Write $K=11a/12$, so $P=KT^4$. With $\rho\sim\Sigma/H$, vertical balance gives
$$
\frac PH\sim\rho\Omega^2H,
\qquad
\boxed{H\sim\frac{KT^4}{\Sigma\Omega^2}}.
$$
At fixed $\Sigma$ and radius, the volumetric viscous heating is
$$
C_+=\frac94\Omega^2\mu
=\frac94\alpha\Omega P\propto T^4.
$$
The neutrino cooling is
$$
C_-=A\rho T^{1/\beta}
\sim A\frac{\Sigma}{H}T^{1/\beta}
\propto T^{1/\beta-4}.
$$
<Thermal stability of an accretion disk> requires the cooling rate to have the larger logarithmic temperature slope:
$$
\frac1\beta-4>4.
$$
Since $\beta>0$, the stable range is
$$
\boxed{0<\beta<\frac18}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-321/1/b/ii/solution