Zero-radiative-flux obstruction for a polytropic ideal gas

For positive finite opacity and nonvacuum ideal gas satisfying radiative diffusion, zero flux implies $dT/dz=0$. Write $P=\mathcal{R}\rho T$, where $\mathcal{R}=k_B/({\mu }_m{m}_p)$ for constant mean molecular weight ${\mu }_m$ in proton-mass units. If a polytropic equation of state $P=K{\rho }^{\gamma }$ with constant $K>0$ is also imposed, then $T=(K/\mathcal{R}){\rho }^{\gamma -1}$. For $\gamma \ne 1$, constant temperature would force constant density, contradicting vertical hydrostatic equilibrium $P^{\prime }=-\rho {\Omega }^{2}z$ on an open interval away from the midplane. The familiar finite polytropic slab for $\gamma >1$ is therefore a solution of the hydrostatic/adiabatic closure, not of this finite-opacity zero-radiative-flux system. For $\gamma =1$, the consistent isothermal profile is $\rho ={\rho }_m\exp [-{\Omega }^2{z}^2/(2\mathcal{R}{T}_m)]$, with no finite surface.