= Solution
The outward <radiative flux> is $F=L/(4\pi r^2)$. A <photon> of energy $E$ carries <momentum> $E/c$, so the momentum transferred per unit volume and time to matter is $\kappa\rho F/c$, when $\kappa$ is the appropriate transport opacity. Momentum conservation makes the radiation stress decrease outward by this amount. In the thin atmosphere, where the radiation stress is treated by its scalar <radiation pressure>,
$$
\boxed{\frac{dP_{\rm rad}}{dr}=-\frac{\kappa\rho F}{c}=-\frac{\kappa\rho L}{4\pi r^2c}.}
$$
This is the first moment of the <radiative transfer equation>. The <grey atmosphere> approximation uses a frequency-independent transport opacity; in an optically thick diffusion regime it becomes the <Rosseland mean opacity>. Near the surface the <plane-parallel atmosphere> approximation removes geometric stress-divergence corrections.
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