Self-Consistent Black Hole Accretion Spectral Models and the Forgotten Role of Coronal Comptonization of Reflection Emission

2.9 MB·

by  James F. Steiner

Draft version January 16, 2017
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
SELF-CONSISTENT BLACK HOLE ACCRETION SPECTRAL MODELS AND THE FORGOTTEN ROLE OF
CORONAL COMPTONIZATION OF REFLECTION EMISSION
James F. Steiner1†, Javier A. Garc´ıa2,3,4‡,
Wiebke Eikmann4, Jeffrey E. McClintock3, Laura W. Brenneman3, Thomas Dauser4, Andrew C. Fabian5
Draft version January 16, 2017
Abstract
Continuum and reflection spectral models have each been widely employed in measuring the spins
of accreting black holes. However, the two approaches have not been implemented together in a
7
photon-conserving, self-consistent framework. We develop such a framework using the black-hole X-
1
ray binary GX 339–4 as a touchstone source, and we demonstrate three important ramifications:
0
(1) Compton scattering of reflection emission in the corona is routinely ignored, but is an essential
2
consideration given that reflection is linked to the regimes with strongest Comptonization. Properly
n
accounting for this causes the inferred reflection fraction to increase substantially, especially for the
a
hard state. Another important impact of the Comptonization of reflection emission by the corona
J
is the downscattered tail. Downscattering has the potential to mimic the relativistically-broadened
3 red wing of the Fe line associated with a spinning black hole. (2) Recent evidence for a reflection
1
component with a harder spectral index than the power-law continuum is naturally explained as

Compton-scattered reflection emission. (3) Photon conservation provides an important constraint on
]
E the hard state’s accretion rate. For bright hard states, we show that disk truncation to large scales
H R >> RISCO is unlikely as this would require accretion rates far in excess of the observed M˙ of
the brightest soft states. Our principal conclusion is that when modeling relativistically-broadened
.
h reflection, spectral models should allow for coronal Compton scattering of the reflection features, and
p when possible, take advantage of the additional constraining power from linking to the thermal disk
- component.
o
Subject headings: accretion, accretion disks — black hole physics — stars: individual (GX 339–4) —
r
t X-rays: binaries
s
a
[
1. INTRODUCTION and the hard X-ray emission is attributed to Compton

1 up-scattering of the thermal seed photons originating in
Nature’s black holes are delineated by mass into two
v primary classes: supermassive (M (cid:38)105M ) and stellar the disk component (e.g., White & Holt 1982; Gierlin´ski
7 mass (M (cid:46)100M ), with any falling in-be(cid:12)tween termed etal.1999). TheComptonemissionfromthecoronahas
7 (cid:12) a power-law spectrum, which (generally) cuts off at high
intermediate mass. For a black hole of any mass, the
7 energies. A portion of this coronal emission returns to
no-hair theorem holds that the black hole is uniquely
3 the disk and irradiates its outer atmosphere. The disk’s
and completely characterized by its mass and its spin
0
heated outer layer is photoionized, resulting in fluores-
. angular momentum. One of the principal challenges in
1 modern astrophysics is to measure and understand the cent line emission as the atoms de-excite. In addition to
0 its forest of line features, the reflection continuum pro-
distribution of black hole spins (a ).
7 ∗ duces a broad “Compton hump” peaked at ∼ 30 keV,
One of the most widely-employed approaches for mea-
1 above the Fe edge.
suringblackholespinisthroughmodelingfluorescedfea-
: Relativity sets Doppler splitting and boosting as well
v tures in the reflection spectrum. The most prominent
as gravitational redshift for the reflection features pro-
i and recognized such feature is the Fe-Kα line complex
X duced across the disk. Each such feature is accordingly
(Fabian et al. 1989; Brenneman & Reynolds 2006; Bren-
imprinted with information about the spacetime at its
r neman2013;Reynolds2014). Theterm“reflection”here
a pointoforigininthedisk. Afeatureofprincipalinterest
refers to the reprocessing of hard X-ray emission illumi-
is the Fe K line whose red wing is used to estimate the
natingthediskfromabove. Theilluminatingagentisun-
disk’s inner radius, which in turn provides a constraint
derstood to be a hot corona enshrouding the inner disk,
on the black hole’s spin.
[email protected] The other primary method for measuring black hole
†EinsteinFellow. spin is via thermal continuum-fitting (Zhang et al. 1997;
‡AlexandervonHumboldtFellow. McClintock et al. 2014). In this case, the blackbody-
1MIT Kavli Institute for Astrophysics and Space Research,
likeemissionfromtheaccretiondiskisthecomponentof
MIT,70VassarStreet,Cambridge,MA02139.
2Cahill Center for Astronomy and Astrophysics, California interest, and spin manifests through the efficiency with
InstituteofTechnology,Pasadena,CA91125. which the disk radiates away the rest-mass energy of ac-
3Harvard-Smithsonian Center for Astrophysics, 60 Garden creting gas. In effect, the spin is estimated using the
Street,Cambridge,MA02138.
4Remeis Observatory & ECAP, Universit¨at Erlangen-Nu¨rn- combinedconstraintprovidedbythedisk’sobservedflux
berg,Sternwartstr.7,96049Bamberg,Germany. andpeaktemperature. Forthismethodtodeliveranes-
5Department of Astronomy, Cambridge University, Mading- timate of spin, it is necessary to have knowledge of the
leyRoad,Cambridge,CB30HA,UK.
2 Steiner et al.
blackhole’smass,theline-of-sightinclinationofthespin correspond to “Box A” in Garc´ıa et al. (2015, here-
axis, and the system’s distance. after,G15), which is comprised of > 40-million X-ray
The reflection method is applied to both stellar-mass countsascollectedbyRXTE’sProportionalCounterAr-
andsupermassiveblackholes,whereascontinuumfitting ray (PCA; Jahoda et al. 2006) with a spectral range
ispredominantlyusefulforstellar-massblackholes. Both ∼3−45 keV.
methodsrely upon asingle crucialfoundationalassump- In G15, a simple spectral model was adopted, con-
tion: that the inner edge of the accretion disk is exactly sisting of a cutoff power-law continuum, a component
matchedtotheradiusoftheinnermost-stablecircularor- of relativistically broadened reflection (relxill), and a
bit (ISCO), which is a monotonic function of both mass narrowcomponentofdistantreflection(xillver;i.e.,lo-
and spin. cated far from the regime of strong gravity and also far
Although there is overlap in the set of BH systems from the corona). In addition, a cosmetic Gaussian ab-
whichhavemeasurementsfromeachapproach,themeth- sorption feature was included in the model near the Fe
odsareoptimizedforopposingconditions;whenasystem complex at ∼ 7.4 keV. For further details on the data
is most amenable for one method, the other is generally and modeling procedure adopted, we refer the reader to
hampered. For instance, for the thermal state in which G15.
continuum-fitting is most adept (e.g., McClintock et al. G15 followed the common practices standard in reflec-
2014; Steiner et al. 2009a), Comptonization and reflec- tion modeling of hard-state data, which do not include
tion are faint compared to the bright disk. Conversely, the self-consistent effects we introduce here. In G15, the
hard states are dominated by the Compton power law reflection fits were found to clearly demand that reflec-
and its associated reflection (e.g., Fabian & Ross 2010), tion originate from a very small inner radius, a radius
andherethethermaldiskisquitecoolandfaint,oftenso of (cid:46) 2 R . And yet no thermal emission was required
g
weak that the thermal emission is undetected. As a re- by the fit, although two reflection components were de-
sult,bothmethodsarenotusuallyappliedfruitfullytoa manded (the broad and narrow components mentioned
singledataset(see,e.g.,Steineretal.2012). Instead,for above). Giventhehighluminosityofthishardstate,can
transient black-hole X-ray binaries, one can apply both the reflection fit be reconciled with the apparent dearth
methods to observations at distinct epochs capturing a of thermal disk emission? What is the effect of includ-
range of hard and soft states as the source evolves. ing Compton-scattering on the reflection emission? Can
As consequence of this phenomenological decoupling a single reflection component, partially transmitted and
between thermal and nonthermal dominance, the spec- partially scattered by the corona, account for the com-
tral models describing thermal disk emission versus plete signal? Our work addresses these questions using
Comptonandreflectionemissionhavelargelyundergone the GX 339–4 data as a touchstone data for examining
independent development. And as a result the cross- the impact of a self-consistent framework on black-hole
couplingbetweenspectralcomponentshasbeensparsely spectral data as a general practice.
studied. One notable effort by Petrucci et al. (2001) ex- Section 2 describes our overall approach. We first fo-
ploredtheeffectofcoronalComptonizationonthereflec- cus on the impact of Compton-scattering on Fe lines in
tion continuum flux and Compton hump, but in general reflection spectra in Section 3, and then discuss a fully
this has gone unexplored. A handful of similar efforts self-consistent approach in Section 4. In Section 5, we
havetakenstepstowardsself-consistenttreatmentofthe apply this prescription to GX 339–4. Finally, a broader
thermal and nonthermal spectral components – notably discussion and our conclusions are given in Sections 6
usingeqpair(Coppi1999;e.g.,Kubota&Done2016)or and 7.
compps (Poutanen & Svensson 1996), and to lesser de-
greeParkeretal.(2016);Basak&Zdziarski(2016);Plant 2. STRUCTURINGASELF-CONSISTENTMODEL
et al. (2015); Tomsick et al. (2014); Steiner et al. (2011); We focus on a self-consistent disk-coronal spectral
Miller et al. (2009). But an advanced self-consistent ap- model,inwhichthethermaldiskemission,reflection,and
proach has not been realized. In this work, we exam- coronal power-law are interconnected in a single frame-
ine several important aspects of a self-consistent model, work. Our approach makes several simplifying assump-
sighted towards measuring black hole spin. tions. We firstly assume that the power-law component
Adopting the usual assumption that the hard power- isgeneratedbyComptonizationinathermalcoronawith
lawphotonsoriginateinahotcoronathatComptonscat- no bulk motion, and that the corona is uniform in the
ters thermal disk photons, then by extension the reflec- sense that its temperature, optical depth, and covering
tion photons emerging from the inner disk will likewise factor8 areinvariantacrosstheinnerdisk(i.e.,weignore
undergo the same coronal Compton scattering. Recog- any radial gradients that may affect the corona-disk in-
nizing this, as a first step towards self-consistency, we terplay). While we do not impose a particular geometry
begin by examining the impact of coronal Comptoniza- on the corona, we make the simplifying assumption that
tion on the reflection spectrum. This is particularly im- thecoronaemitsisotropically. Forachosengeometry,an
portant for the hard state, in which Compton scattering anisotropycorrection(seee.g.,Haardt&Maraschi1993)
is most pronounced. After touching upon this point, we could be applied to the reflection fraction; an investiga-
go on to tackle the larger objective of producing an in- tionofsuchgeometry-specificcorrectionsisleftforfuture
terlinkeddisk-coronalspectralmodelwhichimposesself- work.
consistency. To connect with observations, we apply our We assume that the disk is optically thick and geo-
modeltothepeakbrighthard-statespectrumofGX339–
4, which is described in detail in Garc´ıa et al. (2015). 8 While it is often assumed that optical depth fully determines
These data are among the highest-signal reflection the fraction of emitted photons that scatter in the corona, this is
spectra ever studied. Specifically, the data in question onlytrueforafixedcoveringfactor.
Compton Scattering of BH Disk Reflection 3
106 seed Gaussian seed Gaussian
f =0.63, E =100 keV, Γ=2.30 f =0.63, kT=50 keV, Γ=2.30
sc cut sc e
f =0.63, E =140 keV, Γ=2.00 f =0.63, kT=70 keV, Γ=2.00
105 fsc=0.86, Ecut=100 keV, Γ=1.75 fsc=0.86, kTe=50 keV, Γ=1.75
sc cut sc e
f =0.86, E =140 keV, Γ=1.50 f =0.86, kT=70 keV, Γ=1.50
sc cut sc e
x
u 104
l
F

n
o 103
t
o
h
P 102 E comp
cut
101
100
0.1 1 10 100 1000 00 ..11 11 1100 110000 11000000
Energy (keV) EEnneerrggyy ((kkeeVV))
Fig. 1.—TheshapeoftheCompton-scatteredcontinuumthatresultsfromseedphotonsinputat1keV,forarangeofcoronalproperties.
We show the output of simplcut acting on an input Gaussian line at 1 keV using each of its two kernels: Ecut (left) which we adopt
throughoutthetext, andcomp(right) whichis basedonnthcompanddescribedinAppendixA.The valuesoffSC correspond toτ =1
and τ =2 for uniform coronae with unity covering fraction. Note that here most line photons go into the scattered wings; these appear
faint with respect to the peak because they are very broad even though they contain most of the signal. Of principal importance is the
significantdownscatteredcontributionfromtheline,whichisappreciableandissteeperwhenΓislarge.
50 100
Hard State Soft State seed relxill spectrum
Comptonized relxill spectrum

10
e)
arbitrary scal 105 srCReimloexlmpai lltplci+vtuoxitsn i× lti li zvcxe eildlriln vFeee r p lrionfeile (a*=0.9) Fνν 10
F (ν
ca onFu d0lidg.c11. opu2ll.ad—utshiAbelnryEe5finomlelruriegmsyt pri (coa11ktt00ebeiVonrn)otiaasdlhlyopwlb uien3sg0a nhasoorwruorwcdeorwoeEfn5flnsseecycragtstity ote (enm11kr00eeacVdtoi)mcrepeflroerncoetrino3itnn0s craeosFflmaiegpcg.toir1n3oe.ene—nnctosT(morlhpiedeloxnelffiienlneelct.)tfocoTrofhmaCe1pcosoomomnlEpiepdntnloetebn.rtlgeaTslyccyh ka(tetkrtltreaiiaensnnVncrsetaimst)rntpieistgn1rtahees0ddroo ierrwnceenffsllteaeeccmctttbiiohooirnnrseoosainnodaent)r(ieesiflff.eseeh.cc,ot1ttiw0ohonn0ef
estimating spin. Top panels depict composite spectra modeled as
Compton-scattering on this emission. Dashed and dotted black
either broad and narrow reflection or as a Comptonized narrow
lines show the portions which have respectively scattered or been
reflection component. The bottom panels present the profile of
transmitted through the corona. The spectrum illustrated here
the modeled coronal scattering as acting on a narrow Fe line to
illustrate its shape. For reference, we also overlay the shape of was generated for fSC =0.3, Γ=2, and Ecut =100 keV for the
a relativistically broadened line with spin a∗ = 0.9. Note from cFoerofonrat;hae∗d=isk0.,qTh=e3b,oilo=m4e5tr◦i,clpoghoξt=on2,caonudntaissoildaernatibcuanldfoarncbeotohf
the bottom panels that the broad relativistic line profile matches
curves;theapparentenhancementfortheblackcurveinthe1–100
closelythatoftheCompton-downscatteredwingforthehardstate
(with a steeper Γ=1.5), but for the soft state (Γ=2.5) the red keV range is due to the νFν scaling, and is compensated at the
lowestenergies((cid:46)1keV)atwhichthegreencurveisbrighter(and
wingisnarrower.
whichdominatesthephotoncount).
metrically thin and that it terminates at an inner ra-
commonly termed the “color correction”) describes the
dius R ≥R . The boundary condition at the inner
in ISCO relationship between color temperature T and effective
radius assumes zero torque9. Spectral hardening (also c
temperature T , and it is defined as T = fT . The
eff c eff
correctiontermf isassumedtoscaleasf ∼T1/3 (Davis
9 Basak & Zdziarski (2016) point out that the zero-torque c
condition may not be applicable when the disk is truncated at
Rin > RISCO. A detailed consideration of the precise boundary torqueisbeyondourscope.
4 Steiner et al.
& Hubeny 2006). Radiation emitted by the disk that is post-scattering distribution via a Green’s function. The
bent back and strikes the disk due to gravitational lens- mostbasicimplementationconsistsofaone-ortwo-sided
ing (i.e., “returning radiation”) is not considered here. power law distribution (Steiner et al. 2009a). In this pa-
A fully self-consistent model of the accreting system per we adopt the two-sided version and go one step fur-
should include the following couplings between compo- ther,introducingamoresophisticatedimplementationof
nents: simpl termed simplcut, which we will use throughout
and which we now describe.
• disk-corona: A fraction f of thermal accretion
sc
diskphotonsareCompton-scatteredbythecorona 3.1. simplcut
intothepower-lawcomponent. Reflectionphotons,
simplcut10 is an extension of the simpl model that
which originate at the disk surface, will also scat-
adopts a cut-off power-law shape for the Compton com-
ter in the hot corona. Photon conservation here
ponent. It is governed by four physical parameters: the
ensuresthattheComptonpower-lawphotonsonce
scattered fraction f , the spectral index Γ, the high-
originated as thermal disk emission. SC
energy turnover, and the reflection fraction (R ). We
F
• corona-reflection: The Compton-scattered pho- note that fSC is distinct from the optical depth τ; fSC,
tons which illuminate to the disk give rise to the and Γ are fitted, independent of τ. This operationally
reflection component. The reflection fraction de- allows for a variable (non-unity) covering factor of the
scribes the flux of Compton-scattered photons di- coronaabovethedisk. (τ canbeinferredforadesiredge-
rected back to the disk relative to the Compton ometryusingthespectralindexandhigh-energyturnover
flux that is transmitted to infinity (Dauser et al. terms.) RF isdefinedasthenumberofscatteredphotons
2016). Properly incorporating photon counting en- whichreturntoilluminatethedisk(i.e.,thoseproducing
suresthattheCompton-scattereddiskphotonsare reflection) divided by the number which reach infinity.
conservedandcorrectlyapportionedbetweencom- Noprescriptionforangle-dependenceisapplied,whichis
ponents. For instance, in this case photon con- equivalenttousingtheobservedpower-lawfluxasproxy
servation links the flux of the observed Compton for all flux at infinity.
power-lawcomponenttothefluxofthissamecom- Within simplcut are two options for the scattering
ponent from the vantage point of the disk (i.e., as kernel. The first kernel, which we adopt throughout un-
prescribed by the reflection fraction). lessotherwisespecified,isshapedbyanexponentialcut-
offE anddescribedbythefollowingGreen’sfunction,
cut
• disk-reflection: The inner radius of the disk is normalized at each seed energy E :
0
tracked by two independent spectral characteris-
tics: the red wing of the relativistic reflection
(cid:26)(E/E )−Γ exp(−E/E )dE/E , E ≥E
features and the emitting area of the multicolor- G(E;E )dE ∝ 0 cut 0 0
blackbody disk. A self-consistent treatment of 0 (E/E0)Γ+1dE/E0, E <E0,
these two components is achieved by ensuring that (1)
the radius of the thermal emission in the soft state
andthereflectioncomponentinthehardstatemu- The second kernel is based upon nthcomp (Zdziarski
tuallyconstraintheareaofthethermalemissionfor et al. 1996), with electron temperature kTe taking the
thehard-statedisk. Atthesametime,photoncon- place of Ecut. This kernel and its implementation are
servationprovidesaconstraintonthetemperature described in Appendix A, and its shape is illustrated in
via the disk photon luminosity. therightpanelofFigure1. WeopttousetheEcut kernel
throughoutforthesakeofitsstraightforwardcomparison
• corona/disk-jet: Although self-consistent mod- withpublishedresults(thoughseeAppendixAtoseeour
eling of the radio jet can constrain the hard-state modeling results using both kernels).
geometry(seee.g.,Coriatetal.2011),theradiojet Figure 1 demonstrates simplcut’s effect, illustrating
is beyond our scope is not considered in this work. the net impact of the corona on a narrow 1 keV line
viewed through coronae with a range of settings. The
3. COMPTON-SCATTERINGOFREFLECTIONPHOTONS
graylinesdepictseed1keVphotons. Foracoveringfac-
We begin our investigation of self-consistent modeling torofunity, thefamiliaropticaldepthτ isrelatedtothe
withanin-depthexaminationofthecouplingofthether- scatteredfractionbyf =1−exp(−τ);electrontemper-
SC
mal disk photons and the coronal electrons which give aturekT andhigh-energycutoffE areapproximately
e cut
rise to the Compton power law. The Compton scat- matchedtooneanotherusinganapproximatecorrespon-
tering of thermal disk photons into a power-law compo- denceE =2−3kT (herewehavearbitrarilyselected
cut e
nent has been widely studied (e.g., Poutanen & Svens- E =2 kT for presenting the kernels).
cut e
son 1996; Titarchuk 1994; Dovˇciak et al. 2004; Zdziarski The most important feature to note is the prominent
etal.1996,andreferencestherein). However,theimpact downscattering of the line by the hot corona (e.g., Matt
ofComptonscatteringonthenon-thermalreflectionfea- etal.1997),whichisessentiallyidenticalbetweenthetwo
turesemittedfromthedisk’ssurfaceistypicallyignored, kernels. Thisisworthparticularattentiongiventhatthe
though see Wilkins & Gallo (2015) and Petrucci et al. redwingofreflectionfeaturesarewidelyemployedines-
(2001). timating the inner-disk radius (and thereby the spin) of
As a first-order treatment, we can convolve the re- accreting black holes. Note also that the transmitted
flection spectrum with the Compton-scattering kernel
such as simpl (Steiner et al. 2009b). This model redis- 10 Available at http://space.mit.edu/~jsteiner/simplcut.
tributes a scattered fraction f of seed photons into a html
SC
Compton Scattering of BH Disk Reflection 5
portion of the line remains narrow. In considering the narrow-lineSeyfert1sshowspectrawithΓinthisrange,
Compton-scattering of reflection features, our prescrip- which may indicate that they have also have low f .)
SC
tion using simplcut greatly improves upon the present In the bottom panels, we show as red dashed lines the
status quo, whichisseeninthefigureasthenarrowgray shapes of the corresponding Compton-scattering profile
line. for a narrow Fe line. For each, the resulting lineshape
is comprised of sharp transmitted and broad scattered
3.2. On Compton Downscattered Line Features components. As a reference, a relativistically broadened
GiventhatComptonscatteringproducesemissionthat Fe-line profile for a spin a∗ = 0.9 is overlaid as a blue
can contribute to the red wing of a line profile (notably dotted line. Note the close correspondence between the
the Fe line), this may plausibly impact spin measure- high-spin relativistic line and the Compton wing for the
ments. We now explore this phenomenon in greater de- hard state in the left-bottom panel; note also that the
tail. The principal question before us is whether or not Compton-scattered lineshape for the soft state is not as
Compton downscattering can mimic the effect of strong broad as a high-spin (relativistic) line. In both cases the
gravityonfluorescentlineemission. Thisisanimportant upscattered power law from the Fe line contributes an
consideration given that the hard states in which reflec- excess blueward of the line-center, which is potentially
tioniswidelystudiedisassociatedwithstrongCompton detectable in a general case. But since here the upscat-
scattering. teredlineliessignificantlybelowtheComptoncontinuum
AsdescribedbyEqn.1,theshapeofthedownscattered in flux, the effect has minor impact in these examples.
emissionispurelyafunctionofΓ(specifically,thedown- In Figure 3 we show the net impact of coronal Comp-
scatteringspectralindexis−Γ−1; also, seePozdnyakov tonizationonafullrelativisticreflectionspectrum(mod-
et al. 1983). Meanwhile, the shape of the red wing of eled via relxill). This illustrates the generic case in
a relativistic line is given principally by the spin of the which relativistic reflection features from an accreting
black hole, but it is also affected11 by the line emissivity blackholeinanX-raybinaryorAGNundergoCompton-
q and inclination i. scattering through the corona. The comingling of these
Forreference,usingcanonicalvaluesq =3andi=60◦ effects applies the vast majority of X-ray reflection data
we find the following correspondence between the shape of black holes, and motivates its presentation here. In
of a red wing for a relativistic line (at a given a ) and the figure, a green line shows the naked reflection emis-
∗
the downscattered wing of a narrow line due to Comp- sion from a nonrotating black hole (a∗ = 0), i.e., the
tonization: a = 1 matches Γ ≈ 1.5, a = 0.7 matches reflection component emitted by the disk prior to any
∗ ∗
Γ ≈ 2.5, a = 0.4 matches Γ ≈ 3.5, and a = 0 matches coronal Compton scattering. After passing through a
∗ ∗
Γ (cid:38) 4. It therefore follows that if Compton-scattering corona with fSC = 0.3 and Γ = 2, the scattered spec-
werebeingconflatedwithrelativisticdistortion, thiscan trum is both hardened and also smeared out in energy,
berevealedthoughexaminationofdataspanningarange with net result shown as the solid black curve. Note for
of Γ, or equivalently, a range of spectral hardness. the black curve that the ∼ 30 keV Compton hump and
Ideally, one would examine both hard and soft (or the red wing of the ∼ 6.7 keV Fe line are broader, and
steep-power law; Remillard & McClintock 2006) states that the reflection features appear muted relative to an
to maximize this difference. One expects that any bias enhanced (and harder) continuum.
introduced by this effect would lead a harder spectrum
tofittoahighervalueofspin(or,equivalently,asmaller 4. TOWARDSACOMPOSITEMODEL
innerradius). WenotethatthemostrecentNuSTARre-
4.1. Model Components
flection studies of Cyg X–1 find a =0.93−0.96 for the
∗
soft state and a >0.97 in the hard state (Walton et al. As the primary feature of our full spectral model, we
∗
2016;Parkeretal.2015). Thedirectionofthismismatch ensureaself-consistentlinkingofthefluxesbetweendisk,
in spin (corresponding a factor ∼25% difference in R ) Compton, and reflection components. Incorporating the
in
isconsistentwiththetypeofbiasonewouldexpectfrom effectsofComptonizationbythecoronadescribedabove,
unmodeled coronal Comptonization. weadopttwobasicmodelswhichwewillemployinfitting
We compare relativistic and Compton-scattering ef- GX 339–4 spectral data:
fects in Figure 2 in order to illustrate how the two may Model 1:
be confused. In the top panels, we present illustrative tbabs(simplcut⊗(ezdiskbb+relxill)+xillver),
(simulated) spectra in black comprised of both broad and
and narrow reflection components (analyses using two Model 2:
reflection components being ubiquitous for AGN, and tbabs(simplcut⊗(ezdiskbb+relxill)).
commonplace for stellar BHs). The broad component Model1isareformulationofthetypicalspectralmodel
originates in the regime of strong gravity while the nar- employed for many supermassive and stellar black-hole
row component is produced from further away, where systems (mentioned in Section 3.2), which consists of
relativisticdistortionsarenegligible. Inredwe illustrate bothbroad(relativistic)andnarrowreflection. Here,the
thatclosely-matchingspectracanbeproducedbyinstead narrowreflectionisassumedtobeproducedfarfromthe
Compton-scattering a single narrow reflection compo- black hole, for instance in an outer disk rim. This dis-
nent. The left panels depict a representative hard-state tant emission would not undergo appreciable Compton
spectrumwithspectralindexΓ=1.5,andtherightpan- scattering by a central corona. Because of this, we fix
els a soft spectrum with Γ = 2.5. (We note that many xillver’s setting for reflection fraction to -1 (the nega-
tive sign merely indicates that the continuum power-law
11 We note that increasing q tends to decrease the line central is omitted), and fit for its normalization. Accordingly,
peak,whileincreasinginclinationhastheoppositeeffect. we note that wherever a fit values of R is shown, this
F
6 Steiner et al.
refersspecificallytotherelativisticreflectioncomponent. times solar, was demanded by the relativistic-reflection
Model 2 is identical to Model 1 except that we consider component. At the same time, the Fe abundance of
just a single relativistic reflection component. Both in- thexillvercomponentwasincompatiblewiththislarge
clude a first-order treatment of reflection and Compton valueandwasinsteadconsistentwithasolarsetting. (3)
scattering (but not higher order exchanges12). Given the conclusion that the inner disk extends down
Although omitted from the descriptions above, when to – or very near-to – R , and given the high lumi-
ISCO
we apply these models to GX 339–4 we also include a nosity for these hard-state data, the lack of evidence for
Gaussian absorption line at ∼ 7.4 keV (this line may athermaldiskcomponentinthePCAdataissurprising.
be an instrumental artifact rather than physical in ori- Having incorporated the Compton scattering of inner-
gin, and is discussed in great detail in G1513. tbabs disk reflection into our model, we now reexamine the
(Wilmsetal.2000)describesphotoelectricabsorptionas most luminous data from G15. We explore whether two
X-rays traverse the line-of-sight interstellar gas column distinct reflection components are now required (given
with N =5×1021cm−2 (Fuerst et al. 2016), a quantity the impact of including Comptonization), and then go
H
which we keep fixed throughout our analysis. on to determine whether the fundamental conclusions of
ezdiskbb is a spectral disk model14 for a geometri- G15 hold up to this more holistic approach. That is, we
cally thin multi-color accretion disk with zero torque at will test whether the lack of a detected disk component
the inner-boundary (Zimmerman et al. 2005). Impor- canbereconciledwiththehighspinandmodest-at-most
tantly,ezdiskbbhasbeenvariouslyshowntorecoveres- truncation of the disk in the bright hard state. In addi-
sentiallyconstantradiifromsoftblack-holespectra(e.g., tion, wetestwhetherrelxillaloneprovidesasufficient
Gou et al. 2011; Chen et al. 2016; Peris et al. 2016), modelforthereflectionsignal,orwhethertheaddedxil-
demonstrating its utility here. lver component is still required. As well, we consider
relxill and xillver are leading models of spectral whether the three puzzles are in any way resolved using
reflection (Garc´ıa & Kallman 2010; Garc´ıa et al. 2014; our self-consistent model.
Dauser et al. 2014a). The essential difference is that Parameters common to relxill and simplcut are
relxill describes reflection from the inner-disk where tied to one another, namely, inclination i, Γ, R , and
F
gravitational redshift and Doppler effects are important, E . For Model 1, xillver’s ionization parameter is
cut
and xillver is used for unblurred reflection occurring frozen to log ξ = 0 and the Fe abundance is set to
far from the relativistic domain. unity. Only the normalization for xillver is free (op-
In G15, we concluded that a composite model with a erationally, its setting for R is fixed to -1). As in G15,
F
singlereflectioncomponent,akintoModel2,didnotad- we keep the reflection emissivity frozen to q = 3 and fit
equately fit the bright hard state of GX 339–4. Instead, for inner-disk radius while keeping the spin at its max-
a composite model akin to Model 1 (e.g., using relxill imum value a = 0.998 (see Section 6). As a result of
∗
and xillver together) was very successful. The spec- our enforcing photon conservation, the normalization of
tral fits spanned a range of hard-state luminosities and relxill is not free. Instead, through comparison of the
together demanded that the spin of GX 339–4 must be models, we fix the normalization to a fixed function of
quite high (a > 0.9). The same data also exhibited a the disk and reflection parameters, changing in strength
∗
preference for quite modest disk truncation (R (cid:46)5R ) principally as R is adjusted. Specifically, through ex-
in g F
in the hard state, with the inner radius growing slightly ploration, we empirically derived a scaling relation that
larger at lower luminosities. links the Compton power-law emerging from disk (i.e.,
simplcut⊗ezdiskbb) to the illuminating power law (in
4.2. Parameter Constraints relxill16) to within 5% in flux. This serves as first-
While the spectral fits in G15 were of high statisti- order approach to photon conservation, but again does
cal quality, G15 identified three puzzling aspects of the notaccountforanisotropicredistributioninangledueto
best-fitting model: (1) The inner-disk reflection was as- scattering or other geometric effects.
sociatedwithastartlinglylowreflectionfraction(∼0.2), To anchor the ezdiskbb temperature and normaliza-
whereas values closer to unity are expected15 (see, e.g., tion to values corresponding to RISCO, we turn to the
abundant soft-state spectral data of GX 339–4 whose
Dauseretal.2014b). (2)AverylargeFeabundance,>5
Comptonandreflectioncontributionsarequiteminimal.
12 For instance, we do not consider added contribution from This is similar to the approach adopted by Kubota &
reflectionphotonswhichComptonbackscattertore-illuminatethe Done (2016) in their study of a steep power-law state of
diskandproducefurtherreflection,whichagainscatters,etc. This GX 339–4, which also employed a thermal spectrum as
contributionfallsoffatordernas,roughly,(fSCRF/(1+RF))n. a reference point. We choose a ∼ 2×106 count spec-
13InG15,thelineispositionedcloserto7.2keV;there,7.2keV
trumfromMarch1, 1998(ObsID30168-01-01-00)which
wasdeterminedinafittoalargerdatasetwhichspannedawide
range of luminosities. We are fitting just the highest luminosity has an X-ray flux comparable to that of the G15 Box A
subset–BoxA–fromthatsamplewhichrevealsamodestincrease data (see their Fig. 1), a factor ∼ 3 below GX 339–4’s
inthelineenergycentroidwhenfittedindependently(possiblyre- peak brightness. The temperature and normalization of
latedtothegasbeingmoreionized).
14 ezdiskbb is a nonrelativistic disk model; nevertheless, it has thediskinquestionare, respectivelykTsoft =0.699keV,
oneimportantadvantageoveritsavailablerelativisticcounterparts and Nsoft = 721. Owing to the abundant evidence that
bhspec(Davis&Hubeny2006)orkerrbb(Lietal.2005);namely, soft-state disks reach R (e.g., Steiner et al. 2010),
ISCO
it can allow for disk truncation. By contrast, the relativistic disk and that the stable disk radius can be recovered in soft
modelsmaketheassumptionthatRin=RISCO. states to within several percent, we employ these num-
15 We note that the model definitions of normalization and re-
flection fraction in relxill have been updated in more recent re-
leases(Dauseretal.2016);RF usingtheupdateddefinition(v0.4) 16 This is the power law obtained by setting RF = 0 in the
is∼0.3±0.03. Whilelarger,thisisstillveryfarfromunity. unscatteredrelxillmodel.
Compton Scattering of BH Disk Reflection 7
bers as benchmark values for GX 339–4’s thermal disk isshownasareddottedline. Thesharpcosmeticabsorp-
at R =R . tion line near Fe is strong and pronounced in Model 2,
in ISCO
We then allow for our hard-state spectrum to have a but suppressed by the xillver Fe-Kβ edge in Model 1.
different R which is free to take values R ≥ R . Still, its inclusion in Model 1 is significant at a ∼ 5 σ
in in ISCO
We link the thermal disk properties to (i) the ratio level. Each of Models 1 and 2 produce relatively similar
R /R ,whichisafitparameterinrelxill,andalso goodness-of-fits to their non-self-consistent counterparts
in ISCO
to(ii)thediskphotonluminosityL ascomparedtoour andforbothcasesgivenearlyidenticalpatternsofresid-
ph
reference soft spectral state. uals as shown in the bottom panels. Thus, despite the
In Appendix B, we derive how the disk properties constraints which result from regulating the interplay of
scalewithchangesfromthephotonluminosityandinner- the spectral components, our self-consistent framework
radius. The disk radiative efficiency η ∝ R−1, and ac- verysuccessfullymodelsthisextremelyhigh-signalspec-
in
cordingly, the seed thermal disk in the hard state is de- trum of GX 339–4 in its peak bright hard state.
scribed by As a result of imposing self-consistency, the low value
of R obtained in G15 (i.e., Models 1G and 2G) has
F
increased to several times its original value, now in a
kT =kT (cid:18) Lph (cid:19)3/5(cid:18) Rin (cid:19)−6/5 keV, (2) rangeclosetounitywhichisalignedwithexpectation. In
disk soft L R part,thisincreaseoccursbecausetheappreciableComp-
ph,soft ISCO
tonization by the corona (f ∼ 0.4) dilutes the equiv-
SC
and alent width of the Fe line (and other spectral features)
since scattering a feature acts to blend it into the con-
(cid:18) L (cid:19)−4/5(cid:18) R (cid:19)18/5
N =N ph in . (3) tinuum (e.g., Steiner et al. 2016). Accordingly, in order
disk soft L R for the (scattered) model to match the Fe-line strength
ph,soft ISCO
in the data, R necessarily increases. This same ef-
Equivalently, in terms of the mass accretion rate M˙ , fect was noted pFreviously by Petrucci et al. (2001). For
Model 1, surprisingly, the the best-fitting R decreased
(cid:32) M˙ (cid:33)3/8(cid:18) R (cid:19)−9/8 slightly relative to G15 (though within ∼ 1σinlimits) by
kTdisk =kTsoft M˙soft RISiCnO keV, (4) ∆ferRreind≈so0lu.1ti−on0.e2mRegr.geHsowwietvhera,nfoirnnMerodreald2iuas nRewtphraet-
in
is ∼ 3.8 times larger than for the non-self-consistent
and
Model 2G.
(cid:32) M˙ (cid:33)−1/2(cid:18) R (cid:19)7/2 We find that self-consistency does not ameliorate the
N =N in . (5) problem of high Fe abundance, only slightly affecting its
disk soft M˙soft RISCO value. And based on the significantly better quality-of-
fit for the Model 1 compared to Model 2, we conclude
that the new self-consistent picture has not removed the
5. RESULTS need for two distinct reflection components. In fact, the
For each of Models 1 and 2, we contrast our self- χ2 gap between the two best fits is slightly larger for the
consistent analyses of GX 339–4 against the “standard” self-consistent models.
reflectionformalismasadoptedinG15,i.e.,amodelwith Apart from the question of which model flavor per-
noself-consistentlinkingofdiskandreflection/Compton forms best, there are several interesting differences be-
components. To highlight their tie to the G15 paper, tween Models 1 and 2. For Model 2, because larger R
in
these comparison “standard” models are termed Mod- is preferred, the self-consistent model is matched with
els 1G and 2G. correspondingly higher N and lower kT . Model 2
disk disk
In our analysis, we use XSPEC v12.9.0 (Arnaud 1996) also prefers lower values of the ionization parameter (by
to perform a set of preliminary spectral fits. For con- a factor of ∼ 5) and of f , but a higher inclination i.
SC
sistency with G15, we employed relxill v0.2i, and we Most striking, the M˙ required for Model 1 is less than
excluded the data in the first four channels while ignor- half that of the reference, soft-state spectrum, whereas
ing all data above 45 keV. To ensure that the redis- for Model 2, M˙ is instead four-times larger that for the
tribution of photons was accurately calculated, an ex-
corresponding soft state!
tended logarithmic energy grid was employed that sam- Large R resulting in large M˙ is a generic property
pled from 1 eV to 1 MeV at 1.4% energy resolution via in
of the self-consistent model because (from inspection of
“energies 0.001 1000. 1000 log”. When a best fit was
Eqns 4 and 5) if R is increased for a fixed M˙ , the pho-
found, we estimate the errors through a rigorous ex- in
ton flux diminishes and its peak moves to lower energy;
ploration of parameter space carried out using Markov-
ChainMonteCarlo(MCMC).Weemployedthepython both effects reduce the Compton power law’s amplitude.
packageemcee(Foreman-Mackeyetal.2013)asoutlined This is similar to the argument presented by Dovˇciak
& Done (2016) who detailed how the luminosity of the
in Steiner et al. (2012), with further details provided in
Compton component constrains the interplay between
Appendix C.
the seed photon flux and the corona’s covering factor.
The corresponding results are summarized in Table 1,
For a very luminous hard state (say, (cid:38) 10% of the Ed-
We present our best fits for both self-consistent mod-
dington limit), large-scale truncation (R >> R )
els in Figure 4. The spectral components are shown as in ISCO
solid colored lines and the data are shown in black. The would imply highly super-Eddington values of M˙ .
seed disk emission (i.e., the emission that would be ob- Tobestexaminetheimpactoftheself-consistentmodel
servedfromthebarediskifthecoronaweretransparent) ontheinner-radius(orequivalently,ontheresultingspin
8 Steiner et al.
]
)
1
-V Model 1 Model 2
1.00
e
k

1
-s

2
-m
c
s 0.10
n total
o
t seed thermal disk
o
h Comptonized disk
P
relativistic reflection
(

V 0.01 distant reflection
e
k
[

Fν 1 10 100 1 10 100
Energy (keV) Energy (keV)
5
Model 1 Model 2
0

χ -5
∆ 55
Model 1G Model 2G

00

--55

3 1100 30 3 1100 30
Energy (keV) Energy (keV)
Fig. 4.—Afullyself-consistentmodelcomprisedofaCompton-scatteredthermaldiskcomponentandassociatedreflectionemission,as
appliedtothebrighthard-statedataofGX339–4. Thetoppanelsshowthecompositefitsandmodelcomponentswiththedatainblack.
Model1isshowninleftpanelsandModel2intheright. The∼7.4keVabsorptionlineisevidentineach,butismuchmorepronouncedin
Model2. Bottompanelsshowthefitresidualsforbothself-consistentandnon-self-consistentvariantsofeachmodel. Notethesimilarity
intheresidualsamongbothvariantsforeachofModels1and2. TheassociatedfitsarepresentedinTable1.
TABLE 1
GX 339–4 spectral fit results
Parameter Priora Model1 Model1G Model2 Model2G
Γ F 1.65+0.01 1.583±0.013 1.694±0.011 1.662+0.004
−0.02 −0.007
fsc F 0.42±0.07 ··· 0.32+−00..0198 ···
Ecut (keV) LF 93.±4. 88.±3. 87.±4. 90.±3.
i(deg.) F 48.2±1.2 48.2±1.4 55.4+2.1 58.6+0.8
−1.6 −1.1
Rin/RISCO(a∗=0.998) F 1.52+−00..1284 1.6+−00..26 5.3±0.8 1.4+−00..13
logξ F 3.38+0.09 3.37+0.08 2.783±0.017 2.803±0.021
−0.05 −0.05
AFe (solar) LF 6.3+−11..84 7.8±2.1 3.56±0.20 4.0±0.2
RF F 0.8+−00..42 0.206+−00..001261 0.75+−10..2028 0.492±0.018
N d,LF 0.68±0.12 1.36±0.05 0.88+0.43 1.619±0.022
relxill −0.20
kT (keV) d 0.31±0.03 ··· 0.15+0.05 ···
disk −0.02
N d 4300+2700 ··· 1.3+0.8×105 ···
disk −1300 −0.6
N LF 0.28±0.05 0.24±0.03 ··· ···
xillver
E (keV) F 7.38+0.16 7.39±0.11 7.37±0.04 7.45±0.03
gabs −0.12
τ LF 0.018±0.005 0.019+0.006 0.154±0.019 0.143±0.015
gabs −0.004
L /L F 0.55±0.10 ··· 2.1+1.5 ···
ph ph,soft −0.5
M˙/M˙ d 0.40±0.12 ··· 4.1+5.4 ···
soft −1.6
χ2/ν 69.65/59 68.21/59 113.82/60 105.46/60
Note. —AllfitswereexploredusingMCMC;valuesanderrorsrepresentmaximumposterior-probability
densityandminimum-width90%confidenceintervalsunlessotherwisenoted.
a Priors are either flat (F) or flat on the log of the parameter (LF). Parameters marked “d” have values
derived from the fit parameters. For the self-consistent models, these values are not directly fitted for;
instead,theyaredeterminedbyfitparametersthroughtherelationshipsoutlinedinSection4.
Compton Scattering of BH Disk Reflection 9
60 breadth of line features emitted by the disk. The dif-
Model 1 - 68.2 ferencearisesfromtheirviewofthecoronaasbeingcen-
Model 1G - 68.2
50 Model 2 - 105.5 trally concentrated (though still porous) so that it very
Model 2G - 105.5 efficientlyscattersemissionfromtheinnerdiskwherethe
gravitationally-broadened red wing is produced, while it
40
is much less efficient at scattering emission from further
outinthedisk. Wenotethatthereismixedevidencefor
2 ∆χ 30 acoronawiththisgeometry; itisvariouslysupportedby
microlensing data of distant quasars (e.g., Chartas et al.
20 2009) and reverberation studies which require very com-
pact corona (Zoghbi et al. 2010), but it is contested by
Dovˇciak & Done (2016) on the grounds that a corona so
10
compact would be starved of the seed photons required
to produce the observed X-ray luminosity.
0 The contrast between the results of Wilkins & Gallo
-15 -10 -5 0 (2015)andourshighlightstheimportanceofunderstand-
-R /R (a=0.998) ing coronal geometry and the necessity of investigating
in ISCO *
Fig. 5.—Theimpactofaself-consistentparadigmonthedeter- Compton-scattering effects for a range of geometries in
mination of the inner-disk radius. We show the change in χ2 as order to better understand systematics in employing re-
thediskinner-radiusisvaried. Theradiusdecreasestotherightso flection spectroscopy to estimate disk radius and spin.
thattruncationislargesttotheleft. Model1ispresentedingreen At present, these matters are all but unexamined.
and Model 1G in blue; Model 2 is shown in red and Model 2G in
Basak & Zdziarski (2016) analyze XMM-Newton spec-
orange. ThecloudsofpointsshowrandomdrawsfromtheMCMC
chains. Importantly, self-consistency penalizes small-radius solu- tra of GX 339–4 using a model similar to ours, with the
tions while reducing the penalty on truncated models. However, Compton, reflection and disk components modeled us-
onlyforthedisfavoredModel2isthetruncateddiskpreferredover ing nthcomp, relxill and a standard multicolor disk.
thesmall-radiussolutionfromG15(i.e.,Model2G).
Whiletheycross-linkedcommonparametersbetweenthe
modelcomponents,asignificantshortcomingoftheirap-
determination), we present fit χ2 versus Rin in Figure 5. proach is not tying the disk seed photons to the large
Here, solid lines show interpolations of “steppar” analy- number of photons scattered into the power law. They
ses in xspec (a routine which systematically optimizes alsodidnotaccountforComptonscatteringofreflection
the fit in sequential steps) across a range of Rin. Over- emission in the corona. In contrast to our results, they
laidaspoint-cloudsarerandomsamplingsoftheMCMC foundlarge-scaledisktruncation,withR oftensofR ,
in g
chains. For Models 1 and 2, it is evident that imposing and they attributed a tentative detection of ∼ 0.2 keV
self-consistency on the model has the effect of penaliz- thermalemissiontothethermalizationofpower-lawpho-
ing very low values of Rin, and conversely making high tons in the disk’s atmosphere. Specifically, they claim
Rin solutions more favorable. This effect does not sig- Rin tobeanorderofmagnitudelargerthanwefindhere.
nificantly alter the best-fitting radius determined from WenotethatifsuchlargevaluesofR arecorrect, then
in
Model1, whichsignificantlyoutperformsModel2. How- to produce the observed 1 − 10 keV bright hard-state
ever, self-consistency does alter the fit landscape, which luminosity one would require the associated M˙ to be ex-
is a clear indication that the self-consistent constraints tremely large, several times greater than the peak M˙ of
can impact results.
GX 339–4’s soft state17. This is because for a given ob-
In Model 2, which is statistically disfavored, the ef-
servedluminosity, whenR isincreasedthediskphoton
fect of self-consistency is much more pronounced: The in
luminosity and peak temperature for a given M˙ both
χ2 surface is double-troughed, with one minimum at
diminish. Both effects in turn cause the amplitude of
R ≈1.4R (=1.8 R ) and another at a much larger
in ISCO g the Compton power law to drop, and so match the data
radius of R ≈ 5.3R (=6.6 R ). Thus, imposing
self-consisteinncy on MoIdSCelO2 drasticalgly changes the solu- M˙ must be large. If one makes the reasonable assertion
tion,favoringsubstantialtruncationandpenalizingmild that the soft state peaks close to the Eddington limit,
truncation. Thereasonself-consistencyproducesthisre- thenlarge-scaletruncationR>>RISCO ofabrighthard
sult is two-fold: (1) The downscattered Compton line state would generally require a super-Eddington mass
emission contributes a flux excess in the same spectral accretion rate.
region as the red wing of a relativistically-smeared line, Fu¨rst et al. (2015) modeled low-luminosity hard-state
and (2) the small-radius solution is penalized for requir- NuSTAR spectra of GX 339–4. They found that the re-
ing a disk component that is not observed. As can be flectioncomponenthadaharderpower-lawindexΓthan
gleaned from Figure 4, Models 1 versus 2 could readily the direct continuum component. The modest difference
be distinguished through direct detection of the thermal in the spectral indexes ∆Γ ≈ 0.2−0.3 is plausibly ex-
disk using a pile-up free detector with good low-energy plained in our model by the hardening of the reflection
sensitivity (e.g., NICER; Gendreau et al. 2012). spectrumduetoComptonscatteringinthecorona. This
can be readily seen in the dashed line of Fig. 3 which is
6. DISCUSSION harder than the input Γ = 2. Although difficult to dis-
cern by eye in Figure 4, the Compton-scattered portion
OurresultsdifferwiththoseofWilkins&Gallo(2015)
concerningtheeffectsofComptonizationonreflectionin
the inner disk. These authors find that strong Comp- 17AsdeterminedbyfixingRintomatchtheirvalueswhenfitting
tonization f (cid:38) 0.5 has the effect of reducing the theself-consistentimplementationsofModels1and2
SC
10 Steiner et al.
ofthereflectioncomponentemergesfromthecoronawith From our analysis using the self-consistent models, we
aharderspectralindex, ∆Γ≈0.15. Thisisbecause, like find that a single Compton-scattered reflection compo-
the Comptonized thermal photons, the Comptonized re- nent is not preferred for GX 339–4; as in G15, the dual-
flectionspectrumisboostedinenergywhenscatteringin reflection model is strongly favored. In fact, imposing
the corona. As one would expect, the magnitude of ∆Γ self-consistency on the dual-reflection model and includ-
canbeshowntogrowwithf . Thishardeningexplains ingtheeffectsofComptonscatteringonthebroadreflec-
SC
why the reflection component can be brighter than the tion component has minor effect on the reflection best-
Compton continuum even if R < 1, as it quite appar- fitting parameters aside from the reflection fraction R
F F
entlyisforModel1inFigure4. Thisoccursbecausethe whichislargerbyafactor∼2−4. Atthesametimeand
Compton-scatteredreflectioniseffectivelyboostedbyan importantly,thelargechangeintheinner-diskradiusfor
additional factor of the “Compton y” parameter. the disfavored Model 2 illustrates that self-consistency
In applying our reflection model, we have proceeded has the potential to significantly affect reflection esti-
under the assumption of maximal spin (a = 0.998), mates of black hole spin.
∗
which effectively sets R for the thermal state data. Additionally, we find that bright hard states cannot be
ISCO
If we had instead assumed any lower value for spin, the reconciled with large-scale disk truncation unless either
tensionbetweenthediskcomponent(giventhatthedata (1) the mass accretion rate M˙ is super-Eddington (well
rule out a bright, hot disk) and reflection component in excess of the peak soft-state M˙ ) or else (2) the hard
(which prefers a disk proximate to the horizon) would
state power-law component is attributed to another ra-
have strictly increased, and the fits would have wors-
diation mechanism. This conclusion is a consequence of
ened. In this sense, our approach of adopting maximal
the photon conservation that is a bedrock of our model.
spin provides a conservative estimate of the importance Fora fixed M˙ one canintuitively understandthis result:
of self-consistency on the modeling.
a disk with large R is cool, produces few photons, and
in
has a Compton component that peaks at lower energy.
Atthesametime,thecoronacannotbeverysmall,orelse
7. CONCLUSIONS
it would require an enormously luminous disk to scatter
In summary, we have demonstrated that the Comp-
sufficient photons into the observed power-law compo-
tonization of the Fe line and other reflection features in
nent.
a hot corona can mimic the effects of relativistic distor-
For a multi-epoch study of a source, the effects of
tion – and potentially affect estimates of black hole spin
Comptonscatteringonreflectionfeaturescanbeassessed
– by producing a downscattered red wing. This notion-
asthesourcerangesoversoftandhardstates;i.e.,asthe
ally calls into question the prevalence of dual-reflection
photonindexΓvaries. Ourmodelpredictsthatanybias
component spectral models in which one component is
related to Compton scattering of the reflection emission
in the relativistic domain, and hence strongly blurred,
having been omitted from modeling efforts should cause
while the other one is assumed to occur far from the
soft-state fits to measure lower spins than found in hard
black hole and to be correspondingly narrow. The pre-
statesforthesamesource(e.g., consistentwiththefind-
cise shape of a relativistically-distorted line is strongly
ingsofthemostrecentNuSTARreflectionstudiesofCyg
affected by the spectral index, and hence it changes as
X–1).
a source transitions between hard and soft states. The
As illustrated in Figure 5, enforcing self consistency
Comptonization of reflection features is particularly im-
has the potential to very significantly affect estimates of
portant for hard spectral states because they are domi-
the inner disk radius and spin when modeling a reflec-
nated by Compton-scattered photons from the thermal
tion spectrum in the presence of a thermal component.
diskandtheirbyproduct–reflection. Giventhatthermal
Furthermore, this approach provides new, mutual con-
disk photons are strongly Comptonized in hard states, straints, e.g., on R and M˙ . Self-consistent reflection
the associated reflection emission is inevitably strongly in
models therefore deserve exploration and further devel-
Comptonized as well.
opment.
We have incorporated reflection and Comptonization
Going forward, given that the power-law component
into a self-consistent disk-coronal spectral model that
in hard states is strongly Comptonized, standard prac-
properly conserves photons between the thermal, reflec-
ticeshouldincludeComptonizationofthereflectioncom-
tion,andComptoncomponents. Withinthisframework,
ponent in the corona, particularly for studies aimed at
and using two specific models (Models 1 and 2), we ana-
constraining R or a .
lyzed a bright hard-state RXTE spectrum of GX 339–4 in ∗
containing 44-million counts, which was analyzed previ-
ouslybyG15. Animportantconstraintwasobtainedus- JFS has been supported by NASA Einstein Fellow-
ingasoft(thermaldominant)statespectrumofGX339– ship grant PF5-160144. We thank Ramesh Narayan and
4 at approximately the same luminosity as the G15 CharithPerisforhelpfuldiscussions,andtheanonymous
data to anchor R and the hard-state disk scaling. referee.
ISCO
APPENDIX
A. SIMPLCOMP
In addition to our adopted scattering kernel, which computes the photon redistribution following an exponentially
cutoffpowerlaw,wehavealsoimplementedakernel“comp”usingtheComptonizationmodelnthcomptonumerically
compute the scattered distribution at each E . In xspec, toggling the energy turnover parameter positive or negative
0
switches between kernels. A positive value calls comp (Z˙ycki et al. 1999; Zdziarski et al. 1996) and the value sets the