 Solution

In the limiting model with negligible gas pressure, $p=p_r=4\sigma T^4/(3c)$. Its gradient is $p_z=16\sigma T^3{T}_z/(3c)$. Combining radiative diffusion and hydrostatic equilibrium givesBecause $\kappa$ and $\Omega$ are constant with height at the fixed disk radius,Equating this to the viscous heating rate $9\mu {\Omega }^2/4$ forcesThis proves constant viscosity in radiation-supported vertical balance. It fixes the local effective dynamic viscosity uniquely, independent of height, surface density and radius within this ideal limiting model. It does not fix a height-independent kinematic viscosity: $\nu (z)=4c/(9\kappa \rho (z))$ still depends on the local mass density.