Comparison of the generalized standard disk model with data

We now compare the EPIC pn data for MCG-6-30-15 with spectral models constructed from this generalized standard model of thin-disk accretion. However, we must first choose a functional form for $f(r)$, the fraction of the dissipated energy released in the irradiating X-ray continuum. Here, we choose the function form:

where $r_{\rm out}$ can be considered as a ``coronal truncation radius''. We also examine the situation in which $f(r)$ is a powerlaw in radius (see below).

Using this form for $f(r)$ in eqn. 11, we construct new relativistic smearing functions and hence a spectral model that can be compared with the data. This model (and indeed all models presented in the rest of this paper) assume a near-extremal Kerr black hole (with spin $a=0.998$) and employ the Laor (1991) relativistic transfer function. We shall refer to the most general form of our model, where $\Delta \eta$ and $r_{\rm out}$ are free parameters, as tTORQUED (shorthand for truncated-TORQUED disk).

Figure 4: Fits of physical accretion disk models to the time average EPIC-pn spectrum of MCG$-$6-30-15. The fits displayed here were performed on the 2-10keV data, although only the 3-9keV range is shown for clarity. See Section 3.4 and Table 2 for details of these models and the fits.

Figure 5: Confidence contours on the ($\Delta \eta$,$r_{\rm out}$)-plane for the tTORQUED model applied to the time-averaged 2-10keV EPIC-pn data. From top to bottom, contours are 68%, 90% and 95% for two interesting parameters. See Section 3.4 for details.

The results of fitting model tTORQUED to the EPIC pn data are reported in Fig. 4a, Fig. 5 and Table 2. Examination of the confidence contours in the $(r_{\rm out},\Delta\eta)$-plane shows that, within the context of this model, the data require the disk to be both torqued (i.e., $\Delta\eta>0$) and possess a finite coronal truncation radius at better than the 95% confidence level for two interesting parameters. In fact, the data require a very strongly torqued disk, with $\Delta\eta>22$ (i.e., 2200%) at the 90% confidence level for one interesting parameter. In the language of Agol & Krolik (2000), the data argue for an ``infinite-efficiency disk'', in which the dominant energy source is the black hole spin as opposed to gravitational potential energy of the accretion flow.

We explore the constraints imposed by these data further by restricting parameters of the tTORQUED model and examining the effect on the goodness-of-fit. Firstly, we consider the case in which the disk is subject to a torque at $r=r_{\rm ms}$ but the corona is not truncated (i.e., $r_{\rm out}\rightarrow \infty$; we refer to this as the TORQUED model). From Table 2, it can be seen that the goodness-of-fit parameter increases slightly ($\Delta\chi^2=7$ for one less degree of freedom in both the 2-10keV and 0.5-10keV fits). An application of the F-test suggests that this is a significantly worse description of these data at the 99.2% level. However, Protassov et al. (2002) have pointed out that it is formally incorrect to use the F-test in this case; the TORQUED model lies on one boundary of the parameter space describing tTORQUED (the $1/r_{\rm out}=0$ boundary), and this fact can skew the probability distribution of the goodness of fit parameter. Due to this caveat, we consider that the evidence for coronal truncation is marginal.

Secondly, we assess the evidence for the presence of the inner torque at $r=r_{\rm ms}$. If we impose the restriction that $\Delta\eta=0$, we have an irradiation profile that follows a PT dissipation profile, albeit with an outer truncation radius. We refer to this model as tPTDISK. As reported in Table 2 (also see Fig. 4b), the goodness-of-fit parameter increases by $\Delta\chi^2=13$ upon the removal of this one degree of freedom from the models. The F-test implies that this is a significantly worse description of the data at the 99.97% level. Note that we do not impose the restriction that $\Delta\eta>0$ in our tTORQUED fits and, hence, the restricted model tPTDISK does not lie on the boundary of the parameter space describing tTORQUED. Thus, the restriction on the application of the F-test raised by Protassov et al. (2002) does not apply here and we have no reason to distrust the F-test results. Hence, these data provide strong evidence for the presence of an inner disk torque.

We note that the best-fitting parameters of tPTDISK also might be inconsistent with the overall spectral energy distribution of MCG-6-30-15. The coronal truncation radius in these fits is constrained to be $r_{\rm out}=5.0^{+1.0}_{-0.7}\,r_{\mathrm g}$, the same as the half-light radius of the accretion disk ($r_{1/2}\approx 5r_{\mathrm g}$; Agol & Krolik 2000). Since 30-50% of the total radiative luminosity of this AGN is observed to emerge in the X-ray band (Reynolds et al. 1997), this result would imply an extremely high value of $f(r)$ (i.e., almost unity) in the inner disk.

We can also use this chain of reasoning to eliminate more extreme forms for the coronal dissipation fraction $f(r)$. In detail, we refit the tPTDISK model allowing $f(r)$ to have a power-law form, $f(r)\propto r^{-\lambda}$. The goodness of fit is close to that for the best-fit tTORQUED model. However, these fits require $\lambda>2$ at the 90% confidence level. Noting the trivial fact that $f(r)$ cannot exceed unity, integration of the coronal dissipation across the disk implies that at most 3% of the dissipated energy can be released in the X-ray corona. Again, this violates the constraints on the total energetics of this source by a factor of 5, even once we include the fact that the instantaneous X-ray flux drops by a factor of 2 when the source enters the Deep Minimum state.

Finally, we examine the doubly restricted model PTDISK in which $\Delta\eta=0$ and $r_{\rm out}\rightarrow \infty$. From Table 2 it can be seen that this is a much worse fit to the data, with $\Delta\chi^2=101$ (upon the restriction of two model parameters) compared with the most general tTORQUED model. Indeed, it can be seen in Fig. 4c that the line profile visibly misses the data in the sense that it is insufficiently redshifted. Furthermore, the PTDISK model constrains the inclination to be less than $6^\circ$ ($8^\circ$ if only the 2-10keV data are considered). If we force the inclination to be $28^\circ$ (the value deduced from the long ASCA observation of the ``normal'' state of this object by Tanaka et al. [1995]), the goodness of fit is further decreased by $\Delta\chi^2=38$ and $\Delta\chi^2=63$ for the 2-10keV and 0.5-10keV fits, respectively. In this case, the systematic residuals in the fit are further exaggerated (Fig. 4d).

In summary, we construct relativistic smearing functions weighted by physically-motivated irradiation profiles whose parameters include the extra radiative efficiency $\Delta \eta$ due to the torque that is applied to the $r=r_{\rm ms}$ (via MHD processes within the plunging region) and a coronal truncation radius $r_{\rm out}$. We have found strong evidence that the disk is strongly torqued and, at this instant in time, may well be radiating primarily via the work done by this torque (a so-called infinite-efficiency disk). There is weaker evidence for a radial dependence of $f(r)$ which we model as a truncation of the corona at $r_{\rm out}\approx 6GM/c^2$.