4 Stability of $\beta$-discs

We shall now examine two kinds of instability that an accretion disc can undergo: viscous instability which can lead to disc fragmentation and thermal instability.

4.1 Viscous stability

A general condition for viscous stability is (Frank et al. 1992)

$$\left(\frac{\partial \ln \dot{M}}{\partial \ln \Sigma}\right)_\Omega > 0 .$$ (17)

In a standard $\alpha$-disc, the equilibrium relation $\dot{M}(\Sigma)$takes the famous "S''-shape with the instability occuring on the intermediate branch of negative slope (Cannizzo 1993). On the contrary, according to Eqs. (2) and (7) $\beta$-discs are viscously stable everywhere, since in Eq. (17) is always fulfiled. We illustrate this important property in Fig. 2, with two typical $\dot{M}(\Sigma)$-relations obtained through vertical structure computations for a $\beta$-disc and a $\alpha$-disc (Huré 2000).

Figure 2: Relation between the accretion rate and the surface density, from vertical structure computations, at R=1010 cm from the center and with M=1 $M_\odot$, using the $\alpha$-viscosity prescription (with $\alpha =0.1$) and the $\beta$-prescription ( $\beta =10^{-5}$)

4.2 Thermal stability

Thermal instability occurs when the energy balance between radiative losses and viscous heating can no longer be maintained after perturbation of the equilibrium state. The condition for stability is written (Pringle 1981):

$$\left(\frac{\partial \ln Q^+}{\partial \ln T}\right)_{\Sigma,... ...\partial \ln Q^-}{\partial \ln T}\right)_{\Sigma, \Omega} < 0$$ (18)

where Q⁺ is the heating rate (right-hand-side of Eq. (9)) and Q^- is the cooling rate (with Q⁺=Q^- at equilibrium). The above criterion is quite general and can be applied to non standard viscosity prescriptions, provided that the vertical dynamical time scale $t_z \sim H/c_{\rm s}$ is smaller than the thermal time scale $t_{\rm th} \sim PH/Q^+$. The requirement of a subsonic turbulence discussed in the previous section automatically ensures that since we have the relation

$$\frac{t_{\rm th}}{t_z} \sim \frac{1}{ \beta } \left( \frac{H}{R} \right)^2 = \frac{1}{{\cal M}^2_{\rm t}} \cdot$$ (19)

It is easy to see that $Q^+ \propto \Omega^3 R^2 \Sigma$ with the $\beta$-prescription and so the thermal stability of a $\beta$-disc is only governed by properties of the cooling function. The situation is different in an $\alpha$-disc where the heating rate $Q^+ \propto \Omega^3 H^2 \Sigma$is implicitly a function of T through H. To determine the slope of the cooling function Q^-(T), we have used vertically averaged equations (see Appendix A) and computed the term $(\partial \ln Q^- / \partial \ln T)$ analytically (see Appendix B).

   

Table 1: Quantity $(\partial \ln Q^- / \partial \ln T)_{\Sigma,\Omega}$ and inferred thermal stability for ideal opacity sources (disregarding self-gravitation). Values and comments in parenthesis concern the $\alpha$-prescription. Various cases are considered: $\tilde{\beta}=1$ ( $\tilde{\beta}$ is the ratio of the gas pressure to the total pressure); $\tilde{\beta}=\frac{1}{2}$ and $\tilde{\beta}=0$ (radiative pressure only); Thompson scattering only ( $\kappa \equiv \sigma _{\rm T}$); free-free opacity only ( $\kappa \equiv \kappa _{\rm ff} \propto \rho T^{-7/2}$); both absorption processes (with $x = \frac{\kappa_{\rm ff}}{\kappa_{\rm ff}+\sigma_{\rm T}} \le 1)$

$\beta$	opacity	optically thick limit (see Eq. (B.8))		optically thin limit (see Eq. (B.11))
1	$\sigma_{\rm T}$	-4	stable         (-3, stable)	-4	stable         (-3, stable)
	$\kappa_{\rm ff}$	-8	stable         (-7, stable)	0	stable         (+1, unstable)
	$\sigma_{\rm T} + \kappa_{\rm ff}$	-4(1+x) < 0	stable	$-4(1-x) \le 0$	stable
$\frac{1}{2}$	$\sigma_{\rm T}$	-4	stable          ( $-\frac{2}{3}$, stable)	-4	stable         ( $-\frac{2}{3}$, stable)
	$\kappa_{\rm ff}$	$-\frac{55}{6}$	stable         ( $-\frac{35}{6}$, stable)	$-\frac{1}{8}$	stable          ( $+\frac{5}{8}$, unstable)
	$\sigma_{\rm T} + \kappa_{\rm ff}$	$-4 - \frac{31}{6}x < 0$	stable	$-4+\left( \frac{10-7x}{6+2x} +\frac{7}{2} \right) x <0$	stable
0	$\sigma_{\rm T}$	-4	stable          (+4, unstable)	-4	stable         (+4, unstable)
	$\kappa_{\rm ff}$	$-\frac{23}{2}$	stable         ( $-\frac{7}{2}$, stable)	$-\frac{1}{4}$	stable          ( $+\frac{1}{4}$, unstable)
	$\sigma_{\rm T} + \kappa_{\rm ff}$	$-4 - \frac{15}{2}x < 0$	stable	$-4+\left( \frac{8-7x}{2+2x} +\frac{7}{2} \right) x <0$	stable

The resulting expression Eq. (B.7) may be used to analyze the thermal stability for very simple ideal cooling functions: electron scattering and free-free processes, as considered already by Piran (1978). We have distinguished 4 cases: optically thick, optically thin, gas pressure dominated and radiation pressure dominated. This can easily be handled by suitable settings in Eq. (B.7) (see Eqs. (B.8) and (B.11)). The results are summarized in Table 1. The remarkable point is that $\beta$-discs are always thermally stable for these ideal cooling mechanisms, which is in complete agreement with the conclusions drawn by Piran (1978). Note that the optically thin gas pressure dominated regime with free-free emission is marginally stable whereas it is unstable with the standard viscosity law, as noticed by Pringle et al. (1973).

Figure 3: Right panel: temperature radial profile ( thin line) and thermally instable regions ( bold line) in a "realistic'' $\beta$-disc for M=1 $M_\odot$ and $\dot{M}=10^{-8}$ $M_\odot$/yr. Left panel: the quantity $(\partial \ln Q^- / \partial \ln T)_{\Sigma,\Omega}$ versus T computed from Eq. (B.7)

The situation is quite different when using realistic Rosseland and Planck opacities as well as an accurate equation of state. A typical result is displayed in Fig. 3, where we have considered a disc surrounding a one solar mass central object and $\dot{M}=10^{-8}$ $M_\odot$/yr. We find that the disc is thermally unstable when the midplane temperature is in the range 5000-9000 K (zone 2 in the figure), due to the steep rise in the opacity. Like in a standard disc, this unstable regime corresponds to the partial ionization of hydrogen. A specially interesting case is displayed in Fig. 4 where the accretion rate is $\dot{M}=10^{-12}$ $M_\odot$/yr , typical for an EBS disc in quiescence. There is a (multi-valued) optically thin solution with  K (zone 2bis in the figure). We have checked that this regime is also compatible with both $H/R \ll 1$ and ${\cal M}_{\rm t} < 1$. We see that the hottest branch of this solution, which has already been investigated by Williams (1980), is thermally stable.

Figure 4: Same legend as for Fig. 3 but for $\dot{M}=10^{-12}$ $M_\odot$/yr typical of a quiescent EBS disc. The optically thick/thin limit is reached at $r_{10} \simeq 0.2$ where $T \simeq 6600$ K. The optical thickness $\tau$ goes through a minimum for $T \simeq 4500$ K where $\tau \simeq 4~10^{-4}$

Similar computations have been carried out for many input pairs ( $M, \dot{M}$) corresponding to AGN, EBS and YSO discs and the overall conclusions are the following:

• the hot optically thick solution which extends down to the central object (zone 1 in Figs. 3 and 4) is always thermally stable (this includes the radiative pressure dominated solution);
• the region where hydrogen recombines (zone 2) is always unstable if the disc is optically thick;
• the hottest branch of the optically thin solution with 7000 K  K and ${\rm d}T/{\rm d}R < 0$ (zone 2bis), when it exists, is always stable, even if hydrogen recombination occurs;
• the cold branches of the optically thin solution with  K and ${\rm d}T/{\rm d}R >0$ are always unstable;
• the outermost parts (zone 3) are globally stable. Thin unstable zones can be present.