= Solution
The four equations describe, respectively, vertical <hydrostatic equilibrium> in the central object's gravity, the vertical divergence of <radiative flux> balancing <viscous dissipation>, optically thick <radiative diffusion>, and the <equation of state> combining <ideal gas> <gas pressure> with <radiation pressure>. Here $\mu$ is <dynamic viscosity>, not the mean molecular weight $\mu_m$; $\kappa$ is the <opacity>, $\sigma$ the <Stefan-Boltzmann constant>, and $\Omega$ the local orbital frequency.
In the radiation-pressure limit, $p=4\sigma T^4/(3c)$, so <radiative diffusion> gives
$$
\frac{dp}{dz}=\frac{16\sigma T^3}{3c}\frac{dT}{dz}=-\frac{\kappa\rho F}{c}.
$$
Compare this with vertical <hydrostatic equilibrium>. Wherever $\rho\ne0$,
$$
F(z)=\frac{c\Omega^2}{\kappa}z.
$$
At fixed radius, constant <opacity> makes $dF/dz=c\Omega^2/\kappa$. The <viscous dissipation> equation therefore yields
$$
\boxed{\mu=\frac{4c}{9\kappa},}
$$
independent of height within the optically thick radiation-supported layer. The result assumes the supplied gravity, heating and diffusion equations; optically thin surface corrections are outside that approximation.
Back to article page
