Non-relativistic radiative accretion disks & the $\alpha$-prescription

Initially, we shall discuss non-relativistic models of accretion disks. Here, we follow the presentations and arguments presented in [110], [111], and [112]. Throughout this discussion, we assume cylindrical polar coordinates, $(r,\phi,z)$.

We consider a rotating mass of gas around a point mass $M$, which represents the black hole. We assume that the gas can radiate-efficiently and, hence, that it maintains a temperature well below the virial temperature of the system. Furthermore, we assume that an angular momentum transport mechanism operates and can be characterized by a kinematic viscosity $\nu$, but we suppose that the time scale for redistributing angular momentum is long compared with either the orbital or radiative time scales. Under these assumptions, the gas should form a geometrically-thin disk (with vertical scale height $h$) in which fluid elements are orbiting in almost circular paths with angular velocity $\Omega(r)$. A small radial velocity $v_r(r,t)$ corresponds to the actual accretion flow.

If we denote the surface mass density of the disk as $\Sigma(r,t)$, the conservation of mass and angular momentum can be written as

$$r\frac{\partial \Sigma}{\partial t}+ \frac{\partial}{\partial r}(r\Sigma v_r)=0,$$ 5

and

$$r\frac{\partial}{\partial t}(\Sigma r^2\Omega)+\frac{\partia... ..._r\Omega)= \frac{1}{2\pi}\frac{\partial {\cal G}}{\partial r},$$ 6

where

$${\cal G}(r,t)=2\pi r^3\nu\Sigma\frac{\partial\Omega}{\partial r}$$ 7

is the torque exerted on the disk interior to radius $r$ by the flow outside of this radius. These equations can be combined, using the fact that in Newtonian gravity $\Omega\propto r^{-3/2}$, to obtain

$$\frac{\partial \Sigma}{\partial t}=\frac{3}{r}\frac{\partial... ...r^{1/2}\frac{\partial}{\partial r}(\nu\Sigma r^{1/2}) \right].$$ 8

Noting that $\nu$ may be a function of the local variables in the disk, this has the form of a non-linear diffusion equation governing the behavior of $\Sigma(r,t)$ given some initial state. Given a solution for $\Sigma(r,t)$, the radial velocity is

$$v_r=-\frac{3}{\Sigma r^{1/2}}\frac{\partial}{\partial r}(\nu \Sigma r^{1/2}).$$ 9

We need to know the behavior of $\nu$ in order to close these equations and permit a full determination of the radial structure of the accretion disk. All of the complexities currently ignored (such as the $z$-structure of the accretion disk, and the detailed microscopic physics) enter into the problem via $\nu$. However, in the case of time-independent accretion disks (i.e. $\partial/\partial t = 0$), it is possible to determine some of the most interesting observables independent of our ignorance of $\nu$. In particular, it is readily shown (e.g. Chapter 5 of [112]) that the viscous dissipation per unit disk face area is

$$D(r)=\frac{3GM\dot{M}}{8\pi r^3}\left[ 1-\left(\frac{r_{\rm in}}{r}\right)^{1/2}\right],$$ 10

where $\dot{M}=2\pi r\Sigma(-v_r)$ is the rate at which mass is flowing inwards through the disk, and $r_{\rm in}$ is some inner boundary to the disk where the torque ${\cal G}$ goes to zero. One peculiarity of viscous accretion flows becomes apparent upon an examination of this last expression -- for $r\gg r_{\rm in}$, the rate that energy is lost from the disk is three times the local rate of change of binding energy. That is to say that for a ring of the disk located at $r\gg r_{\rm in}$, two thirds of the dissipation is attributable to energy that is `viscously'' transported from the interior disk, while the remaining one third is attributable to the local rate of change of binding energy. Of course, to compensate for this effect, the innermost regions of the disk radiate less than the local rate of change of binding energy. Integrating over the whole disk, the total luminosity of the disk is $L_{\rm disk}=GM\dot{M}/2r_{\rm in}$. An equal amount of energy remains in the form of kinetic energy of the accreting material as it flows through $r=r_{\rm in}$. If there is a (slowly-rotating) star at the center of the disk (as opposed to a black hole), this remaining energy can be radiated in the disk-star boundary layer.

If we suppose that the viscously dissipated energy is radiated as a black body spectrum, energy conservation [$\sigma_{\rm SB} T(r)^4=D(r)$; where $\sigma_{\rm SB}$ is the Stephan-Boltzman constant] gives the temperature of the disk surface as a function of radius,

$$T(r)=\left(\frac{3GM\dot{M}}{8\pi\sigma_{\rm SB} r^3}\left[ 1-\left(\frac{r_{\rm in}}{r}\right)^{1/2}\right]\right)^{1/4}.$$ 11

For a mass accretion rate that scales with mass (i.e. for a fixed ratio between the source luminosity and the Eddington limit) and a scaled radius $r/M$, the temperature of the disk scales weakly with the black hole mass $T\propto M^{-1/4}$. The accretion disks around supermassive black holes generally possess a characteristic disk temperature of $T\sim 10^5-10^6{\rm\thinspace K}$, making them optical and ultraviolet sources. On the other hand, accretion disks around stellar mass black holes are appreciably hotter with $T\sim 10^7{\rm\thinspace K}$, making them sources of thermal soft X-rays.

For a thin disk, there are no appreciable motions or accelerations in the $z$-direction. Hence the $z$-structure (i.e., the vertical structure) of the disk is determined by a hydrostatic equilibrium condition,

$$\frac{1}{\rho}\frac{\partial p}{\partial z}=\frac{\partial}{\partial z}\left[ \frac{GM}{(r^2+z^2)^{1/2}}\right],$$ 12

where $p$ is the pressure. Noting that the sound speed is $c_{\rm s}\sim \sqrt{p/\rho}$, the hydrostatic equilibrium expression can be used to see that the vertical scale height $h$ is,

$$h\sim \frac{c_{\rm s}r}{v_{\rm K}},$$ 13

where $v_{\rm K}\equiv (GM/r)^{1/2}$ is the local Keplerian velocity. Thus, an accretion disk is geometrically thin ($h/r\ll 1$) if the Keplerian velocity is highly supersonic.

In order to study the detailed physical structure of accretion disks, or any aspect of their time-variability (including the stability of accretion disks), we require a knowledge of the viscosity. In an extremely successful application of dimensional analysis, Shakura & Sunyaev [110] parameterized the viscosity as $\nu=\alpha c_{\rm s}h$, where $c_{\rm s}$ is the sound speed in the disk, $h$ is the vertical scale height and $\alpha$ is a dimensionless parameter which, on general hydrodynamical grounds, is argued to be less than unity. Furthermore, they made the supposition that $\alpha$ is a constant for a given accretion disk. The resulting family of disk models are referred to as $\alpha$-models (or Shakura-Sunyaev models). Given the $\alpha$-prescription, we can solve for the detailed radial disk structure (the results of this exercise are given in [110] and nicely reviewed by [112]). In addition, the assumptions underpinning the basic equations of the radiatively-efficient thin accretion disk can now be checked post-priori. From eqn. 13 it can be seen that the radial velocity is given by

$$v_r\sim \frac{\nu}{R}\sim \frac{\alpha c_{\rm s} h}{r}\ll c_{\rm s}.$$ 14

Thus, the radial inflow is very subsonic. Using the radial force equation, this also implies that the orbital velocity is very close to Keplerian. Furthermore, we can relate the various fundamental time scales governing the disk. Let the dynamical time scale be defined as $t_{\rm dyn}=\Omega^{-1}$. It can be shown that the time scale on which the dissipated energy is radiated from the disk, the thermal time scale, is given by $t_{\rm th}\sim t_{\rm dyn}/\alpha>t_{\rm dyn}$. Finally the time scale on which angular momentum is redistributed, the viscous time scale, is given by $t_{\rm visc}\sim r^2/\nu\sim (r/h)^2t_{\rm th}\gg t_{\rm th}>t_{\rm dyn}$. The ordering of these time scales, together with the fact that the disk material orbits the black hole in almost Keplerian orbits, justifies the assumptions listed at the beginning of this section.

While the $\alpha$-prescription is extremely useful, allowing us to construct simple analytic models of accretion disks, it is important to realize the severe limitations of this approach. As will be discussed in § 3.2.3, we now believe that MHD turbulence is responsible for the angular momentum transport that drives accretion. While the angular momentum transport resulting from MHD turbulence can be approximately parameterized in terms of an ``effective $\alpha$'', there are circumstances where it is incorrect to treat the accretion flow as an unmagnetized gas with an anomalously high kinematic viscosity ([108,113]. Even when used within its domain of validity, the actual value of $\alpha$ applicable to real systems is very unclear with estimates covering full range from essentially zero to unity (see discussion in [112]).