ON THE ROLE OF THE ACCRETION DISK IN BLACK HOLE DISK–JET CONNECTIONS

The AMI arrays are located in Cambridge, UK. They consist of two aperture synthesis telescopes mainly devoted to CMB-related studies (Zwart et al. 2008). The observations described in this paper were made with the Large Array (the reconfigured and reequipped Ryle Telescope), consisting of eight 13 m antennas with a maximum baseline of about 120 m, observing in the band 12–18 GHz. The angular resolution is typically 25 arcsec. Monitoring of small-diameter sources is undertaken in the manner described by Pooley & Fender (1997) for the Ryle Telescope; observations are interleaved with those of a phase-reference calibrator, and after appropriate calibration the data for individual baselines are averaged as vectors. The in-phase component then provides an unbiased estimate of the flux density of the source in question.

The flux-density scale is calibrated by regular observations of 3C48, 3C147, and 3C286. The overall scale is believed to be consistent with that used for the Very Large Array (R. A. Perley 2012, private communication) allowing for the use by AMI of a single linear polarization; all the AMI measurements are of Stokes' I+Q.

Table 1 lists the start times and flux density at 15 GHz for 124 AMI observations made across the time span of our Suzaku observations. These observations are made on a best effort basis, and the observation spacing is sometimes irregular. This small downside is outweighed by the typical high cadence of these observations and their sensitivity.

In order to compare radio flux density to the X-ray spectral properties derived in the fits described above, the start times of the AMI and Suzaku observations were examined. The AMI radio observation most closely following each X-ray observation was identified. In this way, AMI observations were selected for direct comparison to the 20 Suzaku observations. This ordering is based on the expectation that the X-ray-emitting inflow may help to shape the radio jet, but that any influence of the radio jet on the inflow is negligible by comparison.

The mean delay between X-ray and radio observations in this program is 2.3 days. This delay represents innumerable orbital timescales at the innermost stable circular orbit around the black hole, but it is an order of magnitude less than viscous timescales through the entire accretion disk in X-ray binaries, which is expected to set the overall character of the accretion flow geometry (e.g., Esin et al. 1997).

Table 2 lists some basic parameters of the Suzaku observations obtained through this program in 2009, including the XIS spectra utilized from each observation.

Each Suzaku/XIS camera has a single CCD in a square array of 1024 by 1024 pixels. The CCDs are 18 arcmin on each side. The XIS0 and XIS3 cameras are front-illuminated CCDs, while XIS1 is back illuminated. The XIS2 camera is no longer functional due to a micrometeoroid impact. The breadth of the telescope point-spread function (PSF)—approximately 2.0 arcmin (half-power diameter)—is helpful in spreading the flux of a bright incident source over many pixels. In addition, the XIS cameras are equipped with operational modes that can further aid in the observation of bright sources. The key issue is to limit photon pile-up, which can falsely reduce the flux level that is recorded and distort the spectrum by registering multiple incident photons as single events. The key quantity is the photon flux per event box (typically 3 × 3 pixel boxes used to select good X-ray events and reject cosmic rays) per CCD frame time (see, e.g., Miller et al. 2010). Cygnus X-1 is a very bright source, and the instrumental set-up was chosen to limit the effects of photon pile-up on the resultant XIS spectra.

The HXD pointing position was used in all observations; in this configuration, vignetting then reduces the effective area, lowering the photon flux per event box. In the vast majority of observations, each camera was operated with a 1/4 window plus 0.5 s burst option. This has the effect of limiting the frame time, reducing the photons registered in each event box per clock time, and again limiting pile-up. In a few cases, one of the cameras was run in "PSUM" mode, which offers very high time resolution at the expense of imaging information. PSUM mode is not robust against pile-up effects for bright sources, and data taken in this mode are not considered in this work.

All Suzaku data were reduced using version 6.9 of the HEASOFT suite (including FTOOLS and CALDB). The XIS data were reduced in accordance with the Suzaku ABC Guide; these procedures and additional conservative reduction steps are briefly described below.

XIS events can be recorded and telemetered in different editing modes; 2 × 2 and 3 × 3 are the most common, and 5 × 5 is used less frequently. Each unfiltered event file for each camera was reprocessed using the tool "xispi." New "cleaned" event files were then made using the "xisrepro" script, run within xselect. This is often the full extent of data processing for XIS data, but we took additional steps to produce the best possible spectra.

• 1.  
  For each event file, we next ran the "aeattcor.sl" script (Nowak 2009; Nowak et al. 2011), which corrects the data for the large spacecraft wobble endemic to Suzaku. This script generates a new attitude file, which was subsequently applied to each cleaned event list. In every case, this procedure yielded a sharper PSF.
• 2.  
  We then ran the XIS pile-up estimation tool (Nowak 2009; Nowak et al. 2011), and produced images with contours corresponding to different levels of photon pile-up. Extraction regions were then chosen so as to limit photon pile-up. Spectra were extracted from square boxes, 240 pixels on each side, with an inner exclusion circle with a radius of 30 pixels and centered on the source coordinates. This effectively limited photon pile-up in extracted spectra to 5% or less. Backgrounds were extracted using the same region shape, offset from the center of the PSF.

In this work, we only consider spectra obtained outside of the well-known "dipping" intervals in Cygnus X-1. As shown by Balucinska-Church et al. (2000), dipping intervals vary with phase, clustering around ϕ = 0 (where the companion is closer to Earth). This clustering suggests that the dips are partially due to obscuration by inhomogeneities or "blobs" in the focused companion wind. Some dipping is seen at every orbital phase, however (again, see Balucinska-Church et al. 2000), and thus all observations must be screened for dips.

Dipping intervals were identified and excluded using the methods described by Nowak et al. (2011). Briefly, light curves and color–color diagrams were made using the XIS data, and intervals with dip-like properties were identified. These intervals were excluded by specifying additional time selections in the standard good-time interval files. This provides a relatively simple but effective means of identifying and removing dipping intervals and the spectral distortions that they introduce.

For each XIS camera, all event files were then loaded into "xselect" jointly. Regions were defined according to the prescription above. The standard good-time filtering was applied. Spectra, backgrounds, final light curves, and final good-time files were then extracted.

Redistribution matrix files were constructed for each spectrum using the tool "xisrmfgen," and ancillary response files were made by running the "xissimarfgen" tool with an input of 200,000 photons. The XIS0 and XIS3 CCDs are both front-illuminated chips, with very similar response functions. In observations where both of these CCDs were operated in imaging modes, their spectra, backgrounds, and response functions were combined using the tool "addascaspec."

It is nominally the job of the redistribution matrix file to account for oversampling of the detector energy resolution, and to handle it appropriately. This file should be built based on a statistical characterization of how often a photon—with an energy known a priori—ends up in one of N channels that sample the relevant energy window. In practice, this characterization is very difficult; the mutual dependence of flux in different channels is hard to characterize perfectly. As a result, binning can tend to add more continuum into weak and/or narrow line features than is nominally correct, and thereby reduce the significance of such features. Our goal is to accurately characterize reflection in the Fe K band, which requires separating its features from (weak and narrow) ionized absorption features. A stronger binning might legitimately reduce the statistical uncertainty in some continuum and reflection parameters, for instance, but might illegitimately reduce the complicating effects of ionized absorption in the companion wind. Although an even stronger binning would better reflect the intrinsic energy resolution of the XIS cameras, via the FTOOl "grppha" the XIS spectra were grouped to have at least 10 counts per bin for the reasons cited here.

Owing to its different response function, and its different (and generally more severe) calibration uncertainties, XIS1 was excluded from our analysis. The single exception is the observation on April 23, in which both the XIS0 and XIS3 cameras were operated in "PSUM" mode. The use of XIS1 in this case incurs a small but unavoidable systematic uncertainty relative to the other observations.

The XIS cameras suffer from calibration uncertainties at the edges of their spectral bandpass and in the Si region (see, e.g., Nowak et al. 2011). In order to avoid these problems, the XIS spectra were fitted in the 0.8–9.0 keV band. In the front-illuminated XIS cameras, the largest calibration issue in the Si band can be modeled with the addition of a Gaussian absorption line at 1.86 keV. (A negative Gaussian centered at 2.44 keV is required to account for calibration uncertainties in the XIS1 response function.) In a small number of cases, residuals in the Si band were markedly worse than in other observations, with instrumental flux residuals extending up to 3.5 keV. Nowak et al. (2011) noted similar problems; following that work, the 1.5–3.5 keV band was ignored when fitting full relativistic reflection models on May 25 and 29 and June 2 and 4. These observations do not occur at the same binary phase nor do they occur at particularly high or low flux levels. This suggests a short-term change in the instrument response rather than a change in the nature of the spectra observed from Cygnus X-1. In case this interpretation is incorrect, the 1.5–3.5 keV range was included in fits with phenomenological models.

The HXD/pin spectra were reduced in the standard manner recommended by the ABC Guide. This procedure is described here for clarity.

"Tuned" background files for each observation were downloaded from the Suzaku Web site. These files take several weeks to accumulate and are available some time after an observation is made. A common good-time file was made by merging intervals from the cleaned pin event file and the non-X-ray background event file. Time filtering of the cleaned event file was then performed within "xselect" using the merged good-time file, and a pin spectrum was extracted. The resulting spectral file was corrected for dead time using the tool "hxddtcor" and the appropriate housekeeping file.

A non-X-ray background spectral file was also extracted using "xselect," after time filtering with the merged good-time file. As recommended by the ABC Guide, the exposure time of the non-X-ray background spectral file was increased by a factor of 10 using the ftool "fparkey." To create the cosmic X-ray background spectral file, the response file appropriate for the HXD pointing position and gain and calibration epoch of each observation was selected from the HXD calibration database. This file was used to simulate a spectral file within XSPEC, exactly as per the specifications in the ABC Guide. The non-X-ray background and cosmic X-ray background files were then merged using the ftool "mathpha."

In all cases, the HXD/pin spectra were fitted in the 12–70 keV band. This range was found to be relatively free of strong calibration uncertainties.

Source spectra and non-X-ray background spectra from the HXD/GSO were generated and corrected for dead time using the ftool "hxdgsoxbpi." The appropriate raw GSO background file for each observation was downloaded from the public archive. The spectral response files (including the "correction arf") appropriate for each observation were also downloaded from the public archive.

In all cases, the HXD/GSO spectra were fitted in the 70–500 keV band. This range was found to be relatively free of strong calibration uncertainties. The upper limit of 500 keV is the highest energy at which the source is detected in all of the observations.

In order to account for the effects of X-ray absorption in the companion wind, we downloaded and analyzed the longest Chandra/HETG observation in the "low/hard" state available in the public archive. Specifically, we analyzed a 55.85 ks exposure made starting 2003 March 4 at 15:45:02 UT (sequence 400302, ObsID 3815). To prevent photon pile-up, the ACIS-S array was read-out in "continuous clocking" mode, and a gray filter was used on the S3 chip to prevent frame dropping. (For details and additional examples of this instrumental set-up, see Miller et al. 2006b, 2008).

The data were obtained using the Chandra "TGCAT" facility, which archives calibrated up-to-date, level-2 event files, spectral files, and redistribution matrix files. All processing was performed using CIAO 4.2. Using the tool "dmtype2split," we divided the delivered "pha2" file into individual components. Ancillary response files were generated using the tool "mkgarf" and combined the first-order MEG spectra and HEG spectra. These spectra were then analyzed using XSPEC.

As reported in many prior papers, the combined MEG and HEG spectra of Cygnus X-1 contain absorption lines that originate in the companion wind. The wind is not thought to be point symmetric nor even axisymmetric. Rather, comparisons from different points in the binary phase suggest that the wind is "focused" (e.g., Sowers et al. 1998; Miller et al. 2005; Gies et al. 2008; Hanke et al. 2009). The column density in the wind is maximal when the companion star is closest to the observer (ϕ = 0), and is reduced when the black hole primary is closer to the observer (ϕ = 0.5).

At present, no systematic, self-consistent photoionization study of spectra obtained at different binary phase points has been performed. Indeed, treatments in the literature vary widely (though the reader is referred to Hanke et al. 2009 as a thorough treatment). To address the absorption observed in this long Chandra observation, we constructed a large grid of XSTAR models using "xstar2xspec" (version 2.2). The spectral energy distribution (SED) used to create the XSTAR grid was taken from a disk blackbody plus power-law model (kT = 0.19 keV, Γ = 1.8, commensurate with values found in fits to the Suzaku data) used to describe the Chandra continuum. This grid was then included in direct fits to the MEG and HEG spectra as a table model that multiplied a simple disk blackbody plus power-law continuum. In these fits, the ionization parameter of the absorbing gas, its equivalent neutral hydrogen column density, and its velocity shift were all free parameters.

Based on the ephemeris given by Brocksopp et al. (1999), this long Chandra observation started at a binary phase of ϕ = 0.77. Direct fitting suggests a number density of log(n) = 11, an ionization parameter of log(ξ) ≃ 2 (where ξ = L/nr²), and an equivalent neutral hydrogen column density of N_H ≃ 4 × 10²¹ cm⁻², and a modest redshift gives a reasonable description of the data (the strongest lines from abundant elements are fitted well). This is in broad agreement with other results obtained near to this phase (Miller et al. 2005; Hanke et al. 2009). In more detailed treatments, individual ions may require slightly different parameters. However, these values are representative of the results obtained even as the column density varies with binary phase (see Hanke et al. 2009). Last, it is important to note that this model correctly accounts for Fe xxv and Fe xxvi absorption lines that fall on top of the relativistic disk line in Cygnus X-1 (see Figure 1).

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In all fits to the Suzaku spectra, then, this XSTAR table model was included. Although ionized absorption lines are evident in the Fe K band after fitting for disk reflection, the resolution of the XIS makes it difficult to clearly detect weak lines at lower energy, at least at the level required to constrain the ionization of a global model. Therefore, an ionization parameter of log(ξ) = 2.0 was held fixed when fitting the XSTAR table model. Moreover, the redshift of a few × 100 km s⁻¹ found in fits to the Chandra/HETG spectra is insignificant at the resolution of the XIS, so a velocity shift of zero was used in all cases. Thus, only the column density was allowed to vary.

Whether or not CCD spectra could usefully constrain the parameters of an ionized absorption zone was an open question. To this end, the column density measured via phenomenological X-ray spectral fits (see Section 3.1) is plotted versus binary phase in Figure 2. The expected variation is evident in the data: the column density is highest close to ϕ = 0 and reduced close to ϕ = 0.5. This is a strong indication that our modeling is able to correctly account for variations in the spectrum due to the companion wind, and to differentiate those variations from true changes in the low-energy disk flux and relativistic line flux. We remind the reader that dipping intervals were excluded from the spectra in this exercise and in the analysis that follows.

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Figure 3 shows a typical broadband spectrum and the associated data/model ratio that results from fitting with a simple power-law model. The spectrum was fitted by ignoring the spectrum below 3 keV, in the Fe K band (taken to be 5–7 keV here, based on typical broad emission line profiles), and in the 15–45 keV band. The strong soft excess is evidence of thermal emission from an accretion disk, and the curvature seen through the HXD bandpass is strongly indicative of disk reflection.

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Disk reflection models can differ in their assumptions. Some are especially suited to high ionization (e.g., "reflionx"; see Ross & Fabian 2005; also see Dovciak et al. 2004), some are suited to low ionization (e.g., "pexrav"; see Magdziarz & Zdziarski 1995). Most models assume a constant density in the upper levels of the disk where reflection occurs; this might be appropriate if, e.g., magnetic pressure dominates. Other models assume a vertical density gradient determined by hydrostatic equilibrium (e.g., "xion"; see Nayakshin & Kallman 2001).

In order to provide relatively model-independent results, then, we initially fitted every observation with a phenomenological spectral model. The model effectively replicates a relativistically blurred disk reflection spectrum with a simple continuum, but it is easily reproduced and it is free of assumptions regarding disk structure, the nature of the interaction between the corona and disk, and the physical processes at work in the corona. (The results of these fits can be compared to the Comptonization fits made to the same HXD data by Torii et al. 2011.)

The soft excess at low energy was fitted using a disk blackbody model with a modified emissivity law (the "diskpbb" model in XSPEC; e.g., Mineshige et al. 1994). This is a generalization of the better-known "diskbb" model (Mitsuda et al. 1984), in that the disk temperature is assumed to vary with radius according to T∝r^−p far from the black hole. The "diskbb" model assumes the "standard" value of p = 3/4. In light of recent results showing that disks may be strongly irradiated by the corona in the low/hard state (Rykoff et al. 2007; Gierlinski et al. 2009; Reynolds et al. 2010; Reynolds & Miller 2012), we have assumed p = 0.5 as per an irradiated accretion disk. This does not bias the disk radius to be close to the black hole; rather, more of the flux is assumed to originate from larger radii.

Again, the focus of this paper is on how parameters vary, not absolute values. However, the disk continuum does permit constraints on the inner disk radii—often a parameter of interest—and we have derived radii based on the continuum as self-consistently as possible. The flux normalization of "diskbb" and "diskpbb" components is given by K = (r/km)²cos θ(d/10 kpc)⁻². In calculating radii in gravitational units, we have assumed d = 1.86 kpc (Reid et al. 2011), an inclination of θ = 27°, and a mass of M = 14.8 M_☉ (Orosz et al. 2011). This inclination is that of the binary system; the inner disk may be misaligned (for a discussion, see Maccarone 2002), and we are simply assuming that the binary and inner disk inclinations are consistent. Given that gravitomagnetohydrodynamics tends to anchor the inner disk perpendicular to the black hole spin vector (Bardeen & Petterson 1975), and that jet production should be aligned with that vector, the absence of a two-sided jet in radio images of Cygnus X-1 (Stirling et al. 2001) suggests that the inner disk is also viewed at a low inclination.

The "diskbb" model is a standard against which correction factors and calibrations have been calculated. In this analysis, the "true" inner disk radius is calculated via r_in = ηf²_grf²_colr_col/$\sqrt{(}g)$, where η is the ratio of the inner disk radius to the radius at which the emissivity actually peaks, f_gr accounts for relativistic effects, and g accounts for relativistic effects on apparent inclination (we have assumed η = 0.7, f_gr = 0.5, and g = 0.3, as per Zhang et al. 1997, with the assumption that Cygnus X-1 likely has a high spin). The color correction factor, f_col, accounts for radiative transfer (we have assumed f = 1.7 as per Shimura & Takahara 1995 and Merloni et al. 2000). The value of η that we have taken is likely to be marginally too high, in that it assumes a disk where T∝r^−3/4 holds at large radii; a disk following T∝r^−1/2 will dissipate most of its energy at even larger radii and would nominally require a smaller value of η.

The medium- and high-energy spectrum was fitted using a broken power-law component ("bknpow"), modified by an exponential cutoff at high energy ("highecut"; see Wilms et al. 2006 for background on this simple model for the high-energy spectrum of Cygnus X-1).

The Fe K band in Cygnus X-1 is known to be comprised of at least two components: a narrow neutral emission line arising either though irradiation of clumps in the companion wind or the outer disk, and a broad relativistic line that is dynamically broadened in the innermost accretion disk (see, e.g., Miller et al. 2002, 2005; Nowak et al. 2011; Duro et al. 2011). The narrow, neutral line is very narrow even at HETG resolution (see Figure 1), so it was modeled using a zero-width Gaussian component with a centroid energy frozen at 6.40 keV. Although early attempts to constrain the spin parameter in Cygnus X-1 yielded inconsistent results (e.g., Miller et al. 2005, 2009b), more recent efforts using improved models and data have converged on a high spin (Gou et al. 2011; Duro et al. 2011). The broad line was therefore modeled using the "Laor" line model (Laor 1991). A drawback of the Laor model is that it assumes maximal spin; however, it allows a large range of disk radii to be explored, and the focus of this work is to understand relative rather than absolute values. The line centroid energy was bounded between 6.70 and 6.97 keV (Fe xxv–xxvi) as per an ionized disk. The emissivity index was fixed at q = 3 (where J(r)∝r^−q); this was motivated by preliminary phenomenological and reflection fits that all returned values of q consistent with 3.0. The inclination was fixed at 27°, again assuming that the black hole spin vector is aligned with the binary inclination (but see Maccarone 2002). The outer illumination radius was fixed at 400 GM/c² and the inner radius was allowed to vary.

The total phenomenological spectral model used was

$$\begin{eqnarray*} {\rm const} &\times& {\rm tbabs} \times {\rm mtable}\lbrace {\rm wind}\_{\rm abs}\rbrace \times ({\rm gauss}+{\rm laor} \\ &+&{\rm diskpbb} + {\rm bknpow}) \times {\rm highecut}. \end{eqnarray*}$$

To provide a more physical description of the inner accretion flow, we also fitted each of the broadband spectra with the "reflionx" disk reflection model (Ross & Fabian 2005; note that Comptonization models, sometimes invoking multiple Comptonization zones, may also provide a reasonable physical description of the data; see Torii et al. 2011). This model assumes a constant density for the accretion disk, and the emergent spectrum is averaged over viewing angle. However, it is suited to a broad range of ionization parameters; it includes emission lines from C, N, O, Ne, Mg, Si, S, and Fe; and its temperature and ionization calculations do not assume local thermodynamic equilibrium. Unlike some reflection models, "reflionx" is calculated over the 1 eV to 1 MeV range, so it can safely be applied across the entire 0.8–500 keV bandpass of our Suzaku spectra. The broad range of elements and charge states considered in "reflionx" is of importance at low energy, where emission lines from reflection may blur together to create a pseudo-continuum. Fits with "reflionx," then, are crucial to resolving how much of the apparent disk emission is actually due to viscous dissipation in the disk, and how much might be due to reflection.

In fitting "reflionx," the iron abundance was fixed at the default (solar) value, and the source redshift was fixed at zero. The photon index of the power law assumed to irradiate the disk within "reflionx" was linked to the photon index of a separate, simple, unbroken power-law continuum component. The ionization parameter of the accretion disk and the flux normalization of the "reflionx" component were both allowed to float freely. The reflection spectrum was blurred using the "kdblur" model, which is based on the Laor line function. As per the phenomenological model, the inclination was fixed at 27°, the emissivity was fixed at q = 3, the outer illumination radius was fixed at 400 GM/c², and the inner disk radius was allowed to float freely. The narrow Fe K emission line at 6.4 keV was included as before. The soft thermal emission was again modeled using a generalized disk blackbody assuming irradiation ("diskpbb" with p = 0.5), and the total spectral model was modified by a high-energy cutoff ("highecut"). The total spectral model used was

$$\begin{eqnarray*} {\rm const} &\times& {\rm tbabs} \times {\rm mtable}\lbrace {\rm wind}\_{\rm abs}\rbrace \times ({\rm gauss}+{\rm dispkbb} \\ &+&\! {\rm pow}+ {\rm kdblur} \!\times\! {\rm atable}\lbrace {\rm reflionx.mod}\rbrace) \!\times\! {\rm highecut}. \end{eqnarray*}$$

It must be noted that the total spectral model is not entirely self-consistent. The "reflionx" model assumes an exponential cutoff with an e-folding energy fixed at 300 keV. This is generally a factor of two higher than is measured in fits with the phenomenological model (see below). Owing to this disparity, acceptable reflection fits cannot be obtained unless a separate cutoff is applied. This could have the effect of accounting for some curvature that is properly due to reflection instead through a spectral cutoff. The reflection results are driven by the features and sensitivity achieved below 10 keV, and then again in the 20–40 keV range (where the Compton back-scattering "hump" peaks), and less by the nature of the spectrum at and above 300 keV.

It should also be noted that "reflionx" does not measure the "reflection fraction," unlike some prior reflection models. This is partly because there is no longer a simple linear relationship between the incident and reflected flux, when ionization is included and properly treated. Rather, the albedo of the disk increases with ionization at high ionization parameters. This is even more true when the ionization balance within the disk is actually being solved by the reflection model. Moreover, the values obtained in calculating a "reflection fraction" are strongly dependent upon the energy band considered, and values obtained via blurred ionized reflection models fitted over the 0.8–500.0 keV band would not easily map to values obtained with older, un-blurred models fit to a standard RXTE bandpass (e.g., 3–100 keV).

Last, the "Laor" relativistic line model (Laor 1991) and the "kdblur" convolution model based on "Laor" enable relatively quick spectral fits and are sufficient to characterize relative differences in line properties. Newer line models that include spin as a variable parameter are now available (e.g., Dovciak et al. 2004; Beckwith & Done 2004; Brenneman & Reynolds 2006). Although such models might be better suited to measuring absolute values, they can greatly increase the time required to make a fit. New line models appear to give improvements over older models at the 10% level (Beckwith & Done 2004).

Prior to interpreting the results of X-ray spectral fits, and the implications of correlations between X-ray fitting parameters and radio flux density, the X-ray and radio data were checked for consistency with prior results in a model-independent fashion. The HXD/pin count rate was selected for comparison, to best approximate the kind of data that are obtained in typical X-ray monitoring observations. Figure 4 shows light curves of the radio flux density and HXD/pin count rates; the data were divided by their mean values in order to show the relative strength of flux variations. In order to put the X-ray observations in a broader context, the full set of 124 radio observations listed in Table 1 is plotted in Figure 4. It is clear from this figure that the radio and hard X-ray points track each other well, and that the radio flux shows higher fractional variability than the X-ray flux.

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Figure 5 plots radio flux density versus the HXD/pin count rate, for the 20 Suzaku observations and the radio observations that followed most closely after each. The quantities are clearly correlated, though there is scatter. A Spearman's rank correlation test gives a coefficient of ρ = 0.45, corresponding to a 4.4% chance of a false correlation. Thus, the Suzaku and AMI data confirm the radio–X-ray coupling that has been seen previously with X-ray monitor data (see Gallo et al. 2003; Wilms et al. 2006; Zdziarski et al. 2011). This provides a measure of confidence that our program has sampled Cygnus X-1 in a period of typical disk–jet coupling. The prior demonstrations of a positive radio and X-ray flux correlation also show a degree of scatter, despite the higher cadence of X-ray flux points provided by the RXTE/ASM. This suggests that the scatter in Figure 5 is not exclusively due to a lack of strict simultaneity.

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The results detailed in Tables 3 and 4 are generally good fits to the spectra, with χ²/ν ≃ 1. Figure 6 depicts a typical fit to the broadband spectrum with the phenomenological model. Figures 7 and 8 show the amplitude of the broad and narrow Fe K lines, when the data are fitted with a phenomenological continuum. Figure 9 shows a typical fit to the broadband spectrum of Cygnus X-1 with the more physical relativistically blurred disk reflection model.

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Where the fits are not formally acceptable, there are likely three main causes. First, calibration uncertainties produced especially poor fits to the spectra obtained on certain days, for instance on May 29. Second, the "reflionx" reflection model has a hard lower bound of Γ = 1.4 for the power-law index of the hard emission assumed to illuminate the disk. A number of fits pegged at this limit, suggesting that a slightly harder power-law might have been favored by the data. Last, the hard emission may be better described using a broken power law, and "reflionx" only assumes a simple power law at low energy. It is notable that most spectra are better fitted with the phenomenological model than with the blurred reflection model. Other efforts have also found that physical models do not provide better fits to spectra from Cygnus X-1 than phenomenological models (Wilms et al. 2006; Nowak et al. 2011). This is likely the result of the greater freedom afforded by a broken power-law and a separate line component, and the limitations of "reflionx" noted above. Overall, the quality of the spectral fits—both phenomenological and physical—is such that the resulting measurements and correlations are meaningful.

Given the complexities of these observations, including the use of the HXD pointing position for the XIS, the spacecraft wobble, attempts to correct for the wobble, the use of window and burst options, efforts to mitigate photon pile-up in the XIS, and evolution of the HXD performance, it is not clear what the detector cross-normalization factors (captured by the "constants" in both families of spectral models) should be. The use of constant factors to normalize across different detectors when fitting over a broad energy range is well established (see, e.g., Sobczak et al. 2000, concerning multiple spectra of XTE J1550−564 obtained with RXTE). Moreover, some of the relevant effects likely change between observations. The constant factors were therefore allowed to float in each observation. Values for the constant fell in the 0.90–1.10 range in the phenomenological fits, and in the 1.02–1.32 range for the blurred reflection fits (values for the PIN/GSO factor fell in the 0.75–1.05 range and 0.80–1.17 range, respectively). A similar observational set-up was used to observe the black hole candidate MAXI J1836−194 using Suzaku (Reis et al. 2012). The values of the cross-normalization factors found in our fits are broadly consistent with values covered by the error contours derived from fits to the spectra of MAXI J1836−194. Small changes to the factors we recorded (e.g., 0.01) cause small changes in the continuum fits, not the line properties. Thus, although uncertainty in the flux offsets could be assessed as a systematic error, it is not one that has a strong effect on the features of interest. Larger changes (e.g., 0.1), or setting the XIS/PIN factor to a value of 1.17 (common in more standard modes), results in dramatically worse fits in many cases, such that no phenomenological nor physical spectral model is acceptable.

In every sense, the phenomenological X-ray spectral fitting results are typical of Cygnus X-1 in the "low/hard" state. The power-law index above the spectral break is found to be quite hard, generally in the range of 1.4 ⩽ Γ₂ ⩽ 1.5. Similar indices are measured via fits with the blurred "reflionx" model. The cutoff energy measured in the phenomenological fits is generally consistent with kT = 23 keV. This could be indicative of the electron temperature of the hard X-ray corona, but it is likely that much of the curvature fitted by this low-energy cutoff is actually due to disk reflection. This possibility is supported by the fact that Γ₂ is found to be close to the simple power law in the reflection fits (see Tables 3 and 4), and by the fact that the power-law break energy is generally found to be at or above the Fe K line energy, where reflection is expected to produce a similar change in the continuum.

In both families of fits, the temperature of the disk component is found to be quite low, generally at or below kT ≃ 0.2 keV. Again, this is fully consistent with black hole X-ray binaries in the "low/hard" state generally, and Cygnus X-1 in particular (see, e.g., Reis et al. 2010; Reynolds & Miller 2012). The temperature values lie below the energy boundary considered in fits to the XIS spectra; however, simple blackbody functions peak at 2.8 kT, and disk blackbody functions peak at a slightly higher energy. Thus, although the accretion disk is found to be cool in all cases, a good deal of its flux falls within the XIS fitting range and serves to constrain the fitting parameters. Disks in this temperature and flux range cannot be detected in observations obtained with the RXTE/PCA and INTEGRAL/JEM-X cameras, for instance, upon which prior coordinated observing campaigns have been built (Wilms et al. 2006).

The inner disk radius implied by the disk continuum is of interest. Tables 3 and 4 give the normalization of the disk component and the radius implied for d = 1.86 kpc (Reid et al. 2011), M = 14.8 M_☉, and i = 27° (Orosz et al. 2011), as well as various correction factors. Although a couple of observations imply greater deviations, most normalizations differ by less than a factor of ⩽2, which would correspond to radii generally varying by less than 40%. It is immediately apparent that this modest variation is far less than the factor of ∼4 variation seen in the radio flux in Figure 4. The normalizations of the disk components are generally higher in the relativistic reflection fits (r_mean = 3.9 GM/c², σ_r = 0.6 GM/c²) than in the phenomenological fits (r_mean = 3.4 GM/c², σ_r = 0.8 GM/c²), suggesting that a small part of the putative disk continuum could potentially be due to blurred atomic lines from disk reflection. Including all of the relevant correction factors, none of the 40 disk continuum fits implies a radius greater than 6 GM/c². Indeed, 33 of 40 disk continuum fits suggest r_in ⩽ 4 GM/c², which implies a spin parameter of a ⩾ 0.6.

Fits to the relativistic line and blurred reflection spectrum provide an independent check on the innermost extent of the accretion disk and variations in that radius. Phenomenological fits with a separate Laor line component generally find radii between 3–4 GM/c² (r_mean = 3.5 GM/c², σ_r = 0.6 GM/c²), in good agreement with fits to the continuum. All radii inferred using fits with the additive "Laor" line component are found to be significantly below 6 GM/c². Radii measured via the blurred reflection fits give a broader range of values (r_mean = 4.7GM/c², σ_r = 5.2 GM/c²), and less agreement with the disk continuum. A number of fits with the relativistically blurred reflection component imply a radius only marginally larger than 1.2 GM/c², corresponding to maximal black hole spin. A small number of fits imply a larger disk radius, e.g., r = 20  ±  9 GM/c² on May 6. Observations in which larger radii are found generally represent poor measurements: if only those observations with error(r)/r ⩽ 3 are considered, all radii are below 6 GM/c² and values approach maximal spin. Indeed, weighting the values by the inverse of their errors gives a mean radius of 1.7 GM/c². Discarding poor measurements, and assuming that the smallest measured values indicate the actual innermost radius defined by the spin of the black hole, the disk continuum and disk reflection fitting results point to a spin in the range 0.6 ⩽ a ⩽ 0.99.

This program represents the first time that a CCD spectrometer has observed a single source on 20 occasions, each time achieving the sensitivity needed to detect a relativistic line and disk reflection spectrum. Therefore, some simple checks on the choice of a reflection geometry are very much in order. For instance, simple reflection models predict that the flux of a relativistic line should follow the flux of the power-law continuum that illuminates the disk (e.g., George & Fabian 1991). Figure 10 shows a tight correlation between the flux in the Laor line component and power-law component in phenomenological fits. A Spearman's rank correlation test gives a coefficient of 0.958 or a probability of just 3.3 × 10⁻¹¹ of a false correlation. This may represent clear support for the illuminated disk geometry assumed by simple disk reflection models.

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Evidence of feedback between the disk and corona is predicted to take the form of a relationship between the fraction of the incident flux that is reflected by the disk and the incident power-law photon index (the R − Γ correlation; Gilfanov et al. 1999; Zdziarski et al. 2003). Evidence of this correlation has been seen in fits to gas spectrometer data (e.g., Gilfanov et al. 1999; Zdziarski et al. 2003), but this program represents the first opportunity to test the correlation using CCD spectra in which relativistic lines are better revealed and separated from narrow components.

In the case of neutral reflection, the equivalent width of the relativistic line can serve as a proxy for the reflection fraction as the two are closely related (George & Fabian 1991 predict that R ∼ EW/180 eV, where R is the reflection fraction). Figure 11 plots the equivalent width of the Laor line component versus the soft power-law index measured in phenomenological fits to the X-ray spectra. A tight, positive correlation is again observed. A Spearman's rank correlation test gives a coefficient of ρ = 0.921, with a false correlation probability of just 3.9 × 10⁻⁹. The slope of this relationship is clearly greater than unity, and broadly consistent with ≃ 2. This is broadly consistent with the linear relationship expected between these parameters, and thus broadly consistent with feedback between the disk and corona. The fact that the slope is greater than unity may hint at the need to treat reflection more self-consistently, changes to the disk albedo due to ionization, and/or the influence of gravitational light bending, which predicts a slope steeper than unity if the hard X-ray source is very close to a spinning black hole (Miniutti & Fabian 2004).

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Spectral fits with the relativistically blurred disk reflection spectrum allow the flux in the directly observed power-law and the reflection component to be measured independently and self-consistently. It is interesting to note that the reflected flux—not the direct power-law flux—is found to be the dominant flux component in half of the observations (see Table 4). Moreover, taking the ratio of the reflected flux to the total flux as a proxy for the reflection fraction R, there is no significant correlation between R and Γ in the full reflection fits. Together, these results suggest that feedback between the disk and corona, and perhaps the nature of the corona itself, may be more complex than prior studies revealed.

Yet it is difficult to discount the extremely tight correlation between the relativistic line flux and power-law flux (see Figure 10 and the correlations listed in Table 6) based on the phenomenological fits. That correlation is so tight that it calls into doubt the results obtained from the more physical models including a blurred reflection component (see below). There is a simple way in which these apparently contradictory results can be reconciled, however. The power law in the phenomenological fits is a broken power law, meant to mimic a proper reflection spectrum. The flux in this component may not be direct flux, but rather a mixture of direct and reflected flux. Correlating the line and continuum flux would then be a spurious procedure of (at least partly) correlating a component against itself. A very tight relationship should result from such a test. More discussion of the reflection models and their consequences follows in Section 6.

Fits with the relativistically blurred "reflionx" component also suggest a moderately high ionization parameter, generally in the 3 < log(ξ) < 4 range (see Table 4). As with the other parameters measured in fits to these Suzaku observations, this level of ionization is typical of the low/hard state (see, e.g., Reis et al. 2008; Blum et al. 2009). In this regime, He-like and H-like charge states of Fe are expected to be the most prominent. The Laor line centroid energy values were restricted to the range appropriate for these charge states, but many of the fits measured centroids within this range. Thus, the reflection fits and the separate line fits are in broad agreement with regard to the ionization of the disk.

The details of fits to the narrow Gaussian line fixed at 6.40 keV are omitted from Tables 3 and 4 owing to space limitations, and because this feature is not of central importance. The narrow emission line flux was generally found to be consistent with $1.0\times 10^{-3} \ {\rm ph{\rm otons}} \ {\rm cm}^{-2} \ {\rm s}^{-1}$ and an equivalent width of 15–20 eV. These values are consistent with prior detections of this line component with ASCA (Ebisawa et al. 1996) and at high resolution using the Chandra/HETGS (Miller et al. 2002; Nowak et al. 2011). Moreover, these values are consistent with the illumination of cold clumps in the companion wind or the outer accretion disk (Ebisawa et al. 1996).

Figures 12 and 13 plot radio flux density against values of the inner disk radius. Figure 12 plots inner disk radii derived from phenomenological fits, while Figure 13 shows the inner disk radii derived from fits including the relativistically blurred reflection model. In these figures, it is clear that individual measures of the inner disk radius generally lie within a very narrow range of values. It is also clear that variations in the radio flux density occur without changes in the inner disk radius. This lack of a clear correlation is confirmed in rank correlation tests against the line and reflection-derived radii, and the disk normalization from which continuum-based radii derive. There is a single exception that merits additional scrutiny.

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In fits including the relativistically blurred reflection model, the inner disk normalization (radius) measured through continuum fits is anti-correlated with radio flux density: ρ = −0.533, corresponding to a 1.5% chance of a false correlation. In this instance, the correlation is skewed by just two data points, derived in fits to the spectra obtained on June 2 and 4 (see Table 4). There is no evidence of a general trend in the normalization (radius) values in question. As noted previously, data taken at that time are particularly affected by calibration problems in the 1.5–3.5 keV band (for a discussion of similar problems, see Nowak et al. 2011).

Given that (1) only two points have driven the results of the rank correlation test, (2) those two points are derived in spectra that are less reliable than most other spectra, (3) fits to the disk continuum with a more phenomenological model do not find a significant correlation (ρ = 0.312, with a 18% chance of false correlation), (4) only four radius measurements are statistically inconsistent with a very narrow range (3 GM/c² ⩽ r_in ⩽ 4 GM/c²), (5) systematic errors are likely to be much larger than statistical errors, and (6) independent fits to the Fe K line and reflection spectrum find no correlation with radio flux density, this apparent anti-correlation is not robust.

Taken as a whole, the data show that the jet power (as traced by radio flux) is likely not modulated by changes in the inner radius of the accretion disk in the low/hard state.

Although variations in the inner disk radius do not appear to be the source of fluctuations in the jet, there is evidence of disk–jet coupling in our observations. In the phenomenological fits to the X-ray spectra, the disk flux and disk temperature are positively correlated with the radio flux density (ρ = 0.636 and ρ = 0.615, respectively, implying 0.3% and 0.4% probability of false correlation). In the more physical spectral fits, wherein a relativistically blurred disk reflection component is included, the disk temperature and flux are the two parameters that are most strongly correlated with the radio flux (ρ = 0.675 and ρ = 0.640, respectively, implying 0.1% and 0.2% probability of false correlation). Figure 14 plots radio flux density versus the disk continuum flux measured in the phenomenological and more physical models, while Figure 15 plots radio flux density versus the disk temperature in each family of models. It is notable that all of these correlations are stronger than the correlation between radio flux and the HXD/pin count rate, which is indicative of the correlations found when low-resolution X-ray monitoring data or gas spectrometer data are used to study disk–jet connections.

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Simple accretion disk theory and fundamental work on advection-dominated flows show that the mass accretion rate may serve to modulate the disk temperature, flux, and its inner radial extent (e.g., Esin et al. 1997). Above a critical threshold in mass accretion rate, however, it is likely that the disk is always close to the innermost stable circular orbit. This appears to be largely confirmed by recent observations with CCD and dispersive X-ray spectrometers, especially after disk correction factors are considered (e.g., Reynolds & Miller 2012). The critical point seems to be L/L_Edd ≃ 0.001, corresponding to $\dot{m}_{{\rm Edd}} \simeq 0.01$ for a canonical accretion efficiency of 10% (Miller et al. 2006a; Rykoff et al. 2007; Tomsick et al. 2009; Reis et al. 2010).

The luminosity of Cygnus X-1 is over this threshold: the mean bolometric luminosity, based on the values in Table 3, is L = 1.2 × 10³⁷ erg s⁻¹ or L/L_Edd ≃ 0.006. Thus, it is possible that variations in the mass accretion rate primarily affect the disk temperature and flux rather than the inner disk radius. This is supported by the relative constancy of the disk radii that we have derived in both phenomenological and more physical spectral models, using both the disk continuum and the disk reflection spectrum (see Figures 12 and 13). The jet in the low/hard state may be sensitive to the only disk parameters that can vary in this regime. It is particularly appealing that the disk flux—a proxy for the mass accretion rate through the disk—is positively correlated with radio flux. This could indicate a simple coupling between the mass accretion rate in the disk and the power of the jet.

It is also interesting to note that the disk ionization (as measured by the reflection model) shows a weak positive correlation with the radio flux density (ρ = 0.415), indicating a 6.9% chance of false correlation. This parameter is difficult to measure, and errors are typically large. Nevertheless, a plot of radio flux density versus the ionization parameter appears to show a weak positive trend (see Figure 16).

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Again, prior correlations between radio flux density and X-ray flux from monitoring data are effectively a correlation between the radio flux density and the hard X-ray flux. Those correlations make no statement about the origin of the hard X-ray flux nor how the hard X-ray flux might be divided between different components. Table 5 shows that radio flux density is correlated with the indices and flux of the (broken) power-law component used in phenomenological spectral fits to the Suzaku data. This provides a relatively straightforward basis for the correlation between radio flux density and hard X-ray flux (see Figures 5 and 17).

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However, the broken and the cutoff power-law model used in the phenomenological fits is really a proxy for a full reflection model. Correlations using the more physical model parameters give the opposite result: radio flux density is anti-correlated with the direct power-law flux (ρ = −0.414, corresponding to a 7% chance of false correlation). Instead, the radio flux density is positively correlated with the reflected flux (ρ = 0.551, giving a 1.2% chance of false correlation; see Table 7 and Figure 18). In this context, it is also useful to build on the results in Section 4.3 by noting that the reflected flux is found to be anti-correlated with the direct power-law flux (ρ = −0.662, giving a 0.2% chance of false correlation; see Figure 19), in apparent contradiction to simple disk reflection models. As discussed below, this finding is fully consistent with a model involving light bending effects (e.g., Miniutti & Fabian 2004).

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