Grey atmosphere

A grey atmosphere has frequency-independent opacity. The often-used temperature law additionally assumes a plane-parallel atmosphere in radiative equilibrium, a thermal source in local thermodynamic equilibrium, the Eddington closure approximation and a specified surface boundary condition.

Let $J,H,K$ be the frequency-integrated radiation-field moments and let $\tau$ increase inward. Thermal radiative equilibrium gives $J=B=\sigma T^4/\pi$ and constant $H=\sigma T_{\mathrm{eff}}^4/(4\pi )$. The moment equation $dK/d\tau =H$ and closure $K=J/3$ implyThe Eddington surface condition $J(0)=2H$ fixes $q=2/3$, givingThus the surface is cooler than $T_{\mathrm{eff}}$ and $T=T_{\mathrm{eff}}$ at $\tau =2/3$ in this model. The constant $2/3$ follows from the boundary approximation as well as the closure. Frequency-independent opacity by itself does not determine the full temperature profile, and more exact surface transfer changes this approximation.