Theoretical framework

Firstly, we shall review some of the pertinent theory related to the geometrically-thin accretion disks of the type thought to be operating in Seyfert nuclei such as MCG-6-30-15. The standard thin disk model of black hole accretion was developed in a Newtonian setting by Shakura & Sunyaev (1973), and extended into a fully relativistic theory by Novikov & Thorne (1974) and Page & Thorne (1974; hereafter PT). In this model, the accretion disk is assumed to be geometrically-thin, radiatively-efficient, and in a steady-state. Furthermore, it is postulated that the disk experiences zero torque at the radius of marginal stability. With these assumptions, one can compute the dissipation rate, and hence total radiative flux as a function of radius and black hole spin:

$$D_{\rm PT}(r;a)=\frac{\dot{M}}{4\pi r}{\cal F},$$ 1

where we have defined the function ${\cal F}$

with,

This dissipation profile is zero at $r=r_{\rm ms}$ due to the zero-torque assumption, increases to a broad peak at $r\sim 1.5r_{\rm ms}$, and then declines as $\epsilon\propto r^{-3}$ at large radii (thick line in Fig. 3).

It has been realized in recent years that the assumption of zero-torque at the radius of marginal stability may be invalidated by magnetic connections between the Keplerian portion of the accretion disk and either the plunging region (i.e., the region $r<r_{\rm ms}$) or the rotating event horizon itself (Krolik 1999; Gammie 1999; Li 2002, 2003). The formal generalization of the PT disk models including a (arbitrary) torque at $r=r_{\rm ms}$ is given by Agol & Krolik (2000), who parameterized the extra torque via the corresponding enhancement in the radiative efficiency of the disk, $\Delta \eta$. The work done by the torque on the disk produces a new component to the disk dissipation that is very centrally concentrated,

$$D_{\rm tor}(r;a)=\frac{3\dot{M}r_{\rm ms}^{3/2}C_{\rm ms}^{1/2}\Delta \eta}{8\pi r^{7/2}C(r)},$$ 2

where $C(r)=1-3M/r+2a M^{3/2}/r^{3/2}$ and $\Delta \eta$ is the additional disk efficiency induced by the torque (see Fig. 3). When this component is substantial, the overall dissipation profile is so concentrated that one can no longer ignore (even at a crude level) the effects of returning radiation (Cunningham 1975). In the limiting case of an infinite-efficiency disk (i.e., a disk that derives its whole luminosity from work done by the central torque) around a near-extremal Kerr black hole, as much as half of the radiation emitted from the disk can return to the disk via the action of strong light bending. Agol & Krolik (2000) showed that the effect of returning radiation is to produce an extra source of disk illumination described by the expression,

$$D_{\rm ret}(r;a)\approx\frac{3M\dot{M}\Delta\eta R_\infty(a)}{8\pi r^3},$$ 3

where $R_\infty(a)$ is well described by the fitting formula given by Agol & Krolik (2000).

Thus, in this ``generalized standard model'' of thin-disk accretion onto black holes, the energy that is dissipated per unit proper time and per unit proper surface area is

$$D_{\rm tot}(r;a)=D_{\rm PT}(r;a)+D_{\rm tor}(r;a).$$ 4

Suppose that a fraction $f(r)$ of this energy is transported into a disk corona and hence radiated in the hard X-ray continuum (rather than as soft thermal emission from the optically-thick part of the accretion disk). If the corona is geometrically-thin then, with the exception of returning radiation, we need not consider light bending effects when deducing the X-ray flux that irradiates the optically-thick accretion disk (and hence gives rise to the observed reflection spectrum). Assuming that the corona is geometrically-thin and emits isotropically, the optically-thick disk will be irradiated by X-rays with an intensity,

$$I_{\rm X}(r;a)=f(r)\left[D_{\rm PT}(r;a)+D_{\rm tor}(r;a)\right]+\bar{f}D_{\rm ret}(r;a),$$ 5

where $\bar{f}$ is an appropriate averaging of $f(r)$ over the inner radii of the disk that contributes to the returning radiation. Given a functional form for $f(r)$, this irradiation profile can be used to construct the appropriately weighted relativistic smearing kernel that can then be convolved with the rest-frame reflection spectrum, thereby producing a full spectral model of smeared reflection from the disk.