= Adiabatic exponents of a monatomic gas-radiation mixture
{title2=$\Gamma_1=(32-24\beta-3\beta^2)/(24-21\beta)$}
For fixed-composition monatomic <ideal gas> plus equilibrium <radiation pressure>, specific internal energy is $u=3\mathcal RT/2+a_{\rm r}T^4/\rho$. An <isentropic process> satisfies $du=P\,d\rho/\rho^2$, giving $\Gamma_3-1=(8-6\beta)/(24-21\beta)$ and
$$
\Gamma_2=\frac{32-24\beta-3\beta^2}{24-18\beta-3\beta^2},\qquad
\nabla_{\rm ad}=\frac{8-6\beta}{32-24\beta-3\beta^2}.
$$
The <stellar gas-pressure fraction> must be allowed to change along the adiabat. The pure-radiation and pure-gas limits give $\Gamma_1=\Gamma_2=4/3$ and $5/3$, respectively. Partial ionization or different gas heat capacities change these formulas.
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