For a <perfect gas>, $c_s^2\sim kT/(\mu_m m_p)$. At the margin of the <Toomre stability criterion>,
$$
c_s\sim\frac{G\Sigma}{\Omega},
\qquad
T\sim\frac{\mu_m m_p}{k}\frac{G^2\Sigma^2}{\Omega^2}.
$$
Integrating viscous heating through the disk and applying <radiative diffusion> gives
$$
F\sim\alpha P\Omega H\sim\alpha\Sigma c_s^2\Omega,
\qquad
F\sim\frac{\sigma T^4}{\kappa\rho H}
\sim\frac{\sigma T^4}{\kappa\Sigma}.
$$
Equating these fluxes yields
$$
\boxed{
\alpha\sim\frac\sigma\kappa
\left(\frac{\mu_m m_p\Sigma}{k}\right)^4
G^6\Omega^{-7}}.
$$
The <kinematic viscosity> is $\bar\nu\sim\alpha c_s^2/\Omega$, so
$$
\bar\nu\propto\Sigma^4\Omega^{-7},Sigma^2\Omega^{-3}
=\Sigma^6\Omega^{-10}.
$$
Because a <Keplerian accretion disk> has $\Omega\propto r^{-3/2}$,
$$
\boxed{\bar\nu\propto r^{15}\Sigma^6}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 / 1 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-321/1/b/i/solution