= 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 $\gamma\ne1$, constant <temperature> would force constant density, contradicting vertical <hydrostatic equilibrium> $P'=-\rho\Omega^2z$ 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 $\rho=\rho_m\exp[-\Omega^2z^2/(2\mathcal R T_m)]$, with no finite surface.
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