THE REFLECTION COMPONENT FROM CYGNUS X-1 IN THE SOFT STATE MEASURED BY NuSTAR AND SUZAKU

Cygnus X-1 (Cyg X-1)is a bright, high-mass X-ray binary that was discovered in the early days of X-ray astronomy (Bowyer et al. 1965) and was identified with the optical counterpart HD 226868 (Murdin & Webster 1971). It is best known for being the first system with a high enough mass measurement to rule out the possibility that the compact object is a neutron star (e.g., Gies & Bolton 1986), making it the first confirmed black hole (BH) system. The current constraint on the BH mass is 14.8 ± 1.0$\,M_{\mathord \odot }$ (Orosz et al. 2011). Cyg X-1 has been instrumental in improving our understanding of accreting BHs, their spectral states, and the relationship between the accretion disk and the jet (see Remillard & McClintock 2006 for a review of BH binaries).

Currently, a major on-going effort in BH studies is to measure their spins. A non-zero BH spin changes the space time around the BH, requiring the Kerr rather than the Schwarzschild metric to describe the geometry. The spin is also one possible source for powering the relativistic jets seen coming from BHs. One technique for measuring the BH spin involves modeling the multi-temperature thermal component that comes from the accretion disk (McClintock et al. 2006). A major challenge for this technique is that the distance to the system and the inclination of the inner disk must be known. For Cyg X-1, the distance is well established with a parallax measurement of $1.86^{+0.12}_{-0.11}$ kpc (Reid et al. 2011), which is consistent with a measurement using the dust scattering halo (Xiang et al. 2011). The improved distance determination has also led to new constraints on the binary inclination. Combined modeling of optical spectroscopy (i.e., the companion's radial velocity) and photometry over all orbital phases has given a binary inclination of 271 ± 08 (Orosz et al. 2011). Orbital modulations are seen in the optical light curves that depend on the shape of the companion star and the inclination of the system. Although some misalignment between the inner disk inclination and the binary inclination is possible (Maccarone 2002), under the assumption that they are the same, Gou et al. (2011) find that the spin of the Cyg X-1 BH is a_* > 0.95 (3σ limit) and an even higher spin limit (a_* > 0.983 at 3σ) has been recently reported (Gou et al. 2013).

Another technique for measuring BH spin involves modeling the Compton reflection component that is due to hard X-ray emission shining on the inner part of the optically thick accretion disk. The reflection spectrum includes fluorescent emission lines, with the Fe Kα lines typically being the strongest (Fabian et al. 1989), and a broad excess in the ∼10–50 keV energy range (Lightman & White 1988). The reflection spectrum can be distorted by the relativistic effects of Doppler broadening from the fast orbital motion and the gravitational redshift due to the BH's gravitational field (Fabian et al. 1989). The emission lines can also be broadened when photons are Compton scattered out of the narrow line core (Ross & Fabian 2005). This implies that broad emission lines are not necessarily an indication of relativistic effects. However, the Compton broadening is symmetric, so modeling the asymmetric component is the key to using this technique to constrain BH spin (Reynolds & Nowak 2003; Miller 2007).

For both thermal and reflection component modeling techniques, the BH spin measurement is actually inferred from the measurement of the location of the inner radius of the optically thick and "cold" (i.e., not fully ionized) accretion disk. The BH spin measurement then comes from identifying the inner radius with the innermost stable circular orbit (ISCO). For a non-rotating BH (a_* ≡ Jc/G(M_BH)² = 0, where J is the angular momentum of the BH, c is the speed of light, G is the gravitational constant, and M_BH is the mass of the BH), the ISCO is at six gravitational radii (R_g = GM_BH/c²) and, for a maximally rotating BH (a_* = 1), the ISCO approaches 1 R_g.

For Cyg X-1, most of the reflection studies have used X-ray spectra from times when the source was in the hard state. In this state, it is unclear whether the assumption about the inner disk radius being at the ISCO holds. For BH transients, studies allow for the possibility that the disk recedes when the source is in the faint hard state at an Eddington fraction (L/L_Edd) of ∼0.1–0.01% (Nowak et al. 2002; Tomsick et al. 2009; Cabanac et al. 2009), but there is evidence that the disk remains close to or at the ISCO during the bright part of the hard state (Miller et al. 2006; Reis et al. 2010). Historically, Cyg X-1 has been in the bright part of the hard state, making the ISCO assumption plausible. Using hard-state observations, Nowak et al. (2011) did not report a spin measurement but put an upper limit on disk recession. Other reflection-based measurements constrained the BH spin to be 0.6 ⩽ a_* ⩽ 0.99 (Miller et al. 2012), $a_{*} = 0.88^{+0.07}_{-0.11}$ (Duro et al. 2011), and $a_{*} = 0.97^{+0.014}_{-0.02}$ (Fabian et al. 2012a).

The reflection fits in the hard state provide evidence for high BH spin consistent with the limit on the BH spin from thermal modeling in the soft state. In this paper, we report on the details of reflection modeling in the soft state using observations with the Nuclear Spectroscopic Telescope Array (NuSTAR; Harrison et al. 2013) and Suzaku (Mitsuda et al. 2007). NuSTAR covers the 3–79 keV bandpass, which is ideal for reflection studies. Its detectors give unprecedented energy resolution in the hard X-ray band and provide high throughput without the photon pile-up that occurs for charge-coupled device (CCD) observations of bright sources. NuSTAR has already been used for reflection studies of the supermassive BH NGC 1365 (Risaliti et al. 2013), as well as the Galactic BH GRS 1915+105 (Miller et al. 2013). In this paper, we provide details of the observations, instrument capabilities, and the data reduction methods in Section 2. The results of the spectral fitting are reported in Section 3 and the results are discussed in Section 4. Finally, we present conclusions in Section 5.

We observed Cyg X-1 with NuSTAR and Suzaku on 2012 October 31 and November 1 (MJD 56,231 and 56,232). Figure 1 shows the soft X-ray light curve from the Monitor of All-sky X-ray Image (MAXI; Matsuoka et al. 2009), indicating how this observation fits into the ∼4 yr history of this source. At the time of the observation, the MAXI 2–4 keV count rate (normalized by effective area) was 1.83 ± 0.04 s⁻¹ cm⁻² and the 4–10 keV count rate was 0.55 ± 0.02 s⁻¹ cm⁻² (obtained from the MAXI Web site¹⁸), demonstrating that the source was in the soft state based on the MAXI count rate and hardness criteria determined by Grinberg et al. (2013).

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2.1. NuSTAR

2.2. Suzaku

Figure 2 shows the 3–79 keV NuSTAR and 0.5–9 keV XIS light curves. There is good overlap in the coverage between the two satellites; however, their Earth occultations are not exactly in phase and the Suzaku coverage extends somewhat beyond that of NuSTAR's. Flaring, which is typical of Cyg X-1 in this state, is more evident in NuSTAR's hard X-ray band than in the softer X-ray regime covered by XIS. Perhaps the most notable feature in the XIS light curves are brief drops in the count rate. It is possible that these are absorption dips due to material in the massive donor star's stellar wind. This is plausible because the observations occurred at a binary orbital phase of 0.85–0.97 (where 1.0 corresponds to superior conjunction when the donor star is between the observer and the BH) based on the ephemeris of Brocksopp et al. (1999). Absorption dips are typically seen in this range of orbital phase (Bałucińska-Church et al. 2000; Poutanen et al. 2008).

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In order to determine the level of spectral variation during the observations, we extracted the 3–10 keV and 10–79 keV NuSTAR count rates and produced a plot of hardness, which is the 10–79 keV count rate divided by the 3–10 keV count rate, versus time (Figure 2(c)). Even during the flares, we see little variability in the hardness. Given the relatively low level of spectral variability, we combined all of the data into a single spectrum.

Due to the high count rate for Cyg X-1, the XIS spectra show features that we suspect are related to photon pile up. An upturn in the spectra above ∼9 keV is observed and is readily explained by pile up. The spectrum below 1.2 keV shows features that appear to be absorption lines; however, we cannot rule out the possibility of some distortion due to instrumental effects and we defer a detailed study to a later paper. In addition, there are known calibration uncertainties in the 1.7–2.1 keV band related to the Si K edge. After these considerations, for XIS0 and XIS1, we used the 1.2–1.7 keV and 2.1–9 keV bands for spectral analysis and binned the data based on the instrumental energy resolution (see Nowak et al. 2011). For PIN and GSO, we used the 15–68 keV and 50–296 keV energy ranges, respectively. For NuSTAR, we used 3–79 keV energy range and binned the spectra for FPMA and FPMB separately, requiring that each bin have a signal-to-noise ratio of at least 30 (after background subtraction).

We used the XSPEC software package (Arnaud 1996) to fit the combined NuSTAR plus Suzaku spectrum with a model consisting of a multi-temperature "disk blackbody" thermal component (Mitsuda et al. 1984) plus a power law (model 1). These continuum components were subject to absorption with the tbabs model and we used Wilms et al. (2000) abundances and Verner et al. (1996) cross sections for this interstellar absorption. We included a multiplicative constant as a free parameter for each instrument to account for differences in overall normalization. Figure 3 shows the XIS and NuSTAR residuals for this fit in terms of the data-to-model ratio, revealing a strong reflection component with a broad iron Kα emission line and a reflection hump above ∼15 keV. Figure 3(b) illustrates the complexity of the iron line, which has an absorption line at 6.7 keV in addition to the broad line in emission.

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The fit can be significantly improved with the addition of a Gaussian emission line and a cutoff at high energies (model 2), using highecut, which provides an exponential cutoff with a folding energy of E_fold for energies greater than a cutoff energy, E_cut. If the Gaussian parameters are allowed to take any values, the line centroid is near 5.3 keV, which is well below the 6.4–7.1 keV iron regime, and the line is extremely broad (σ = 1.57 keV). In addition to the Gaussian parameter values being unphysical, this model does not give a formally acceptable fit with a reduced χ² ($\chi ^{2}_{\nu }$) of 2.00 for 1149 degrees of freedom (dof). The continuum parameters (e.g., a best-fit inner disk temperature of kT_in = 0.62 keV and a power-law photon index of Γ = 2.5) are consistent with the source being in the soft state. This model gives absorbed and unabsorbed 0.5–100 keV fluxes of 4.33 × 10⁻⁸ erg cm⁻² s⁻¹ and 6.09 × 10⁻⁸ erg cm⁻² s⁻¹, respectively. For a source distance of 1.86 kpc, this implies a luminosity of 2.5 × 10³⁷ erg s⁻¹, which, for a BH mass of 14.8$\,M_{\mathord \odot }$, gives an Eddington-scaled luminosity of 1.3%.

As shown in Figure 4(a), the largest residuals for model 2 are in the 6–8.5 keV part of the spectrum. In addition to the fact that we are still not modeling the 6.7 keV absorption line, which is due to the photoionized wind of the massive companion star, a Gaussian is too simple to fit the broad emission feature and the absorption edge that are present in the reflection component. Thus, we removed the Gaussian and added a simple ionized absorber and a reflection component (model 3).

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For absorption due to the wind, we constructed a grid of table models using XSTAR version 2.2.1bg (Kallman & Bautista 2001). Solar abundances were assumed for all elements, the number density was fixed at n = 10¹² cm⁻³, and the turbulent velocity of the gas was fixed at v_turb = 300 km s⁻¹ (e.g., Miller et al. 2005; Hanke et al. 2009). We used an input spectrum consistent with model 1 described above in order to construct a grid spanning 2 ⩽ log(ξ) ⩽ 5, where ξ is the ionization parameter in units of erg cm s⁻¹ and 1.0 × 10²¹ cm⁻² ⩽ N_H ⩽ 5.0 × 10²² cm⁻², where N_H is the column density of the absorber. In total, 400 grid points were calculated and summed into a multiplicative table model that was included in XSPEC analysis, with N_H, ξ and v/c as variable parameters. Although v/c was originally left as a free parameter, we found a 90% confidence upper limit of <0.0004 and we fixed it to zero in the fits described below. This parameter is driven by the strong absorption line at 6.7 keV, which is due to Fe xxv.

For the reflection, we used the reflionx model (Ross & Fabian 2005). This model includes the hard X-ray bump, the absorption edges, and the emission lines, so that the full reflection component is physically self-consistent. In addition, the emission lines are Compton broadened (see Section 1). The version that is available online²¹ has the folding energy for its exponential cutoff fixed at 300 keV, but, for our fits, a new model, reflionx_hc, was produced with E_fold as a free parameter. For the direct component, the highecut parameters were set to be consistent with reflionx_hc: E_cut was set to zero and E_fold was forced to have the same value as the free parameter in the reflection model. One other difference between reflionx and reflionx_hc is that the ionization parameter was extended to higher levels based on early fits to the Cyg X-1 spectrum. While this is a more realistic physical model than using the Gaussian to fit the iron line, model 3 provides a worse fit ($\chi ^{2}_{\nu }$ = 2.72 for 1148 dof) than model 2 and large residuals are still present in the 5–9 keV regime.

A major improvement in the fit (to $\chi ^{2}_{\nu }$ = 1.21 for 1143 dof) comes from convolving the reflection component with a relativistic blurring model (model 4). For blurring, we used the relconv model (Dauser et al. 2010), which is based on the physics described in Fabian et al. (1989) and Laor (1991), but relconv allows for a range of spin values. For these fits, we assume that the accretion disk extends to the ISCO and the blurring, which is most apparent in its effect on the iron line shape, depends on the BH spin (a_*), the disk inclination (i), and the radial dependence of the emissivity of reflected flux. The emissivity is assumed to have a power-law shape (L∝r^−q, where L is the luminosity illuminating the reflecting material, r is the radial distance from the BH, and q is the emissivity index) or broken power-law shape. The fit parameter values for the broken power-law emissivity (model 4) and for the power-law emissivity (model 5) are given in Table 2 and Figure 5(a) shows the components of the former model.

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Table 2. Fit Parameters for Models Assuming a Disk Blackbody

Parameter	Unit/Description	Model 4 Valuea	Model 5 Valuea	Model 7 Valuea
Interstellar absorption
NH	1021 cm−2	6.0 ± 0.3	6.2 ± 0.2	6.2 ± 0.2
Disk blackbody
kTin	keV	$0.558^{+0.004}_{-0.002}$	0.558 ± 0.003	$0.557^{+0.004}_{-0.002}$
NDBB	Normalization	20,800$^{+1200}_{-800}$	19,600$^{+800}_{-600}$	19,600$^{+1000}_{-900}$
Cutoff power law
Γ	Photon index	$2.589^{+0.005}_{-0.022}$	2.66 ± 0.02	2.672 ± 0.014
Npl	Normalizationb	6.0 ± 0.4	$6.8^{+0.4}_{-0.2}$	7.4 ± 0.3
Efold	keV	$120^{+20}_{-10}$	$190^{+20}_{-10}$	$200^{+50}_{-20}$
Simple ionized absorber
NH	1022 cm−2	$3.45^{+0.14}_{-0.23}$	$3.31^{+0.29}_{-0.10}$	2.86 ± 0.17
log ξ	erg cm s−1	$5.0^{+0.0}_{-0.2}$	$4.84^{+0.16}_{-0.02}$	$5.00^{+0.00}_{-0.03}$
Reflection component (reflionx_hc)
ξ	erg cm s−1	$18{,}100^{+1900}_{-4200}$	$20{,}000^{+0}_{-500}$	$20{,}000^{+0}_{-800}$
Fe/solar	Abundance	2.9 ± 0.4	1.9 ± 0.2	$1.93^{+0.13}_{-0.23}$
Nref	Normalization (× 10−6)	$5.955^{+0.003}_{-0.327}$	$6.6^{+0.5}_{-0.4}$	$6.0^{+0.5}_{-0.3}$
Relativistic blurring (relconvc)
qin	Emissivity index	$10.0^{+0.0}_{-0.4}$	$10.0^{+0.0}_{-0.5}$	3.0d
qout	Emissivity index	$-1.2^{+1.1}_{-4.6}$	10.0	3.0d
Rbreak	Index break radius (Rg)	$10^{+15}_{-3}$	⋅⋅⋅	⋅⋅⋅
a*	BH spin	0.9882 ± 0.0009	$0.91^{+0.01}_{-0.02}$	0.75 ± 0.05
i	Inclination (deg)	$69.2^{+0.5}_{-0.9}$	$59.3^{+0.5}_{-1.3}$	42.4 ± 0.5
Cross-normalization constants (relative to FPMA)
CXIS0	⋅⋅⋅	1.081 ± 0.005	1.082 ± 0.005	1.081 ± 0.005
CXIS1	⋅⋅⋅	1.038 ± 0.005	1.039 ± 0.005	1.038 ± 0.005
CFPMB	⋅⋅⋅	1.001 ± 0.001	1.001 ± 0.001	1.001 ± 0.001
CPIN	⋅⋅⋅	1.202 ± 0.006	1.204 ± 0.006	1.207 ± 0.007
CGSO	⋅⋅⋅	1.23 ± 0.06	1.17 ± 0.05	1.16 ± 0.05
χ2/ν	⋅⋅⋅	1388/1143	1501/1145	1610/1146

Notes. ^aWith 90% confidence errors. A value of zero for the positive error indicates that the parameter's error range reached the upper limit of values provided for the model. ^bIn units of ph s⁻¹ cm⁻² keV⁻¹ evaluated at 1 keV. ^cTwo other parameters in this model are the inner and outer radii from where the reflected emission is coming: R_in is set to be at the ISCO and R_out = 400 R_g. ^dFixed.

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The fit parameters indicate reflection off highly ionized material (ξ > 13,900 erg cm s⁻¹ for model 4 and ξ > 19,500 erg cm s⁻¹ for model 5) and a steep emissivity index (q > 9.5). In addition, we find a high BH spin of a_* = 0.9882  ±  0.0009 for model 4 and $a_{*} = 0.91^{+0.01}_{-0.02}$ for model 5. These are 90% confidence statistical errors and it is important to note that they do not include any systematic component. The inclinations obtained are $i = 69.2^{+0.5}_{-0.9}$ deg and $i = 59.3^{+0.5}_{-1.3}$ deg for models 4 and 5, respectively, both of which are significantly different from the value of 271 measured for the binary (Orosz et al. 2011). If we fix the inclination to the binary value and refit the spectrum, we obtain a very poor fit with $\chi ^{2}_{\nu }$ = 2.45 for 1144 dof even for the case of broken power-law emissivity; the residuals are shown in Figure 4(d, model 6). Furthermore, we made error contours²² for the spin and inclination parameters for model 4 (see Figure 6). Although there is some correlation between these parameters and a nearby local minimum exists, the 3σ contours do not extend below i ∼ 658.

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For all the models presented thus far, if q_in is left as a free parameter, we obtain values close to 10, which is the maximum of the allowed range. We also explored the implications of lower emissivity index by fixing it to q = 3. This gives $\chi ^{2}_{\nu }$ = 1.40 for 1146 dof, which is significantly worse than the high-q (and high-i) fit, but the inclination is $42.4^{+0.4}_{-0.5}$ deg, which is much closer to the binary value. The residuals for this model are shown in Figure 4(e, model 7) and the parameters are given in Table 2. Although the BH spin is somewhat lower for model 7, the relatively poor fit suggests that this value is not reliable. For model 4, we left q_out as a free parameter and a value of –$1.2^{+1.1}_{-4.6}$ is obtained, indicating that, beyond 10 R_g, the flux incident on the disk is actually increasing with radius. Although such a rising profile could occur over some range of radii, we note that it is non-physical for the emissivity to continue to increase with radius indefinitely.

While the parameter and BH spin constraints above rely only on modeling the reflection component, a previous Cyg X-1 spin measurement obtained by fitting the soft state spectrum relied primarily on modeling the thermal component (Gou et al. 2011, 2013). Rather than using the disk blackbody model, they used the model kerrbb, which is a multi-temperature thermal accretion disk model that accounts for changes in the inner disk (e.g., the inner radius) due to the BH spin. Also, instead of adding a power law, they used the convolution model simpl (Steiner et al. 2009), which is different from the disk blackbody plus power-law model described above because it uses the kerrbb component as the seed photon input to the Comptonization region. With this model, we obtain $\chi ^{2}_{\nu }$ = 1.32 for 1146 dof. The residuals are shown in Figure 4(f, model 8) and the model components are shown in Figure 5(b). The parameter values from the fit are given in Table 3. The constraint on the spin parameter, a_* = 0.838 ± 0.006, comes from both the thermal component and the reflection component and the inclination (i = 539 ± 04) is still significantly higher than the binary value. Model 8 uses a single power law for the emissivity with an index of q = 7.8 ± 0.5. Figure 6 shows the error contours for spin and inclination for model 8.

Table 3. Fit Parameters for Models with kerrbb

Parameter	Unit/Description	Model 8 Valuea	Model 9 Valuea
Interstellar absorption
NH	1021 cm−2	6.6 ± 0.2	6.5 ± 0.1
Thermal component (kerrbbb)
i	Inclination (deg)	53.9 ± 0.4	40.4 ± 0.5
a*	BH Spin	0.838 ± 0.006	0.973 ± 0.004
$\dot{M}$	Accretion rate (1018 g s−1)	$0.203^{+0.004}_{-0.006}$	$0.127^{+0.005}_{-0.007}$
Comptonization (simpl)
Γ	Photon index	2.66 ± 0.02	2.65 ± 0.02
fscat	Scattering fraction	0.104 ± 0.004	$0.10499^{+0.00344}_{-0.00001}$
Efold	keV	$190^{+40}_{-20}$	$180^{+30}_{-10}$
Simple ionized absorber
NH	1022 cm−2	$3.46^{+0.12}_{-0.20}$	$3.27^{+0.08}_{-0.41}$
log ξ	erg cm s−1	$5.00^{+0.00}_{-0.13}$	$5.0^{+0.0}_{-0.2}$
Reflection component (reflionx_hc)
ξ	erg cm s−1	$20{,}000^{+0}_{-1200}$	$19{,}200^{+800}_{-3700}$
Fe/solar	Abundance	$1.99^{+0.11}_{-0.22}$	$2.00^{+0.18}_{-0.14}$
Nref	Normalization (× 10−6)	$6.45^{+0.01}_{-0.23}$	$5.9^{+1.3}_{-0.3}$
Relativistic blurring (relconvc)
q	Emissivity index	7.8 ± 0.5	$2.48^{+0.09}_{-0.05}$
Gaussian blurring (gsmooth)
σ	keV	⋅⋅⋅	$0.28^{+0.02}_{-0.04}$
Cross-normalization constants (relative to FPMA)
CXIS0	⋅⋅⋅	1.081 ± 0.005	$1.081^{+0.002}_{-0.004}$
CXIS1	⋅⋅⋅	1.038 ± 0.004	$1.038^{+0.002}_{-0.004}$
CFPMB	⋅⋅⋅	1.001 ± 0.001	1.001 ± 0.001
CPIN	⋅⋅⋅	1.205 ± 0.007	1.205 ± 0.006
CGSO	⋅⋅⋅	1.17 ± 0.06	1.16 ± 0.06
χ2/ν	⋅⋅⋅	1512/1146	1510/1145

Notes. ^aWith 90% confidence errors. A value of zero for the positive error indicates that the parameter's error range reached the upper limit of values provided for the model. ^bFixed parameters and their values include η = 0.0, which corresponds to the zero torque inner boundary condition, M_BH = 14.8$\,M_{\mathord \odot }$, d = 1.86 kpc, and a spectral hardening factor of 1.7. ^cTwo other parameters in this model are the inner and outer radii from where the reflected emission is coming: R_in is set to be at the ISCO and R_out = 400 R_g. The inclination and spin parameters (i and a_*) are free, but they are forced to take the same values as for kerrbb.

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Although we do not focus on calibration details in this paper, there is excellent agreement between FPMA and FPMB with the relative normalization being consistent to within 0.1% for all the spectral models described above, which is actually better than expected. Relative to the NuSTAR/FPMA, we find normalization constants of 1.081 ± 0.005 for XIS0, 1.038 ± 0.004 for XIS1, 1.205 ± 0.007 for PIN, and 1.17 ± 0.06 for GSO. These numbers are for model 8, but Tables 2 and 3 show very similar relative normalizations for all models. Thus, there is very good agreement between NuSTAR and XIS and the fact that PIN and GSO are somewhat higher is expected.²³

The combination of NuSTAR and Suzaku provides a measurement of the Cyg X-1 reflection spectrum with unprecedented quality. While NuSTAR measures the entire reflection component (iron line, absorption edges, and hard X-ray bump), the XIS provides an extension to lower energies that is essential for constraining the thermal component. NuSTAR and XIS agree to a remarkable extent on the shape of the iron line (see Figure 3(b)) and NuSTAR provides a huge improvement in the statistical quality of the data, while alleviating some systematic concerns such as pile up.

We have presented fits to the spectrum with several different models and, while some parameters show significant differences, others agree about the properties of the system. It is clear that the source was in the soft state with a prominent thermal component and a power law with a photon index between Γ = 2.59 and 2.67, which meets the Γ > 2.5 criterion for Cyg X-1 to be in the soft state (Grinberg et al. 2013). There is clear evidence for absorption due to highly ionized material, which is consistent with the findings of Yamada et al. (2013). Also, the fits agree that the ionization state of the disk material that leads to the reflection component is high and that iron is overabundant by a factor of 1.9–2.9 relative to solar.

The BH spin and inclination measurements vary from model to model by more than the 90% confidence statistical errors, indicating that there is significant systematic uncertainty. For the inclination, it is also necessary to compare our values of i = 42°–69° with the value of i = 271 ± 08 that is obtained by modeling optical photometric and spectroscopic measurements (Orosz et al. 2011), but the optical measurement is of the inclination of the binary while the reflection component measures the inclination of the inner part of the disk. We have shown (see Figure 4(d)) that the reflection model simply cannot reproduce an inclination as low as 271.

However, we must also keep the limitations of the spectral model in mind. While the relconv calculation is for a specific inclination angle, reflionx_hc calculates the spectrum of the reflection component by averaging over angles. Another consideration is that the ionization parameter (ξ) is at the top of the available range for reflionx_hc. The model already includes Compton broadening of the lines, which increases with increasing ξ, but it is possible that some extra Compton broadening is necessary to account for a higher ionization. Also, surface turbulence may cause some symmetric line broadening that is not taken into account by reflionx_hc. To test this, we added a Gaussian convolution model (gsmooth), which acts on the reflection component along with relconv. The results are shown in Table 3, where this is listed as model 9, and the inclination decreases significantly from 539 to 404. The value obtained for the gsmooth σ parameter is $0.28^{+0.02}_{-0.04}$ keV. Using kT = (1/2)m_ec²(σ/E)², where m_e is the electron rest mass, this value of σ corresponds to a temperature of kT = 0.4–0.6 keV at E = 6–7 keV, which is in line with the inner disk temperatures we obtain from the disk blackbody fits.

While adding symmetric smoothing of the iron line and reflection component (i.e., extra Compton broadening) causes a drop in i, we emphasize that asymmetric relativistic broadening is required by the data. For our original Gaussian fit to the iron line, we obtained a best-fit centroid value of 5.3 keV, which shows that the line has the low-energy tail expected for a gravitational redshift. Also, we obtained a very poor fit with reflionx_hc (Figure 4(b)), where the Compton broadening was already included. Adding the relativistic broadening provided a very large improvement to the fit (Figure 4(c)).

Although conclusions about the BH spin depend on the different possibilities for the inclination, the models that provide good fits to the data (models 4, 5, 8, and 9) all have a_* > 0.83, indicating at least relatively high spin. The best fit (model 4) also has the highest spin a_* = 0.9882 ± 0.0009, but this either requires a very large warp in the accretion disk or that the binary inclination is somewhat higher than the best-fit value found in Orosz et al. (2011). We note that Table 1 in Orosz et al. (2011) reports that some of their models give significantly higher inclinations, but the χ² values for the higher inclination models are worse.

Another potentially interesting result that comes from this spectrum is the constraint on the emissivity profile. A comparison of models 4–6 indicate that a broken power-law emissivity is preferred as it is a very steep profile in the inner part of the disk (q_in ∼ 10). For active galactic nuclei (AGNs), relatively steep profiles (q = 4.3–5.0) were reported for MCG–6-30-15 (Wilms et al. 2001) and steep profiles are discussed in Wilkins & Fabian (2012). Walton et al. (2013) studied a large sample of AGNs and found that steep profiles are common. This has been taken as evidence that the irradiating source comes from very close to the BH and Fabian et al. (2012b) conclude that it must lie within 1 R_g of the BH event horizon. While this may also be the case for Cyg X-1, our fits with very steep profiles (models 4, 5, and 8) also have inclinations between 539 and 692. Our fit with gsmooth (model 9) included in the model gave a much flatter index of $q = 2.48^{+0.09}_{-0.05}$, leaving open the possibility that the profile is relatively flat, in which case the source is at a height of 5–10 R_g or more.

While we cannot conclude anything definitive about the slope of the emissivity profile, if it is very steep, this might point to a "lamppost" geometry (Dauser et al. 2010), where the emission actually comes from the base of a collimated jet. This geometry may not be relevant for the soft state because there is no evidence for a jet. Despite this, if we start with model 8 but replace relconv with relconv_lp (model 10), we find i = 415 ± 05 and a_* = 0.953 ± 0.006 (see Figure 6 for the error contours), with only small changes in the other parameters. However, the quality of the fit is somewhat worse ($\chi ^{2}_{\nu } = 1.44$) for model 10 compared with the models reported in Tables 2 and 3.

After exploring different continuum models, emissivity geometries, and conditions for the material in the accretion disk, we only find inner disk inclinations that are >13° higher than the binary value measured by Orosz et al. (2011) and, as discussed above, one explanation is that there is a warp in the accretion disk. Analytical calculations as well as numerical simulations have shown that disk warps can occur (Bardeen & Petterson 1975; Schandl & Meyer 1994; Fragile et al. 2007) and that they should occur if the BH spin is misaligned with the orbital plane (and outer disk). As the alignment time for an accreting BH can be longer than the lifetime of a high-mass system (Maccarone 2002), if the Cyg X-1 BH formed with a misaligned spin, it would remain misaligned. If jets are aligned with the BH spin, then there is evidence for misalignment in systems like Cyg X-3, V4641 Sgr, and GRO J1655–40 (Maccarone 2002). It should be noted that Fragile (2009) has shown that, under certain assumptions about the thickness of the accretion disk, BH spin measurements using the inner radius of a warped disk can be incorrect. While a disk warp may not be the only possibility for Cyg X-1, further investigations to determine if the disk is really warped have important implications for the BH spin measurement.

We have presented a detailed study of the ∼1–300 keV spectrum of Cyg X-1 in the soft state. The spectrum is complex and consists of multi-temperature blackbody, power law, and reflection components along with absorption from highly ionized material in the system. Although the observation was of moderate duration (∼29 ks), NuSTAR provides a very high-quality and high-statistics measurement of the reflection spectrum, including an iron complex with broad emission and narrow absorption lines. We find that the reflecting material has a high ionization state, is overabundant in iron relative to solar, and requires broadening of the iron line that is well described by a relativistic blurring model.

While all models that provide a good fit to the spectrum indicate a rapidly rotating BH with a_* > 0.83 and our best-fitting model has a_* = 0.9882 ± 0.0009 (90% confidence statistical errors only), we were not able to obtain a good fit with the inclination fixed to the Orosz et al. (2011) binary value. This may indicate a misalignment between the orbital plane and the inner accretion disk (by >13°), missing physics in the spectral models, or it may possibly motivate work to confirm the measurement of the binary inclination. Regardless of which of these possibilities is correct, it is clear that the combination of NuSTAR's high throughput and energy resolution provides a major advance in reflection studies, allowing for strict tests of the models, which we expect to lead to improved constraints on the physical processes at work in Cyg X-1 and other accreting BH systems.

This work was supported under NASA Contract No. NNG08FD60C and 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). J.A.T. acknowledges partial support from NASA Astrophysics Data Analysis Program grant NNX13AE98G. L.N. wishes to acknowledge the Italian Space Agency (ASI) for financial support by ASI/INAF grant I/037/12/0-011/13. J.A.T. thanks L. Brenneman, G. Matt, and D. Ballantyne for useful discussions about reflection modeling. This work made use of IDL software written by N. Barrière for rebinning the NuSTAR spectra. This research has made use of the MAXI data provided by RIKEN, JAXA, and the MAXI team.