3 Application range of the $\beta$-viscosity

Let us now examine under which conditions the $\beta$-prescription allows for subsonic turbulence in geometrically thin accretion discs in active galactic nuclei (AGN), evolved binary systems (EBS) and young stellar objects (YSO). For this first exploration, we will be content with the usual one-dimensional steady state approach to compute the sound speed and the aspect ratio, the vertical structure being reduced to vertical averages. The reader can find the relevant equation set in Appendix A.

In a stationary Keplerian disc characterized by a steady accretion rate $\dot{M}$, the surface density $\Sigma$ is determined by the conservation of mass and momentum:

$$\dot{M} = 3 \, \pi \nu \Sigma ,$$ (7)

ignoring a correcting factor which is negligible sufficiently far from the center (see Frank et al. 1992). Applying the $\beta$-prescription (2), we find

 

$$\Sigma \simeq 2.43~10^5 \; \beta_{-5}^{-1}\, f_{\rm edd} \, x^{-1/2} \qquad {\rm g\,cm}^{-2}$$

$$\simeq 92 \; \beta_{-5}^{-1} \, \dot{m}_{16} \, M_0^{-1/2} \, r_{10}^{-1/2} \qquad {\rm g\,cm}^{-2}$$

$$\simeq 70 \, \, \beta_{-5}^{-1} \, \dot{M}_{-8} \, M_0^{-1/2} \, r_{\rm AU}^{-1/2} \qquad {\rm g\,cm}^{-2}$$ (8)

where $f_{\rm edd} \simeq 4.5 \; \dot{M}_0 \; M_{8}^{-1}$ is the Eddington factor (around 0.1 typically), M = M₈ 10⁸ $M_\odot = M_0$ $M_\odot$, $\dot{M} = \dot{M}_0$ $M_\odot = \dot{m}_{16}~10^{16}$g/s $= \dot{M}_{-8}~10^{-8}$ $M_\odot$/yr, $R= x \, R_{\rm S} = R_{10}~10^{10}$ cm $=r_{\rm AU}$ AU ( $R_{\rm S} = 2\,GM/c^2$being the Schwarzschild radius of the black hole), and $\beta_{- 5}=\beta~10^{5}$. These dimensionless variables are well suited to describe the three families of discs quoted above. Note that Eq. (8) is quite general and does not depend on the vertical structure.

Thermal equilibrium between radiative cooling and viscous heating provides another equation. Assuming that the disc is optically thick, we have (Frank et al. 1992):

$$\frac{16 \, \sigma T^4}{3 \, \kappa_{\rm R} \Sigma} = \frac{3 \, \Omega^2 \dot{M}}{8 \, \pi}$$ (9)

where $\sigma$ is the Stefan constant and $\kappa_{\rm R}$ the Rosseland mean opacity.

3.1 Regimes compatible with subsonic turbulence

In gas pressure supported discs, the hydrostatic equation yields directly the mid-plane temperature T:

$${P_{\rm g} \over \rho} = \Omega ^2 H^2 = {k T \over \mu {\rm m}_{\rm H}} ,$$ (10)

with the usual notations for the gas pressure $P_{\rm g}$, the density $\rho$, the Boltzmann constant k, the molecular weight $\mu$ and the mass of the hydrogen atom $m_{\rm H}$. It then follows from Eqs. (6), (8), (9) and  (10) that the turbulent Mach number in those discs is

 

$${\cal M}_{\rm t} \simeq 3.1 \; \beta_{-5}^{5/8} \, \kappa_{\rm R}^{-1/8} \mu^{1/2} \, f_{\rm edd}^{-1/4} \, M_8^{1/8} x^{-1/16}$$

$$\simeq 0.3 \; \beta_{-5}^{5/8} \, \kappa_{\rm R}^{-1/8} \mu^{1/2} \, \dot{m}_{16}^{-1/4} \, M_0^{7/16} r_{10}^{-1/16}$$

$$\simeq 4~10^{-2} \; \beta_{-5}^{5/8} \, \kappa_{\rm R}^{-1/8} \mu^{1/2} \, \dot{M}_{-8}^{-1/4} \, M_0^{7/16} r_{\rm AU}^{-1/16} .$$ (11)

Note the relatively strong dependence of the Mach number on the viscosity parameter and its weak dependence on the opacity and specially the radius.

We conclude that the $\beta$-prescription is certainly applicable to gas pressure dominated accretion discs around forming stars, white dwarfs, neutrons stars and stellar black holes, but not to such discs around supermassive black holes, where this prescription would imply supersonic turbulence.

The innermost part of accretion discs may be dominated by radiation pressure if they reach a sufficiently high temperature; this occurs for instance in standard discs surrounding black holes or compact objects. In such a case, the hydrostatic equation is written

$${P_{\rm r} \over \rho} = \Omega ^2 H^2 = {8 \sigma \over 3 c} {T^4 H \over \Sigma} ,$$ (12)

and therefore, according to (9), H depends only on the accretion rate, and not on the viscosity prescription:

$$H = {3 \kappa_{\rm R} \over 16 \pi c} \dot{M} .$$ (13)

Thus the Mach number is given by

$${\cal M}_{\rm t} \simeq 3~10^{-3} \, \beta_{-5}^{1/2} \, \kappa_{\rm R}^{-1}\, f_{\rm edd}^{-1} \, x$$

$$\simeq 5~10^{-2} \, \beta_{-5}^{1/2} \, \kappa_{\rm R}^{-1}\, \dot{m}_{16}^{-1} \, M_0 \, x$$ (14)

meaning that turbulence is expected to be subsonic in the inner part of radiation pressure dominated discs.

Figure 1: Disc thickness versus radius for a $\beta$-disc with $\beta =10^{-5}$ ( solid line) and for a standard disc with $\alpha =0.01$ ( dashed line), for =1 $M_\odot$ and $\dot{M}=10^{-8}$ $M_\odot$/yr

3.2 Thin Keplerian discs assumption

In this paper, we restrict ourselves to Keplerian discs, which are necessarily geometrically thin, i.e. $H/R \ll 1$. Eq. (6) shows that the turbulent Mach number increases as the aspect ratio decreases; therefore the disc cannot be thinner than $\sqrt{\beta} \times R$. One easily finds from Eq. (11) that

 

$$\left(\frac{H}{R}\right)_\beta \simeq 1.5~10^{-3} \, \kappa_{\rm R}^{1/8} \, \beta_{-5}^{-1/8}\, \mu^{-1/2} m_0^{- 7/16} \, \dot{m}_{16}^{1/4} \, r_{10}^{1/16}$$

$$\simeq 0.08 \, \kappa_{\rm R}^{1/8} \, \beta_{-5}^{-1/8} \mu^{-1/2} \, m_0^{-7/16} \, \dot{m}_{-7}^{1/4} \, r_{\rm AU}^{1/16}$$ (15)

within the gas pressure dominated regions. It is interesting again to make the comparison with standard discs. Using power law solutions of $\alpha$-discs (Huré 1998) and Eq. (15), we obtain

$$\frac{{\rm H}_\beta}{{\rm H}_\alpha} \simeq 0.9 \; \beta_{-5}^{1/8} \alpha^{1/10} ,$$ (16)

indicating that $\beta$-discs globally should have about the same shape as standard $\alpha$-discs. This is confirmed in Fig. 1, where we have plotted the disc thickness computed from vertically averaged equations with realistic opacities and equations of state, for the two viscosity laws and for M=1 $M_\odot$ and $\dot{M}=10^{-8}$$M_\odot$/yr which could represent both a YSO disc and a EBS disc. In the innermost part of AGN, where the radiation pressure dominates, the disc thickness is independent of the viscosity prescription, as we have already pointed out, and therefore ${\rm H}_\beta={\rm H}_\alpha$. We thus conclude that, despite the flaring, $\beta$-discs are expected to be thin and Keplerian, provided that the accretion rate does not reach too large values. Note that $\alpha$-discs are subject to the same constraint.