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

Vertical hydrostatic equilibrium is established by sound waves over the disk scale height, soThe local heating-cooling balance is established on the thermal timeThe surface density of a disk evolves only as angular momentum is redistributed, on the viscous timeFor a thin disk, $H/r\ll 1$, so $t_{\mathrm{vert}}\lesssim t_{\mathrm{th}}\ll t_{\nu }$. The vertical and thermal equations can therefore be treated as local equilibria while $\Sigma$ evolves.

Write $K=11a/12$, so $P=K{T}^4$. With $\rho \sim \Sigma /H$, vertical balance givesAt fixed $\Sigma$ and radius, the volumetric viscous heating isThe neutrino cooling isThermal stability of an accretion disk requires the cooling rate to have the larger logarithmic temperature slope:Since $\beta >0$, the stable range is

Local thermal equilibrium givessoSubstitute this into $dP/dz=-z{\Omega }^2\rho$ and use $P=K{T}^4$. The result integrates towithTherefore, for $|z|\le H$,The exponent requested for the temperature is $m=\beta$, and the midplane relations are