{c}

Analyze a local temperature perturbation at fixed radius and fixed <surface density of a disk> $\Sigma$. The <thermal timescale of an accretion disk> is shorter than its <viscous timescale>, so mass redistribution is negligible during the perturbation. Let $Q^+$ be viscous heating and $Q^-$ radiative cooling, measured consistently through one face or both faces, and define net cooling by $\dot Q=Q^--Q^+$. A temperature increase is unstable if it decreases net cooling.

With <gas pressure> dominant, vertical <hydrostatic equilibrium> and the <ideal gas> law give
$$
c_s^2\propto T_c,\qquad H\propto T_c^{1/2},\qquad
\rho_c\sim\frac{\Sigma}{2H}\propto T_c^{-1/2}.
$$
For the <alpha disk> prescription with temperature-independent $\alpha$, $\nu=\alpha c_sH\propto T_c$, and the <Keplerian rotation> heating law is
$$
Q^+=\frac98\nu\Sigma\Omega_K^2\propto T_c.
$$
One must not hold the steady-state $\dot m$ fixed while perturbing the temperature: the stress and heating respond through $\nu$, even though $\Sigma$ is temporarily fixed.

At an equilibrium temperature $T_0$ where $Q^+=Q^-=Q_0$, if $Q^-\propto T_c^q$ and $Q^+\propto T_c$, then
$$
\left.\frac{d\dot Q}{dT_c}\right|_{T_0}
=(q-1)\frac{Q_0}{T_0}.
$$
The cooling exponent therefore decides the sign.

For hot, ionized, <optically thin> gas, the usual intended cooling model is <thermal bremsstrahlung>, whose volume emissivity scales as $\rho^2T_c^{1/2}$. Integrated vertically,
$$
Q^-_{\rm ff}\propto H\rho_c^2T_c^{1/2}
\propto\frac{\Sigma^2}{H}T_c^{1/2}\propto T_c^0.
$$
Thus $q=0$, and
$$
\boxed{\frac{d\dot Q}{dT_c}=-\frac{Q_0}{T_0}<0:
\quad\text{the optically thin bremsstrahlung branch is unstable}.}
$$
A hotter annulus expands vertically, lowering its density enough to offset the direct increase in bremsstrahlung emission; viscous heating still rises.

For an <optically thick medium> with the <Kramers opacity law>, $\kappa\propto\rho_cT_c^{-7/2}\propto T_c^{-4}$ at fixed $\Sigma$. Thermalized <radiative diffusion> gives
$$
Q^-\propto\frac{T_c^4}{\kappa\Sigma}\propto T_c^8,
\qquad
\boxed{\frac{d\dot Q}{dT_c}=7\frac{Q_0}{T_0}>0:
\quad\text{the gas-pressure Kramers branch is stable}.}
$$
Its falling opacity lets a hotter annulus radiate much more efficiently.

For an <optically thick medium> with dominant <electron-scattering opacity>, $\kappa$ is nearly independent of temperature. Provided enough true absorption is present to thermalize the radiation,
$$
Q^-\propto T_c^4,
\qquad
\boxed{\frac{d\dot Q}{dT_c}=3\frac{Q_0}{T_0}>0:
\quad\text{the gas-pressure electron-scattering branch is stable}.}
$$
This conclusion differs from the <thermal instability of a radiation-pressure-dominated alpha disk>, where the heating has a different temperature dependence. The question explicitly excludes that pressure regime.

The first conclusion needs its cooling assumption stated. $T>10^4\,\mathrm K$ alone does not guarantee that <thermal bremsstrahlung> dominates; atomic-line cooling, ionization changes, external illumination, or energy advection can change the answer. For a general optically thin emissivity $\rho^2\Lambda(T_c)$, the corresponding exponent is
$$
q=\frac{d\log\Lambda}{d\log T_c}-\frac12.
$$
Accordingly, under the same fixed-$\Sigma$ alpha prescription,
$$
\boxed{\text{optically thin thermal stability requires }
\frac{d\log\Lambda}{d\log T_c}>\frac32.}
$$
The unstable/stable/stable classification is the standard free-free/diffusion answer, rather than a universal result specified by temperature and optical depth alone.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 347 / 3 / b / Solution

{title2=$t_{\rm th}$}

The thermal timescale compares the internal energy stored in a disc annulus with its local heating or cooling. In a gas-pressure-supported <alpha disk>, $t_{\rm th}\sim(\alpha\Omega_K)^{-1}$, whereas the <viscous timescale> is longer by approximately $(R/H)^2$. A local thermal perturbation is therefore studied at fixed <surface density of a disk>, rather than fixed steady-state <mass accretion rate>.

Codex @codex  0

 Incoming links: Thermal timescale of an accretion disk