The role of magnetic fields in the innermost disk

While stating the assumption of zero-torque at the radius of marginal stability, Page & Thorne [115] noted that significant magnetic fields might allow torques to be transmitted some distance within the radius of marginal stability. This idea has been rejuvenated by recent work on MRI driven turbulence.

Figure 5: Model dissipation profiles for torqued time-independent accretion disks (from [126]). The thick line shows the dissipation profile for a standard non-torqued disk (i.e., one in which the ZTBC applies) around a near-extremal Kerr black hole with spin parameter $a=0.998$. The thin solid lines show the torque-induced dissipation component $D_{\rm tor}$ for spin parameters of (from left to right) $a=0.998,0.99,0.9,0.5,0.3$, and $0$. Note how the torque-induced dissipation profile becomes more centrally concentrated as the spin parameter is increased. The dashed lines show the torque-induced component of the emitted flux, including the effects of returning radiation.

Significant insight can be gained from analytic considerations, assuming that the inner regions of the accretion disk are time-independent and axisymmetric. In separate treatments by Julian Krolik and Charles Gammie [126,127,128], it was shown that magnetic fields might be effective at transporting angular momentum from material in the plunging region into the main body of the accretion disk. Accompanying this angular momentum transport, we would expect either local dissipation/radiation of the liberated binding energy, or energy transport into the radiatively-efficient portions of the disk. Either way, the principal consequence is to enhance the efficiency $\eta$ of the disk above the value predicted by the Page & Thorne models. The simplest analytic approach (followed by Agol & Krolik [126]) is to model this effect as a finite torque applied at the radius of marginal stability, i.e, an explicit violation of the ZTBC. In addition to modeling the magnetic connection between the accretion disk and the plunging region, this torque can also model any magnetic connection between the accretion disk and the rotating black hole itself [129]. This torque performs work on the accretion disk, thereby adding to the power ultimately dissipated in the disk and increasing the disk efficiency.

This salient aspects of this model are illustrated in Fig. 5 which displays the dissipation rate in the disk per unit disk face area as a function of Boyer-Lindquist radius, $D(r)$. The thick line in this plot shows the dissipation law for a standard Page & Thorne accretion disk around a black hole with spin parameter $a=0.998$ [$D_{\rm PT}(r;a=0.998M)$ given by eqn. 19]. Also shown are the additional dissipation profiles resulting from non-zero torques applied at the radius of marginal stability $D_{\rm tor}(r;a)$. It can be shown that [126]

$$D_{\rm tor}(r;\hbox{$a^*$})=\frac{3\dot{M}r_{\rm ms}^{3/2}C_{\rm ms}^{1/2}\Delta \eta}{8\pi r^{7/2}C(r)}$$ 16

where $C(r)=1-3M/r+2\hbox{$a^*$}M^{3/2}/r^{3/2}$ and $\Delta\eta$ is the additional disk efficiency induced by the torque. Strictly, this is true if the additional torque does not directly affect the dynamics of the disk. In this case, the total dissipation is just the sum $D(r;\hbox{$a^*$})=D_{\rm PT}(r;\hbox{$a^*$})+D_{\rm tor}(r;\hbox{$a^*$})$. The torque-induced component to the dissipation is much more centrally concentrated than the standard disk dissipation profile. In fact, it is so centrally concentrated that a significant fraction of the radiative flux emitted from the disk surface will return to the disk surface due to strong light bending effects. This ``returning radiation'' will be absorbed or scattered from the disk surface and, hence, adds to the local energy budget relevant for determining the observed amount of radiative flux from a given radius. Including this effect, the total emitted flux per unit area of the disk can be written as

$$D_{\rm tot}(r;\hbox{$a^*$})=D_{\rm PT}(r;\hbox{$a^*$})+D_{\rm tor}(r;\hbox{$a^*$})+D_{\rm ret}(r;\hbox{$a^*$}),$$ 17

where $D_{\rm ret}(r;\hbox{$a^*$})$ is the term accounting for returning radiation. From the point of view of an observer riding on the disk at a large radius, the returning radiation produces a ``mirage'' of the central disk raised to some height above the disk plane. Thus, at large radii, irradiation due to returning radiation scales as $r^{-3}$ while the dissipated flux dominates at small radii. Thus, as shown in [126], the term $D_{\rm ret}$ is approximately given by

$$D_{\rm ret}(r;\hbox{$a^*$})\approx\frac{3M\dot{M}\Delta\eta R_\infty(\hbox{$a^*$})}{8\pi r^3},$$ 18

where $R_\infty(\hbox{$a^*$})$ is well described by the fitting formula given by Agol & Krolik [126]. The function $D_{\rm tot}(r;\hbox{$a^*$})$, which includes the effects of returning radiation, is also shown in Fig. 5.

The analytic models of Gammie, Agol and Krolik [128,126] suggest that these torqued accretion disks can become super-efficient ($\eta>1$). In this case, some fraction of the radiative power must be extracted from the spin of the black hole. This is a particular realization of the Penrose process, with the accreting material in the innermost parts of the plunging region being placed on negative energy orbits by magnetic connections with the accretion flow at larger radii.

Simulations are required to move beyond the restrictive geometry and physics of the analytic models. To date, almost all of the relevant MHD simulations have been inherently non-relativistic and have mocked up the effects of General Relativity via the use of the Paczynski-Wiita [130] pseudo-Newtonian potential ,

$$\Phi=\frac{M}{r-2M},$$ 19

where, to recall, we have set $G=c=1$. Newtonian computations using this modified point-source potential serve as remarkably effective toy-models for examining dynamics in the vicinity of a Schwarzschild black hole. Simulations of MHD accretion disks within this potential have been presented in a number of papers[124,125,131,132,133]. These works support the hypothesis that energy and angular momentum can be transported out of the plunging region, although the universality of these processes is still the subject of debate [131,132].

We note in brief that, at the time of writing, fully relativistic MHD codes capable of studying the evolution of the MRI and MHD turbulence around rapidly rotating black holes are just starting to become available [134]. Simulations performed with such codes will be extremely important in assessing the true nature of the innermost regions of black hole accretion disks.