A MULTIWAVELENGTH STUDY OF CYGNUS X-1: THE FIRST MID-INFRARED SPECTROSCOPIC DETECTION OF COMPACT JETS

We used the SL2 (5.20–7.70 μm), SL1 (7.40–14.50 μm), LL2 (14.00–21.30 μm), and LL1 (19.50–38.00 μm) modules at two nodding positions for sky subtraction. The total exposure times were each time set to 60 s in SL1 and SL2, 120 s in LL2, and 2 × 120 s in LL1. Basic Calibration Data (BCD) were reduced following the standard procedure given in the IRS Data Handbook.⁶ The basic steps include bad pixel correction with IRSCLEAN v1.9, sky subtraction, as well as extraction and calibration (wavelength and flux) of the spectra—with the Spectroscopic Modeling Analysis and Reduction Tool software SMART v8.1.2—for each nodding position. Note that in LL2 and LL1, where the sky appeared weakly non-uniform between both nods, the subtraction residuals were fitted by a first-order polynomial. Spectra were then nod-averaged to improve the signal-to-noise ratio (S/N).

The flux calibration appeared consistent between the different modules at Obs. 1 and Obs. 2, with differences less than 5%, and this allowed us to combine all the spectra by scaling them to SL1, which has the highest flux level with respect to SL2 and LL2. However, at Obs. 3, SL2 had a flux level about 14% lower than that of SL1, while no such problem was detected with LL2 nor LL1. We were not able to definitively assess the reason for such a discrepancy, but a misalignment of the SL2 slit is the most reasonable explanation. We therefore scaled the SL2 flux level to that of SL1.

In the reduction of the data from the two pointed instruments PCA and HEXTE, we follow the procedure used in previous studies of Cygnus X-1 (e.g., Wilms et al. 2006). Data were extracted using HEASOFT 6.9 for all times at least 10 minutes after passages of the South Atlantic Anomaly. During the observations, two (first two RXTE observations) and three (third RXTE pointing) proportional counter units were in operation.

Our Cygnus X-1 observations show a strong absorption feature around 9.7 μm. At the distance to the source (≈2 kpc), we expect a considerable column of interstellar material, so this feature is most likely from silicate dust in the ISM along the line of sight. Several consistent measurements of the visible extinction are found in the literature (see Caballero-Nieves et al. 2009, and references therein). In the companion paper (J. C. Lee et al. 2011, in preparation), we derive an absolute visual extinction A_V ≈ 2.95 from the opacity of the ISM silicate absorption feature τ_9.7, defined by A_V = 18.5 × τ_9.7 (Draine 2003), consistent with that of Wu et al. (1982), A_V = 2.95 ± 0.22 with R_V = 3.1.

We therefore use this value to deredden the spectra with the extinction laws given in Chiar & Tielens (2006), one of the most recent works covering the entire IRS spectral range. In their paper, the authors derived the A_λ/A_K ratio rather than the usual A_λ/A_V one. To express the extinction ratio in the standard way, we assign to A_K the value derived from the law given in Fitzpatrick (1999) for the diffuse ISM (R_V = 3.1), which is A_K = 0.111 × A_V. Moreover, these authors find that beyond 8 μm, where silicate absorption dominates, the MIR extinction is different for the diffuse ISM and the Galactic center. They consequently propose two extinction laws, and we explored both options for dereddening our spectra of Cygnus X-1. We find that the Galactic center case clearly overestimates the silicate absorption depth, and we therefore adopt the corrections based on the extinction due to the diffuse ISM. We note that a single extinction correction (i.e., A_V = 2.95 ± 0.22) produces equally good results for all three epochs. This gives us further confidence that the observed silicate absorption features are caused by dust in the diffuse ISM as opposed to material directly associated with the Cygnus X-1 system.

The resulting dereddened spectra displayed in Figure 1 show unambiguously that the MIR emission of Cygnus X-1 is variable. Indeed, the MIR continuum was brighter during Obs. 1 and Obs. 2, the difference with respect to Obs. 3 increasing with wavelength, from no difference at 6 μm to an excess of 15–20 mJy beyond 10 μm.

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The RXTE spectra are fitted between 3 and 200 keV in XSPEC v.12.5.1. We tested several continuum models such as the standard α −disk (Mitsuda et al. 1984) plus a Comptonization component (Titarchuk 1994) modified by photoelectric absorption, but we find that all the spectra are best described by the phenomenological model combining a broken power law with a high energy cutoff and a Gaussian to account for the iron emission, both modified by photoelectric absorption. These results are consistent with previous studies (see, e.g., Wilms et al. 2006). The best-fit parameters for all the observations are listed in Table 2 and the fits are displayed in Figure 2.

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Based on the classification of Cygnus X-1's spectral states as given by Wilms et al. (2006), these fit results show unambiguously that the source is in the HS during Obs. 1 and Obs. 3, with Γ₁ photon indices of about 1.83 and 1.75, respectively. The 3–200 keV unabsorbed flux is nevertheless about 50% larger during Obs. 1 than during Obs. 3 which, considering the X-ray/radio correlation observed in microquasars in the HS, suggests a stronger contribution of compact jets to the overall emission of Cygnus X-1 during Obs. 1. In contrast, the spectrum of the source during Obs. 2 is softer, with Γ₁ ≈ 2.08. Such a photon index is consistent with an IS. But similarly, the high 3–200 keV unabsorbed flux suggests a stronger contribution of compact jets than during Obs. 3.

In Figure 3, we display the Cygnus X-1 radio (15 GHz, 5 minute time resolution) and ASM (1.2–12 keV, day-averaged) light curves, as well as the ASM HR2 (C/A) hardness ratio, between MJD 53500 and MJD 53580, bracketing our observations. Clearly, it is seen that the count rate is lower and the hardness ratio higher during Obs. 3, i.e., when the MIR continuum appears to be at its lowest level (see Figure 1, blue spectrum). Consistent with the spectral fitting, the ASM light curve and HR2 hardness ratio show that the source during Obs. 2 is in a softer state and exhibits a slightly higher flux than during Obs. 1 (see also Figure 4). This is in good agreement with the light curve trend shown in Figure 3 between MJD 53510 and MJD 53530, which shows a sharp decrease in the hardness ratio and a slow increase in the count rate. This probably points to a higher accretion disk activity becoming unsteady around MJD 53520. Then, the count rate drops between MJD 53530 and MJD 53550 before stabilizing, while the hardness ratio follows the opposite path. This behavior is characteristics of a failed-state transition as characterized by Wilms et al. (2006).

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Moreover, the Ryle light curves during Obs 1. and Obs. 2 exhibit an average radio flux between 12 and 25 mJy, while the radio flux is on average lower and less steady during Obs. 3. This behavior is shown by the 20 s time resolution Ryle light curves, displayed in Figure 5, which cover the exact time range of each IRS observation. It is clear that Obs. 1 and Obs. 2 are consistent with compact jets. In contrast the radio activity during Obs. 3 is very unstable, covering a range from non-detection to 48 mJy. This behavior is not by itself inconsistent with the presence of compact jets, and possible explanations of this flickering could be attributed to some instabilities internal to the jet or some peculiar accretion flow properties.

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Finally, it is worth mentioning that there have been indications, from the timing behavior, of two different HS regimes (Pottschmidt et al. 2003; Axelsson et al. 2008). One of them is characterized by a very high hardness level, a low X-ray flux, and characteristic frequencies shifted to the lower edge in the power density spectrum (PDS). This is clearly the case of the HS detected during Obs. 3 (see V. Grinberg et al. 2011, in preparation, for the full PDS), which is actually the beginning of the hardest X-ray activity ever observed for Cygnus X-1 (Nowak et al. 2011). In contrast, the HS properties during Obs. 1 are consistent with what is expected in the canonical picture.

Considering the very unstable radio activity of Cygnus X-1 during Obs. 3, a jet contribution to the MIR spectrum is very unlikely. We therefore only consider the emission from the star. To model the photosphere, we simply use a blackbody. Cygnus X-1's companion star, HD 226868, is an O9.7ab supergiant and its unabsorbed emission peaks in the ultraviolet (UV). The MIR alone is therefore not sufficient to constrain the star's temperature, so we fix it to the most recent derived value, i.e., 28000 K as given in Caballero-Nieves et al. (2009), based on theoretical fitting to the ultraviolet spectrum. In their seminal papers, Wright & Barlow (1975) and Panagia & Felli (1975) showed that the bremsstrahlung from a homogenous spherically expanding wind would be responsible for a MIR and radio excess ∝ν^0.6 in supergiant stars. We therefore add a power law to take the wind emission into account. This simple model is summarized by Equation (1):

$$\begin{equation} F(\nu)=\left(\frac{R_\ast }{D_\ast } \right)^2 \times B(\nu, 28000)+N_{\rm ff}\left(\frac{\nu }{15\;{\rm GHz}} \right)^{\alpha _{\rm ff}}, \end{equation} \tag{ 1 }$$

where R_* and D_* are the companion star's radius and distance, B is the blackbody emission, and N_ff and α_ff are the bremsstrahlung flux density at 15 GHz and spectral index.

The best fit is displayed in Figure 6 and the best-fit parameters are listed in Table 3. The ratio R_*/D_* is consistent with the one expected from Cygnus X-1's companion star, since the derived stellar radius for a typical 2 kpc distance is about 19.9 R_☉, the typical value for a typical late O supergiant. Moreover, the power-law spectral index is characteristic of bremsstrahlung from an expanding stellar wind, with α_b ≈ 0.55, consistent with the 0.6 theoretical value within the 90% uncertainties. The MIR continuum of Cygnus X-1 during Obs. 3 can therefore be described by that of HD 226868. This alone does not rule out the possible presence of compact jets. Nevertheless, both the average X-ray and radio fluxes are at least 50% lower than during Obs. 1 and 2. So, even if they are present, either the compact jets are too faint to be detected in the MIR or they break to the optically thin regime beyond 30 μm. Indeed, in this regime, the IR flux is expected to vary with respect to the radio flux as F_IR ≈ F^21/17_radio (Heinz 2004). Considering the low radio flux level, this very probably excludes a significant contribution of a compact jet to the MIR spectrum of Cygnus X-1 during Obs. 3.

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The radio behavior does exhibit compact jets that may be responsible for the observed continuum increase. We consider two cases to model their contribution, (1) the spectral break does not occur in the IRS spectral range so the optically thick synchrotron emission can be described by a simple power law or (2) the spectral break is located in the IRS spectral range and the emission of the compact jets can be modeled by a broken power law.

To better constrain the compact jets' emission parameters, we include the average 15 GHz flux densities of Cygnus X-1 during the IRS/LL observations. We then fit these MIR/radio SEDs by fixing the blackbody parameters and bremsstrahlung spectral index to those of Obs. 3.

4.2.1. Case 1: Simple Power Law

The overall equation describing the Cygnus X-1 emission is

$$\begin{eqnarray} F(\nu)&=&\left(\frac{R_\ast }{D_\ast } \right)^2 \times B(\nu, 28000)+N_{\rm ff}\left(\frac{\nu }{15\;{\rm GHz}} \right)^{\alpha _{\rm ff}}\nonumber\\ &&+N_{\rm j}\left(\frac{\nu }{15\;{\rm GHz}} \right)^{\alpha _{\rm j}}, \end{eqnarray} \tag{ 2 }$$

where N_j and α_j are the synchrotron flux density at 15 GHz and spectral index, respectively.

Here, N_ff was left free to mimic a possible change in the bremsstrahlung intensity due to a variation of the star's mass loss rate. The best-fit parameters are listed in Table 4, and the corresponding fitted SEDs displayed in Figure 7. Both fits are consistent with a contribution of compact jets to the MIR emission of Cygnus X-1 during Obs. 1 and Obs. 2. Indeed, the derived spectral indices are 0.01 ± 0.02 and 0.05 ± 0.01, respectively, consistent with typical values. Moreover, the bremsstrahlung flux density at 15 GHz during Obs. 1 is found to be similar to that during Obs. 3 (the no-jet case, see Section 4.1), 0.66 ± 0.06 mJy compared to 0.74^+0.10_{− 0.07} mJy. In contrast, N_ff during Obs. 2 is lower (0.44^+0.06_{− 0.05} mJy).

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Table 4. Best Parameters from the Fits of Obs. 1 and Obs. 2 Radio/MIR SEDs of Cygnus X-1: Case 1

Obs. No.	T*	R*/D*	αff	Nff	αj	Nj	χ2/dof
⋅⋅⋅	(K)	(R☉ kpc−1)	⋅⋅⋅	(mJy at 15 GHz)	⋅⋅⋅	(mJy at 15 GHz)	⋅⋅⋅
1	28000 (frozen)	9.94 (frozen)	0.55 (frozen)	0.66+0.05− 0.05	0.01+0.02− 0.02	15.60+5.41− 4.24	0.96/301
2	=	=	=	0.44+0.06− 0.05	0.06+0.02− 0.02	22.00+5.28− 4.32	0.72/301

Notes. The model used is a combination of a blackbody and two power laws. The error bars are given at the 90% confidence level.

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4.2.2. Case 2: Broken Power Law

Here, the overall equation describing the Cygnus X-1 emission is

$$\begin{eqnarray} F(\nu)&=&\left(\frac{R_\ast }{D_\ast } \right)^2 \times B(\nu, 28000)+N_{\rm ff}\left(\frac{\nu }{15\;{\rm GHz}} \right)^{\alpha _{\rm ff}}\nonumber\\ &&+\left\lbrace \begin{array}{@{}ll@{}}N_{\rm j}\left(\frac{\nu }{15\;{\rm GHz}} \right)^{\alpha _{\rm j1}}&\mbox{if } \nu \le \nu _{\rm c} \\ \ms {N_{\rm j}}\left(\frac{\nu _{\rm c}}{15\;{\rm GHz}} \right)^{\alpha _{\rm j1}-\alpha _{\rm j2}}\left(\frac{\nu }{15\;{\rm GHz}} \right)^{\alpha _{\rm j2}}&\mbox{if } \nu \ge \nu _{\rm c}, \\ \end{array} \right. \end{eqnarray} \tag{ 3 }$$

where N_j, α_j1, α_j2, and ν_c are the jets' synchrotron flux density at 15 GHz, optically thick and thin spectral indices, and cutoff frequency, respectively.

When all the parameters of the broken power law and the bremsstrahlung normalization N_ff are free, the resulting fits are poorly constrained due to several degeneracies. We therefore proceeded with three scenarios, each time fixing one parameter only: (a) α_j2 is fixed to −0.6, i.e., the theoretical spectral index expected from the optically thin synchrotron emission of the compact jets, (2) N_ff is fixed to the values found in case 1, (3) N_ff is fixed to the value found during Obs. 3, i.e., we assume that the bremsstrahlung intensity is the same for all the observations. The best-fit parameters are listed in Tables 5, 6, and 7, and the SEDs are displayed in Figure 8.

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Table 5. Best Parameters from the Fits of Obs. 1 and Obs. 2 Radio/MIR SEDs of Cygnus X-1: Case 2, Scenario (a)

Obs. No.	T*	R*/D*	αff	Nff	αj1	νc	αj2	Nj	χ2/dof
⋅⋅⋅	(K)	(R☉ kpc−1)	⋅⋅⋅	(mJy at 15 GHz)	⋅⋅⋅	(THz)	⋅⋅⋅	(mJy at 15 GHz)	⋅⋅⋅
1	28000 (frozen)	9.94 (frozen)	0.55 (frozen)	0.75+0.03− 0.04	$\hphantom{${-}$}$0.01_{-0.02}^{+0.02}$$	27.05+2.44− 1.78	−0.6 (frozen)	15.40+4.43− 3.34	0.81/300
2	=	=	=	0.78+0.02− 0.04	−0.01+0.01− 0.01	14.80+1.16− 1.07	−0.6 (frozen)	21.30+3.39− 2.45	0.69/300

Notes. The model used is a combination of a blackbody, a power law, and a broken power law. The error bars are given at the 90% confidence level.

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Table 6. Best Parameters from the Fits of Obs. 1 and Obs. 2 Radio/MIR SEDs of Cygnus X-1: Case 2, Scenario (b)

Obs. No.	T*	R*/D*	αff	Nff	αj1	νc	αj2	Nj	χ2/dof
⋅⋅⋅	(K)	(R☉ kpc−1)	⋅⋅⋅	(mJy at 15 GHz)	⋅⋅⋅	(THz)	⋅⋅⋅	(mJy at 15 GHz)	⋅⋅⋅
1	28000 (frozen)	9.94 (frozen)	0.55 (frozen)	0.66 (frozen)	0.04+0.01− 0.01	29.43+2.38− 2.29	−0.58+0.27− 0.21	15.20+0.11− 0.09	0.85/300
2	=	=	=	0.44 (frozen)	0.06+0.01− 0.01	42.61+3.73− 1.79	1.21+0.80− 0.36	22.04+0.28− 0.27	0.69/300

Notes. The model used is a combination of a blackbody, a power law, and a broken power law. The error bars are given at the 90% confidence level.

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Table 7. Best Parameters from the Fits of Obs. 1 and Obs. 2 Radio/MIR SEDs of Cygnus X-1: Case 2, Scenario (c)

Obs. No.	T*	R*/D*	αff	Nff	αj1	νc	αj2	Nj	χ2/dof
⋅⋅⋅	(K)	(R☉ kpc−1)	⋅⋅⋅	(mJy at 15 GHz)	⋅⋅⋅	(THz)	⋅⋅⋅	(mJy at 15 GHz)	⋅⋅⋅
1	28000 (frozen)	9.94 (frozen)	0.55 (frozen)	0.74 (frozen)	0.01+0.01− 0.01	29.27+1.90− 1.91	−1.14+0.44− 0.34	15.30+1.65− 1.41	0.78/300
2	=	=	=	=	0.01+0.01− 0.01	15.28+1.24− 1.16	−0.55+0.04− 0.10	21.60+2.80− 2.30	0.69/300

Notes. The model used is a combination of a blackbody, a power law, and a broken power law. The error bars are given at the 90% confidence level.

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4.2.3. Interpretation

This, to our knowledge, is the first measurement of the spectral break of a compact jet based on spectral fitting instead of photometry, and only the second detection of such a break in a black-hole X-ray binary after GX 339−4 (Corbel & Fender 2002). We can use the theoretical expression of the expected cutoff frequency, to see if our value is consistent with that of GX 339−4, scaled with respect to the BH mass, distance, and accretion rate. In the HS, if F_ν is the flux density at a given frequency ν, $\dot{M}$ the accretion rate, and M_X the BH mass, we have for a radiatively inefficient jet, from Falcke & Biermann (1995) and Heinz & Sunyaev (2003):

$$\begin{eqnarray} \dot{M} &\propto & F_\nu ^{12/17}\nonumber \\ \nu _{\rm b} &\propto & \frac{\dot{M}^{2/3}}{M_{\rm X}}, \end{eqnarray} \tag{ 4 }$$

where ν_b is the cutoff frequency.

For GX 339−4, we consider a distance of 8 kpc, a BH mass of 10 M_☉, and a flux density at 15 GHz of 15 mJy (Zdziarski et al. 2004; Fender et al. 1997; Corbel et al. 2003). For Cygnus X-1  the numbers are 2 kpc, 11 M_☉ (Caballero-Nieves et al. 2009), and 15 mJy, respectively. Using Equation (4), we find ν_GX ≈ 4.06 × ν_CYG, where ν_GX and ν_CYG are the GX 339−4 and Cygnus X-1 cutoff frequencies, respectively. For our measured ν_CYG ≈ 2.9 × 10¹³ Hz, this relation gives ν_GX ≈ 1.15 × 10¹⁴ Hz, which is in good agreement with the values derived in Corbel & Fender (2002). Therefore, despite the intrinsic uncertainties associated with the MIR spectral fitting, we believe that we see the cutoff frequency in the Cygnus X-1 MIR spectrum during Obs. 1.

We can use the presence of the synchrotron break to constrain the jet geometry. As pointed out in Heinz (2006), the radio photosphere is located at a de-projected distance of about 5 × 10¹³ cm from the black hole, based on the fact that about 50% of the flux is unresolved at an angular resolution of 3 mas in the VLBA observations reported by Stirling et al. (2001).

For a Blandford–Koenigl jet (Blandford & Koenigl 1979), the location of the photosphere along the jet, z_{τ = 1}, is inversely proportional to the frequency, z_{τ = 1, ν}∝ν⁻¹. Given the marginally higher radio flux in our observation compared to the Stirling et al. (2001) observations, the radio photosphere should be located at a distance of about 6 × 10¹³ cm. At a frequency of ν_b ≈ 2.9 × 10¹³ Hz, the IR photosphere would be located at a distance of $z_{\tau =1,2.9\times 10^{13}\,{\rm Hz}} \approx 1.7\times 10^{10}\,{\rm cm}$. Since the break from optically thick to thin occurs when the photosphere reaches the base of the jet, our result would suggest that the base of the jet has a scale of about half a light-second. For a BH of mass M ≈ 10 M_☉, this corresponds to about 10,000 gravitational radii, which is fairly large compared to size scales typically assumed for the base of the jet, usually assumed to occur on scales of a few tens to hundreds of gravitational radii. For example, the jet in M87 is resolved down to scales of less than 100 gravitational radii.

A simple Blandford–Koenigl model is a severe oversimplification over the almost four orders of magnitude in frequency and thus photospheric scale between IR and radio. Deviations are almost certainly to be expected. However, any change would still have to obey the observed flat SED. As argued in Heinz (2006), simple geometric changes to the Blandford–Koenigl model do not alter the behavior of the jet significantly. For example, allowing the jet to be continually collimated (with R_jet∝z^ζ) instead of being conical (ζ = 1) and requiring energy conservation such that p∝z^−2ζ results in a slight altered dependence of z_{τ = 1, ν}∝ν^−1/ζ, and a change of the spectral index from completely flat to $\alpha = 1 - \frac{1}{\zeta }$ (Heinz 2006). Bringing the base of the jet inward (toward the BH) would require a ζ > 1, implying a steeper spectrum, while microquasars tend to have flat to slightly inverted spectra. A simple geometric explanation for the large implied scale for the base of the jet is thus not plausible.

It is clear that the jet must be accelerated between the base of the jet and the radio photosphere, changing the Doppler parameter δ along the jet. Since the optical depth to synchrotron self-absorption is proportional to δ², a significant increase in δ from the base of the jet, where the IR emission originates, to the radio photosphere, would boost the radio optical depth, effectively moving the photosphere outward by a factor of δ^2/3. However, given that the jet in Cygnus X-1 is likely not ultrarelativistic (Gleissner et al. 2004), it is unlikely that this could bring the base of the jet inward by more than a factor of a few.

Finally, it is possible that the estimated resolved flux fraction of the radio of ∼50% was overestimated in Heinz (2006). For example, if instead of 50%, only 25% of the flux were actually resolved in that observation, it would decrease z_{τ = 0} by an order of magnitude, again moving the size of the IR photosphere and thus the base of the jet toward scales more typically associated with jet acceleration. However, very long baseline observations tend to underestimate the resolved flux, rather than the unresolved flux.

If the size scale over which the jet forms is really this large, it would imply that large scale outflows (i.e., winds) from the disk are fundamentally involved in jet formation. Indeed, a recent study of GRS 1915+105 has shown that the accretion disk wind and the jets were competing for the same matter supply (Neilsen & Lee 2009). Moreover, it has long been speculated that winds support jet collimation by providing lateral pressure support. This would make the jet in Cygnus X-1 qualitatively different from AGN jets which have been imaged down to scales significantly smaller than this in terms of gravitational radii. It would also imply that the jet originates over a region much larger than the X-ray emitting corona that is typically assumed to be the base of the jet. This underscores the need for further confirmation of the nature of the outflow responsible for the observed radio emission and, by extension, the large scale diffuse radio and emission line nebula around Cygnus X-1.

Figure 10 displays the contributions of the stellar photosphere (blue), the bremsstrahlung from the expanding stellar wind (magenta), and the compact jet extrapolated to the high-energy (green), from our fit of Cygnus X-1's SED in the case 2, scenario (a). The simultaneous Ryle, Spitzer, and RXTE data are also superimposed, as well as the non-contemporaneous INTEGRAL/IBIS data that were recently published (Laurent et al. 2011). The authors detected for the first time polarized γ-ray emission beyond 400 keV, and they argue that it is due to compact jets. Our result is fully consistent with this scheme, as the optically thin part of the jet perfectly describes the INTEGRAL/IBIS data, providing strong evidence that we actually detect the spectral break around 2.9 × 10¹³ Hz. Nevertheless, it is clear that the RXTE spectrum of Cygnus X-1 cannot be described by the compact jet alone, and one or several extra components are needed. The spectral similarity between the spectral indices of the IR and the X-ray spectra might suggest that one of these components is synchrotron self-Compton (SSC) emission in the base of the compact jet itself (Markoff & Nowak 2004; Markoff et al. 2005). However, this would require a large compactness of that region, in direct contradiction to our discussion regarding the implied large size of the IR photosphere and thus the base of the jet above. The more natural interpretation would be Comptonization of disk photons by the corona, independent of the optically thin synchrotron spectrum from the jet. Moreover, whether or not this result implies a disk origin of the X-ray emission in other sources like GX 339−4, whose NIR spectra are more consistent with a synchrotron origin of the X-ray, remains an open question.

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Bremsstrahlung is found to be variable and to contribute significantly to the MIR emission of Cygnus X-1, more than the compact jet itself. The fact that its intensity is similar in the HS (Obs. 1 and Obs. 3) and lower in the failed-transition state (Obs. 2) is consistent with an anti-correlation of the mass-loss and accretion rates, as was first shown by Gies et al. (2003), who derived a mass-loss rate $\dot{M_{\rm w}}$ of about 2.57 × 10⁻⁶ M_☉ yr⁻¹ and 2.00 × 10⁻⁶ M_☉ yr⁻¹ for the HS and SS, respectively. We can also derive $\dot{M_{\rm w}}$ from the theoretical expression which links it to our fit-derived bremsstrahlung flux density S_ν at 15 GHz, including clumping effects (Lamers & Waters 1984; Scuderi et al. 1998):

$$\begin{eqnarray} \dot{M_{\rm w}}&=&2.26\times 10^{-7}\times \sqrt{f_\infty }\times \left[ S_{\nu } \left(\frac{\nu }{10\;{\rm GHz}} \right)^{-0.6} \left(\frac{T_{\rm e}}{10^{4}\;{\rm K}} \right)^{-0.1} \right.\nonumber\\ &&\left. \times \, \left(\frac{D_\ast }{1\;{\rm kpc}} \right)^2 \right]^{\frac{3}{4}} \left(\frac{\mu v_\infty }{100\;{\rm km \;s}^{-1}} \right), \end{eqnarray} \tag{ 5 }$$

where f_∞ is the clump filling factor of the wind, T_e is the electron temperature far from the photosphere, which is well approximated as 0.5T_*, μ is the mean atomic weight of the expanding plasma per electron, usually taken as about 1.3 for an O supergiant (Lamers & Leitherer 1993), and v_∞ is the terminal velocity of the wind (≈1600 km s⁻¹; Gies et al. 2008). When clumping is not included (i.e., f_∞ = 1), we derive mass-loss rates of about 8.79 × 10⁻⁶ M_☉ yr⁻¹ and 5.95 × 10⁻⁶ M_☉ yr⁻¹ for Obs. 3 (HS) and Obs. 2 (IS), respectively. Neglecting clumping clearly leads to overestimating the mass-loss rate by a factor of 3–4, as shown by Lamers & Waters (1984). This is therefore a hint that the Cygnus X-1 wind is clumped, and we need f_∞ ≈ 0.09–0.10 in Equation (5) to match the mass-loss rates given in Gies et al. (2008). This value is consistent with that derived from the study of the X-ray dips (M. Hanke et al. 2011, in preparation).