Alternative Explanations for Extreme Supersolar Iron Abundances Inferred from the Energy Spectrum of Cygnus X-1

The bright black hole (BH) high-mass X-ray binary Cygnus X-1 has been key for many of the advances in our knowledge of accreting BHs. A mass measurement of the compact object showing that it is well above the maximum mass allowed for a neutron star (>3 M_⊙) provided the first evidence that such compact objects exist (Gies & Bolton 1986). The fact that BHs enter into different spectral states, such as the soft and hard states, was first demonstrated with observations of Cyg X-1 (Tananbaum et al. 1972). Cyg X-1 is one of two sources where it has been possible to show that the compact jet, which is seen in the hard state, is an extended radio feature (Stirling et al. 2001). Cyg X-1 is also known to have a strong power-law component in its spectrum extending to at least a few megaelectronvolts (McConnell et al. 2002) with a very high level (∼70%) of polarization (Laurent et al. 2011; Jourdain et al. 2012), possibly indicating that the emission above 400 keV is dominated by synchrotron emission from the jet.

The properties of Cyg X-1 are very well-constrained compared to most BH binaries. It has a parallax distance measurement of ${1.86}_{-0.11}^{+0.12}$ kpc (Reid et al. 2011), which is consistent with a dust-scattering measurement (Xiang et al. 2011). Optical and near-IR observations made over its 5.6 day orbit constrain the BH mass to be 14.8 ± 1.0 M_⊙ and its binary inclination to be 271 ± 08 (Orosz et al. 2011). The spin of the BH has been constrained using two techniques: modeling the thermal continuum and the reflection component. The measurements agree that the spin of the BH is high. The continuum technique gives a value for the spin (a_* > 0.983, 3σ limit) that is consistent with being maximal or very slightly below maximal (Gou et al. 2014), while the reflection measurements allow for a somewhat more modest spin (Duro et al. 2011; Fabian et al. 2012; Miller et al. 2012; Tomsick et al. 2014; Walton et al. 2016). For example, using four observations, Walton et al. (2016) find a range of a_* values from 0.93 to 0.96. One reason that the Walton et al. (2016) values are a little lower than the Gou et al. (2014) values is that Gou et al. (2014) assume that the inner disk is aligned with the orbital plane.

The Gou et al. (2014) and Walton et al. (2016) spin measurements occur when the source is in the soft state and rely on the assumption that the disk extends to the innermost stable circular orbit (ISCO). Measurements of Cyg X-1 and other systems, such as LMC X-3 (Steiner et al. 2010), provide good evidence that the disk extends to the ISCO in the soft state. However, the situation is less clear in the hard and intermediate states. For the hard state, it is predicted that thermal conduction of heat from the corona will cause the inner disk to evaporate (Meyer et al. 2000), leading to an optically thick and geometrically thin disk that is truncated at some significant distance away from the BH. While there is evidence for truncation at luminosities near 0.1% of the Eddington limit (L_Edd) for GX 339–4 (Tomsick et al. 2009), measurements of the reflection component in the bright hard state ($\gtrsim 5 \% \,{L}_{\mathrm{Edd}}$) often lead to estimates of inner radii very close to the ISCO for several sources, including GRS 1739–278 (Miller et al. 2015). The luminosity of Cyg X-1 is typically 0.8%–2.8% L_Edd, using 0.1–500 keV unabsorbed fluxes (Yamada et al. 2013), and even with high quality data from the Nuclear Spectroscopic Telescope Array (NuSTAR) and Suzaku, there have been conflicting results concerning whether the disk in the hard state is truncated or not (Parker et al. 2015; Basak et al. 2017).

Given that the binary inclination for Cyg X-1 has been measured to high precision, it is also interesting to compare the inner disk inclinations inferred from reflection measurements to the binary value (271 ± 08). In the soft state, Tomsick et al. (2014) found an inner disk inclination >40°, and Walton et al. (2016) reported values between 38° and 41° for four different soft state measurements. Taken at face value, the data suggest a disk warp of 10°–15°, which has important implications for BH formation since it is very likely that the BH would need to be formed with its spin misaligned from the binary plane to produce this warp (Bardeen & Petterson 1975; Schandl & Meyer 1994; Fragile et al. 2007). Another implication is that the Cyg X-1 BH would need to be born with its rapid rotation rate (as described above, a_* = 0.93–0.96 for a warped disk or a_* > 0.983 if the entire disk is aligned with the orbital plane).

Another surprising result from the reflection fits for Cyg X-1 as well as GX 339–4 is the measurement of high iron abundances. For example, Walton et al. (2016) find a value of A_Fe = 4.0–4.3 times solar for Cyg X-1, and Parker et al. (2015) find A_Fe = 4.7 ± 0.1. For GX 339-4, values of A_Fe = 5 ± 1 and ${6.6}_{-0.6}^{+0.5}$ have been found (García et al. 2015; Parker et al. 2016). Fürst et al. (2015) also obtain high values of A_Fe for GX 339–4, but they find that the abundances change significantly for different assumptions about the continuum fitting. V404 Cyg is another BH system where reflection fits have resulted in supersolar abundances of A_Fe = 2–5 (Walton et al. 2017).

While there have been previous reports of NuSTAR and Suzaku observations of Cyg X-1 in the soft state and the hard state (Tomsick et al. 2014; Parker et al. 2015; Walton et al. 2016; Basak et al. 2017), the work in this paper presents the first analysis of contemporaneous observations of Cyg X-1 in the intermediate state with these satellites. As the name implies, the intermediate state has luminosities and spectral parameters (such as disk temperatures and power-law slopes) that are in between the soft and hard states, and it may provide clues to how sources in general and Cyg X-1 in particular make transitions between soft and hard states. The Cyg X-1 stellar wind can cause significant absorption of the X-ray spectrum, including distortion of the iron line (Tomsick et al. 2014; Walton et al. 2016), but the time of the intermediate state observation was carefully chosen to occur at the orbital phase when the BH is in front of the companion to minimize the absorption. In Section 2, we describe the observations and how the data were analyzed. Section 3 includes the results of the spectral fitting, and the results are discussed in Section 4.

We observed Cyg X-1 with NuSTAR (Harrison et al. 2013) and Suzaku (Mitsuda et al. 2007) on 2015 May 27–28 (MJD 57,169–57,170). Light curves measured by the Monitor of All-sky X-ray Image (MAXI; Matsuoka et al. 2009) and the Swift Burst Alert Telescope (BAT; Krimm et al. 2013) in the 2–20 and 15–50 keV energy bands, respectively, are shown in Figure 1. They cover the time of the observation that we are focusing on in this work as well as the earlier hard state observation reported on in Parker et al. (2015) and Basak et al. (2017). The BAT count rate was similar for the two observations, but the 2–20 keV MAXI count rate was higher during the 2015 May observation. Grinberg et al. (2013) used Cyg X-1 data from all-sky monitors, including MAXI, to define count rates for different states. The weighted average of the count rates for MAXI measurements within 3 days of the 2015 May observation gives 2–4 and 4–10 keV rates of 1.032 ± 0.011 and 0.654 ± 0.006 s⁻¹, respectively, and the ratio of the 4–10 to 2–4 keV rates is 0.633 ± 0.009. These values indicate that Cyg X-1 was in the intermediate state during the 2015 May observation. The states are indicated in Figure 1.

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Table 1 provides information about the NuSTAR and Suzaku observations used in this work, the observation identifier numbers (ObsIDs), the start and stop times of the observations, and the exposure times obtained. The middle of the observation is close to the orbital phase corresponding to the superior conjunction of the supergiant, and the range of orbital phases covered is 0.42–0.56. This was planned in order to minimize absorption due to stellar wind material. In the following, we describe the reduction of the data from each mission in more detail.

2.1. NuSTAR

We processed the data for the two NuSTAR focal plane modules (FPMA and FPMB) using HEASOFT v6.21, NUSTARDAS v1.7.0, and version 20170503 of the NuSTAR calibration data base (CALDB). We produced cleaned event files using nupipeline and then light curves and spectra using nuproducts. For light curves and spectra, we used a circular source extraction region with a radius of 180'' and a circular background region with a radius of 90''. Due to the brightness of Cyg X-1, the background region was placed on a corner of the NuSTAR field of view away from the source. We rebinned the spectra with the requirement that the source is detected at the 30σ level or higher in each bin.

The 3–79 keV light curve (Figure 2(a)) shows variability in the FPMA+FPMB count rate with the lowest 50 s bin having a rate of 493 s⁻¹ and the highest having 834 s⁻¹. However, the hardness (Figure 2(b)), which is the ratio of the 10–79 to the 3–10 keV rates, shows very little variation, indicating that the energy spectrum is relatively stable during the observation. While NuSTAR observed the source for ∼36 ks, the actual livetime was ∼20 ks (Table 1).

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2.2. Suzaku

We initially fitted the NuSTAR+Suzaku energy spectrum with a model consisting of a multi-temperature thermal disk (diskbb) component (Mitsuda et al. 1984) and a power law. We also included interstellar absorption using v2.3.2 of the tbnew¹⁸ model and a multiplicative constant to allow for differences in normalization between the six spectra (XIS0+XIS3, XIS1, FPMA, FPMB, PIN, and GSO). For modeling the absorption, we used Wilms et al. (2000) abundances and Verner et al. (1996) cross sections. A very poor fit is obtained with χ² = 29,415 for ν = 2759 degrees of freedom (dof). Figure 3 shows the data-to-model ratio residuals for this model. In each case, the Suzaku and NuSTAR spectra were fitted simultaneously, but we show two panels per model so that the data from each instrument are visible. The residuals for the diskbb+powerlaw model (see Figures 3(a)₁ and (a)₂) show the hallmarks of relativistic Compton reflection with positive features at the Fe Kα transition energies and in the 20–100 keV range. In the reflection scenario, these features are identified as fluorescence of iron in the accretion disk (Fabian et al. 1989) and the "Compton hump" (Lightman & White 1988). The negative residuals in the ∼8–15 keV range are partly due to the surrounding positive features and partly caused by a relativistically smeared Fe absorption edge.

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Replacing the power law with a relxill component leads to a huge improvement in the fit (to χ²/ν = 3715/2751). The relxill model includes a direct power law with an exponential cutoff as well as the reflection component predicted when a cutoff power law is incident on an optically thick accretion disk. The calculation of the reflection component is based on the xillver model (García et al. 2013, 2014), which includes continuum, emission lines, and absorption edges. This component is then relativistically smeared, including the effects of the relline convolution calculation (Dauser et al. 2013). In this paper, we used v1.0.0 of the relxill model.¹⁹ Although the residuals shown in Figures 3(b)₁ and (b)₂ are much improved, there is evidence for a narrow iron emission line, which has been seen previously (Miller et al. 2002; Walton et al. 2016), and is likely due to X-ray irradiation of the companion star, the stellar wind material, or the outer disk.

Assuming that the narrow line is caused by irradiation of some cool material in the system, it is physically reasonable to assume that it is part of a full reflection component, motivating the addition of a xillver component to the model. A model consisting of diskbb+relxill+xillver significantly improves the quality of the fit to χ²/ν = 3566/2750. Here, the only additional free parameter is the xillver normalization. We assume that the cool material producing the xillver component is neutral, and we fix the log of the ionization parameter (ξ) to zero. The other xillver parameters are tied to the relxill values. All the parameters for this fit are shown in Table 2. While most parameters are free, some are fixed or restricted to avoid searching physically unreasonable regions of parameter space. We fixed the interstellar column density, which has been measured many times (e.g., Tomsick et al. 2014), to N_H = 6 × 10²¹ cm⁻². Also, we restricted the spin of the BH, a_*, to be >0.93 based on previous measurements (Gou et al. 2014; Walton et al. 2016). The relxill model assumes that the source emissivity has a broken power-law dependence on radial distance from the BH with an index of q_in inside the break radius (R_break) and q_out outside. We fix q_out to a value of 3.0, and we restrict q_in to be greater than q_out. Although the GSO data shows that there is a cutoff to the spectrum, the errors on the GSO points are large compared to NuSTAR, and this causes problems with constraining the exponential cutoff energy (E_cut) and the cross-normalization constant (C_GSO). Thus, we fix the value of E_cut, and the value we use is 300 keV in order to allow more direct comparisons to results from the reflionx model (Ross & Fabian 2005), which are described below.

Table 2.  Fit Parameters for Reflection Models with Free Iron Abundance

Parameter	Unit/Description	Valuea	Valuea
		(relxill-based)	(reflionx-based)
Interstellar Absorption
NH	1021 cm−2	6.0b	6.0b
Disk-blackbody
kTin	keV	0.378±0.002	0.376±0.002
NDBB	Normalization	44,600±1100	$46,{600}_{-1400}^{+1500}$
Direct Component Plus Partially Ionized/Relativistic Reflection
Γ	Photon Index	1.826±0.004	1.712±0.006
Ecut	keV	300b	300b
logξ	ξ in erg cm s−1	3.374 ± 0.013	${3.588}_{-0.035}^{+0.044}$
AFe	Abundance Relative to Solar	>9.96	${10.6}_{-0.9}^{+1.6}$
Ω/2π	Covering Fraction	0.87 ± 0.04	⋯
qin	Emissivity Index	${9.1}_{-0.4}^{+0.5}$	9.3 ± 0.3
qout	Emissivity Index	3b	3b
Rbreak	Index Break Radius (RISCO)	${2.24}_{-0.22}^{+0.07}$	${2.41}_{-0.22}^{+0.06}$
Rin	Disk Inner Radius (RISCO)	${1.33}_{-0.06}^{+0.03}$	${1.26}_{-0.02}^{+0.03}$
Rout	Outer Radius for Reflection (Rg)	400b	400b
a*	Black Hole Spin	>0.987	>0.988
i	Inclination (°)	37.5 ± 0.7	35.6 ± 1.0
Nrelxill	Normalization	$({3.74}_{-0.05}^{+0.04})\times {10}^{-2}$	⋯
Nreflionx	Normalization	⋯	$({1.88}_{-0.15}^{+0.13})\times {10}^{-4}$
Ncutoffpl	Normalization	⋯	${26.2}_{-1.3}^{+0.7}$
Neutral Reflectionc
$\mathrm{log}\xi$	ξ in erg cm s−1	0.0b	1.0b
Nxillver	Normalization	$({1.77}_{-0.25}^{+0.22})\times {10}^{-3}$	⋯
Nreflionx	Normalization	⋯	(3.5 ± 0.4) × 10−4
Cross-normalization Constants (Relative to XIS0+XIS3)
CXIS1	⋯	0.794 ± 0.003	0.794 ± 0.003
CFPMA	⋯	1.005 ± 0.004	1.002 ± 0.004
CFPMB	⋯	1.007 ± 0.004	1.003 ± 0.004
CPIN	⋯	1.247 ± 0.007	1.244 ± 0.007
CGSO	⋯	1.13 ± 0.03	1.15 ± 0.03
Goodness of Fit
χ2/ν	⋯	3566/2750	3360/2750

Notes.

^aWith 90% confidence errors. ^bFixed. ^cThe model assumes that there is a single direct component that produces both the partially ionized/relativistic and neutral reflection components. Also, i and A_Fe are assumed to be the same for the partially ionized/relativistic and neutral reflection components.

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While the addition of the xillver component improves the fit and eliminates the residuals near 6.4 keV, there are still significant residuals in the NuSTAR spectrum (see Figures 3(c)₁ and (c)₂). Specifically, negative residuals can be seen in the iron edge region near 7 keV, and there are positive residuals at the high-energy end. Although the high-energy NuSTAR residuals could be reduced somewhat (but not completely eliminated) by using a larger value of E_cut, this would worsen the fit to the GSO data. Looking at the model parameters in Table 2, another concern is the very high iron abundance of nearly 10 times solar abundances (A_Fe > 9.96).²⁰ Although supersolar iron abundances have been seen previously when fitting the reflection spectra of Cyg X-1 in the hard and soft states (Tomsick et al. 2014; Parker et al. 2015; Walton et al. 2016; Basak et al. 2017), those values (A_Fe between 2 and 5) were significantly lower than the value of 10 that we are seeing for the intermediate state spectrum. We also fit the spectrum with the direct component being a Comptonization continuum (relxillCp) instead of a cutoff power law, but the best-fit value for A_Fe is still at its maximum value of 10.

Although some X-ray binaries may have supersolar abundances, it is difficult to explain why A_Fe would be variable for Cyg X-1. For GRS 1915+105, short-term (∼10–100 s) variations in A_Fe have been claimed (Zoghbi et al. 2016), and it was suggested that these may be caused by levitation of the iron atoms by radiation pressure (Reynolds et al. 2012; Zoghbi et al. 2016). However, the luminosity of Cyg X-1 (∼1%–2% of the Eddington limit) is much lower than for GRS 1915+105, indicating that the effect should be weaker for Cyg X-1. Also, even for GRS 1915+105, the periods where the iron abundances were found to be high (A_Fe ∼ 3) lasted only for tens of seconds, and the duty cycle for supersolar abundances was small. Thus, the fact that our reflection fits suggest such a high iron abundance requires us to consider what other physics is missing from the reflection models that could cause an artificial increase in A_Fe.

One assumption that is made in the constant density reflection models, such as relxill, is that the actual density does not have a large effect on the predicted spectrum. The rationale for this is that although the radiation will penetrate more deeply (in physical length units) into a low-density disk than a high-density disk, the penetration is not different in terms of optical depth. However, García et al. (2016) have recently looked at possible density effects, including the fact that free–free absorption and heating of the disk both depend quadratically on density. As density increases, the rise in free–free absorption leads to an increase in temperature, causing extra thermal emission from the outer layers of the disk (García et al. 2016). Thus, a BH binary that has a disk with n_e ∼ 10²⁰ cm⁻³ will have a soft X-ray excess compared to a disk with n_e = 10¹⁵ cm⁻³, which, motivated by values expected for active galactic nuclei (AGNs), is the density assumed for relxill.

In order to investigate the effects of higher disk density, we switch from relxill to reflionx (Ross & Fabian 2005) because a version of reflionx that extends the range of n_e to above 10²⁰ cm⁻³ has been produced (Ross & Fabian 2007), and we have the capability to make additional customized models using the same computer code. While a high-density version of relxill (called relxillD) is available, the maximum density considered in relxillD is only 10¹⁹ cm⁻³. Prior to switching to the high-density reflionx model, we refitted the Cyg X-1 spectrum with the standard reflionx model (Ross & Fabian 2005), which assumes a density of 10¹⁵ cm⁻³. The model is diskbb+reflionx+relconv*(cutoffpl+reflionx), where the direct component, cutoffpl, and the reflection component, reflionx, are convolved with relconv, which is the same relativistic convolution model that is part of relxill. In addition to using reflionx for the relativistic reflection component, we also replaced xillver with reflionx. We fix log ξ to 1.0, which is the minimum value for reflionx, and verified that this component has a narrow emission line at 6.4 keV, implying that it is still a valid way to model reflection from cool material. An advantage of switching to the reflionx model is that the range of A_Fe values goes to 20, allowing us to see how much this parameter increases beyond 10.

Table 2 shows that the parameter values are very similar for the relxill-based and reflionx-based models. In particular, the reflionx model still requires a very high iron abundance, which is well-constrained at ${A}_{\mathrm{Fe}}={10.6}_{-0.9}^{+1.6}$. However, some differences are also apparent, and these can be seen in Figure 4. With reflionx, the reflection component is somewhat stronger relative to the direct component. The extra emission from the reflection component in the soft X-ray band in the model allows the power-law index to harden slightly (from Γ = 1.826 ± 0.004 to 1.712 ± 0.006), and the harder power-law provides a somewhat better fit (see the χ² values in Table 2).

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To improve our understanding of why the models require such high iron abundances, we fixed A_Fe to 1.0 (solar abundances) and refit the spectrum. In both the relxill-based and reflionx-based models, the fit is very poor and, perhaps surprisingly, the largest differences between the models and the data are at the high-energy end rather than in the iron line region (see Figure 5). In these models, we fixed the cross-normalization constant for GSO to the values of C_GSO given in Table 2 because otherwise C_GSO would rise to unreasonably high levels. While both models demonstrate that low density models with solar abundances cannot fit both the iron line and the high-energy part of the spectrum, the details of why the two models fail is different. For the relxill-based model, changing to A_Fe = 1 causes a drop in the reflection component's soft X-ray emission, leading to a softening of the power-law index to Γ = 1.963 ± 0.002, which causes the model to drop well below the data at high energies. For the reflionx-based model, the Fe absorption edge is not as deep as for relxill, and the reflection component increases to fit the Fe edge in the data. However, the reflection hump turns over at an energy that is much too low, leaving a large excess at high energies.

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Comparing the spectra with A_Fe ∼ 10 (Figure 4) to those with A_Fe = 1 (Figure 5), it may be somewhat surprising that the apparent difference in the iron line strength is not larger. However, while increasing A_Fe does cause the emission line flux to rise, the absorption also increases, and this can be most easily seen by the fact that the iron absorption edge is much deeper for the spectra shown in Figure 4. The continuum absorption also increases, and we have checked that if we increase A_Fe in the relxill model while leaving all other parameters fixed, the model flux drops above and below the iron line, leading to an increase in the iron line equivalent width (EW). The details of how the EW depends on A_Fe as well as the ionization parameter ξ have been well-documented for the xillver model (García et al. 2011). The EW increases with A_Fe but decreases as the disk becomes more ionized.

We produced a high-density model called reflionx_hd, which is an XSPEC table model using the same code described in Ross & Fabian (2007). The free parameters in the model are the power-law photon index (Γ), the ionization parameter (ξ), and the density (n_e). Models are calculated for 10 values of Γ between 1.4 and 2.3, 10 values of ξ between 100 and 10,000 erg cm s⁻¹, and 16 values of n_e between 10¹⁵ and 10²² cm⁻³. An important difference between reflionx_hd and the model described in Ross & Fabian (2007) is that the Ross & Fabian (2007) model includes blackbody irradiation that is an additional heat source for the disk, but we do not include this source for reflionx_hd. We emphasize that A_Fe = 1 for this model, so the high-density effects can be seen by comparing the results to the fits shown in Figures 5(b)₁ and (b)₂.

Using reflionx_hd provides a very large improvement in the fit. The parameters for the fit are provided in Table 3, and the quality of the fit is the best that we have obtained thus far (χ²/ν = 3020/2752). This is also demonstrated in Figures 3(d)₁ and (d)₂, which show that this model removes the positive residuals at the high-energy end of the NuSTAR bandpass as well as the dip near 7 keV. The density is well-constrained to be n_e = (3.98^+0.12_−0.25) × 10²⁰ cm⁻³ (see Table 3), and the components for this model are shown in Figure 6. The additional soft X-ray emission caused by the higher density is apparent. The soft excess peaks at an energy above the diskbb temperature, and we note that a change in kT_in does not lead to a model that fits both the thermal disk component and the soft excess. Although we originally left q_in and R_break as free parameters in the model, they were not well-constrained, and we fixed q_in to be equal to q_out (so they were both fixed to 3). At least in part, the emissivity parameters became poorly constrained because the inner disk radius parameter increased (ultimately, we obtained ${R}_{\mathrm{in}}={7.3}_{-1.9}^{+4.6}$ R_ISCO), but the emissivity is assumed to drop to zero inside R_in.

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Table 3.  Fit Parameters for the High-density Reflection Model

Disk-blackbody

Parameter	Unit/Description	Valuea	Valuea
		(Power-law Emissivity)	(lamppost)
Interstellar Absorption
NH	1021 cm−2	6.0b	6.0b
kTin	keV	0.317 ± 0.002	0.317 ± 0.002
NDBB	Normalization	$100,{500}_{-3500}^{+3700}$	$100,{400}_{-3400}^{+2500}$
Direct Component Plus Partially Ionized/Relativistic Reflection
Γ	Photon Index	1.779 ± 0.007	${1.777}_{-0.006}^{+0.009}$
Ecut	keV	300b	300b
$\mathrm{log}\xi$	ξ in erg cm s−1	${3.302}_{-0.006}^{+0.011}$	${3.303}_{-0.002}^{+0.014}$
AFe	Abundance Relative to Solar	1.0c	1.0c
qin	Emissivity Index	3b	⋯
qout	Emissivity Index	3b	⋯
h	Lamppost Height (RISCO)	⋯	${20.0}_{-1.0}^{+15.9}$
Rin	Disk Inner Radius (RISCO)	${7.3}_{-1.9}^{+4.6}$	${2.9}_{-0.8}^{+3.5}$
Rout	Outer Radius for Reflection (Rg)	400b	400b
a*	Black Hole Spin	>0.93	${0.949}_{-0.019}^{+0.013}$
i	Inclination (°)	${18}_{-5}^{+4}$	<20
ne	Density in 1020 cm−3	${3.98}_{-0.25}^{+0.12}$	${3.98}_{-0.33}^{+0.13}$
Nreflionx	Normalization	${97.9}_{-1.5}^{+2.9}$	${97.8}_{-1.5}^{+4.0}$
Ncutoffpl	Normalization	24.8 ± 0.4	${24.8}_{-0.3}^{+0.4}$
Neutral Reflectiond
$\mathrm{log}\xi$	ξ in erg cm s−1	1.0b	1.0b
Nreflionx	Normalization	(8.6 ± 0.8) × 10−4	$({8.3}_{-0.7}^{+0.2})\times {10}^{-4}$
Cross-normalization Constants (Relative to XIS0+XIS3)
CXIS1	⋯	0.793 ± 0.003	${0.793}_{-0.003}^{+0.001}$
CFPMA	⋯	1.002 ± 0.004	${1.002}_{-0.001}^{+0.004}$
CFPMB	⋯	1.004 ± 0.004	${1.004}_{-0.001}^{+0.004}$
CPIN	⋯	1.244 ± 0.007	1.244 ± 0.007
CGSO	⋯	1.02 ± 0.03	1.02 ± 0.03
Goodness of fit
χ2/ν	⋯	3020/2752	3016/2751

Notes.

^aWith 90% confidence errors. ^bFixed. ^cThe reflionx_hd model is for solar abundances. ^dThe model assumes that there is a single direct component that produces both the partially ionized/relativistic and neutral reflection components. Also, i and A_Fe are assumed to be the same for the partially ionized/relativistic and neutral reflection components.

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In order to check on whether the source geometry has a significant impact on the results, we also fit the spectrum with a lamppost geometry by changing the relconv convolution model to relconv_lp, while continuing to use the high-density reflection model (reflionx_hd). With the lamppost, the quality of the fit is equivalent (χ²/ν = 3016/2751), and most parameters, including the density, show very little or no changes. The lamppost height is h = 20.0^+15.9_−1.0 R_ISCO, and the inner disk radius is ${R}_{\mathrm{in}}={2.9}_{-0.8}^{+3.5}$ R_ISCO (see Table 3). Although this still suggests a slightly truncated disk, changes to the lamppost model cause the 90% confidence lower bound on R_in to move from 5.4 R_ISCO to 2.1 R_ISCO (considering errors).

We have focused on a broadband spectrum of Cyg X-1 in the intermediate state with relatively high statistical quality to investigate current uncertainties related to accreting BHs at intermediate luminosities. These include questions about the disk geometry, including understanding when disks become truncated and whether they are warped, as well as about the supersolar iron abundances that have been inferred from reflection fits. To place the intermediate state observation in the context of the previous reports of NuSTAR and Suzaku observations of Cyg X-1, we compile some key parameters in Table 4. As expected, the values of the disk-blackbody temperature and Γ are intermediate between the hard state value from Basak et al. (2017) and the soft state values from Tomsick et al. (2014) and Walton et al. (2016). The 0.5–100 keV luminosity for the intermediate state observation is 1.8 × 10³⁷ erg s⁻¹, which corresponds to L/L_Edd = 0.94% for a 14.8 M_⊙ BH. In comparison, the hard- and soft-state values are 0.62% and 1.72%, respectively.

Table 4.  Key Parameters for Cyg X-1 in Different Spectral States

	Tomsick et al. (2014)	Walton et al. (2016)	This Work	This Work	Basak et al. (2017)
	(NuSTAR and	(4 NuSTAR	relxill	reflionx_hd	(NuSTAR and
	Suzaku)	Observations)		(lamppost)	Suzaku)
Spectral State	soft	soft	intermediate	intermediate	hard
kTin (keV)	0.56	0.40–0.47	0.378 ± 0.002	0.317 ± 0.002	0.14–0.15
Γ	2.6–2.7	2.56–2.74	1.826 ± 0.004	${1.777}_{-0.006}^{+0.009}$	1.70–1.71
Rin/RISCO	1	1	${1.33}_{-0.06}^{+0.03}$	${2.9}_{-0.8}^{+3.5}$	13–20 Rga
a*	>0.83	0.93–0.96	>0.987	${0.949}_{-0.019}^{+0.013}$	⋯
i (°)	>40	38.2–40.8	37.5 ± 0.7	<20	24–43
AFe	1.9–2.9	4.0–4.3	>9.96	1.0	2.2–4.6
Fluxb	8.0 × 10−8	⋯c	4.4 × 10−8	⋯d	2.9 × 10−8
Luminositye	3.3 × 1037	⋯c	1.8 × 1037	⋯d	1.2 × 1037
L/LEdd	1.72%	⋯c	0.94%	⋯d	0.62%

Notes.

^a13–20 gravitational radii, R_g, corresponds to 13–20 R_ISCO for a maximally rotating BH, 6.5–10 R_ISCO for a BH rotating at a_* = 0.94, and 2.2–3.3 R_ISCO for a non-rotating BH. ^bUnabsorbed flux in the 0.5–100 keV band in erg cm⁻² s⁻¹. ^cWe do not quote fluxes or luminosities for these four observations. Only NuSTAR was used in the analysis, and the disk-blackbody component is not constrained well enough to extrapolate down to 0.5 keV. ^dThe fluxes and luminosities are not significantly different for reflionx_hd compared to relxill. ^eIn erg s⁻¹ using the 0.5–100 keV unabsorbed flux and a distance of 1.86 kpc.

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As described above, previous observations of Cyg X-1 have shown supersolar abundances, but the intermediate state observation marks the first time that values as high as 10 times solar have been seen in Cyg X-1, prompting us to look for other explanations. A possible explanation that we study in some detail here is that the high A_Fe is related to high-density effects, and our results show that a high-density model allows A_Fe to drop to solar abundances while actually improving the fit to the spectrum. The improvement in the fit occurs because the higher density produces extra soft X-ray emission, allowing for a harder direct power law, which provides a better match to the high-energy part of the spectrum. Figure 7 shows that the soft excess caused by the increase in free–free absorption only begins to be significant in the NuSTAR bandpass when densities of ${n}_{e}\gtrsim {10}^{20}$ cm⁻³ are reached. While García et al. (2016) describe this effect, that work also emphasizes that the atomic physics that goes into the reflection models is only known to be accurate up to densities of 10¹⁹ cm⁻³. Thus, a caveat to our results is that more accurate determinations of quantities such as the rates of atomic transitions could have an impact on the high-density reflection models.

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While the high-density explanation for Cyg X-1 makes sense because estimates show that BH binary disks should have densities that are much higher than 10¹⁵ cm⁻³ (Svensson & Zdziarski 1994; García et al. 2016), it is not a unique explanation for the high A_Fe values. We have also produced a model with n_e = 10¹⁵ cm⁻³ but with blackbody irradiation of the disk, and this also provides a good fit to the spectrum with A_Fe = 1. However, the good fit is obtained for a blackbody temperature of 0.7 keV, which is hotter than the inner disk temperature of kT_in = 0.3–0.4 keV that we measure directly. Thus, this model is not physically self-consistent, but it does show another way that the addition of a soft excess component can provide a good fit to the spectrum with solar iron abundances.

In other papers on fits to Cyg X-1 spectra in the hard or intermediate state, models with two Compton continuum components have been used. Yamada et al. (2013) and Basak et al. (2017) both fit broadband Cyg X-1 spectra with hard and soft Comptonization components. We have applied such models to our spectrum, and they also provide good fits with solar iron abundances. However, they have many more free parameters than our reflionx_hd model, and the physical picture is more complicated as it is unclear that coronae with temperatures in the 1–5 keV range (e.g., as found by Basak et al. 2017) exist. There are other physical scenarios that might produce multiple high-energy components such as a disk corona and a jet, and these have previously been applied to Cyg X-1 spectra (e.g., Nowak et al. 2011), but they are also more complicated and have many more free parameters than the reflionx_hd model.

Regardless of how the extra emission below the iron line is produced, it has an impact on the constraints on the inner radius of the disk because it is the gravitationally redshifted wing of the line that provides the power to measure R_in. This can be seen in our spectral fits since the high-A_Fe fits give inner radii of R_in = 1.3 R_ISCO (see Table 2), which are very close to the ISCO, while the high-density fits show larger inner radii. Table 3 lists values of ${R}_{\mathrm{in}}={7.3}_{-1.9}^{+4.6}$ R_ISCO for the broken power-law emissivity and ${R}_{\mathrm{in}}={2.9}_{-0.8}^{+3.5}$ R_ISCO for the lamppost model.

The high-density models also affect the values obtained for the inner disk inclination. While we have previously found values that are significantly larger than the binary inclination (Tomsick et al. 2014; Walton et al. 2016), and our high-A_Fe fits in this work also give high inclinations, the high-density fits give ${18}_{-5}^{{+4}\circ}$ for the broken power-law emissivity and a 90% confidence upper limit of <20° for the lamppost model. Using the lamppost model, we performed an additional fit to the data with the inner disk inclination fixed to the binary inclination of 27°. While this does degrade the quality of the fit somewhat (χ²/ν = 3036/2752 compared to 3016/2751), the residuals are still relatively small.

As the density of AGN disks is estimated to be close to the standard value of n_e = 10¹⁵ cm⁻³, we would expect the AGN iron abundances to be closer to the true values. However, there are several examples of supersolar iron abundances obtained by fitting their spectra with reflection models. The AGN abundances range from moderately supersolar values, such as A_Fe = 2–4 for NGC 3783 (Reynolds et al. 2012) and A_Fe ∼ 3 for NGC 1365 (Risaliti et al. 2013) to values of A_Fe = 9 for 1H 0707–495 (Fabian et al. 2009) or even higher for IRAS 13224–3809 (Fabian et al. 2013). In at least some cases, the supersolar abundances may be real. Looking at optical line properties of quasars, Wang et al. (2012) show evidence for abundances up to 7 times solar, and Reynolds et al. (2012) hypothesize that radiative levitation can enhance iron abundances. We discuss this possibility in Section 3 and argue that this effect would cause rapid changes in A_Fe for X-ray binaries. However, this would probably not be a problem for AGNs due to the longer timescales.

In addition to the caveats discussed above about the atomic physics being uncertain above 10¹⁹ cm⁻³, about the high-density model not being a unique solution, and about the high iron abundances in AGNs, it is also important to point out that we have only looked at the high-density model for the case of solar abundances. While we can rule out large changes in A_Fe, implying that abundances of 10 times solar are unreasonable for Cyg X-1, we cannot rule out supersolar abundances at lower levels. Thus, we caution against overinterpretation of the inner radius and inclination values. In the near future, it is important to improve the atomic physics in the high-density models and also to make reflection models that consider a range of A_Fe values. With such models, it would be worthwhile to expand the analysis to more Cyg X-1 spectra as well as other sources where fit parameters suggest the possibility of supersolar abundances.

This work made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. We thank the NuSTAR Operations, Software and Calibration teams for support with the execution and analysis of these observations. This research has made use of the NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA). This research has made use of data obtained from the Suzaku satellite, a collaborative mission between the space agencies of Japan (JAXA) and the USA (NASA). J.A.T. acknowledges partial support from NuSTAR Guest Observer grant NNX15AV23G. A.C.F. acknowledges support from ERC Advanced Grant 340442. D.J.W. acknowledges support from STFC in the form of an Ernest Rutherford fellowship. J.A.T. thanks K. Hamaguchi for help with the XIS data reduction and T. Dauser, K. Koljonen, and A. Zoghbi for useful discussions. This research has made use of the MAXI data provided by RIKEN, JAXA, and the MAXI team.