Re-estimating the Spin Parameter of the Black Hole in Cygnus X-1

Three typical components might be included in the spectrum of a black hole X-ray binary, namely the thermal, power-law, and reflection components (refer to Figure 2 in GOU11 for a schematic illustration of these different components).

The model we employed in this paper is the most advanced and physically realistic model used in GOU11 and GOU14. The structure of the model, naming all the components that it comprises, with a concise description of each one, are as follows (the detailed parameter configuration is the same as GOU11 and GOU14, unless otherwise indicated):

$$\begin{eqnarray*}&&\begin{array}{l}\mathrm{CRABCOR}\ast \mathrm{CONSTANT}\ast \mathrm{TBABS}\,\left[\mathrm{SIMPLR}\otimes \mathrm{KERRBB}{\rm\small{}}2\right.\\ +\left.\,\mathrm{KERRDISK}+\mathrm{KERRCONV}\otimes (\mathrm{IREFLECT}\otimes \mathrm{SIMPLC})\right].\end{array}\end{eqnarray*}$$

Thermal component: the thermal radiation is emitted from the accretion disk. kerrbb2 (McClintock et al. 2006) is a fully relativistic thin-disk model, including the effects of the returning radiation and the limb darkening. This model reads in and uses a pair of lookup tables for the spectral hardening factor f, computed using bhspec (Davis et al. 2005; Davis & Hubeny 2006) and kerrbb (Li et al. 2005), corresponding to two representative values of the viscosity parameter: α = 0.01 and α = 0.1 (here we adopt α = 0.1). kerrbb2 only has two principal fit parameters: the dimensionless spin parameter a_* and the mass accretion rate $\dot{M}$.

Power-law component: the thermal seed photons supplied by the accretion disk are inverse Compton-scattered into the power-law component by hot electrons in the corona. The convolution model simplr, a modified version of simpl (Steiner et al. 2009b), is an advanced empirical Comptonization model including three main fit parameters: the photon power-law index Γ, the scattered fraction f_sc, and the reflection constant parameter χ. As we mentioned, we adopt ${f}_{\mathrm{sc}}\lesssim 25 \%$ as our spectrum selection standard.

Reflection component: a fraction of the scattered seed photons are emitted downward toward the disk, illuminating the disk and generating the reflection component. kerrdisk + kerrconv ⨂ (ireflect ⨂ simplc)] models the entire reflection component including the Compton hump, the absorption edge, and the emission lines. simplc represents the part of the power-law component that strikes the disk (Steiner et al. 2009b), which is regarded as the input spectrum for the convolution model ireflect (Magdziarz & Zdziarski 1995). Photons emitted near the black hole suffer various relativistic effects, leading to a blurring of the observed spectrum. kerrconv is utilized with the aim of smearing the spectrum (Brenneman & Reynolds 2006). kerrdisk is a model for the iron emission line with the spin allowed to be a free parameter (Brenneman & Reynolds 2006). Here it should be noted that the effect on the spin measurements of using different reflection models is almost negligible, as demonstrated in Section 5.3 of GOU11.

In addition, we use three multiplicative models. (1) crabcor is used for correcting calibration deviations of RXTE. (2) constant is an energy-independent multiplicative factor in order to reconcile calibration discrepancies between different detectors. (3) tbabs is an ISM absorption model (Wilms et al. 2000).

Our model provides a good fit to all six spectra, with reduced χ² values varying from 1.12 to 1.49. The fit parameters are shown in Table 2. A representative plot of the unfolded spectrum is given in Figure 1. Note that there is a larger residual between 8 and 15 keV, which will be discussed in detail later, and should not significantly affect the spin results.

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Table 2. Best-fit Results for Our Adopted Model ^a

Number	Model	Parameter a	Spectral Number						Case 1 b
			SP1	SP2	SP3	SP4	SP5	SP6
1	tbabs	NH	0.74 ± 0.01	0.82 ± 0.02	0.83 ± 0.03	0.80 ± 0.02	0.79 ± 0.02	0.81 ± 0.01	0.72 ± 0.01
2	simplr	Γ	2.39 ± 0.01	2.38 ± 0.04	2.46 ± 0.03	2.69 ± 0.02	2.62 ± 0.03	2.50 ± 0.01	2.45 ± 0.01
3	simplr	fsc	0.247 ± 0.005	0.106 ± 0.016	0.156 ± 0.012	0.211 ± 0.010	0.163 ± 0.009	0.093 ± 0.002	0.092 ± 0.002
4	kerrbb2	a*	${0.9999}_{-0.0087}^{+0.0000}$	${0.9999}_{-0.0137}^{+0.0000}$	${0.9999}_{-0.0155}^{+0.0000}$	${0.9999}_{-0.0130}^{+0.0000}$	${0.9999}_{-0.0115}^{+0.0000}$	${0.9999}_{-0.0015}^{+0.0000}$	0.9696 ± 0.0034
5	kerrbb2	$\dot{M}$	0.17 ± 0.02	0.20 ± 0.03	0.19 ± 0.04	0.19 ± 0.03	0.18 ± 0.03	0.18 ± 0.01	0.23 ± 0.01
6	kerrdisk	EL	6.57 ± 0.06	6.74 ± 0.08	6.55 ± 0.07	6.40 ± 0.13	6.55 ± 0.11	6.70 ± 0.08	6.46 ± 0.06
7	kerrdisk	q	2.89 ± 0.08	3.00 ± 0.09	2.73 ± 0.11	1.64 ± 0.40	2.66 ± 0.17	2.95 ± 0.06	3.45 ± 0.10
8	kerrdisk	NL	0.009 ± 0.001	0.024 ± 0.003	0.029 ± 0.003	0.012 ± 0.002	0.011 ± 0.002	0.014 ± 0.001	0.026 ± 0.001
9	kerrdisk	EW	0.11	0.36	0.29	0.14	0.21	0.29	0.61
10	ireflect	XFe	6.3 ± 0.3	6.0 ± 1.1	3.9 ± 0.5	3.5 ± 0.3	5.9 ± 0.6	4.1 ± 0.6	3.5 ± 0.6
11	ireflect	ξ	62 ± 10	335 ± 186	137 ± 45	21 ± 7	57 ± 18	381 ± 51	412 ± 64
12	constant	NASCA	0.955 ± 0.003	⋯	⋯	⋯	⋯	⋯	⋯
13	constant	NMEG	⋯	0.688 ± 0.009	0.575 ± 0.007	0.704 ± 0.007	0.713 ± 0.005	0.682 ± 0.006	0.690 ± 0.011
14	constant	NHEG	⋯	0.863 ± 0.011	0.889 ± 0.012	0.887 ± 0.009	0.923 ± 0.007	0.889 ± 0.008	0.903 ± 0.014
15		Reduced χ2 (χ2/degree)	1.49 (678.09/454)	1.12 (705.15/627)	1.15 (656.82/572)	1.32 (823.61/624)	1.25 (929.81/744)	1.24 (1241.40/998)	1.09 (1087.79/998)
16		f	1.6	1.6	1.6	1.6	1.6	1.6	1.5
17		l	0.02	0.03	0.03	0.02	0.02	0.02	0.02

Notes.

^a Names of the different model components are listed in column 2. In column 3, the parameters from top to bottom are (1) equivalent hydrogen column density N_H in units of 10²² atoms cm⁻²; (2) the photon power-law index Γ; (3) the scattered fraction f_sc; (4) the dimensionless spin parameter a_* (the upper limit is set to 0.9999); (5) the mass accretion rate $\dot{M}$ in units of 10¹⁸ g s⁻¹; (6) the rest-frame central line energy ${E}_{{\rm{L}}}$, in units of keV; (7) the emissivity index q; (8) the line flux N_L in units of photons cm⁻² s⁻¹; (9) the equivalent width EW of the line in units of keV; (10) the iron abundance X_Fe, relative to the solar abundance; (11) the disk ionization parameter ξ in units of erg cm s⁻¹; (12) the detector normalization constant N_ASCA of ASCA relative to RXTE PCU2; (13) the detector normalization constant N_MEG of Chandra MEG relative to RXTE PCU2; (14) the detector normalization constant N_HEG of Chandra HEG relative to RXTE PCU2; (15) the reduced chi-squared value, total chi-squared value, and number of degrees of freedom; (16) the spectral hardening factor f; and (17) the bolometric, Eddington-scaled X-ray luminosity of the thermal disk component, l = L/L_Edd, where L = $L({a}_{* },\dot{M})$ and L_Edd = 2.8 × 10³⁹ erg s⁻¹ for M = 21.2 M⊙. The uncertainties are quoted at the 90% confidence level. ^b Case 1: fit to SP6 with the inclination angle i fixed at 4247. The parameter configuration is exactly the same as that in columns 5–9.

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As a caveat, in all six cases, the values of a_* reached the maximum value of the XSPEC model kerrbb2, 0.9999. This value is larger than the physical upper limit of a_*, which we will discuss in the next section. The preliminary error in a_* computed with XSPEC and listed in Table 2 is due to counting statistics, while the final adopted error due to the combined uncertainties in D, M, and i is acquired via our MC analysis.

Requiring the spectrum to be dominated by an accretion disk component for reliable application of the continuum-fitting method, we adopt f_sc ≲ 25% as the selection standard. The values of this parameter in our analysis, 24.7% ± 0.5%, 10.6% ± 1.6%, 15.6% ± 1.2%, 21.1% ± 1.0%, 16.3% ± 0.9%, and 9.3% ± 0.2%, satisfy this standard.

It is equally essential to select spectra with bolometric Eddington-scaled luminosities l < 0.3 (McClintock et al. 2006, 2014; Zhu et al. 2012), in order to ensure a geometrically thin disk. This parameter is quite stable and low in our analysis, ranging from 0.01 to 0.03, which meets this criteria without difficulty.

Previous work on estimating a_* found that the combined observational uncertainties in M, i and D dominate the error budget (GOU11, McClintock et al. 2014). This part of the error is usually calculated through MC simulations (Liu et al. 2008; Gou et al. 2011; McClintock et al. 2014). In order to do so, for each individual spectrum, the procedure is as follows. (1) Assuming that each one of these three parameters is independent and Gaussian-distributed, we generate 3000 sets of (M, i, D). (2) Then we calculate lookup tables of f for these parameter sets. (3) Finally, we fit the spectrum to our adopted model to determine a_*. With 3000 simulations run, we obtain the histogram of a_*.

After repeating the procedure expressed above for all six spectra, we obtain six histograms of a_*. The histogram for each spectrum is shown in Figure 2, and the summed histogram is plotted in Figure 3. The spin value in SP2 is lowest, therefore we report the final result of a_* > 0.9985(3σ) based on a conservative estimation from this spectrum.

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In this paper, we have obtained a much more stringent limit on the spin of the black hole in Cygnus X-1 than found from previous work: a_* > 0.9985 at the 3σ level of confidence. This very high spin parameter confirms that Cygnus X-1 hosts an extreme Kerr black hole.

However, Thorne (1974) pointed out that the maximum expected spin once the absorption of hot photons from the disk is taken into account is 0.998 (hereafter, the Thorne limit), as the absorption will produce a counteracting torque when the spin is extremely high. We note that our spin constraint is slightly higher than the Thorne limit. This is mainly because the upper limit in our model kerrbb2 is set at 0.9999, and our obtained spin parameter is based purely on the theoretical model.

It should be noted that although the Thorne limit has generally been adopted by the community as a reasonable guess for an astrophysical upper limit, there is still some debate as to the exact value of the upper limit. Sadowski et al. (2011) suggested that black holes with supercritical accretion flows could reach an equilibrium spin value up to 0.9994, which exceeds the Thorne limit. They showed that the equilibrium spin value depends strongly on the assumed value of the viscosity parameter, and also proved that for high accretion rates the impact of captured radiation on spin evolution is negligible. Cygnus X-1 is not believed to be in the supercritical regime, and hence these considerations may not apply. However, a number of other factors might affect the precise spin measurement, and they are discussed in the following sections. Regardless, the spin of Cygnus X-1 appears to be very high, as found previously by both continuum-fitting (GOU11, GOU14) and iron-line fitting studies (Fabian et al. 2012; Tomsick et al. 2014, 2018; Parker et al. 2015; Duro et al. 2016; Walton et al. 2016).

Figure 4 shows the effect on a_* of separately varying the input parameters M, i, and D. a_* takes the extreme value of 0.9999 when all three parameters are fixed on their newly obtained, best-fitting values.

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In applying the continuum-fitting method, we assume that the binary orbital angular momentum vector is aligned with the spin axis of the black hole. Cygnus X-1 is suggested to have formed with little to no natal kick (Mirabel & Rodrigues 2003; Rao et al. 2020), so the spin axis of the black hole should still be aligned with the orbital angular momentum vector (as discussed by Miller-Jones et al. 2020, who estimated a likely maximum misalignment angle of order 10°).

However, this assumption is controversial. Some previous work on fitting the reflection component of Cygnus X-1 suggested a different inner disk inclination angle compared to the binary orbital inclination, which implies a possible spin–orbit misalignment (also suggested in some other black hole binary systems, Fragos et al. 2010; Dong et al. 2020). Tomsick et al. (2014) evaluated the lower limit of this misalignment to be 13°. Walton et al. (2016) estimated the magnitude of a disk warp to be about 10°–15°. However, these fits often require very high iron abundances, and Tomsick et al. (2018) recently found that a much lower disk inclination of <20° could be accommodated if the electron density was free to vary, even suggesting that a misalignment might not even be required in this scenario. Thus, while the possibility of a misalignment should be considered, this scenario would appear to remain under debate.

Nonetheless, we investigated this possibility, and as shown in Figure 4(a), the value of a_* monotonically decreases with increasing i. We find that even given a misalignment angle of 15° (equivalent to i = 4247), the spin value is 0.9696 (see Case 1 in Table 2), which is still very high.

The spectral hardening factor f in kerrbb2 is precalculated with a table model bhspec, which is related to the metallicity Z. Our fit results are all given for Z = 1 ${Z}_{\odot }$. However, the metallicity of Cygnus X-1 is slightly supersolar (Shimanskii et al. 2012), as also implied by the fitted supersolar values of Fe abundance in our reflection model ireflect.

Unfortunately, the current model bhspec only contains two possible metallicity cases (Z = 1 Z_⊙ and Z = 0.1 Z_⊙), and does not yet have an option for supersolar abundance. However, if the metallicity increases, the value of the spin increases slightly. In addition, substantial changes in metallicity only produce a negligible effect on spin. For example, as stated in Gou et al. (2009), the spin of the black hole in LMC X-1 increases from 0.937 ± 0.020 to 0.938 ± 0.020 if Z increases from 0.1 Z_⊙ to 1 Z_⊙.

Throughout this work, in order to get a more conservative constraint, we fixed the viscosity parameter α to 0.1, because a_* increases slightly as α decreases (Gou et al. 2009). However, we reanalyzed our data using α = 0.01, and found that the results remain unchanged, as they have already reached their maximum value of 0.9999.

The model utilized in the continuum-fitting method is the general relativistic Novikov–Thorne model that assumes a razor-thin disk. However, a real accretion disk will have finite thickness, which will affect measurements of the black hole spin. Kulkarni et al. (2011) evaluated the error introduced by the deviations from the Novikov–Thorne model. They computed disk spectra obtained from three-dimensional general relativistic magnetohydrodynamic (GRMHD) simulations, and then fitted these simulated spectra with kerrbb. By comparing the spin results with the spin used in the GRMHD simulations, they concluded that the model errors become obvious only at low spins and high inclinations. In the case of i = 30° and a_* = 0.98 (close to the values estimated for Cygnus X-1), the spin calculated by fitting the simulated spectra was 0.985 ± 0.001 and the value of $\bigtriangleup \,{a}_{* }$ is 0.005. Moreover, they might have overestimated the error due to the deviation from the Novikov–Thorne model because their simulation results are for a thicker disk with a bolometric Eddington-scaled luminosity l = 0.4–0.7, while the bolometric Eddington-scaled luminosity of Cygnus X-1 in this paper is only 0.02 (as stated in Section 6 of GOU11).

We notice that there is a larger residual between 8 and 15 keV, which is not fitted with the current model. However, it should be emphasized that the spin parameter is determined from the disk thermal component (which dominates the spectra, peaks around 1.5 keV, and is concentrated below 6 keV; see Figure 3 of GOU11 for reference), and thus the residual between 8 and 15 keV should not significantly affect the spin results.

Previous work has proposed many theories to explain similar residual structures. Fabian et al. (2020) found an 7–9 keV excess emission component during the soft state in spectra of MAXI J1820+070. They explained this additional component by including another blackbody whose temperature is slightly higher than that of the disk itself. Given the higher temperature, the additional emission is linked to the innermost region of the disk and with the start of the plunging region. In this region, the density is still high enough to heat the gas, possibly emitting blackbody radiation. We utilized an addition model BBODY to describe this component, found that the fit is poor (>2), and the spin could not be determined certainly.

Another possibility is that the observed residual structure is an additional Comptonization component. In our fits, we only utilized Comptonization models that have a pure nonthermal power-law electron distribution. However, some previous works of the fast time variability seen in the high/soft state prefer a stratified Comptonization spectrum (Grinberg et al. 2014). It is likely that there exist hybrid thermal/nonthermal electron distributions, resulting in an additional low-temperature thermal Comptonization component (Gierliński et al. 1999; Zdziarski et al. 2002; Kawano et al. 2017). This component then produces a higher-energy power-law tail, which would also lead us to overestimate the black hole spin. For instance, as mentioned in Kawano et al. (2017), the spin of the black hole in Cygnus X-1 is 0.95 ± 0.01 for a simple nonthermal Comptonization; and decreases to 0.80${}_{-0.30}^{+0.08}$ for hybrid thermal/nonthermal Comptonization. A similar residual structure between 5 and 10 keV was found in the spectra of the black hole candidate MAXI J1828-249 in a possible intermediate state (Oda et al. 2019). They modeled this excess with a Comptonization of disk photons by thermal electrons with a relatively low temperature and inferred it could be generated in a region where the disk inner edge intrudes into the hot inner flow, or a Comptonizing region above the disk surface. We used an addition model comptt to describe the Comptonization of soft photons, and reflected two Comptonization components. The model is reconstructed as follows:

$$\begin{eqnarray*}&&\begin{array}{l}\mathrm{CRABCOR}\ast \mathrm{CONSTANT}\ast \mathrm{TBABS}\,[\mathrm{KERRBB}{\rm\small{}}2\\ +\mathrm{SIMPL}\otimes \mathrm{COMPTT}+\mathrm{KERRDISK}\\ +\,\mathrm{KERRCONV}\otimes (\mathrm{IREFLECT}\otimes \mathrm{SIMPL}\otimes \mathrm{COMPTT})].\end{array}\end{eqnarray*}$$

The fit parameters are listed in Table 3. For SP1, the fit is better than former model with the reduced χ² decreasing from 1.49 to 0.94. The spin parameter a_* decreases from 0.9999 to 0.9947 ± 0.0014. The photon power-law index decreases from 2.39 ± 0.01 to 1.99 ± 0.01. The scattered fraction f_sc is larger, ranging from 0.247 ± 0.005 to 0.399 ± 0.007. For SP2-SP6, the values of a_* are unaffected, which are still at 0.9999. However, the errors quoted here (estimated in XSPEC) are obviously larger. Additionally, for these five spectra, this model cannot constrain the parameters of comptt well probably because the component is weak. The scattered fraction f_sc are somehow increasing from ∼0.15 to ∼0.40. For SP5, the fit is slightly worse with reduced χ² increasing from 1.25 to 1.27. While for SP2, SP3, SP4, and SP6, the fit is better with reduced χ² ranging from 1.12, 1.15, 1.32, 1.24 to 0.93, 1.02, 0.94, 1.12, respectively. We also try the same setup of Kawano et al. (2017), fixing T₀, kT, and τ at 0.44 keV, 3.4 keV and 1.66, respectively. Details are shown in Case 1 of Table 3. The spin parameter a_* is still pegged at 0.9999.

Table 3. Best-fit Results for a Hybrid Thermal/Nonthermal Comptonization Model

Number	Model	Parameter	Spectral Number						Case 1 a
			SP1	SP2	SP3	SP4	SP5	SP6
1	tbabs	NH	0.73 ± 0.01	0.73 ± 0.05	0.71 ± 0.06	0.65 ± 0.06	0.71 ± 0.03	0.75 ± 0.04	0.76 ± 0.02
2	kerrbb2	a*	0.9947 ± 0.0014	${0.9999}_{-0.1587}^{+0.0000}$	${0.9999}_{-0.2105}^{+0.0000}$	${0.9999}_{-0.1798}^{+0.0000}$	${0.9999}_{-0.0903}^{+0.0000}$	${0.9999}_{-0.1449}^{+0.0000}$	${0.9999}_{-0.0169}^{+0.0000}$
3	kerrbb2	$\dot{M}$	0.18 ± 0.01	0.14 ± 0.06	0.22 ± 0.02	0.25 ± 0.03	0.13 ± 0.10	0.29 ± 0.05	0.16 ± 0.03
4	simplr	Γ	1.99 ± 0.01	2.40 ± 0.04	2.46 ± 0.02	2.53 ± 0.02	2.43 ± 0.05	2.52 ± 0.02	2.39 ± 0.02
5	simplr	fsc	0.399 ± 0.007	0.447 ± 0.440	0.431 ± 0.359	0.393 ± 0.198	0.361 ± 0.234	0.361 ± 0.311	0.492 ± 0.176
6	comptt	T0 a	0.45 ± 0.06	0.45 ± 0.44	0.40 ± 0.23	0.41 ± 0.36	0.39 ± 0.16	0.42 ± 0.26	0.44(fixed)
7	comptt	${kT}$ b	5.77 ± 2.51	${3.87}_{-3.77}^{+125.44}$	${4.00}_{-3.90}^{+113.39}$	${4.11}_{-4.01}^{+164.92}$	${4.06}_{-3.96}^{+96.8}$	${3.95}_{-3.85}^{+109.05}$	3.4(fixed)
8	comptt	$\tau$ b	2.08 ± 0.74	${0.81}_{-0.80}^{+37.60}$	${0.63}_{-0.62}^{+26.64}$	${0.33}_{-0.32}^{+23.50}$	${0.80}_{-0.79}^{+26.65}$	${0.43}_{-0.42}^{+19.5}$	1.66(fixed)
9	comptt	$\mathrm{Norm}$ b	1.08 ± 0.48	${2.50}_{-2.50}^{+88.45}$	${3.64}_{-3.64}^{+110.73}$	${3.86}_{-3.86}^{+165.12}$	${2.15}_{-2.15}^{+55.37}$	${2.41}_{-2.41}^{+70.82}$	2.12 ± 0.07
10	kerrdisk	${E}_{{\rm{L}}}$	6.59 ± 0.04	6.56 ± 0.07	6.51 ± 0.08	6.49 ± 0.08	6.57 ± 0.07	6.55 ± 0.08	6.54 ± 0.07
11	kerrdisk	q	2.74 ± 0.03	2.51 ± 0.19	2.50 ± 0.21	2.52 ± 0.22	2.53 ± 0.19	2.57 ± 0.19	2.38 ± 0.16
12	kerrdisk	NL	0.019 ± 0.001	0.020 ± 0.004	0.023 ± 0.004	0.020 ± 0.004	0.014 ± 0.003	0.013 ± 0.002	0.019 ± 0.002
13	kerrdisk	EW	0.21	0.29	0.25	0.23	0.29	0.26	0.26
14	ireflect	XFe	2.2 ± 0.1	5.3 ± 1.2	3.8 ± 0.6	4.5 ± 0.8	4.1 ± 1.0	3.9 ± 0.8	4.4 ± 0.5
15	ireflect	ξ	198 ± 11	107 ± 48	80 ± 31	92 ± 37	111 ± 48	128 ± 53	89 ± 19
16	constant	NASCA	1.012 ± 0.003	⋯	⋯	⋯	⋯	⋯	⋯
17	constant	NMEG	⋯	0.824 ± 0.010	0.664 ± 0.008	0.813 ± 0.009	0.803 ± 0.007	0.800 ± 0.008	0.820 ± 0.009
18	constant	NHEG	⋯	1.037 ± 0.012	1.029 ± 0.012	1.026 ± 0.012	1.039 ± 0.009	1.045 ± 0.010	1.031 ± 0.011
19		Reduced χ2 (χ2/degree)	0.94 (430.62/451)	0.93 (581.00/624)	1.02 (580.58/572)	0.94 (582.79/621)	1.27 (941.07/741)	1.12 (1114.26/994)	0.93 (583.43/627)
20		f	1.6	1.6	1.5	1.6	1.6	1.6	1.6
21		l	0.02	0.02	0.01	0.02	0.02	0.02	0.02

Notes.

^a Case 1: fit to SP2 with T₀ fixed at 0.44 keV, kT fixed at 3.4 keV and τ fixed at 1.66. The parameter configuration is exactly the same as that in columns 5–9. ^b The names of the different parameters in model comptt are as follows: (1) the input soft photon Wien temperature T₀ in units of keV; (2) the plasma temperature kT in units of keV (the lower limit is set to 0.1 keV); (3) the plasma optical depth τ (the lower limit is set to 0.01); (4) the normalization norm (the lower limit is set to 0).

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Whether using a hybrid thermal/nonthermal Comptonization model or a single nonthermal Comptonization model, we all get an extremely high spin. So we can conclude that including comptt is not having the same effect as in Kawano et al. (2017). However, it is interesting to note that the reduced ${\chi }^{2}$ decreases significantly (e.g., for SP1 it goes to 0.94 from 1.49), which means that adding this Comptonization model has a large impact on the high-energy Compton component rather than on the disk, which is the foundation of our spin measurements.

Another widely used technique of measuring the spin is the iron-line profile fitting method (Fabian et al. 1989; Reynolds & Nowak 2003). As mentioned above, some portion of the power-law component is reflected by the disk, generating the reflection spectrum whose main prominent features include the fluorescent emission lines (particularly, the iron Kα line between 6.4 and 6.97 keV, depending on ionization state) and the Compton hump at 20–30 keV (Ross et al. 1999). The intrinsically narrow iron Kα line is then broadened and smeared by the combined effects of Doppler shift, special relativity, and general relativity (Fabian et al. 2000), and by fitting the iron-line profile, one can deduce the spin parameter.

The spin results from the continuum-fitting method both in the previous work and in this work are consistent with the spins derived from the reflection method (Duro et al. 2011; Fabian et al. 2012; Tomsick et al. 2014, 2018; Parker et al. 2015; Duro et al. 2016; Walton et al. 2016), although Miller et al. (2002) found a near-zero spin of Cygnus X-1, which is mainly because the lack of adequate energy coverage (≲10 keV) made it difficult to subtract the underlying continuum properly.

It should be noted that the highly sensitive NuSTAR, launched in 2012, provides unprecedentedly precise measurements of the iron-line region and reflection component. In the soft state, Tomsick et al. (2014) reported a rapid spin of a_* > 0.83 and Walton et al. (2016) estimated a higher spin of 0.93 ≲ a_* ≲ 0.96. Parker et al. (2015) analyzed observations taken in the hard state and found a high spin of a_* > 0.97. In the intermediate state, Tomsick et al. (2018) restricted the spin to be a_* > 0.93 for the high-density reflection model.

In addition, Duro et al. (2016) analyzed a spectra of four simultaneous hard intermediate state observations, and found a spin of ≃0.9. They also suggested that the different inclination angles inferred in the different states (i.e., the reported inclination angle of ≃30° in the hard intermediate state, as compared to the higher inclinations found from the reflection measurements in the soft state) could imply a warp in the inner region of the disk. However, as discussed by Tomsick et al. (2018), these results can be somewhat degenerate with the iron abundance and with the disk electron density.

We found that the spin of Cygnus X-1 is constrained to be a_* > 0.9985, which is the highest among all the stellar-mass black hole systems with a measured spin parameter. Although it may be less extreme due to some systematics, this value would be still very high. If we look through the spin parameters measured for the stellar-mass black hole systems using the continuum-fitting method, it shows that almost all the black holes in high-mass X-ray binaries have a relatively high spin (Cygnus X-1: a_* > 0.9985, this work; LMC X-1: ${a}_{* }={0.92}_{-0.07}^{+0.05}$, Gou et al. 2009; M33 X-7: ${a}_{* }=0.84\pm 0.05$, Liu et al. 2010; IC10 X-1: a_* ≳ 0.8, Steiner et al. 2016). However, the low mass X-ray binaries have a wide range of spins between zero and unity, which may imply different evolutionary paths for these systems (Bambi 2019; Reynolds 2019).

Note that, since the first detection of GW150914 in 2015, LIGO and Virgo have made 3 runs, and also made 50 detections of gravitational-wave events with the majority of stellar-mass binary black hole systems. The results from the first two runs have been cataloged in Abbott et al. (2019), which shows the effective spin χ_eff ⁸ of the binary system before the merger is quite small with the distributions mean μ ≈ 0, and with a narrow distribution, which is confirmed by Miller et al. (2020). Miller et al. (2020) further explored what these ensemble properties imply about the spins of individual binary black hole mergers, reanalyzing existing gravitational-wave events with a population-informed prior on their effective spin, and found that, under this analysis, the binary black hole GW170729, which previously excluded χ_eff = 0, is now consistent with zero effective spin at ≈10% credibility.

The companion star of Cygnus X-1 is around 40 M_⊙, and based on our understanding to the stellar evolution, it may eventually produce a second black hole of the system, forming a binary black hole system in the future. In addition, the current analysis of the system shows that the black hole spin–orbit misalignment is small, within 10° (Miller-Jones et al. 2020). If the black hole of Cygnus X-1 and the companion star were formed from the same cloud initially, then the companion star is usually aligned with the orbital plane. In this case, we should expect the effective spin of the system to be quite large, unless there is a large asymmetric kick during the black hole formation of the companion star, and it will be quite different from the spin distribution obtained from the GW results. This may imply that the Cygnus X-1 system will likely follow a different formation channel (primordial one) compared to the GW black hole system, although it is not clear whether the GW results preferentially support a dynamical or field binary origin (or even other possibilities) for stellar-mass binary black holes detected by GWs. Furthermore, Miller-Jones et al. (2020) shows that, given the current orbital separation, we do not expect Cygnus X-1 or systems with similar properties to yield binary black hole mergers in the age of the universe. So we may have not detected any GW signal from primordial binary black hole systems yet.

As for the origin of the high spin in these high-mass systems, it is usually concluded that the spin is natal (Podsiadlowski et al. 2003; Axelsson et al. 2011; Qin et al. 2019), at least for those high-mass X-ray binaries, mainly because in the whole lifetime of a stellar-mass black hole, it is certainly impossible to spin up from zero to a highly spinning black hole via standard (as opposed to hypercritical) accretion (Wong et al. 2012). One possible evolution channel to explain the high spin is that the BH progenitor undergoes mass transfers while still in the main sequence (see Case-A MT in Qin et al. 2019). Hence, the core of the BH progenitor contains enough angular momentum to form a highly rotating black hole (many observational features predicted by this path, such as unusual surface abundances of the secondary, have been found in Cygnus X-1, as mentioned in Miller-Jones et al. 2020).