X-RAY REFLECTION SPECTROSCOPY OF THE BLACK HOLE GX 339–4: EXPLORING THE HARD STATE WITH UNPRECEDENTED SENSITIVITY

In the thermal state, there is abundant evidence that the accretion disk is truncated near the innermost stable circular orbit (ISCO) (e.g., Gierliński & Done 2004; Penna et al. 2010; Steiner et al. 2010; Zhu et al. 2012). The standard paradigm for the faint hard state is that as the luminosity decreases the inner edge of the disk recedes from the ISCO, leaving a hot ADAF or other coronal flow (Narayan & Yi 1994; Narayan & McClintock 2008). While there is good evidence that at very low luminosities the disk is grossly truncated (for a review, see Narayan & McClintock 2008), the location of the inner edge relative to the ISCO for luminosities in the range ∼0.1%–10% of Eddington is a hotly debated topic. With GX 339–4 as a principal test bed, two methods have been widely used to estimate the radius R_in of the inner edge of the disk in the hard state: (1) modeling the component of emission reflected from the disk, principally the Fe K line; and (2) fitting the continuum spectrum of the accretion disk. The former method, which is addressed in the following section, is the central topic of this paper.

Efforts to estimate the inner edge of the accretion disk in the low/hard state via disk reflection go back farther, but the first strong indication that disks may remain close to the ISCO in bright phases of the low/hard state was made by Miller et al. (2006b). Based on fits to the thermal component, a number of papers claim that there is an optically thick disk that extends inward to the ISCO in the hard state (Miller et al. 2006a, 2006b; Rykoff et al. 2007; Reis et al. 2009, 2010; Reynolds et al. 2010). This claim is strongly contested by Done et al. (2007) and Done & Diaz Trigo (2010); the claim is all the more questionable when one considers that self-consistent disk coronal models (e.g., Steiner et al. 2009) return larger values of the inner-disk radius. More recently, Miller and coworkers have invoked extreme values of the spectral hardening factor in making the case for an untruncated hard-state disk (Reynolds & Miller 2013; Salvesen et al. 2013). This evidence for the presence of such a disk does not appear to us compelling given the difficulties of obtaining accurate estimates of R_in by modeling a faint, cool (kT ≲ 0.2 keV) thermal component that is strongly Comptonized and cut off by interstellar absorption.

The reflection spectrum results from the reprocessing of high-energy coronal photons in the optically thick accretion disk. The result is a rich spectrum of radiative recombination continua, absorption edges, and fluorescent lines, most notably the Fe K complex in the 6–8 keV energy range. This reflected radiation leaves the disk carrying information on the physical composition and condition of the matter in the strong fields near the black hole. The Fe K emission line (and other fluorescent lines) are broadened and shaped by Doppler effects, light bending, and gravitational redshift. By modeling the reflection spectrum, one can estimate both the disk inclination and the dimensionless spin parameter a_* = cJ/GM² (−1 ≤ a_* ≤ 1). In measuring a_*, one estimates the radius of the inner edge of the accretion disk and identifies it with the radius of the ISCO, R_ISCO, which simply and monotonically maps to a_* (Bardeen et al. 1972). For the three canonical values of the spin parameter, a_* = +1, 0 and −1, R_ISCO = 1 M, 6 M and 9 M (for c = G = 1), respectively.

The reflection model most widely used in the past for both general application and measuring black hole spin is reflionx (Ross & Fabian 2005). Recently, an improved reflection model has been developed, relxill,⁴ which is based on the reflection code xillver (García & Kallman 2010; García et al. 2011, 2013, 2014a), and the relativistic line-emission code relline (Dauser et al. 2010, 2013, 2014). Compared to reflionx, relxill incorporates a superior treatment of radiative transfer and Compton redistribution, and it allows for the angular dependence of the reflected spectrum. Furthermore, by implementing the routines of the photoionization code xstar (Kallman & Bautista 2001), relxill provides an improved calculation of the ionization balance. At the same time, limitations of the model include assuming that the density of the disk is independent of vertical height, that the illuminating radiation strikes the disk at a fixed angle of 45°, and that apart from Fe all the elemental abundances are assumed to be solar. The results presented in this paper were derived using relxill to model the relativistically blurred reflection component from the inner disk and xillver to model a distant reflector.

It is important to appreciate the faintness of the reflected features that are crucial for probing effects in the regime of strong gravity, the features that one relies on for estimating R_in and constraining black hole spin. For example, in the spectrum of GX 339–4, even the most prominent feature, the Fe K line, has a typical equivalent width of ∼0.1 keV, and the peak intensity of the line is only about 10% of the local continuum (Section 3). Sensitivity to such faint features requires both high-count spectra and a well-calibrated detector.

The principal detector on board the Rossi X-ray Timing Explorer (RXTE) was the Proportional Counter Array (PCA), which was comprised of five nearly identical Proportional Counter Units (PCUs), each with an effective area of 1600 cm² and with sensitivity from 2–60 keV. Despite the limited spectral resolution of the instrument (≈17% at 6 keV) the archive of PCA data amassed during the RXTE mission (1995–2012) continues to be preeminent for the synoptic study of stellar-mass black holes. A few-dozen bright black holes were observed daily during their outburst cycles with typical exposure times of a few ks. Some 15,000 individual spectra were obtained with a net total exposure time of 30 Ms (1 year). In this paper, we report the results of our analysis of six hard-state spectra of GX 339–4, each a summation of dozens of individual exposures (Section 3). For the spectrum obtained at maximum luminosity (L/L_Edd = 17%) with an exposure time of 46 ks, the total number of counts is 40 million and the counts-per-keV in the continuum at 6.4 keV is 4.4 million, while the total number of counts in the Fe K line region (3–10 keV) is 28 million.

A limitation of the PCA, which has not allowed the implied statistical precision to be realized in modeling data, has been the appreciable ∼1% uncertainties in the detector response (Jahoda et al. 2006; Shaposhnikov et al. 2012). We have overcome this limitation by developing a calibration tool, called pcacorr, that increases the sensitivity of the RXTE PCA detector to faint spectral features—such as the Fe K line/edge–by up to an order of magnitude (García et al. 2014b). By applying pcacorr to a large number of spectra for three black holes, we found that the tool improved the quality of all the fits, and that the improvement was dramatic for spectra with ≳10⁷ counts. The tool allows one to achieve a precision of ∼0.1% rather than ∼1%, thereby making full use of spectra of bright sources with ∼10⁶ counts per channel.

Consequently, our study of the reflection spectrum of GX 339–4 greatly improves on earlier studies using the PCA, such as that by Plant et al. (2015). A limitation of PCA data is its modest resolution, while its major advantage is its freedom from the problematic effects of pileup, which is commonly a serious problem in analyzing and interpreting data for bright sources obtained using CCD detectors (see Section 6.1.3). Another advantage of the PCA, which has only recently been matched by NuSTAR, is its high-energy coverage, which allows observations of both the Fe K region and the Compton hump using a single detector.

This paper is organized as follows: Section 2 describes the observations and data reduction, and Section 3 outlines our procedure for combining the individual spectra into six composite spectra. The luminosities of these spectra, which we refer to throughout as Spectra A–F, range over an order of magnitude. Fitting the spectra individually, while emphasizing the importance of correcting the data using the pcacorr tool, is the subject of Section 4. Our key results appear in Section 5. Therein, we describe how we fit Spectra A–F simultaneously, first fixing the spin parameter and letting the inner-disk radius vary, and then allowing the spin parameter to vary while fixing the inner radius at the ISCO. We discuss our results in Section 6 and offer our conclusions in Section 7.

We first consider the likely possibility that the absorption feature is largely an artifact related to the uncertain energy resolution of the PCA. We then discuss the one plausible physical explanation for the feature known to us, namely that it is produced by absorption in a highly ionized wind. Finally, we consider the role of the xillver component not only in modeling the residual features near the Fe K line, but also its role in improving the fit quite generally.

4.1.1. On the Accuracy of the PCA Energy Resolution

The presence of residual absorption features bracketing the Fe line at ∼5.6 and ∼7.2 keV suggests the possibility that the PCA resolution may be better than assumed in generating the PCU-2 response. We have explored this possibility for Spectrum A. We test the effects of slight changes in the value assumed for the detector resolution by moderately smoothing the data, with the results shown in Figure 4. The smoothing is accomplished using a Gaussian kernel operating over the detector channels; the parameter f specifies the width of the Gaussian as a percent of the channel width. Accordingly, the curve in Figure 4 labeled f = 0 is unsmoothed, while the curves labeled f = 40 and f = 50 correspond to degrading the resolution of the data by 0.9% and 1.5% at 6.4 keV.

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Although this approximate approach to artificially tuning the detector resolution does not eliminate the residuals flanking the Fe line, it does significantly reduce their strength. The test demonstrates that at this extreme level of statistical precision the fit to a line feature is very sensitive to the value assumed for the detector resolution. Specifically, if one assumes that the nominal value of resolution for the unsmoothed case (f = 0) at 6 keV is 17.0%, then the net resolution for f = 50 is 17.3% (i.e., the additional blurring has a width of one-half channel, equivalent to ∼0.2 keV at 6.4 keV, which is combined in quadrature with the nominal resolution width). Meantime, the resolution of the PCA is not known to sufficient accuracy to discriminate such fine differences (N. Shaposhnikov 2015, private communication). This suggests that the residuals near the Fe line may result from the PCA resolution being slightly better than assumed in modeling the detector response. However, this test is inconclusive. To properly assess the importance of tuning the resolution, one must carry out a systematic analysis using the PCA calibration software, which is beyond the scope of this paper.

4.1.2. On the Possibility that the Feature Originates in a Highly Ionized Wind

If the 7.2 keV feature is not an instrumental artifact, a potentially plausible explanation is that it originates in a highly ionized wind that envelops the primary source. Disk winds have been observed in many black hole binary systems, particularly at high accretion rates (e.g., Miller et al. 2006c; Neilsen et al. 2012; Ponti et al. 2012). We investigated this possibility by replacing the gabs component by the photoionized warm absorber model (warmabs). We forced the Fe Lyα line at ∼6.9 keV to be the dominant feature by setting the ionization parameter to its maximum value (log ξ = 5). We linked the Fe abundances of warmabs and relxill while the abundances of all the other elements remain at solar. The fitted blueshift of the Fe Lyα required to model the 7.2 keV feature is z = 0.0576 ± 0.0101, which corresponds to an outflow velocity of v = 1.7 × 10⁴ km s⁻¹. The model provides a good fit (${\chi }_{\nu }^{2}=1.17$), which is very comparable to that achieved using Model 3 (see the top-left panel of Figure 5 for details and a comparison of the residuals). However, this interpretation seems unlikely on physical grounds due to the extreme column density required by the warm absorber, namely, ${N}_{{\rm{H}}}^{{\rm{abs}}}=(7.7\pm 0.2)\times {10}^{23}$ cm⁻². If one links the warmabs Fe abundance to that of the xillver component (i.e., A_Fe = 1), the fit pegs at the hard limit of the warmabs model (10²⁴ cm⁻²).

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4.1.3. On the Inclusion of the Unblurred Reflection Component

As discussed in Section 3, we combined the individual spectra in order to increase our sensitivity to the reflection features while minimizing the effects of jitter in the power-law index. To verify the procedures we used in combining spectra, we compare the results of fitting our Spectrum A to those obtained by fitting a spectrum created by summing directly all the spectra in Box A.⁶ Applying pcacorr and fitting both spectra with our adopted model (i.e., Tbabs∗(relxill+xillver)∗gabs), we find that the model parameters are all consistent. However, the fit to the summed spectrum is of significantly lower quality (Δχ² = 62.97) than the fit to Spectrum A created using the procedures described in Section 3. Furthermore, as shown in the bottom-left panel of Figure 5, the residuals for the summed spectrum are larger in almost every energy channel. These results demonstrate that our method of combining the individual spectra significantly improves the quality of the fit.

As fully described in García et al. (2014b) and discussed in Section 1.3, the pcacorr tool greatly reduces the effects of instrumental features in PCA spectra, thereby making it possible to achieve good fits to high-count spectra at the 0.1% level of statistical precision. The bottom-right panel in Figure 5 demonstrates the importance of applying the tool to Spectrum A, with its 4 × 10⁷ counts. The figure compares residuals for a fit to uncorrected data to one using pcacorr-corrected data. To most clearly illustrate the power of pcacorr, we use Model 1 (Tbabs∗relxill) and set the systematic errors to zero. Concerning the ∼5.6 keV and ∼7.2 keV features flanking the Fe K line, we note that they are present prior to the application of pcacorr, which confirms that they are not introduced by the correction. The key message of the bottom-right panel in Figure 5 is the degree to which pcacorr diminishes these features and others, especially those below 10 keV and the one near 30 keV, a feature that is likely related to the detector Xe K-edge (Shaposhnikov et al. 2012; García et al. 2014b).

A principal goal of our study is to track the radius of the inner edge of the disk R_in as the luminosity varies by an order of magnitude (i.e., over the range 1.6%–17% of Eddington; Table 1). As discussed in Section 1.1, a question of great interest is whether the inner disk is truncated in the hard state at low luminosities and, if so, to what extent. In order to be maximally sensitive to a disk that is only slightly truncated, we fix the spin to its maximum allowed value, namely a_* = 0.998. In so doing, our focus is on determining how R_in trends with luminosity rather than obtaining accurate estimates of this parameter. We note that most spin determinations in the literature (which assume R_in = R_ISCO) suggest that the spin is high (see Section 6.1.3).

Figure 6 shows the fit residuals for jf-i for two cases: (1) The top panel shows a data-to-model ratio for a fit to Model 0, i.e., the simple absorbed power-law model used to produce the residual plot for a fit to Spectrum A only (which is shown in the left-top panel of Figure 3). This simple fit prominently displays the reflection features, which are strong for all six spectra. The profile of the Fe K line shows moderate variations among the spectra. As the luminosity increases, so does the intensity of the line (cf. Figure 2). Additionally, the blue wing of the line extends to higher energies for the high luminosity spectra, which could be evidence for a shift in the Fe K edge caused by an increase in the ionization of the gas. There are obvious changes at high energies among the spectra, which are likely due to the evolution of the high-energy cutoff with luminosity. (2) Of chief interest, the middle and bottom panels of Figure 6 show the residuals for our adopted Model 3 (for a_* = 0.998).

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Model 3 performs remarkably well for all spectra as indicated by the goodness of fit, ${\chi }_{\nu }^{2}=1.06,$ and also by the uniformly ergodic appearance of the residuals across the energy band. The fit results are summarized in Table 3. A key result, which is discussed in detail in the following section, is evidence that the disk is moderately—but significantly—truncated at the lowest luminosities that can be effectively explored using RXTE (i.e., at a few percent of Eddington).

In order to obtain constraints on black hole spin using either leading method, reflection spectroscopy or continuum fitting, one must assume that R_in = R_ISCO (e.g., McClintock et al. 2014; Reynolds 2014). In performing jf-ii, we make this assumption for all six spectra in order to constrain the spin of the black hole. Doing so allows us to obtain a precise estimate of spin: ${a}_{*}={0.95}_{-0.05}^{+0.03}$ at 90% confidence. As the summary of results in Table 4 shows, the other parameters are quite close to those obtained in jf-i (Table 3), and the goodness of fit is of very comparable quality: ${\chi }_{\nu }^{2}=1.09.$ Given the extreme statistical precision, the residual spectra (data/model and χ²) for jf-ii (not shown) are essentially indistinguishable by eye from the spectra for jf-i, which are shown in Figure 6.

Table 4.  Results for jf-ii: Fit Parameters for Model 3, const∗Tbabs∗(relxill+xillver)∗gabs, with a_* free and R_in = R_ISCO

Model	Parameter	Spectrum A	Spectrum B	Spectrum C	Spectrum D	Spectrum E	Spectrum F
Tbabs	NH (cm−2)	$({5.9}_{-1.9}^{+0.6})\times {10}^{21}$
relxill	a*	${0.95}_{-0.05}^{+0.03}$
relxill	i (deg)	${47.8}_{-1.4}^{+0.9}$
relxill	AFe	${5.4}_{-0.5}^{+1.9}$
relxill	Nr	${1.44}_{-0.08}^{+0.04}$
gabs	E (keV)	7.23 ± 0.08
Constant		1	${0.90}_{-0.04}^{+0.03}$	${0.71}_{-0.03}^{+0.02}$	0.37 ± 0.02	0.17 ± 0.01	0.08 ± 0.01
relxill	Rin (RISCO)	1
relxill	Γ	${1.604}_{-0.027}^{+0.010}$	1.658 ± 0.018	${1.651}_{-0.022}^{+0.015}$	${1.62}_{-0.04}^{+0.02}$	${1.578}_{-0.013}^{+0.009}$	${1.637}_{-0.013}^{+0.009}$
relxill	logξ	3.33 ± 0.03	3.35 ± 0.04	${3.16}_{-0.05}^{+0.10}$	${3.05}_{-0.02}^{+0.04}$	${1.96}_{-0.21}^{+0.12}$	${2.0}_{-0.2}^{+0.2}$
relxill	Ecut	${92}_{-6}^{+2}$	118 ± 8	${160}_{-16}^{+12}$	${440}_{-110}^{+230}$	>830	>940
relxill	Rf	0.20 ± 0.01	${0.20}_{-0.01}^{+0.02}$	0.20 ± 0.02	${0.27}_{-0.05}^{+0.03}$	${0.25}_{-0.03}^{+0.05}$	${0.28}_{-0.04}^{+0.02}$
xillver	Nx	${0.23}_{-0.02}^{+0.05}$	0.26 ± 0.04	0.24 ± 0.05	0.13 ± 0.06	0.08 ± 0.06	<0.09
gabs	Strength	${0.024}_{-0.007}^{+0.009}$	${0.028}_{-0.011}^{+0.018}$	${0.037}_{-0.012}^{+0.016}$	${0.04}_{-0.02}^{+0.03}$	0.025 ± 0.014	${0.020}_{-0.009}^{+0.017}$
	L/LEdd(%)	17.3	14.2	11.9	7.9	3.9	1.6
	χ2	418.66
	ν	384
	${\chi }_{\nu }^{2}$	1.09

Note. For the given model components, the parameters from top to bottom are: hydrogen column density (N_H); dimensionless spin parameter (${a}_{*}={cJ}/{{GM}}^{2},$ where J is the angular momentum of the black hole); inclination of the inner disk (i); iron abundance with respect to its solar value (A_Fe); normalization of the blurred reflection component plus power-law continuum (N_r); energy of the absorption Gaussian centroid (E); constant multiplicative factor between spectra; inner-disk radius (R_in); power-law photon index (Γ); log of the ionization parameter (ξ = 4πF_x/n, where F_x is the ionizing flux and n is the gas density); high-energy cutoff (E_cut); reflection fraction (R_f, ratio of the reflected flux to that in the power-law, in the 20–40 keV band); normalization of the distant (unblurred) reflection (N_x); strength of the absorption Gaussian; X-ray luminosity in terms of Eddington (see notes in Table 1); goodness of the fit (χ²); number of degrees of freedom (ν); goodness of the fit per degree of freedom (${\chi }_{\nu }^{2}={\chi }^{2}/\nu$). Uncertainties are based on a 90% confidence level.

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Concerning our spin estimate and the fundamental assumption that R_in = R_ISCO, we again note that jf-i provides evidence for disk truncation at our lowest luminosities.⁷ Meantime, these low-luminosity spectra are included in computing our single, tied jf-ii estimate of spin. The incorporation of disk truncation effects (which we have ignored) would imply an even higher spin value than is quoted above.

We first discuss four parameters that are global for GX 339–4 (i.e., the same for all six spectra), namely, the Galactic hydrogen column density N_H, the spin parameter a_* of the black hole, the inclination i, and the Fe abundance A_Fe. The entries for these (and all other) parameters estimated in the MCMC analysis given in Tables 3 and 4 are 90% minimum-width confidence intervals about the posterior maxima.

6.1.1. Hydrogen Column Density

6.1.2. Inclination of the Inner Disk

Previous estimates of inclination, which have been obtained by modeling the reflected component, run the gamut (see Table 5). Three papers using the same XMM-Newton and RXTE data uniformly report low values: Miller et al. (2006b) and Reis et al. (2008) obtained $i={20}_{-15}^{+5}$ deg and i ≲ 20°, respectively, while Done & Diaz Trigo (2010), using a different strategy for reducing the data, found i ∼ 20°–27°. A much larger inclination, i = 46 ± 8 degrees, was determined by Shidatsu et al. (2011) using Suzaku data. All of these results were obtained using the reflionx models (Ross & Fabian 2005). Recently, Plant et al. (2015) fitted simultaneously XMM-Newton and Suzaku data sets using both reflionx and xillver and reported two estimates of inclination: $i={36}_{-6}^{+3}$ deg and $i={42}_{-6}^{+11}$ deg. In a different work, Plant et al. (2014b) analyzed three low/hard state observations of GX 339–4 using a recent version of the relxill model and found $i={30}_{-4}^{+5}$ deg, a result that may be biased because it relies solely on low-energy XMM-Newton data.

Table 5.  Compilation of Literature Estimates of R_in Obtained by Fitting Reflection Models to Hard-state Spectra of GX 339–4

Satellite	Instrument	L/LEdd (%)	Rin (Rg)	i (deg)	q	High-energy?	References
XMM-Newton	EPIC-pn (TM)a	1.42	${684}_{-378}^{+301}$	${42}_{-6}^{+11}$	3	Yes	(1)
	⋯	⋯	${972}_{-643}^{+28}$	${36}_{-6}^{+3}$	3	Yes	(2)
	⋯	⋯	${110}_{-40}^{+80}$	60	3	No	(3)
	⋯	⋯	${150}_{-50}^{*}$	60	3	No	(4)
	EPIC-MOS	3.25	5 ± 0.5	${20}_{-10}^{+5}$	3	Yes	(5)
	⋯	⋯	${2.04}_{-0.02}^{+0.07}$	${20}_{-1.3}$	3.16 ± 0.5	Yes	(6)
	EPIC-pn	⋯	10 ± 2	27 ± 3	3	Yes	(7)
	⋯	⋯	${60}_{-20}^{+40}$	60	3	Yes	(8)
	EPIC-pn (TM)a		${318}_{-74}^{+165}$	${42}_{-6}^{+11}$	3	Yes	(1)
	⋯	⋯	${125}_{-51}^{+21}$	${36}_{-6}^{+3}$	3	Yes	(2)
	⋯	⋯	${89}_{-23}^{+55}$	60	3	No	(3)
	⋯	⋯	${128}_{-43}^{+73}$	60	3	No	(4)
	⋯	10.2	${155}_{-53}^{+139}$	${42}_{-6}^{+11}$	3	Yes	(1)
	⋯	⋯	${72}_{-21}^{+42}$	${36}_{-6}^{+3}$	3	Yes	(2)
	⋯	⋯	2.2*	60	3	No	(3)
	⋯	⋯	${47}_{-7}^{+10}$	60	3	No	(4)
	⋯	<0.05	${21}_{-9}^{+17}$	${30}_{-4}^{+5}$	3	No	(14)
	⋯	<0.05	${27}_{-6}^{+6}$	${30}_{-4}^{+5}$	3	No	(14)
	⋯	<0.05	${16}_{-4}^{+7}$	${30}_{-4}^{+5}$	3	No	(14)
Suzaku	XIS0,1,3/PIN	0.14	>65	18	2–3	Yes	(9)
	⋯	⋯	>798	${42}_{-6}^{+11}$	3	Yes	(1)
	⋯	⋯	>745	${36}_{-6}^{+3}$	3	Yes	(2)
	⋯	⋯	${190}_{-90}^{+170}$	50	2.3	Yes	(10)
	⋯	⋯	>180	20	3	Yes	(11)
	⋯	0.13	>30	20	3	Yes	(11)
	⋯	0.19	>10	20	3	Yes	(11)
	⋯	0.25	>70	20	3	Yes	(11)
	⋯	0.60	${32}_{-19}^{+33}$	20	3	Yes	(11)
	⋯	0.91	${7.0}_{-1.3}^{+1.1}$	20	3	Yes	(11)
	⋯	2.0	${13.3}_{-6.0}^{+6.4}$	46 ± 8	2.3 ± 0.1	Yes	(10)
Swift	XRT	1.33	${3.6}_{-1.0}^{+1.4}$	20	3.2 ± 0.6	Yes	(12)
	⋯	⋯	${6.7}_{-2.3}^{+9.1}$	20	3	Yes	(13)
	⋯	0.46	<10	20	3.1 ± 0.4	Yes	(12)
	⋯	⋯	${3.6}_{-0.9}^{+1.9}$	20	3	Yes	(13)
	⋯	1.08	${12.8}_{-9.0}^{+19.8}$	20	3	Yes	(13)
	⋯	1.12	${6.9}_{-3.7}^{+9.1}$	20	3	Yes	(13)
	⋯	1.68	${19.5}_{-8.5}^{+25}$	20	3	Yes	(13)
	⋯	2.23	${16.3}_{-5.2}^{+11.7}$	20	3	Yes	(13)
	⋯	5.21	${19.7}_{-6.5}^{+12.1}$	20	3	Yes	(13)

Note. (1) Plant et al. (2015) implementing xillver, (2) Plant et al. (2015) implementing reflionx, (3) Kolehmainen et al. (2014), (4) Kolehmainen et al. (2014) including a notch feature at ∼9 keV, (5) Miller et al. (2006b), (6) Reis et al. (2008), (7) Done & Diaz Trigo (2010), (8) Done & Diaz Trigo (2010) with fixed inclination, (9) Tomsick et al. (2009) (10) Shidatsu et al. (2011), (11) Petrucci et al. (2014), (12) Tomsick et al. (2008), (13) Allured et al. (2013), (14) Plant et al. (2014b).

^aTM = Timing Mode.

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In jf-i, we obtained a tight constraint on inclination, i = 484 ± 11 (Table 3), a result that is consistent with that obtained in jf-ii (Table 4). Note that the relxill and xillver models used here properly treat the angular distribution of the reflected radiation, unlike earlier reflection models, which only provided an angle-averaged solution (García et al. 2014a).

As an aside, if one makes the usual assumption that the spin of the black hole is aligned with the orbital angular momentum vector (Fragos et al. 2010; Steiner & McClintock 2012), then the lower estimates of inclination discussed above imply implausibly large values of black hole mass based on the Hynes et al. (2003) estimate of the mass function: e.g., $M\sim 5.8/{\mathrm{sin}}^{3}i\sim 100\;{M}_{\odot }$ for i ∼ 25°. Meanwhile, our inclination implies M ∼ 15 M_⊙, consistent with the range of values observed for stellar-mass black holes (Özel et al. 2010; Farr et al. 2011).

The inclination angle is largely determined by the shape and position of the blue wing of the Fe K line, which in our fits is somewhat affected by the inclusion—or exclusion—of the Gaussian absorption feature. The energy of this feature was linked in all six spectra and constrained to 7.23 ± 0.08 keV (Table 4), while the normalization was free to vary. Figure 7, based on our MCMC analysis, shows that while the strength of the gabs component does increase by a factor of ∼2 with decreasing luminosity, it only weakly interplays with inclination. This implies that this component, whose origin is uncertain (Section 4.1), has at most a modest affect on our estimate of inclination.

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6.1.3. Black Hole Spin

The spin of the black hole has been estimated via reflection modeling using three independent data sets: a_* = 0.939 ± 0.004 using XMM-Newton/EPIC-MOS plus RXTE spectral hard-state data (Miller et al. 2006b; Reis et al. 2008); a_* = 0.93 ± 0.02 using XMM-Newton/EPIC-pn plus RXTE spectral data in the very high (or steep power-law) state (Miller et al. 2004; Reis et al. 2008); and a_* = 0.93 ± 0.01 (statistical) ±0.04 (systematic) using Suzaku data (Miller et al. 2008). The corresponding estimates of inclination were all low (i ∼ 10°–20°) implying implausibly high estimates of black hole mass assuming spin–orbit alignment (Section 6.1.2).

There is considerable uncertainty associated with these estimates of spin and inclination because of the effects of pileup, i.e., the arrival of two or more photons in the same or adjacent CCD pixel within a single frame time. For example, Done & Diaz Trigo (2010), analyzing precisely the same hard-state XMM-Newton/EPIC-MOS data as Miller et al. (2006b) and Reis et al. (2008), conclude that the high spin reported by Miller et al. and Reis et al. is the result of severe pileup effects; using PN Timing-mode data (presumably unaffected by pileup), Done & Diaz Trigo (2010) report evidence for a narrow Fe line and a truncated disk. Meantime, Miller et al. (2010) rebut the conclusions of Done and Díaz Trigo. As a second example, the high spin reported by (Miller et al. 2008) based on their analysis of Suzaku data (see above), is challenged by Yamada et al. (2009) who find—using the same data set—evidence for a truncated disk and no need to invoke a rapidly spinning black hole.

Kolehmainen & Done (2010) applied the alternative continuum-fitting method to disk-dominated RXTE data collected during three different outbursts of GX 339–4. This method relies on accurate knowledge of the mass, distance, and inclination of the system, all of which are highly uncertain for GX 339–4. Using approximate bounds on these parameters, Kolehmainen & Done obtained "a strict upper limit" on the spin of a_* < 0.9, which they claim is inconsistent with the spin estimates obtained by modeling the reflection spectrum.

A chief virtue of the PCA data upon which we rely is its freedom from the confusing effects of pileup. A further virtue is the abundance of data, which allows us to track the behavior of GX 339–4 over a range of luminosity, as well as to reach extreme (∼0.1%) levels of statistical precision. By assuming that the inner radius of the disk always remains at the ISCO (jf-ii; Section 5.2), we established a firm constraint on the spin at ${a}_{*}={0.95}_{-0.05}^{+0.03}$ (90% confidence) while obtaining a precise estimate of the inclination, $i={47.8}_{-1.4}^{+0.9}$ degrees. Our spin result is in accord with the earlier Fe-line estimates, but our inclination estimate is distinctly different, and more in line with expectation (Section 6.1.2).

The results of our MCMC analysis allow us to search for possible degeneracies of the spin parameter with other fit parameters. Figure 8 shows for jf-ii spin probability distributions for three key parameters: (1) the inclination angle, which affects the blue wing of the line; (2) the Fe abundance, which affects the strength of both the line and edge; and (3) the strength of the Gaussian absorption, which also could affect the blue wing of the line. While there is no evidence for a substantial correlation between the spin and the inclination or the strength of the Gaussian, there is indication of a moderate positive correlation with the Fe abundance. From this positive correlation, it follows that an increase in Fe abundance will produce a diminished inner radius for fits performed with a fixed spin and variable R_in (Section 6.3).

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6.1.4. Fe Abundance

The Fe abundance for the blurred reflection component (relxill) is surprisingly high: ${4.6}_{-0.3}^{+0.5}$ and ${5.4}_{-0.5}^{+1.9}$ in solar units for jf-i and jf-ii, respectively (Tables 3 and 4). Most studies have merely assumed that the abundance is solar, while Allured et al. (2013) fitted Swift/RXTE hard-state data and also found a super-solar abundance: ${A}_{\mathrm{Fe}}={2.4}_{-0.62}^{+1.47}.$ The high abundance results directly from the remarkable strength of the Fe K line/edge relative to the Compton hump (top panel of Figure 6), which is usually the highest-amplitude feature in the reflection spectrum. A lower Fe abundance underpredicts the strength of the line/edge required to fit the Compton hump. Thus, our ability to constrain the Fe abundance is likely a consequence of our broad bandpass that provides high-sensitivity coverage of all the principal reflection features, from the Fe K line on through complete coverage of the Compton hump. This quality of coverage is not provided by XMM-Newton data (even when RXTE data with a floating normalization are included) which may explain why others have not reported a super-solar Fe abundance.

Quantitatively, forcing the Fe abundance to the solar value (A_Fe = 1) results in a grossly unacceptable fit with ${\chi }_{\nu }^{2}=9.9$ and large residuals across the PCA band (Figure 9), while the inner-disk radius grows by about a factor of 10 compared to the fit with variable Fe abundance. We tried several alternative models (e.g., varying the emissivity index) in an unsuccessful attempt to find an acceptable model with lower Fe abundance. We note that fits to NuSTAR data for the most recent outburst of GX 339–4 likewise require a large Fe abundance (Fuerst et al. 2015).

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The Fe abundance has an influence on the shape of the Fe K line, and it may therefore in turn affect such parameters of interest as the inclination or inner-disk radius. Moreover, the abundance also alters the continuum photoelectric opacity at higher energies, which modifies the depth of the Fe K edge and the red side of the Compton hump. These effects, which are subtle, are driving our fits because of the unprecedented signal-to-noise we have achieved. We shall return to this point in Section 6.3.

We conclude that super-solar Fe abundance is a strong and inescapable requirement of these data. At the same time, the data also require the unblurred (distant) reflection to have moderate—near-solar—Fe abundance. In our tests, fitting the spectrum with the most counts (Spectrum A), we found that adopting a single abundance for both blurred and unblurred components (i.e., relxill and xillver) results in A_Fe ∼ 3.6, an intermediate value between relxill (A_Fe ∼ 5), and the low value required for the xillver component. Though this approach may be intuitively more satisfying, it is strongly rejected by the data, with an increase in χ² of ∼55. The effect of linking the abundances is to decrease the importance of the xillver component by reducing its normalization parameter from ∼0.2 to ∼0.07. An inspection of the residual contributions to χ² (lower panel of Figure 9) reveals that the quality of the fit is degraded not only in the Fe K region but over the entire energy band. In summary, we find strong empirical evidence for the presence of an unblurred reflection component whose Fe abundance is much less than that of the blurred component.

At the same time, there is no obvious reason why the inner-disk abundances should be so high. We note that similar physical processes may be occurring in AGN, since large Fe abundances are likewise found in many cases when fitting relativistic reflection models (Fabian 2006), with 1 H 0707–495 being a prime example (Dauser et al. 2012; Kara et al. 2015). Possibly, an unknown physical effect is being overlooked in current models that is artificially driving the Fe abundance to high values. For example, Reynolds et al. (2012) proposed that radiative levitation of Fe ions in the accretion disk atmosphere could cause an apparent enhancement of their abundance.

Setting aside the Gaussian absorption component (already discussed in Section 4), for jf-i there are six important parameters that are fitted separately for Spectra A–F: the inner-disk radius R_in; the photon index Γ; the ionization parameter ξ; the high-energy cutoff E_cut; the reflection fraction R_f; and the normalization of the unblurred reflection component (xillver). In this section, we show how these parameters depend on luminosity, and we discuss the causes of these dependencies.

Figure 10 illustrates our MCMC results for jf-i (Table 3), the case of fixed spin. The probability distribution for each parameter is shown plotted versus the floating constant factor, which can be regarded as a proxy for the luminosity. The luminosity ranges over somewhat more than an order of magnitude. Each Spectrum is color-coded (see legend in top-left panel). The breadth of a distribution is a measure of uncertainty, while its shape indicates the degree of correlation of that particular parameter with luminosity. We now discuss in turn the behavior of each parameter.

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The evolution of the inner-disk radius R_in with luminosity is shown in the top-left panel of Figure 10. Each spectrum delivers a good constraint on R_in, allowing us to conclude that the inner edge of the disk moves outward by a factor of a few as the luminosity decreases by an order of magnitude, from a nominal value of 17% of Eddington to 1.6% of Eddington (Table 1).

This is a principal result of our paper because R_in and its dependence on luminosity is a matter of central importance for the study of black hole binaries in the hard state (Section 1.1). In Table 5 we summarize estimates of R_in in the literature for GX 339–4 in the hard state, while considering only those results obtained via reflection spectroscopy. The compilation includes results obtained using a wide variety of data and over a large range in luminosity (∼0.1%–20% of Eddington). At a glance, one notes the extreme range of values reported for R_in. The most notable conflict is two grossly disparate values reported for the same XMM-Newton observation: Reis et al. (2008) analyzed MOS and RXTE PCA data and reported ${R}_{\mathrm{in}}={2.04}_{-0.02}^{+0.07}$ R_g, while Plant et al. (2015) analyzed EPIC-pn timing-mode data and reported ${R}_{\mathrm{in}}={318}_{-74}^{+165}$ R_g.

Figure 11 shows all the values of R_in that appear in Table 5 plotted as a function of the Eddington-scaled luminosity. Several studies report results for multiple observations over a range of luminosity; in these cases, the individual data points are highlighted in the left panel of Figure 11 using colored tracks. Meanwhile, individual measurements are shown in the right panel. Our results are shown in both panels with a solid red track connecting the data points. As noted above, we find that R_in increases modestly with decreasing luminosity: Best-fit values trend upward from 2.1 R_g to 4.6 R_g as the luminosity decreases from 17% to 1.6% of Eddington. (Note that the values of R_in in Table 3 are in units of R_ISCO = 1.237 R_g.)

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This trend is consistent with that found in previous studies except for that of Allured et al. (2013) (yellow track), who fitted Swift and RXTE data using the reflionx model. Despite the general agreement that the inner radius shrinks with increasing luminosity, our estimates of R_in at comparable values of luminosity are much smaller than those reported by others.

For example, Plant et al. (2015) found the disk to be extremely truncated based on fits to a Suzaku and three XMM-Newton RXTE spectra using xillver and reflionx (light and dark blue tracks). Kolehmainen et al. (2014) reported a similar trend but smaller values of radius (green track) by analyzing the same three XMM-Newton spectra (excluding the RXTE data) using the rfxconv model (based on the reflionx tables; Kolehmainen et al. 2011). For their highest-luminosity data, they report two values of inner radius: One is a lower limit that is consistent with our results, R_in > 2.2R_g, while the other (which includes a ∼9 keV instrumental feature in the fit; orange track) is reasonably consistent with the results of Plant et al. (2015), R_in ≈ 47 R_g. Overall, the results of Kolehmainen et al. (2014) and Plant et al. (2015) are similar, with the former authors reporting somewhat smaller values of R_in. Interestingly, our results appear to be in reasonable agreement with an extrapolation of the low-luminosity values of R_in reported by Petrucci et al. (2014) (purple track), which is not necessarily expected since these authors assumed solar Fe abundance.

Though the gross disparities in the reported values of R_in may be partially due to differences in the models, this should be a secondary effect since, e.g., tests show that the models xillver and reflionx perform similarly (García et al. 2013). The more likely reason for the inconsistent results is limitations of the data. One of the most severe of these is the effects of pileup, especially for the crucial XMM-Newton data (see Section 6.1.3). Another effect leading to major differences in results is whether or not high-energy data were used. For example, Plant et al. (2015) and Kolehmainen et al. (2014) used the same EPIC-pn data in timing mode, but while Plant et al. (2015) used simultaneous RXTE data to extend the energy coverage, Kolehmainen et al. (2014) eschewed its use because of their concern over the cross-calibration of the two detectors. As a consequence of employing XMM-Newton data only, the results of Kolehmainen et al. (2014) are highly sensitive to calibration issues associated with the rapidly falling and uncertain response of the EPIC-pn detector at energies ≳9 keV.

The PCA has important advantages despite its limited energy resolution and lack of coverage below 3 keV. Most notably, the PCA data are free from the contentious effects of pileup that are inherent to CCD observations of bright sources. Meanwhile, the use of a single detector eliminates problems associated with cross-calibrating a pair of detectors. The much higher effective area of the PCA around the Fe line and Compton hump–and the many dozens of observations—yields spectra with orders-of-magnitude more counts than CCD spectra (Section 5). Moreover, one can now fully utilize these many millions of counts per spectrum to detect subtle effects in reflection features because the response of the PCA has been successfully calibrated to ∼0.1% precision (Shaposhnikov et al. 2012; García et al. 2014b). These virtues of the PCA data are attested to by our success in fitting our reflection models (${\chi }_{\nu }^{2}\sim 1$) to six extremely high signal-to-noise spectra, which individually contain between 3 and 28 million total counts in the 3–10 keV band. Finally, the great abundance of data makes the PCA database unrivaled for synoptic studies of Galactic black holes.

We now return to the question of the grossly discrepant results reported for R_in (Table 5; Figure 11) while reminding the reader that the Fe abundance affects the Fe K line profile and other reflection features at a detectable level given our signal to noise (Section 6.1.4). In turn, the Fe abundance affects other parameters, notably the inner-disk radius and spin parameter, which correlates positively with Fe abundance (Section 6.1.3). We now show that values of R_in found by others are significantly biased by either the low signal-to-noise of their data or inadequate high-energy coverage. Such data make it difficult to distinguish between small R_in with large A_Fe and large R_in with solar abundances, as we illustrate in Figure 12, which compares fits to two relxill models, one with A_Fe = 1 and the other with A_Fe = 5. The model with solar abundance (black curve) can only fit the data when the disk is strongly truncated, which serves to minimize the relativistic effects that blur the line profile. Note, however, that this model then fails to reproduce the depth of the Fe K edge and underpredicts the continuum above ∼30 keV. Figure 12 should be compared directly with Figure 9, where the limitations of the A_Fe = 1 model are apparent from the fit residuals. Unlike most other data sets, the extreme signal in our data clearly discriminates between the two models.

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The truncation of the inner disk and the decrease in R_in with increasing L/L_Edd, which we find, is a prediction of the ADAF model (see Section 1.1). In this paradigm, the inner disk evaporates becoming a very hot and optically thin accretion flow that fills the inner region (see, e.g., Meyer-Hofmeister et al. 2009). Our results are in line with this model, although our observations do not extend to the lower luminosities at which extreme truncation likely occurs.

However, we note that these results are apparently at odds with our non detection of a thermal disk component of emission. Making the usual assumption that all the observed power-law photons are generated by Compton up-scattering of disk photons, we would have expected to detect a thermal component for a disk that extends to such small radii (see Section 7).

We find that the power-law photon index Γ is relatively constant despite the order of magnitude increase in luminosity (top-right panel of Figure 10), a result that has been previously reported for GX 339–4 (e.g., Wilms et al. 1999; Zdziarski et al. 2004; Plant et al. 2014a). Its average value is 1.640 ± 0.035 for jf-i and 1.625 ± 0.030 for jf-ii (std. dev., N = 6; Tables 3 and 4), firmly in the range for the hard state (1.4 < Γ < 2.1; Remillard & McClintock 2006).

In contrast to the constancy of the power-law index, the cutoff energy E_cut systematically decreases with increasing luminosity from >890 keV for Spectrum F down to 97 ± 4 keV for Spectrum A (right-middle panel of Figure 10). This lower value of E_cut for our highest luminosity spectrum is of the same order of magnitude as the 58.5 ± 2.2 keV value reported by Droulans et al. (2010), which is based on their analysis of simultaneous RXTE and INTEGRAL data obtained during another bright hard state of GX 339–4.

Our model achieves good constraints for all six spectra. Remarkably, this is true for even the lowest-luminosity data (Spectrum F) for which the cutoff energy of >890 keV is far beyond the 45 keV limit of the PCA bandpass. This surprising result is a consequence of the detectable effects that are imprinted on the reflected component in the 3–45 keV band by photons with energies of hundreds of keV. We discuss the capability of the relxill model to probe the spectrum at extreme energies in Garcia et al. (2015).

In a Comptonized and isothermal corona, the high-energy cutoff is set by the electron temperature: E_cut ∼ (2–3)kT_e. In such a plasma, thermal disk photons are Compton up-scattered, thereby cooling the coronal electrons while producing the observed power-law continuum. The slope of the power law depends on the interplay between the electron temperature and the optical depth τ_e,

$$\begin{eqnarray}&&{\rm{\Gamma }}=-\displaystyle \frac{1}{2}+\sqrt{\displaystyle \frac{9}{4}+\displaystyle \frac{1}{{\theta }_{{\rm{e}}}{\tau }_{{\rm{e}}}(1+{\tau }_{{\rm{e}}}/3)}},\end{eqnarray} \tag{ 1 }$$

(Lightman & Zdziarski 1987), where ${\theta }_{{\rm{e}}}={{kT}}_{{\rm{e}}}/{m}_{{\rm{e}}}{c}^{2}$ and m_ec² = 511 keV is the electron rest mass. Values of these parameters for Spectra A–F, which are consistent with previous determinations (e.g., Wilms et al. 1999), are summarized in Table 6 for our nominal value of the photon index (Γ = 1.6). We find, as predicted by Equation (1), that the coronal temperature decreases with increasing luminosity, while the optical depth increases.

Table 6.  Coronal Properties^a

Box	L/LEdd	Ecut	Te	τe
	(%)	(keV)	(109 K)
A	17.3	97	0.45	3.03
B	14.2	129	0.60	2.50
C	11.9	179	0.83	1.99
D	7.9	660	3.06	0.72
E	3.9	840	3.90	0.59
F	1.6	890	4.13	0.56

Note.

^aAssumes ${kT}=\frac{2}{5}{E}_{\mathrm{cut}}$ and Γ = 1.6 (See Equation (1)).

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The reflection fraction R_f is a third parameter (in addition to Γ and E_cut) that provides information on the structure of the corona. In relxill, the parameter is empirically defined as the ratio of the reflected flux to the power-law flux in the 20–40 keV band. The results of jf-i show that the reflection fraction of the relativistically blurred component (relxill) ranges from 0.2 ≲ R_f ≲ 0.3, decreasing modestly with increasing luminosity (bottom-left panel, Figure 10). This trend is surprising given that, at the same time, R_in is decreasing so that the area of the reflector should increase. As a further wrinkle, one expects R_f ≳ 1 based on simple arguments (Dauser et al. 2014).

There are several scenarios that can plausibly account for values of R_f < 1. We mention four and then discuss a new, alternative explanation. (1) An obvious explanation is a severely truncated disk (more specifically, a disk with truncation radius large compared to the size of the corona). We discard this possibility as inconsistent with the small values we find for R_in (Section 6.3). (2) Another option is for the corona to be continuously outflowing at relativistic speeds, beaming the bulk of its emission away from the disk (e.g., Miller et al. 2015; Keck et al. 2015). While this is a possible explanation for R_f < 1, one outcome of this scenario is a resultant low value of the emissivity index, which is not obviously required by our data. (3) The value of R_f may be depressed by our assumption of a constant-density disk atmosphere, as Ballantyne et al. (2001) have shown in their studies of hydrostatic atmospheres. The hotter gas layer at the surface of a hydrostatic atmosphere additionally scatters and blurs reflection features (see also Nayakshin & Kallman 2001) thereby diluting the reflection signal relative to a constant density model. (4) The apparent strength of reflection features may also be reduced by the Comptonization of these features in an extended corona, as Wilkins & Gallo (2015) recently proposed. However, for such a corona to be effective in reducing R_f appreciably, it must have a large covering fraction which may interfere with detection of blurred reflection features from the inner disk.

We propose an alternative explanation for R_f < 1 based on the strong dependence of the reflected spectrum on the angle at which an illuminating photon strikes the disk. This angle crucially determines the characteristic depth in the disk at which the photon interacts; this in turn affects the limb-darkening/brightening of the disk (Svoboda et al. 2009; García et al. 2014a). A deficiency of reflionx, relxill, xillver and other widely used reflection models is the simplifying assumption of a fixed incidence angle of 45°. However, a larger angle of incidence (measured with respect to the normal to the disk plane), for example, results in a hotter surface layer and therefore a weaker reflection signature (see Figure 5 in Dauser et al. 2013).

To test whether the assumption of near-grazing illumination substantially increases fitted values of R_f, we produced a new table of xillver reflection models with a fixed incidence angle of 85° and merged them with relline to create a new high-incidence-angle version of relxill (Section 1.2). Fitting Spectra A–F as in Section 5.1 (i.e., jf-i), the fit is slightly worse (Δχ² = 9.71) but statistically comparable and still quite reasonable (${\chi }_{\nu }^{2}=1.09$). Notably, for the 85° model we find that R_f increases with luminosity and that R_f > 1 for the three most luminous spectra (A–C). Meanwhile, all the other parameters are consistent with those for jf-i (Table 3). Importantly, the Fe abundance remains unchanged.

Figure 13 compares the reflection factors computed for the two models. The large-angle model is more in accord with expectation, namely, the reflection fraction trends upward with luminosity and the values at the higher luminosities (with R_in near the ISCO and with correspondingly large reflector area) are sensibly ≳1. Thus, our results qualitatively suggest that the accretion disk in GX 339–4 is illuminated at near-grazing angles with respect to the surface of the disk.

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Within a few gravitational radii of the horizon, the extreme bending of light rays causes photons to strike the disk over a wide range of angles (Dauser et al. 2013; see in particular the middle panel of their Figure 5). Given the strong dependence of R_f on the angle of incidence, reaching firm conclusions concerning the reflection factor will require building a new generation of models, a task beyond the scope of this paper that will be addressed in future work.

As expected, both the normalization N_r and the ionization parameter ξ of the blurred reflection component (relxill) increase with luminosity (Table 3; Figure 10, middle-left panel). In particular, the ionization parameter changes from ξ = 112.2 to ξ = 2041.7, which traces very well the ten-fold increase in luminosity, from 1.6% L_Edd to 17% L_Edd. For the strongly illuminated portion of the disk, the variations in ξ and L deviate mildly from the simple relation ξ = L/nD², where n is the density of the gas (fixed at n = 10¹⁵ cm⁻³ for the relxill and xillver models used here)⁸ , and D is the distance from the coronal source to the strongly heated portion of the disk. Thus, D increases only modestly as the luminosity decreases by an order of magnitude in passing from Spectrum A to Spectrum F:

$$\begin{eqnarray}&&\displaystyle \frac{{D}_{F}}{{D}_{A}}=\sqrt{\displaystyle \frac{{L}_{F}{\xi }_{A}}{{L}_{A}{\xi }_{F}}}\sim 1.3.\end{eqnarray} \tag{ 2 }$$

This small change is reasonable given the correspondingly mild increase in the inner radius obtained for jf-i:

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{F}}{{R}_{A}}=2.2.\end{eqnarray} \tag{ 3 }$$

Likewise, the normalization N_x of the unblurred reflection component (xillver) increases with luminosity (Figure 10, bottom-right panel); although presumably the ionization parameter of this component also increases with luminosity, we approximate the state of the gas in the distant reflector as cold and neutral.