= Solution
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.
Back to article page
