THE EXTREME SPIN OF THE BLACK HOLE IN CYGNUS X-1

Our adopted model that is featured below, and upon which all of our results are based, was constructed by working in detail through a progression of seven preliminary models. We now briefly comment on these models, which are presented in full in Appendices A and B.

Nonrelativistic models. The central component of our three nonrelativistic models, Models NR1–NR3, is the familiar accretion-disk model component diskbb¹⁸ (Mitsuda et al. 1984; Makishima et al. 1986), which does not include any relativistic effects or the effects of spectral hardening, and which has an inappropriate boundary condition at the disk's inner edge (Zimmerman et al. 2005). We nevertheless employed diskbb as an exploratory tool because it has been widely used for decades, and it therefore allows us to compare our reduction/analysis results (for a spectrum of interest) to published results. Furthermore, this familiar model returns a direct and useful estimate of the temperature at the inner edge of the disk, which we use in order to make comparisons between Cygnus X-1 and other black hole binaries. The details of this nonrelativistic analysis and the satisfying results obtained for Model NR3—namely, consistent values of the inner-disk radius and temperature for our three spectra—are presented in Appendix A.

Relativistic models. Similarly, in addition to our adopted relativistic model (Section 3.2), in Appendix B we present our analysis and results for four relativistic models, Models R1–R4, that are built around our fully relativistic accretion-disk model component kerrbb2, which we describe below. This component is a direct replacement for diskbb, returning two fit parameters, namely, the spin and the mass accretion rate (instead of the temperature and radius of the inner disk). The four models progress sequentially in the sense that Model R1 is the most primitive and Model R4 is the most advanced. This sequence builds toward our adopted model. We have chosen to present our results for these preliminary relativistic models, in addition to those for our adopted model, because doing so demonstrates that our modeling of the critical thermal component, and the extreme spin it delivers for Cygnus X-1, are insensitive to the details of the analysis.

The model we employ is a culmination of Models R1–R4 in the sense that it is the most advanced and physically realistic model. The schematic sketch of the X-ray source in Figure 2 illustrates the various model components and their interplay. The structure of our adopted model, naming all the components that comprise it, is expressed as follows:

$$\begin{eqnarray*} &&{\rm CRABCOR*CONST*TBABS [SIMPLR \otimes KERRBB{2}}\\ &&{\rm +KERRDISK+ KERRCONV\otimes (IREFLECT \otimes SIMPLC)]}. \end{eqnarray*}$$

As described in detail below, simplr generates the power-law component using the seed photons supplied by the single thermal component kerrbb2, while the reflection component is likewise generated in turn by ireflect acting solely on the power-law component (i.e., ireflect does not act on the thermal component). Furthermore, the model fits for a single value of a_*, which appears as the key fit parameter in three model components: kerrbb2, kerrdisk, and kerrconv.

We now discuss in turn the model's three principal components—thermal, power law, and reflected—and their interrelationships.

Thermal component. the centerpiece of our adopted model is our accretion-disk model kerrbb2, which includes all relativistic effects, self-irradiation of the disk ("returning radiation"), and limb darkening (Li et al. 2005). The effects of spectral hardening are incorporated into the basic model kerrbb via a pair of look-up tables for the hardening factor f corresponding to two representative values of the viscosity parameter: α = 0.01 and 0.1 (McClintock et al. 2006). Motivated by observational data obtained for dwarf novae (Smak 1998, 1999) and soft X-ray transients (Dubus et al. 2001), and the results of global general relativistic magnetohydrodynamic (GRMHD) simulations (Penna et al. 2010), throughout this work we adopt α = 0.1 as our fiducial value; meanwhile, in Section 5.4 we examine the effects on our results of using α = 0.01 in place of α = 0.1. The entries in the look-up tables for f were computed using both kerrbb and a second relativistic disk model bhspec (Davis et al. 2005; Davis & Hubeny 2006). We refer to the model kerrbb plus this table, and the subroutine that reads it, as kerrbb2 (McClintock et al. 2006). As noted above, the model kerrbb2 has just two fit parameters, namely, the black hole spin parameter a_* and the mass accretion rate $\dot{M}$ (or equivalently, a_* and the Eddington-scaled bolometric luminosity, $l \equiv L_{\rm bol}(a_*,\dot{M})/L_{\rm Edd}$). For the calculations reported in this paper, we included the effects of limb darkening and returning radiation. We set the torque at the inner boundary of the accretion disk to zero (as appropriate when D, M, and i are held fixed), allowed the mass accretion rate to vary freely, and fitted directly for the spin parameter a_*.

Power-law component. The first term in the sum, simplr⊗kerrbb2, models the power-law component and the observed thermal component in combination. This dominant part of the spectrum (see Figure 3) is computed by convolving kerrbb2, which describes the seed photon distribution (i.e., the thermal component prior to being scattered), with simplr. The convolution model simplr is a slightly modified version of simpl (Steiner et al. 2009b) that is appropriate when including a separate and additive reflection component (Steiner et al. 2011). The parameters of simplr (and simpl) are the power-law photon index Γ and the scattered fraction, f_SC, which is the fraction of the seed photons that are scattered into the power-law tail. As used here, simplr describes a corona that sends approximately half the scattered seed photons outward toward the observer and the remainder downward toward the disk, thereby generating the reflected component (see below). Thus, we assume that the power-law component illuminating the disk is the same as the component we observe.

Reflected component. The second and third terms in the sum model the reprocessed emission from the disk that results from its illumination by the power-law component. The model for the illuminating power-law component itself (the term on the far right) is simplc, which is equivalent to simplr⊗kerrbb2 minus the unscattered thermal component (Steiner et al. 2011). This power-law component is acted on by ireflect, which is a convolution reflection model with the same properties as its widely used parent, the additive reflection model pexriv (Magdziarz & Zdziarski 1995). Concerning ireflect, we (1) free the reflection scaling factor¹⁹s while restricting its range to negative values, thereby computing only the reflected spectrum, while assuming that the corona is a thin slab that hugs the disk and emits isotropically; (2) set the elemental-abundance switch to unity, which corresponds to solar abundances, while allowing the iron abundance to be free; (3) link the photon index to the value returned by simplr; and (4) fix the disk temperature at 6.0 × 10⁶ K, the temperature T_in returned by diskbb (see Appendix A). The model ireflect⊗simplc returns a reflected spectrum that contains sharp absorption features and no emission lines. To complete the model of the reflected component, we follow Brenneman & Reynolds (2006) and employ the line model kerrdisk and the convolution smearing model kerrconv, both of which treat a_* as a free fit parameter.²⁰ These models allow the emissivity indices to differ in the inner and outer regions of the disk. For simplicity, and because this parameter is unknown with values that vary widely from application to application, we use an unbroken emissivity profile with a single index q. We tie together all the common parameters of kerrdisk and kerrconv, including the two principal parameters, namely, a_* and q.

The three multiplicative model components are, respectively, (1) crabcor, which corrects for detector effects (see Section 2); (2) const, which reconciles the calibration differences between the detectors (throughout the paper, we fix the normalization of the RXTE/PCU2 detector and float the normalization of the ASCA GIS and Chandra HETG/ACIS detectors); and (3) tbabs,²¹ which models low-energy absorption (Wilms et al. 2000). Concerning tbabs, throughout the paper we allow N_H to vary because the column density is well determined by the data and N_H is known to vary by several percent for all three well-studied supergiant black hole binaries, namely, Cygnus X-1 (Hanke et al. 2009; also see Section 4), M33 X-7 (Liu et al. 2008), and LMC X-1 (Hanke et al. 2010).

For all three spectra, we refitted the data excluding the energy range 5.0–10.0 keV while omitting the component kerrdisk. This excised energy range contains the relativistically broadened Fe Kα line and edge, as well as a significant residual feature near 9 keV (Figure 3). For all three spectra, we find that our results are essentially unchanged, apart from small shifts in the parameters of the reflection component. The results for spectrum SP1 (only) are shown in Table 3 (Case 2) where they are compared to our standard results (Case 1). Thus, we find that the values of the spin parameter returned by the fits are completely determined by the temperature and luminosity of the thermal component and are unaffected by the presence of the line. This conclusion is reasonable given that the line is a minor feature with an equivalent width of ≈0.15 keV or only ≈0.015 keV relative to the continuum at the peak of the thermal component (see Figure 3).

Table 3. Effects of Excluding the Fe Kα Line and Increasing Bandwidth (SP1 Only)^a

Number	Model	Parameter	Case 1	Case 2	Case 3
1	kerrbb2	a*	0.9985+0.0005−0.0008	0.9983+0.0005−0.0037	0.9984+0.0001−0.0003
2	kerrbb2	$\dot{M}$	0.115 ± 0.004	0.115 ± 0.007	0.115 ± 0.003
3	const	⋅⋅⋅	1.000 ± 0.002	0.999 ± 0.002	1.000 ± 0.002
4	tbabs	NH	0.705 ± 0.006	0.708 ± 0.009	0.706 ± 0.005
5	simplr	Γ	2.282 ± 0.010	2.288 ± 0.013	2.287 ± 0.008
6	simplr	fSC	0.225 ± 0.006	0.221 ± 0.010	0.221 ± 0.005
7	kerrdisk	EL	6.56 ± 0.09	0.00 ± 0.00	6.56 ± 0.05
8	kerrdisk	q	2.82 ± 0.02	2.82 ± 0.23	2.81 ± 0.02
9	kerrdisk	NL	0.015 ± 0.001	0.000 ± 0.000	0.015 ± 0.001
10	kerrdisk	EW	0.154	⋅⋅⋅	0.153
11	ireflect	XFe	5.34 ± 0.15	3.93 ± 0.52	4.72 ± 0.12
12	ireflect	s	0.98 ± 0.04	1.00 ± 0.10	0.94 ± 0.03
13	ireflect	ξ	153.1 ± 15.7	189.6 ± 23.7	153.3 ± 11.6
14		χ2ν	1.24(561/454)	0.82(297/361)	1.25(616/493)

Notes. ^aThe parameter set is exactly the same as for Table 2. Case 1: standard 0.7–45 keV fit results for our adopted model, which are copied from column SP1 in Table 2. Case 2: fit to SPI over the range 0.7–45 keV excluding 5.0–10.0 keV (i.e., excluding the Fe Kα line and Fe absorption edge). The value of q is that returned by kerrconv. Case 3: fit to SP1 including HEXTE data that covers the full energy range from 0.7 to 150 keV.

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In Section 2, we asserted that the coverage of the PCA to 45 keV was sufficient to adequately constrain the power-law and reflection components. We now demonstrate that this is true by refitting the data for SP1 (only) while including the HEXTE data spanning the energy range 20 keV to 150 keV (where the source counts are negligible). For the HEXTE data, we do not add a systematic error to the count rates. However, we do correct the detector response to the Crab spectrum of Toor & Seward (see Section 2) using the value of the Crab's photon index as measured using HEXTE:²² Γ = 1.93 ± 0.003. The slope correction is ΔΓ_TS = −0.17. The results obtained using the HEXTE data, which are given in Table 3 (Case 3), are almost identical to our standard results for SP1 (Case 1). This is not surprising because the PCA coverage to 45 keV is more than adequate to determine the slope of the power-law component, and the reflection component is quite weak and dying rapidly at 45 keV (Figure 3).

We replace ireflect⊗simplc+kerrdisk with reflionx²³ (Ross & Fabian 2005), which is widely used in measuring spin via the Fe Kα line. The merit of this alternative reflection model is that it self-consistently calculates the line feature and the reflection component, whereas ireflect models only the absorption edges. The downside of reflionx is its description of the seed photon distribution as a simple power law, which unphysically diverges at low energies (for further comparison of the two models see Steiner et al. 2011). We include relativistic blurring by convolving reflionx with kerrconv. Compared to the results given in Table 2, the values of the parameters returned by the fits are generally different, although reasonable, and the fits are somewhat poorer (χ²_ν = 1.35–1.45). Importantly, the effects on the spin parameter are very small: the value of a_* increases slightly for SP1 (0.9985–0.9999) and is unaffected for SP2 and SP3, which are at their maximum values.

We refitted the spectra using α = 0.01 in place of our fiducial value, α = 0.1. Again, the spin parameter for SP1 increases slightly (0.9985–0.9988), while the values for SP2 and SP3 remain unchanged. Concerning possible metallicity effects, we do not have nonsolar-metallicity table models for computing spectral hardening for such extreme values of spin. However, for three sources (M33 X-7, Liu et al. 2008; LMC X-1, Gou et al. 2009; and A0620–00, Gou et al. 2010), we have found that substantial changes in metallicity produce very small changes in a_*. For example, reducing the metallicity from solar to a tenth solar decreases the spin parameter of a high-spin black hole like LMC X-1 (a_* = 0.938 ± 0.020) by only Δa_* = 0.001. The effect of this same change in metallicity for the slowly-spinning black hole A0620–00 (a_* = 0.135 ± 0.029) is larger, but it is still quite small: Δa_* = 0.014. (The small errors on a_* quoted here come directly from fitting the X-ray spectra, and they do not take into account the uncertainties in D, M, and i, which dominate the error budget.) We note that the super-solar iron abundances implied by our fits using the reflection models, ireflect (Table 2) and pexriv (Appendix B), suggest that the metallicity of the Cygnus X-1 system is possibly enhanced. Accounting for the supersolar abundances will result in a slightly increased estimate of a_*, so our conclusions are robust to enhanced metallicity.

We now describe three technical exercises that artificially examine the effects of varying three parameters of kerrbb2, namely, a_*, D, and the normalization constant N_K (which we have elsewhere fixed to unity, as is appropriate when D, M, and i are specified). Our motivation is to examine the consequences of relaxing the spin parameter away from the extreme values returned by the fits (Table 2).

First, for spectrum SP1 only, we leave N_K fixed at unity and allow the distance to vary (keeping M and i fixed at their fiducial values). For our parallax distance of D = 1.86 kpc, we of course have our standard result, a_* = 0.9985 (Table 2). We now arbitrarily and successively fix the spin parameter at two lower values, a_* = 0.95 and a_* = 0.90, and refit spectrum SP1. The best-fit values of D are then substantially greater than our measured value of 1.86 kpc: 2.43 ± 0.03 kpc and 2.78 ± 0.02 kpc, respectively. Meanwhile, the corresponding values of reduced chi-square are respectively 1.34 and 1.40, which are significantly greater than our standard value of 1.24 (Table 2). Thus, the best fit is achieved for the observed and extreme value of spin.

Secondly and alternatively, we obtain similar results by varying N_K while leaving the distance fixed at its fiducial value of 1.86 kpc. For the same pair of forced values of the spin parameter given above (0.95 and 0.90), we find for spectrum SP1 that the fitted values of N_K are about half the standard value of unity: 0.65 ± 0.01 and 0.45 ± 0.01, respectively (where a smaller value corresponds to a weaker thermal component). Meanwhile, we find values of reduced chi-square that are very similar to those given in the previous example, with values that increase as the spin parameter decreases. That is, we again find that the best fit is achieved for the observed and extreme value of spin.

In a final experiment, we assess the consistency of our spin values for the three spectra, which cannot be adequately judged by considering the extreme values given in Table 2. We make this assessment by artificially setting N_K = 0.5 in our adopted model and refitting the three spectra. In this way, we find best-fit spin values that are in reasonable agreement: a_* = 0.937^+0.001_−0.001 for SP1, a_* = 0.934^+0.021_−0.020 for SP2, and a_* = 0.953^+0.007_−0.006 for SP3.

Based on an analysis of the Fe Kα line profile, Miller et al. (2005) found marginal evidence for high spin using the same ASCA spectrum of Cygnus X-1 that we use. Subsequently, however, Miller et al. (2009) reported a spin of a_* = 0.05 ± 0.01 based on an analysis that combines Fe Kα/reflection models and continuum models, including kerrbb, a relativistic disk model similar to the one we use. That is, the authors jointly applied the continuum-fitting and Fe Kα methods and fitted simultaneously for a single value of a_*. The data analyzed in this case were a pair of 0.5 ks XMM-Newton/EPIC-pn spectra obtained in the "burst" timing mode.

While it is not possible for us to account in detail for the gross difference between the near-zero spin reported by Miller et al. (2009) and our near-extreme value, we note the following. First, the round values of D, M, and i that Miller et al. used as input to kerrbb differ significantly from ours, and these differences all serve to drive the spin-down, e.g., using their values of these parameters and our spectrum SP1, we find a_* = 0.74 ± 0.01. Second, and in addition, Miller et al. fitted for the kerrbb normalization constant, obtaining N_K = 0.31, whereas the standard procedure, which we follow, is to set N_K to unity when D, M, and i are fixed. Fitting SP1 using N_K = 0.31 (and the values of D, M, and i adopted by Miller et al.), we find that the spin drops from a_* = 0.78 to the retrograde value a_* = −0.53 (and the fit is poor, χ²_ν = 2.01.)

The near-zero spin result of Miller et al. (2009) is further called into question because no data above 10 keV were used. This lack of high energy coverage, in the presence of a strong Compton component, seriously compromises results obtained using the continuum-fitting method, as we stress at the outset of Section 2 (and as can be deduced by an examination of Figure 3). Likewise, results obtained using the Fe Kα/reflection method are compromised by the failure to observe the Compton reflection hump around 20–30 keV (e.g., see Larsson et al. 2008).

Very recently, after posting our paper on the astro-ph archive and while revising it for resubmission, a paper appeared reporting another estimate of the spin of Cygnus X-1 via an analysis of the Fe Kα profile (Duro et al. 2011). Assuming that the emissivity profile of the disk can be described by a single power law with index 3.0, these authors conclude (p. 1) that "the black hole is close to rotating maximally," which is in agreement with our result. We note that Duro et al. do not discuss Miller et al. (2005, 2009) or mention the near-zero spin result reported in the latter paper.

Based on an analysis of low-frequency (0.01–25 Hz) QPOs, Axelsson et al. (2005) obtained a spin for Cygnus X-1 of a_* = 0.49 ± 0.01 for M = 8 M_☉. Their result is based on the relativistic precession model of Stella et al. (1999). In this model, the QPO is produced as a result of emission from an orbiting bright spot that is undergoing relativistic nodal and periastron precessions in a slightly tilted and eccentric orbit. The spin parameter is predicted to vary with mass as a_*∝M^−1/5 (see Equation (4) in Axelsson et al. 2005), and therefore the corrected value of spin is a_* = 0.43 for our adopted black hole mass M = 14.8 M_☉. The large discrepancy between this moderate value of spin and the extreme value we find may be a consequence of a fundamental assumption of their model, namely, that the black hole is rotating slowly (a_* ≪ 1). On the other hand, the precession model, with its assumption of geodesic motion, may not apply in this instance.

In Orosz et al. (2011), results are presented for four dynamical models, Models $\cal A\hbox{--}D$. We disregard Models $\cal A$ and $\cal B$, which assume a circular orbit, because these models give poor fits compared to the eccentric orbit models (Δχ² > 50) and because there is clear evidence that the orbit is eccentric. Throughout this paper, we have used M = 14.8 ± 1.0 M_☉ and i = 271 ± 08 from Model $\cal D$, an asynchronous model with a rotational frequency for the O-star that is 40% greater than the orbital frequency. As an alternative to Model $\cal D$, we now consider Model $\cal C$, which assumes synchronous rotation. Model $\cal C$ gives a poorer fit to the data (Δχ² ≈ 13), and it results in disharmony between the light curve and velocity data on the one hand, and the radius and rotational velocity of the O-star on the other (see Table 1 in Orosz et al. 2011). Because of this disharmony, the uncertainties in M and i for Model $\cal C$ are significantly larger than those for our favored model, although the central values of these parameters differ only modestly: M = 15.8 ± 1.8 M_☉ and i = 285 ± 22. Using these values for Model $\cal C$ and spectrum SP1 (which gives the lowest value of spin), we repeated our error analysis (Section 6) and obtained the following 3σ lower limit on the spin parameter: a_* > 0.92.

There remains one uncertainty that calls into question the reliability of the continuum-fitting method, namely, whether the inner X-ray emitting portion of the disk (which will align with the black hole's spin axis) is aligned with the binary orbital plane. For a discussion of this question, see Section 2.2 in Li et al. (2009). As an extension of this discussion, we note the following. First, recent population synthesis studies predict that the majority of systems will have rather small (≲ 10°) misalignment angles (Fragos et al. 2010). Second, in the case of Cygnus X-1, there is reason to believe that the misalignment angle is especially small because of the binary system's low peculiar velocity, which indicates that the system did not experience a large "kick" when the black hole formed (Mirabel & Rodrigues 2003; Reid et al. 2011). As demonstrated in Figure 5, even if there exists a misalignment angle as large as, e.g., 16°, the spin value is still >0.95.

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One might expect that the accretion disk of a fast-spinning black hole like Cygnus X-1 would be hot (T_in > 1 keV) because its inner edge is relatively close to the event horizon and deep in the gravitational potential well. Two principal factors are responsible for the depressed disk temperature (T_in ≈ 0.5 keV; Table 7). The first of these is the low rate of mass accretion through the disk, which is manifested by its low luminosity, only ≈2% of Eddington (Table 2). The second is that relativistic effects (e.g., beaming and light bending) are muted for low disk inclinations. Using our fiducial values of D, M, and i, and fixed values of the fit parameters taken from Table 2, we used kerrbb2 to simulate a pair of spectra, one for the inclination of Cygnus X-1, i = 271, and the other for i = 660 (which is the inclination of GRS 1915+105; McClintock et al. 2006). The luminosities of the two spectra are the same, L/L_Edd ≈ 0.018, while the disk temperatures are T_in = 0.45 keV and 0.86 keV, respectively. Thus, in this comparison, the temperature of the low-inclination disk is depressed by nearly a factor of two because the relativistic effects are weak.

Although careful modeling of warm absorbers (WAs) is usually crucial in determining the spins of supermassive black holes via the Fe Kα method (e.g., Brenneman & Reynolds 2006), we find that the effects of WAs are unimportant in estimating the spin of Cygnus X-1 via the continuum-fitting method. We examined all of the available soft-state Chandra HETG spectra of Cygnus X-1 (ObsIDs 2741, 2742, 2743, 12313/SP2, and 12314/SP3) at E < 1 keV, and we find no evidence for the blend of absorption lines near 0.76 keV (due, e.g., to Fe X–Fe XV, O VII, and O VIII), which is a telltale signature of a WA. (The line WAs produce above 1 keV are mostly discrete and weaker and therefore have an accordingly much smaller effect on the continuum shape and our spin estimates.)

Our preliminary analysis of the low-energy portion (E < 0.8 keV) of a 50 ks hard-state Chandra HETG spectrum (Hanke et al. 2009) did reveal the presence of a single WA. Its two most relevant parameters are its column density, N_H = 7.2 × 10²⁰ cm⁻², and ionization parameter, ξ = 44. (Its turbulent broadening and redshift are, respectively, v = 47.6 km s⁻¹ and z = −0.00086; we assume solar abundances.) This tenuous WA does not affect the continuum shape, but it does produce line blends, the strongest of which is centered at 0.76 keV. Although such a WA is not present in soft-state spectra, we nevertheless tested its effects by reanalyzing SP1, SP2, and SP3 by introducing the additional model component WARMABS, fixing its parameters to the values given above; we obtained values of spin that are essentially identical to those given in Table 2. In one further test, we likewise reanalyzed our three spectra including a second thicker and hotter WA that generates a complex of absorption lines between 0.9 and 1.0 keV (N_H = 1.0 × 10²¹ cm⁻²; ξ = 300; v = 100 km s⁻¹; z = −0.00086). Again, the effects on our spin estimates are negligible.

The spin of Cygnus X-1 is extraordinarily high, placing it in the company of the microquasar GRS 1915+105, estimated using the continuum-fitting method (McClintock et al. 2006; Blum et al. 2009), and the supermassive black hole in MCG-6-30-15, estimated using the Fe-line method (Brenneman & Reynolds 2006). The spins of both of these black holes are reported to exceed a_* = 0.98. As Penrose (1969) first demonstrated, the enormous "flywheel" energy of a fast-spinning black hole can in principle be tapped. For Cygnus X-1, the potentially tappable energy is >2.8 M_☉ c² = 5.0 × 10⁵⁴ erg (Christodoulou & Ruffini 1971); in comparison, the energy radiated by the Sun over its entire ∼10 billion year lifetime is ≲ 0.001 M_☉c². It has been widely suggested that spin-down energy powers the relativistic jets observed for at least some quasars and microquasars (Blandford & Znajek 1977), such as GRS 1915+105 (Mirabel & Rodríguez 1994).

As the spin a_* of a black hole approaches unity, the radius of its ISCO approaches the radius of the event horizon R_EH (Bardeen et al. 1972). For Cygnus X-1, with spin a_* > 0.95 and M = 14.8 M_☉, R_ISCO < 42 km while R_EH < 29 km. The Keplerian velocity of the gas at the ISCO is approximately half the speed of light, and its orbital frequency is >598 Hz. Meanwhile, the frequency of rotation of the black hole itself, which is the spin frequency of space time at its horizon, is >790 Hz (the maximal spin frequency for a_* = 1 is 1091 Hz).

What is the origin of the spin of Cygnus X-1? Was the black hole born with its present spin or was it torqued up gradually by the gas it has accreted over its lifetime? To achieve a spin of a_* > 0.95 via disk accretion, an initially nonspinning black hole must accrete >7.3 M_☉ from its donor (Bardeen 1970; King & Kolb 1999) in becoming the M = 14.8 M_☉ that we observe today. However, to transfer this much mass even at the maximum (Eddington-limited) accretion rate requires >31 million years,²⁵ whereas the age of the binary system is between 4.8 and 7.6 million years²⁶ (Wong et al. 2011). (Even for a_* > 0.92 obtained for our less probable model, the accretion timescale is >25 million years.) Thus, it appears that the spin of Cygnus X-1 must be chiefly natal (also, see Axelsson et al. 2011; Wong et al. 2011), although possibly the high spin could be achieved during a short-lived evolutionary phase of hypercritical mass accretion (Moreno Méndez 2011).

With diskbb as the principal component, we analyzed our three spectra in turn using three composite models of increasing sophistication, the Models NR1–NR3 listed in Table 4. In addition to diskbb, each model includes (1) a Compton power-law component, either compbb (Nishimura et al. 1986) or simplr (Steiner et al. 2009b, 2011) convolved with diskbb; and (2) a reflection component pexriv (Magdziarz & Zdziarski 1995),²⁷ which models the absorption edges, plus an Fe Kα emission-line feature gaussian or laor (Laor 1991). As in the case of our adopted model, also included are the three pre-factors crabcor, const, and tbabs (see Section 3.2). In the case of Models NR2 and NR3, relativistic blurring effects are included using kdblur (Laor 1991).

Model NR1: Cui et al. (1998) analyzed spectrum SP1 using this model, and we do likewise. For the thermal component, our values of the parameters (disk temperature T_in and inner disk radius R_in) are quite similar to those found by Cui et al.: respectively, T_in = (0.449 ± 0.005) keV (our Table 5) and T_in = (0.436 ± 0.004) keV (Table 2 in Cui et al. 1998); and R_in = (2.54 ± 0.06)r_g versus (2.68 ± 0.06)r_g (for D = 1.86 kpc, M = 14.8 M_☉, and i = 271; r_g ≡ GM/c² = 21.9 km). For most of the other fit parameters, however, the differences between our results (Table 5) and theirs are quite significant: for example, for the width of the Fe Kα line we find σ = (1.22 ± 0.04) keV versus σ = (0.35 ± 0.07) keV, and for the ionization parameter ξ ≈ 0 versus ξ > 11911. We attribute these differences to our use of response files that have been significantly updated and improved (Section 2). These new response files not only allowed us to extend the upper energy bound to 45 keV (compared to the 30 keV bound used by Cui et al.), but also greatly improved the fit: χ²_ν = 0.88 versus χ²_ν = 1.47. Furthermore, this comparison is understated because Cui et al. included 1% systematic error in the PCU count rates whereas we used 0.5%; for a systematic error of 1%, our χ²_ν = 0.68.

Applying Model NR1 to SP2 and SP3 also gives very good fits (Table 5, Columns 5 and 6). However, comparing the results for all three spectra one finds a wide variation in the values of the disk temperature T_in (0.27–0.54 keV) and inner disk radius R_in (1.95–8.30 r_g). (We focus here on the parameters for the thermal component because this component ultimately delivers our key result, the spin of the black hole.) We find the performance of this model unsatisfactory for two principal reasons. First, it returns unlikely and near-zero values of the ionization parameter ξ (which we have here fixed to zero); this is probably because both the reflection component and line feature are improperly modeled (see below). Second, this model implies an unrealistically strong line feature (EW = 0.4–0.5 keV). Therefore, we now consider an improved model.

Model NR2. For the line feature, we replace the symmetric Gaussian profile with a skewed, relativistic profile via the laor model, while relativistically blurring the three other additive model components using kdblur (Table 4). The laor model and its companion convolution model kdblur have the serious limitation of assuming a fixed and extreme value of the spin parameter, a_* = 0.998, and they therefore do not give a proper description of black holes with moderate or low spins. Nevertheless, we consider these models adequate for the present purposes because Cygnus X-1 is a rapidly spinning black hole and because the Fe Kα line is only a cosmetic feature relative to the thermal continuum (Section 4). The merit of using laor/kdblur here is that the model is simple and the values of the line strength and ionization parameter returned by the fits are reasonable (Table 6). In the end, however, as in the case of the previous model, we find Model NR2 wanting because the parameters it returns for the crucial thermal component (T_in and R_in) differ quite significantly for the three spectra. We now discuss our most physical nonrelativistic model.

Model NR3. In addition to our criticisms of Models NR1 and NR2, we find these models structurally unsatisfactory because they incorporate two different and discrepant thermal temperatures: diskbb's T_in ∼ 0.5 keV and comptt's T₀ ∼ 1.0 keV. In Model NR3, we solve this problem by replacing compbb with simplr, which generates the Compton power-law component via a convolution by operating on an arbitrary source of seed photons, which in this case is the thermal component diskbb prior to Compton scattering. Meanwhile, we retain the line model laor and the convolution model kdblur. Model NR3 is specified in Table 4, and our fit results are given in Table 7. The fits are good and very comparable to those obtained using the other two models, even though Model NR3 uses one less fit parameter. We note that achieving these fits requires iron abundances that are ≈3.2–5.5 times solar. The benefit of using this more self-consistent model is that it harmonizes the fitted values of the parameters of the thermal component: T_in = 0.532, 0.539, and 0.543 keV and R_in = 2.06, 2.30, and 2.01r_g for SP1, SP2, and SP3, respectively. Meanwhile, the smallness of the inner disk radius R_in is suggestive of the high spin revealed by our relativistic analysis (see Section 3).