Vertical <hydrostatic equilibrium> is established by sound waves over the <disk scale height>, so
$$
t_{\rm vert}\sim\frac H{c_s}\sim\Omega^{-1}.
$$
The local heating-cooling balance is established on the thermal time
$$
t_{\rm th}\sim(\alpha\Omega)^{-1}.
$$
The <surface density of a disk> evolves only as angular momentum is redistributed, on the viscous time
$$
t_\nu\sim\frac{r^2}{\bar\nu}
\sim\frac1{\alpha\Omega}\left(\frac rH\right)^2.
$$
For a <thin disk>, $H/r\ll1$, so $t_{\rm vert}\lesssim t_{\rm th}\ll t_\nu$. The vertical and thermal equations can therefore be treated as local equilibria while $\Sigma$ evolves.

Solution

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

 Solution

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