reflection
8
2 / spectra,reverberationandmicrolensing(Fabianetal2012;Karaet
Δχ 40 al2013;Morganetal2012)indicatethatthecoronainatleastsome
sourcesiscompact,meaningconfinedtor∼10rg orless(Fabian
2012;Reis&Miller2013).Thisseemsplausibleforrapidlyspin-
ningsourceswheremostofthediscenergyreleaseoccurswithin
20
thatradius3.Thecoronaispresumablypoweredbymagneticfields
anchoredinthediscsoanyquasi-stationarycoronaislikelytohave
aheighth<10−30rg,withsmallervaluesapplyingtohigherspin.
0 Even in such cases, the effects of the inevitable strong light
0 0.2 0.4 0.6 0.8 1
bendingmeanthatthesourceappearsanisotropictotheobserver.
a
Thisistheessenceofthelight-bendingmodel(Miniutti&Fabian
2004)wheretheintensityofthereflectionspectrumchangeswith
Figure4.ModelfittingusingthecorrectemissivityprofilefromRELLINE-
sourceheightrelativelylittlewhereastheobservedprimarycoronal
LPtodemonstratethesensitivitytospinanddiscionizationparameter.The
emissionchangesmarkedly.
highestsensitivityariseswhenthesourceheightislowandthusbestirradi-
atestheISCOandalsoifthediscisionized. If the source is extended uniformly in height then the lower
regions,whichgivetheleastdirectemission,willgivethemostir-
radiationtotheinnermostpartsofthediscneartheISCO,whereas
theoppositeappliesfortheupperparts.IftheComptonizationspec-
model and the fluxes of the continuum and reflected line corre-
trumof thecorona changes withheight (theelectrontemperature
spond to those found in the spectrum of the narrow line Seyfert
andopticaldepthareunlikelytobeconstantwithheight)thenitis
1galaxy1H0707-495(Fabianetal2009),butwithaphotonindex
G =3 and disc inclination of 30 deg. A 500ks observation made possiblethattheinnermostpartsneartheISCOareirradiatedbya
power-lawofdifferentphotonindexG tothatwhichcharacterizes
withXMM Newton was assumed using the FAKEITcommand in
themeanobservedpower-lawspectrum.Similareffectsshouldoc-
XSPEC.Ineachcase,thespinwassettobeeither0.998(Fig.4.
upper)or0.5(Fig.4,lower).Wealsoincludeexampleswherean curforaradially-extendedsource.
They do not necessarily make spin difficult to measure4 but
ionizeddiscmodel(Ross&Fabian2005),withionizationparam-
eterx =2000, isassumed. Thereisnow adeepsmeared edgein
thespectrumwhichprovidesadditionalsensitivityandthustighter
constraintonspin. 3 Halfthediscpowerisemittedwithin5rgforahighlyspinningblackhole
andwithin∼33rgatlowspin,Thorne(1974).Ifh∼rinradio-quietobjects,
Just as for real observations, the spin is determined by fit- thenh∼5+28(1−a).
ting the profile of the iron Ka line in the fake spectra over the 4 On the basis of the emissivity profile, Wilkins & Fabian (2012) show
energy range 0.3–10 keV, using either the RELLINE-LPmodel or that thecorona in1H0707-495 is extended. Thereis however significant
RELCONVconvolved with REFLIONXpluspower-lawcontinuum, reflectionfromtheinnerdiscsothespinmeasurementisrobust.
(cid:13)c 0000RAS,MNRAS000,000–000
Reflectionon truncationandspin 5
400 Assumed q=3 do mean that the spectral models may need to be more sophisti-
Assumed q=5,3 Rbreak = 3rg cated than those commonly adopted in which G for the incident
Assumed q=7,3 Rbreak = 5rg
power-lawisusuallyforcedtobethesameasthatforthereflection
component.Theobservedspectralturnoverinthepower-lawcon-
300
tinuummayalsobedifferentfromthatdeducedfromthereflection
spectrum.
Propagating spectral variations produced as power flows
2Δχ200 through a spatially-extended corona, can lead to low-frequency
spectral lags as discussed by Kotov et al (2001) and Wilkins &
Fabian (2013). The common occurrence of such lags (Miyamoto
100 &Kitamoto1989; Nowaketal1999;Gilfanovetal1999;Pouta-
nen2001;McHardyetal2007)isgoodevidencethatcoronaeare
extendedandinhomogeneous.
The inner parts of extended coronae could produce a softer
00 0.2 0.4 0.6 0.8 1 spectrum(higherG )thantheouterparts.Theenergydensityinsoft
a diskphotonswhichareupscatteredbythehotelectronstomakethe
150 Assumed q=3 coronal continuum should peak within a few rg for rapidly spin-
Assumed q=0,3 Rbreak = 5rg ningblackholes.Itsratiototheenergydensityoftheelectronsis
125 an important factor in determining G . Consequently, estimates of
spinfromthesoftexcessinthereflectioncomponent,suchaspre-
sented by Walton et al (2013), could give higher spin values for
100
thesameobjectwhencompared withthoseobtained fromiron-K
measurements.Weassumethatthesimplestmodelisused,where
2Δχ 75 the reflection and power-law components have the same G and a
singlesourceheight,i.e.emissivityprofile.IFtheinnerregionsof
50 thecoronaproduceaharderspectrumthenofcoursetheopposite
conclusioncouldapply.
25
0 4 SYSTEMATICSINTHEX-RAYREFLECTION
0 0.2 0.4 0.6 0.8 1
a METHOD
100 Assumed q=3 Wenow outlinesomeofthesystematicsinherent inthemeasure-
Assumed q=0,3 Rbreak = 10rg ment of spin and inner radius using the X-ray reflection method.
Besidethelevel at whichtheinner radii areilluminated, relevant
80 issuesincludetheionization,density,elementalabundances,level
ofsurfaceturbulenceanduniformityofthedisc,theextenttowhich
thecoronaisextendedandthuswhetheracorrectemissivitymodel
60 isadopted.TheimportantissueofwhethertheISCOandrincoin-
cidehasalreadybeendiscussedintheIntroduction.
2Δχ The ionization state of the surface of the disc, which deter-
minestheionizationparameterx inthereflectionmodel(e.g.either
40
REFLIONX(Ross&Fabian2005)orXILLVER(Garcia&Kallman
2010);Garciaetal2013a),affectstheoverallshapeofthereflection
spectrum.Iftheionizationparameterishigh,the(vertically)outer-
20
mostdiscgasishighlyionizedandmuchoftheironlinescatters
outfrombelowtheouterThomsondepth,leadingtoadditionalline
broadeningduetoComptonscattering(Ross&Fabian2007).The
0 ionizationstatedependsupontheilluminationlevelandpattern(i.e.
−1 −0.5 0 0.5 1
a radialdependence)andonthediscdensityprofile.If,asexpected,
thepower for thecorona isextractedmagnetically fromthedisc,
Figure 5. Model fitting using broken power-law approximations for the thenitisnotobviousthattheradialdependenceofthatextraction
emissivityprofile(seee.g.Wilkins&Fabian2012).Maximumspinisas- shouldmatchthatoftheanalyticdissipationrelationofaShakura-
sumedwithrin=rISCO.Theinnerandouterpower-lawindexes,q,arelisted Sunyaev(1973)orNovikov-Thorne(1974)disc,nor,therefore,that
together with the break radius. If only one index is given then a single thedensityprofileisknownorcanbefoundotherthanbyobserva-
power-lawisassumed.Upperpanel:h=3rg,Middlepanel:h=5rg,Lower tion.Svobodaetal(2012)showsomeeffectsproducedbyvarying
panel:h=10rg.Poorapproximationsleadtounderestimationofspin.(The theionizationparameteracrossthedisc.Anextendedcoronaexac-
truespininthesimulationis0.998.)
erbatestheuncertaintiesintheionizationprofile.
Svoboda et al (2009) discuss the effects produced by limb
darkening,orbrightening,oftheemergentreflectedradiation.Note
thattheverticalstructureofthedisksurfacedependsonthepres-
sures involved, which in most cases are dominated by magnetic
(cid:13)c 0000RAS,MNRAS000,000–000
6 A.C. Fabianet al
fields(Blaes2013),sosimplethermalhydrostaticsupporthaslittle needs to be eliminated before disc truncation can be reliably de-
relevance.Therearecurrentlynodetailedsimulationsofirradiated duced.Independentevidencefordisctruncation,fromthermaldisc
discs to inform us of the expected density structure. Commonly emissionforexample,isnotaffectedbytheheightofthecorona.
used constant density models are an approximation. The vertical Subject to the uncertainty in the relation between the ISCO
density profile of the disc surface affectsthe outer ionization pa- andr ,excellentestimatesofspinandinnerdiscradiusarebeing
in
rameter.Adecreaseddensityleadstoanincreasedionizationwhich madewithX-rayreflectionusingthemethodsoutlinedabove.The
canaffectthedegreeoflimbdarkening. robustmeasurementsofspinfromthereflectionmethodwhichhave
Disc inclinationleads tofurther uncertainties. Inprinciple it yieldedhighspinareperfectlyvalidandsimplyindicatethatmuch
is measurable by the blue wing of the line, which is sensitive to oftheirradiatingradiationoriginatesfromwithin10rg.Theeffects
theincreaseddoppler shiftsproduced byhigher inclination.Mea- discussedabovesuggestthatlesscertainresults(duetothecorona
surements of spin and inclination are correlated, but in a model beinghigher or outflowing, forexample) indicatelower limitson
dependent manner;forexamplesseeTomsickatal(2014). Inad- spin.
dition,thereareintrinsicinclinationeffectsonthereflectionspec-
trum,nowmodelledbyRELXILL(Garciaetal2013b).
Numericalmodelsoftheinnerregionsofthindiscsshowthem
ACKNOWLEDGEMENTS
tobeturbulent(Armitage&Reynolds2003;Beckwithetal2008).
Thiscouldleadtothesurfacedensityofthediscbeingpatchy(see Wethankthereferee,JavierGarcia,forhelpfulcomments.
e.g. Dexter & Quataert 2012), thus leading to a range of ioniza-
tionparameterateachradius.Theremightmoreoverbeadditional
doppler broadening due to the turbulent motions (Tomsick et al
REFERENCES
2014). This is presumably subsonic, which means that it should
notbeimportantforobjectswherealowdiscionization,andthus ArmitageP.J.,Reynolds,C.S.,2003,MNRAS,341,1041
lowdiscsurfacetemperature,isinferred. BeckwithK.,ArmitageP.J.,SimonJ.B.,2011,MNRAS,416,361
Finally we note that the geometry of the corona is probably Beloborodov,A.,1999,MNRAS,510,L123
notknownapriorisoamodel-independentapproachisrequiredfor BlaesO.,2013,SSRev,inpress,arXiv:1304.4879
determiningtheemissivityprofile.Singlyordoublybrokenpower- BrennemanL.,ReynoldsC.S.,ApJ,652,1028
BrennemanL.,2013,arXiv:1309.6334
lawmodelsareagoodguide(Wilkins&Fabian2012).
BrennemanL.,etal,2011,MNRAS,736,103
ChiangC-Y.,ReisR.C.,WaltonD.J.,FabianA.C.,MNRAS,425,2436
Dabrowski Y., Fabian A.C., Iwasawa K., Lasenby A.N., Reynolds C.S.,
5 SELECTIONEFFECTS 1997,MNRAS,288,L11
DauserT.,WilmsJ.,ReynoldsC.S.,BrennemanL.W.,2010,MNRAS,409,
Observational selection effectsmean that the brightest sources in 1534
theSkywilltendtohavebothhighspinandinclination,evenifthe Dauser,T.,Garcia,J.,Wilms,J.,Bck,M.,Brenneman,L.W.,Falanga,M.,
underlying distributionfavoursnospinorinclination.Ifthemass Fukumura,K.,Reynolds,C.S.,2013,MNRAS,430,1694
accretionrateisindependentofblackholespin,thentheincreasein DexterJ.,QuataertE.,2012,MNRAS,426,L71
efficiencyofenergyreleasewithspinmeansthatthemostrapidly DoneC.DiazTrigoM.,2010,MNRAS,407,2287
DovciakM.,KarasV.,YaqoobT.,2004,ApJS,153,205
spinningblackholeswillbethemostluminousandthusdominate
EsinA.,McClintockJ.E.,NarayanR.,1997,ApJ,489,865
flux-limitedsamples(Brennemanetal2011;Reynoldsetal2012).
FabianA.C.,RossR.R.,2010,SSRev,157,167
Similarly,ifthecoronacorotateswiththeblackholeorinnerdisc,
FabianA.C.,etal,2009,Nature,459,540
thenspecialrelativisticaberrationwillmakethesourcebrightness
FabianA.C.,etal2012,MNRAS,424,217
increasewithinclination;thesamewillbetrueofreflectionorin- FabianA.C.,etal,2013a,429,2917
trinsicdiscemission(Sun&Malkan1991; Waltonetal2013).A FabianA.C.,2013b,IAUS,290,3
puffed-updisc,anequatorialaccretionflow(oroutflow),awarped GarcaJ.,KallmanT.R.,2010,ApJ,731,131
disc or a surrounding torusmay however prevent sources at very Garca,J.,Dauser,T.,Reynolds,C.S.,Kallman,T.R.,McClintock, J.E.,
highinclinationsfrombeingprevalent. Wilms,J.,Eikmann,W.2013a,ApJ,768,146
Notethatthedegreetowhichthecoronaiscorotatingabove Garca,J.etal2013b,arXiv:1312.3231
GilfanovM.,ChurazovE.,RevnivtsevM.,1999,A&A,352,182
theinnerdiscofarapidlyspinblackholewillaffecttheapparent
KaraE.,Fabian A.C.,Cackett E.M.,Uttley P.,Wilkins D.R.,ZoghbiA.,
reflection fraction. Thisprovides in principle amethod for deter-
2013,MNRAS,434,1129
miningtherotationrateofthecorona.
KotovO.,ChurazovE.,GilfanovM.,2001,MNRAS,327,799
KulkarniA.,etal,2011,MNRAS,414,1183
LohfinkA.,etal2012,ApJ,758,67
LohfinkA.,etal2013,ApJ,772,83
6 DISCUSSION
MalzacJ.,BeloborodovA.M.,PoutanenJ.,2001,MNRAS,326,417
Robustmeasurementsofblackholespinanddisctruncationfrom MarkoffS.,NowakM.A.,WilmsJ.,2005,ApJ,635,1203
X-rayreflectioninaccretingblackholesusingcurrentdatarequire MarscherA.P.,JorstadS.G.,GomezJ-L.,AllerM.F.,TerasrantaH.,Lister
M.L.,StirlingA.M.,2002,Nature,417,625
that at least part of the coronal emission source lies below about
McClintockJ.E.,NarayanR.,SteinerJ.F.,2013,SSRev,73
10rg.ThisisnecessaryinorderthattheISCOisilluminatedwell. McHardyI.,ArevaloP.,UttleyP.,PapdakisI.E.,SummonsD.P.,Brinkmann
Narrowingoftheironlinecouldbeassociatedwithdisctruncation,
W.,PageM.J.,2007,MNRAS,382,985
coronalelevationormildrelativisticoutflowofthecoronalsource.
MillerJ.M.,etal2002,AoJ,570,L69
Sincejetted radio emission iscommonly seen inthe low stateof MillerJ.M.,etal2004,ApJ,606,L131
stellar mass black holes, it is plausible that either of the last two MillerJ.M.,etal2006,ApJ,653,525
effects,ortheircombination,couldberesponsibleandtheireffect MillerJ.M.,etal2013,ApJ,775,L45
(cid:13)c 0000RAS,MNRAS000,000–000
Reflectionon truncationandspin 7
MillerJ.M.,2007,ARAA,45,441
MillerJ.M.,etal2004,601,450
MiniuttiG.,FabianA.C.,2004,MNRAS,349,1435
MiniuttiG.,etalMNRAS,398,255
MiyamotoS.,KitamotoS.,1989,Nature,342,773
MorganC.W.,etal,2012,ApJ,756,52
NobleS.C.,KrolikJ.H.,SchnittmanJ.D.,HawleyJ.F.,2011,ApJ,743,115
NovikovI.D,ThorneK.S.,1973,inC.DeWitt&B.S.,DeWitted.Black
Holes343-450
NowakM.A.,VaughanB.A.,WilmsJ.,DoveJ.B.,BegelmanM.C.,1999,
ApJ,510,874
PennaR.F.,McKinneyJ.C.,NarayanR.,TchekovskoyA.,ShafeeR.,Mc-
ClintockJ.E.,2010,MNRAS,408,752
PetrucciP-O.,etal2013,arXiv:1310.3039
Plant D.S., Fender R.P., Ponti G., Muoz-Darias T., Coriat M., 2013,
arXiv:1309.4781
PoutanenJ.,2001,AIPC,599,310
ReisR.C.,FabianA.C.,RossR.R.,MiniuttiG.,MillerJ.M.,ReynoldsC.S.,
2008,MNRAS,387,1489
ReisR.C.,FabianA.C.,MillerJ.M.,2010,MNRAS,402,836
ReisR.C.,etal2010,MNRAS,410,2497
ReisR.C.,MillerJ.M.,2013,ApJ,769,L7
ReynoldsC.S.,FabianA.C.,2008,ApJ,675,1048
ReynoldsC.S.,etal2012,AIPC,1427,157
ReynoldsC.S.,2013,SSRev,inpress
RossR.R.,FabianA.C.,2005,MNRAS,358,211
RossR.R.,FabianA.C.,2007,MNRAS,381,1697
ShakuraN.I.,SunyaevR.A.,1973,A&A,24,337
SteinerJ.F.etal2011,MNRAS,416,941
ShafeeR.,McKinneyJ.C.,NarayanR.,TchekhovskoyA.,GammieC.F.,
McClintockJ.E.,2008,ApJ,687,L25
SunW-H.,MalkanM.A.,1989,ApJ,346,68
SvobodaJ.,DovciakM.,GoosmannR.W.,KarasV.,2009,A&A,507,1
SvobodaJ.,DovciakM.,GoosmannR.W.,JethwaP.,KarasV.,MiniuttiG.,
GuainazziM.,2012,A&A,545,A106
ThorneK.S.,1974,ApJ,191,507
TomsickJ.A.,etal2014,ApJ,780,78
VaughanS.,FabianA.C.,BallantyneD.R.,DeRosaA.,PiroL.,MattG.,
2004,MNRAS,351,193
WaltonD.J.,NardiniE.,FabianA.C.,GalloL.C.,2013,MNRAS,428,2901
WilkinsD.R.,FabianA.C.,2011,MNRAS,414,1269
WilkinsD.R.,FabianA.C.,2012,MNRAS,424,1284
WilkinsD.R.,FabianA.C.,2013,MNRAS,430,247
(cid:13)c 0000RAS,MNRAS000,000–000