DISK EMISSION FROM MAGNETOHYDRODYNAMIC SIMULATIONS OF SPINNING BLACK HOLES

Our simulations were carried out with the Harm3d code (Noble et al. 2009) and were designed to resemble closely the ThinHR Schwarzschild simulation of Noble et al. (2010). All of our simulations used the same cooling function and target temperature profile, designed to keep the disk aspect ratio of $H/r\simeq 0.05$. All of our simulations also began with the same initial condition, a hydrostatic torus with a pressure maximum at $r=35M$ and an inner edge at $r=20M$ (hereafter, we use units in which $G=c=1$). Threading these tori was a dipolar magnetic field with mean initial plasma $\beta =100$. The spatial grids for the three Kerr spacetime simulations were very similar to the one utilized for the Schwarzschild case; the only alteration was to extend the radial grid to smaller Kerr–Schild radii in order to follow the shrinkage of the event horizon with increasing spin. As in the earlier work, our radial grid was logarithmic (i.e., constant ${\rm{\Delta }}r/r\simeq 0.004$) and stretched from just inside (five cells) the event horizon to $r=70M$. Consequently, there were 912, 960, 1020, and 1056 radial cells in the simulations with $a/M=0$, 0.5, 0.9, and 0.99, respectively. In all four simulations, there were 160 cells in the polar angle direction, and these cells were strongly concentrated at the equatorial plane. Likewise in all cases, the simulation domain included only the one-quarter circle from $\phi =0$ to $\phi =\pi /2$, with 64 cells across that range and periodic boundary conditions that mapped cells at $\phi =0$ to $\phi =\pi /2$. Boundary conditions on the inner and outer radial surfaces were outflow; the polar-angle boundaries at the axis ($\theta =0,\pi$) were continuous across the pole.

The ThinHR simulation was run to a time 15,000M; the Kerr simulations were run to ≃17,000–18,000M. As is generally the case, inflow equilibrium, to the degree it is achieved, begins at the ISCO and gradually moves outward. It can be characterized in (at least) two ways: by a statistical steady state in the accretion rate onto the black hole and by constancy in the accretion rate as a function of radius when time averaged over the period of interest. Here we show both measures.

Figure 1 shows how the accretion rate through the event horizon varied over time in all four simulations. The peak accretion rates were similar in all four simulations, but, particularly at late times, there is a tendency for the accretion rate to diminish with increasing spin. This tendency was also seen in earlier global simulations with thicker disks and shorter durations (Krolik et al. 2005). Because the period before 10,000M tends to be dominated by transients associated with initial channel solutions, we have restricted our analysis of disk output to periods after that time, 10,000–15,000M for $a/M=0$, 0.5, and 0.9, and 12,000–15,000M for $a/M=0.99$.

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Despite large variations in the time-dependent accretion rate at the horizon, the time-averaged accretion rate had relatively little variation in radius, a sign that overall there was little change in the disk's mass distribution. To illustrate that fact, in Figure 2, we show the ratio $\langle \dot{M}(r)\rangle /\langle \dot{M}({r}_{H})\rangle$ as a function of $r/{r}_{{\rm{ISCO}}}$. Both axes are logarithmic in order to emphasize that ratios are what matter most in this context. Inflow equilibrium is best judged in terms of fractional departures in the accretion rate; radial variation is best considered in units of ${r}_{{\rm{ISCO}}}$ because the greatest energy release takes place at or near the ISCO and, in the Newtonian limit, the potential is a power law in radius. In three of the four cases ($a/M=0$, 0.9, and 0.99), the time-averaged accretion rate changes by at most a few tens of percent for $r\lesssim 3{r}_{{\rm{ISCO}}}$. In the case of $a/M=0.5$, this criterion is achieved only out to $r\simeq 2{r}_{{\rm{ISCO}}}$, beyond which it rises to about twice the accretion rate at the horizon.

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Over these time periods, the magneto-rotational instability (MRI) quality factors for all four simulations were quite good. We measured the quality factors Q_z (the mean number of cells per fastest growing vertical wavelength) and Q_ϕ (the mean number of cells per fastest growing toroidal azimuthal wavelength) by constructing mass-weighted averages on spherical shells and then time averaging over the same periods for which inflow equilibrium was evaluated. When $a/M=0,{Q}_{z}\simeq 25$ and $15\lesssim {Q}_{\phi }\lesssim 20$ across the entire range of radii; when $a/M=0.5$, $15\lesssim {Q}_{z}\lesssim 30$ and $10\lesssim {Q}_{\phi }\lesssim 25$, with comparable quality factors for $a/M=0.9$ and $a/M=0.99$. These numbers compare very favorably with the standard set by Hawley et al. (2011) and Sorathia et al. (2012) that for proper description of the nonlinear development of MHD turbulence in accretion disks if ${Q}_{\phi }\gtrsim 20$, then Q_z should be $\gtrsim 10$.

Given the density, cooling rate, and four-velocity of the fluid at each point in the simulation, we construct a disk surface-temperature map as in Noble et al. (2011). For the purpose of ray tracing to infinity, we remove the geometric effects of finite disk thickness, assuming that all the thermal flux comes from a thin disk near the midplane, while still preserving the time and azimuthal variability in the simulation data. Kulkarni et al. (2011) do the former, but not the latter, leading to systematically softer spectra. The radiation is emitted from this surface with the limb-darkening and polarization properties appropriate for a scattering-dominated, optically thick atmosphere (Chandrasekhar 1960).

Since we are in this paper primarily focused on the thermal state, in which the coronal Compton-y parameter is expected to be small, we ignore any scattering in the corona (to a large degree, in the thermal state, the corona can simply be thought of as an extension of the accretion disk atmosphere). The photons therefore follow geodesic paths from the disk until they reach the distant observer or are captured by the event horizon. However, because we are also interested in polarization signatures, we include scattering of returning radiation off the optically thick disk body, as in Schnittman & Krolik (2009).

As a test of this method, we have also tried using emission models that include only the disk proper, as defined by the region between electron-scattering photospheres, as in Schnittman et al. (2013). This approach requires specifying a physical accretion rate in order to fix the density scale in cgs units. For $\dot{m}=0.03$ in Eddington units (roughly the lower limit of the thermal-dominant state), we find that, for the $a/M=0$ run, roughly 80% of the dissipation takes place in the disk proper, with that number rising to 90% for the spinning cases. For all cases, we find that the coronal and disk radial-emissivity profiles are nearly identical. Thus, if we were to include only the disk emission, the thermal luminosity would decrease slightly, but the spectral shape remains unchanged.

One of the most basic results that comes from these global MHD simulations is the total radiative efficiency of the thin accretion disk. Astrophysically, this number is critical for connecting the accretion history and growth of supermassive black holes in the universe with the observed luminosity and mass distribution functions (Soltan 1982). While it is almost impossible to measure the efficiency of an individual black hole, the shape of the thermal spectrum (which of course is observable) is empirically related to the radiative efficiency because higher efficiency is typically due to higher emission at small radii, producing higher temperatures and harder spectra.

In Table 1 we show several measures of the radiative efficiency. We define η (radiated) to be the ratio between the time-averaged luminosity emitted by the disk and the time-averaged rest-mass accretion rate (photon energy and observer time both measured at infinity). Similarly, η (captured) is the ratio of the time-averaged luminosity crossing the horizon to the time-averaged accretion rate (for photons crossing the horizon, the energy at infinity is easy to measure: ${E}_{\infty }=-{p}_{t}$). Thus the net radiative efficiency measured at infinity is η(radiated)−η(captured).

Table 1.  For a Range of Spins, the Emitted Bolometric Radiative Efficiency η (Radiated) and the Fraction of Flux Captured by the Black Hole η (Captured), for Both NT Disks and Harm3d Simulations

a/M	${\eta }_{{\rm{NT}}}$	${\eta }_{{\rm{NT}}}$	${\eta }_{{\rm{Harm3d}}}$	${\eta }_{{\rm{Harm3d}}}$	${\eta }_{{\rm{Harm3d}}}$
	(radiated)	(captured)	(radiated)	(captured)	(advected)
0.0	0.0572	0.0001	0.0607	0.0020	0.0080
0.5	0.0821	0.0005	0.0833	0.0016	0.0120
0.9	0.1558	0.0027	0.1600	0.0048	0.0115
0.99	0.2640	0.0087	0.2461	0.0111	0.0220

Note. Also listed is the fraction of advected thermal and magnetic energy in the MHD simulations.

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Here, ${\eta }_{{\rm{NT}}}$ is the classical Novikov–Thorne (NT) efficiency, and ${\eta }_{{\rm{Harm3d}}}$ is calculated from our simulations. Note that ${\eta }_{{\rm{NT}}}$ (radiated) is identical to the specific binding energy of a test particle orbiting at the ISCO. Finally, for the simulation data, we also show ${\eta }_{{\rm{Harm3d}}}$ (advected), a measure of the accreted energy carried into the hole in the form of heat and magnetic field. In principle, this energy could be liberated via additional radiation mechanisms.

As first shown in Thorne (1974), we find that very little radiation is actually captured by the black hole. Although the additional flux emitted near and inside the ISCO leads to somewhat higher capture fractions than the NT model would predict, it is still less than 5% of the total emitted flux even for $a/M=0.99$. In the end, the net radiative efficiency is quite close to ${\eta }_{{\rm{NT}}}$, as the additional MHD dissipation is negated by capture and advection, consistent with earlier work (Kulkarni et al. 2011; Noble et al. 2011). As shown in Noble et al. (2011), much of the increase in radiative efficiency relative to NT is due to the enhanced emission near the ISCO, both immediately inside and outside that radius.

In a certain sense, our figures for the efficiency are conservative estimates. By adopting an initial condition with nested dipolar field loops, we ensure that there is some net magnetic flux trapped on the event horizon, but in terms of the ϒ measure introduced by Gammie (1999), it is smaller than the maximum that can be achieved. Here we use the definition of McKinney et al. (2012), in which ${\rm{\Upsilon }}\equiv 0.2{\rm{\Phi }}/({r}_{g}\sqrt{\dot{M}c})$, for accretion rate $\dot{M}$, gravitational radius r_g, and Φ the integral of the absolute value of the radial magnetic field around a spherical surface containing the black hole.

When ${\rm{\Upsilon }}\simeq 6$ (roughly three times the value in our simulation), Avara et al. (2015) found that the radiative efficiency of a disk with somewhat greater aspect ratio ($h/r\simeq 0.1$) orbiting in an $a/M=0.5$ spacetime was roughly double what we found for the same spin parameter. This contrast suggests that the magnetic augmentation to radiative efficiency rises very sharply over the range in ϒ between these values. However, other sorts of field topology—higher-order multipoles in the poloidal field or intrinsically toroidal field—may exist, and how the efficiency behaves as a function of net toroidal flux, and so on, remains to be explored. All that can be said at the moment is that for ϒ comparable to or smaller than in our simulations, the internal magnetic stresses depend only weakly on the field topology (Beckwith et al. 2008).

We see significant variability of the accretion rate and radiative efficiency within each simulation, as well as qualitative differences from one simulation to the next (e.g., differences in the radial fluid properties that do not appear to vary monotonically with spin), representative of the stochastic nature of the turbulent MHD disks. In principle, we could carry out an ensemble of runs for each spin to measure both the mean radiative efficiency and the variance of the observed efficiency due to the random fluctuations in the simulations. However, even the four simulations we did carry out were extremely expensive computationally, and any more would be very difficult to justify.

In Figure 3 we plot the radial luminosity profile dL/dr for each of the black hole spins. In each plot, the solid red curve represents the time- and angle-averaged emitted flux from the Harm3d simulation data. The red dashed curve is the flux that reaches infinity, and the red dotted curve is the flux captured by the black hole. The solid black curve is the emitted flux from an NT disk.

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All four cases share a number of qualitative features. The emission per unit radial coordinate peaks a short distance outside the ISCO, then decreases slightly before leveling off in the plunging region. Inside a point roughly halfway between the ISCO and the horizon, the flux emitted by the inward-flowing gas is almost entirely captured by the black hole. Contrasts also exist. For example, compared to the others, the fraction of the luminosity emitted in the $a/M=0.5$ plunging region is relatively small. The fluctuations alluded to earlier are large enough that these contrasts may be due to our comparatively short averaging times.

Because the ISCO position in Boyer–Lindquist coordinates is a function of spin, a detailed comparison of the radial profiles of the four cases requires defining a new radial coordinate relative to which the emission profiles are the most similar. Beckwith et al. (2008) found that simply scaling the radial coordinate by ${r}_{{\rm{ISCO}}}$, that is, using $r^{\prime} \equiv r/{r}_{{\rm{ISCO}}}$, gives a roughly universal emissivity profile for NT disks, but not MHD disks. To unite MHD disks around black holes of different spins, we have instead developed a coordinate transformation based on the differential proper radial distance $d\tilde{r}={g}_{{rr}}^{1/2}{dr}$. This coordinate transformation requires a constant of integration, or alternatively a free parameter that allows us to choose the radius at which $\tilde{r}=r$. We do this at $r=\frac{3}{2}{r}_{{\rm{ISCO}}}$, corresponding closely to the radius of peak emissivity. Finally, we scale by ${r}_{{\rm{ISCO}}}$ to make the radial coordinate dimensionless: $r*\;\equiv \;\tilde{r}/{r}_{{\rm{ISCO}}}$. Note that, in these coordinates, the horizon is located at negative r^*, but the vast majority of the flux is emitted outside $r*\;=\;0$, where $r\approx 1.15{r}_{{\rm{hor}}}$ for all values of a/M. Details of the coordinate transformation and corresponding emissivity law are presented in the Appendix.

In Figure 4 we show the emission profiles of the four different MHD simulations posed in terms of the dimensionless radial coordinate r*. As expected, they all peak around $r*\;\approx \;1.5$, and at large r*, they all approach the Newtonian large-radius limit ${dL}/{dr}\propto {r}^{-2}$. We also show a simple analytic fit to the emissivity profile, which takes the form of a smoothly broken power law with inner slope of ${dL}/{dr}*\;\propto \;{r}^{*7/4}$ (SBPL; dashed curve), as well as the corresponding profiles for Novikov–Thorne disks ($a/M=0$, dotted curve; $a/M=0.9$, dot-dashed curve). The local flux from the disk is related to ${dL}/{dr}*$ through the relation

$$\begin{eqnarray}&&F(r*)=\displaystyle \frac{1}{4\pi r*}\displaystyle \frac{{dr}*}{{dr}}\displaystyle \frac{{dL}}{{dr}*}.\end{eqnarray} \tag{ 1 }$$

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Along with the emissivity profile, a simple disk model requires a prescription for the fluid velocity. Outside the ISCO, we can use circular, planar geodesic orbits as in Novikov & Thorne (1973). Inside the ISCO, we ignore the fluid's continuing loss of energy and angular momentum, assuming it follows plunge trajectories with constant specific energy and angular momentum ${\varepsilon }_{{\rm{ISCO}}}$ and ${{\ell }}_{{\rm{ISCO}}}$.

In Figure 5 we show how well these geodesic orbits agree with the time- and azimuth-averaged velocities from the $a/M=0.5$ simulation data sampled in the midplane. To emphasize the behavior in the plunging region, we plot ${u}^{\mu }$ versus r–${r}_{{\rm{horizon}}}$ (Boyer–Lindquist coordinates). The $a/M=0.5$ case demonstrates that we accurately capture spin effects while also including a nonnegligible plunge region; for $a/M=0.99$ almost the entire disk (described in Boyer–Lindquist coordinates) is outside the ISCO and thus on circular orbits.

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To determine quantitatively how use of our new model affects parameter inference, we adopt a simple fitting procedure in which the MHD spectrum is treated as observational data with an error on each energy bin proportional to the square root of the number of photons in that bin, then carry out a least-squares fit with a suite of NT and SBPL spectra. Rather than specify a photon count rate and detector response function, we simply assume a total of 10⁶ photons, deposited appropriately through 10 energy channels. Initially, we hold the observer inclination fixed, but the black hole mass, accretion rate, and distance are allowed to vary freely to minimize ${\chi }^{2}$. The results are shown in Table 2 for target parameters of $M=10{M}_{\odot }$ and $\dot{m}=0.1$ and under the assumption that all of the cooling power is efficiently converted into thermal flux at the disk surface.

These results are in agreement with the findings of Noble et al. (2011), finding a modest but consistent overestimate of the spin parameter when fitting to a NT disk model. Kulkarni et al. (2011) also found a systematic error that was due to assuming the NT model, but of slightly smaller magnitude than both Noble et al. (2011) and the new results presented here. Some possible explanations for these differences were outlined in detail in Noble et al. (2011). One potential distinction mentioned there was the difference between averaging over t and ϕ before or after calculating the spectra. We will discuss this issue in more detail in Section 4.2, but for the purposes of facilitating direct comparison with Kulkarni et al. (2011), the fits in Table 2 were done after averaging the simulation data over time and azimuth, then carrying out a single ray-tracing spectral calculation.

The fits quoted in Table 2 were carried out over the energy range 2–10 keV, appropriate for the large body of archival data from black holes in the thermal state. From Figure 6, it is clear that this energy range is also ideal for distinguishing between different spins for systems with comparable masses and luminosities to the fiducial $M=10\;{M}_{\odot }$ and $\dot{m}=0.1$. When performing the fits over a wider energy range 0.1–10 keV, the outer disk contributes greater weight, yet this part of the spectrum is nearly independent of black hole spin, so the net effect is almost negligible, and we find nearly identical values for the best-fit spins.

Not surprisingly, the SBPL model does a better job at recovering the spin parameter for the target spectra, particularly for smaller values of the spin, where emission from the ISCO region is relatively more important. The small variations from perfect fitting are likely due to the stochastic nature of the MHD simulations, as described above. There does not appear to be any clear trend for systematic offsets with observer inclination. Taken together, the deviations between the target spins and the best-fit SBPL spins can be taken as a rough estimate for the systematic error in the spin-fitting method. Comparing with Table 1, it is clear that the radiative efficiency alone is not a sufficient measure of the thermal emissivity: the MHD simulations give efficiencies that are very similar to NT, but the shapes of the observed thermal spectra show deviations from the classic thin disk model. And ultimately, it is the spectral shape that is observed, not the radiative efficiency.

As shown in Figure 6, the shape of the thermal spectrum in the 2–10 keV band is a function of the black hole spin. For a fixed bolometric luminosity, a larger spin leads to a higher disk temperature because the total radiative power is emitted from a smaller region. However, relativistic beaming and Doppler blue-shifting of photons emitted from a highly inclined disk have a similar hardening effect on the observed spectrum. Together, there is an almost perfect degeneracy between the black hole spin and the observer inclination angle. To demonstrate this, we plot in Figure 7 contours of ${\rm{\Delta }}{\chi }^{2}$ between a simulated target spectrum generated from the MHD data with $i=60^\circ$ and $a/M=0.5$, and a template of fitting spectra based on the SBPL model. The simulated spectral data have Poisson error bars consistent with an observation of roughly 10⁶ photons divided into 10 energy bins, typical of what might be expected from a first-generation X-ray polarimeter (Weisskopf et al. 2013; Jahoda et al. 2014).

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This degeneracy is a well-known problem with the continuum-fitting method and is generally resolved by assuming a specific inclination derived from the optical light curve of the distorted stellar companion (Orosz 2002; Remillard & McClintock 2006). One systematic problem with this approach is that the inclination of the inner disk may very well be misaligned with the inclination of the binary orbit (Li et al. 2009).

Another important issue in fitting observations is properly treating the scattering atmosphere of the accretion disk. Because the electron scattering opacity dominates over free–free absorption for the temperatures and densities expected in the inner disk, we expect a Wien spectrum for the local emission (Rybicki & Lightman 2004). For fitting our simulated spectra to NT or SBPL models, motivated by Shimura & Takahara (1995) we used a simple hardening factor of 1.8, as in Noble et al. (2011). When fitting real observational data, McClintock et al. (2014) find that more detailed atmospheric models are required to achieve a good fit (Davis & Hubeny 2006).

As mentioned above in Section 3, there is also the issue of t and ϕ averaging the data before carrying out the ray-tracing calculation. This averaging has the effect of softening the observed spectrum. Consider an annulus of the disk with a uniform temperature T₀. If instead the total thermal flux were emitted from a series of discrete hotspots of temperature T_h and covering fraction C_h, while maintaining the same total flux, the new spectrum is a higher-temperature blackbody with color-correction factor ${f}_{c}={C}_{h}^{-1/4}$.

As a test of this effect, we generated spectra both before and after averaging in time and azimuth, and we find that by first averaging the simulation data and using a color-correction factor of f_c = 1.8, we can exactly match the spectra that result from using the full time-varying 3D data with f_c = 1.6 and then integrating the light curves in time and azimuth. Of course, Nature uses the latter method, so if the observational data fit an azimuthally symmetric model best with a certain hardening function, one must keep in mind that this in fact corresponds to a somewhat softer local spectrum coming from an inhomogeneous disk. Careful analysis of X-ray timing properties may be one promising way of breaking this degeneracy and lead to greater understanding of the properties of the accretion disk atmosphere.

As discussed in detail in Schnittman & Krolik (2009), X-ray polarization observations of black holes in the thermal state provide a powerful new way to probe the dynamics of the inner accretion flow and thereby measure the black hole spin. Chandrasekhar (1960) showed that the photons emitted from a plane-parallel, scattering-dominated atmosphere are polarized in a direction parallel to the emitting surface. The degree of polarization ranges from zero for face-on observers up to 11% for edge-on observers. Because of relativistic effects such as beaming and gravitational lensing, the net polarization of light from the inner regions of an accretion disk is reduced in magnitude, and the angle of polarization is rotated. Because higher-energy photons come from the inner regions, these relativistic effects are energy dependent, and thus the polarization spectrum can be used to probe the inner disk.

This technique was first proposed over 30 years ago by Connors et al. (1980), who considered only direct radiation from the accretion disk. Schnittman & Krolik (2009) included the effects of returning radiation, photons emitted from the disk, deflected by the black hole, and reflected off the far side of the disk before reaching the distant observer. These photons are strongly polarized by the large-angle scattering geometry and impart a significant polarization signature to the total radiated flux.

For the purpose of analyzing the X-ray polarization, we define the black hole spin axis to be along the vertical direction, so a polarization angle of $\psi =0^\circ$ is parallel to the disk. This is the case for the low-energy photons emitted at large radii, where relativistic effects are negligible. The high-energy photons emitted close to the black hole are more likely to contribute to the returning radiation and scatter into the vertical polarization mode. Since the disks around rapidly spinning black holes extend closer to the horizon, they should exhibit more vertical polarization, and the transition from horizontal to vertical polarization occurs at a lower energy.

In Figure 8 we plot the angle and degree of polarization from the MHD simulation data as a function of photon energy for our fiducial parameters $M=10\;{M}_{\odot }$, $L=0.1\;{L}_{{\rm{Edd}}}$, and $i=75^\circ$. The spectropolarization signal is clearly sufficient to distinguish between different spins with even modest energy resolution. The near overlap of the $a/M=0$ and $a/M=0.5$ curves is due to the anomalously low (high) flux from the $a/M=0.5$ ($a/M=0$) simulations, as described above. Real observations would cover a much longer duration of time, smoothing out the stochastic variability seen in the much shorter simulations.

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Polarization also provides a complementary observable that can break the spin-inclination degeneracy inherent in the traditional spectral continuum fitting method (Li et al. 2009). To demonstrate its power, we perform the same sort of spectral fitting shown in Figure 7, but using the broadband polarization information displayed in Figure 8. Instead of the intensity spectrum, we use the energy-dependent Stokes parameters Q_ν and U_ν, as in Schnittman & Krolik (2009). Again, we use a simulated spectrum from a target disk with $a/M=0.5$ and $i=60^\circ$. For a broadband photoelectric polarimeter like PRAXyS (Jahoda et al. 2014), 10⁶ photons divided roughly equally between 10 energy bands give a 1σ uncertainty on the polarization degree of about 1% per band. Fitting the polarization information with the SBPL model in the range 2–10 keV, we get the confidence contours shown in Figure 9. At 2 keV there is still significant returning radiation, so the polarization signal is not a perfect probe of disk inclination. A soft X-ray polarimeter sensitive over the 0.1–0.5 keV band would significantly improve sensitivity to disk inclination.

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The slight spin-inclination degeneracy seen in Figure 9 can be understood as follows. From Figure 8 it is clear that the location of the transition from horizontal to vertical orientation occurs at lower energy for increasing spin. This transition is due to a small number of highly polarized returning photons dominating over the vast majority of weakly polarized direct photons. At higher inclinations, the direct signal is more highly polarized, so it is harder to overcome with returning photons, and thus the transition point shifts to higher energy where the contribution from returning radiation is higher. Therefore, to maintain a constant transition energy, increasing inclinations require a higher spin.

Comparing the contours in Figure 9, we see that the degeneracy due to continuum fitting and that due to polarization intersect at a significant angle. As a result, the range of both parameters consistent with a good fit is substantially reduced. Clearly, X-ray spectropolarimetry observations have the potential to be of great benefit to measurements of black hole spin and inclination.

For all of the contours in Figure 9, we used the SBPL model for the radial emissivity profile. When redoing them with an NT model as in Noble et al. (2011), not surprisingly, we get very similarly shaped contours for ${\rm{\Delta }}{\chi }^{2}$, but with a systematic offset toward higher spin. However, we find that the actual minimum in ${\chi }^{2}$ is consistently lower for the SBPL model than for NT. For the fiducial observations used above with 10⁶ photons over the 2–10 keV band, the combined spectral and polarization fits to the MHD data prefer the SBPL model at roughly 90% confidence. For a photon-limited (as opposed to background-limited) detector, this significance would simply grow with more and longer observations. Thus, X-ray polarimetry combined with X-ray spectroscopy can also be a significant aid in distinguishing different emissivity profiles, thereby creating a new diagnostic of accretion dynamics in the innermost regions of accretion disks around black holes.

Of course, some caveats are in order. The results of Figure 9, while based on a more physically realistic emissivity profile, still include some important simplifying assumptions. We assume the disk is razor thin and in a single plane perpendicular to the black hole spin, and we also ignore contributions from the corona. For a disk with finite aspect ratio H/r, Newtonian geometric effects alone will lead to opposing sides of the disk having a direct line of sight with each other, in turn affecting the polarization signature. While these effects should not be energy dependent, and thus not directly connected to the properties of the inner disk, they would still complicate the interpretation of any polarization measurement. As to the question of a tilted disk, the Bardeen–Petterson effect is expected to closely align any gas with the equatorial plane in the inner regions where the X-ray flux is greatest (Bardeen & Petterson 1975; Li et al. 2009).

There is evidence for a hot, optically thin corona above the disk in numerous observations of the thermal state (Remillard & McClintock 2006). The inverse-Compton scattering of thermal photons through a sandwich corona also leads to a swing from horizontal to vertical polarization (Schnittman & Krolik 2010). Thus there is some degeneracy between spin effects and coronal scattering, but this complication can be mitigated by focusing on the subset of observations with an overwhelming thermal component in the spectrum (McClintock et al. 2011).

Disk corrugation or potentially strong magnetic fields could also complicate the polarization signal (Davis et al. 2009). However, polarization is most sensitive to the global geometry, so even if the disk surface is highly corrugated on a small scale, the scattering-induced polarization signal described by Chandrasekhar (1960) should still dominate when integrating over the entire disk. Similarly, returning radiation is a global effect and should be largely insensitive to small-scale structure (Schnittman & Krolik 2009). Finally, the dilution of the polarization signal due to Faraday rotation by propogation through a turbulent magnetic field will most strongly affect photons below the thermal peak of ∼1 keV (Davis et al. 2009), outside the range of proposed first-generation detectors.