= Solution
The <stellar adiabatic exponents> are fixed-composition, constant-<specific entropy> derivatives:
$$
\boxed{\Gamma_1=\left(\frac{\partial\log P}{\partial\log\rho}\right)_s,\qquad\frac{\Gamma_2-1}{\Gamma_2}=\left(\frac{\partial\log T}{\partial\log P}\right)_s,\qquad\Gamma_3-1=\left(\frac{\partial\log T}{\partial\log\rho}\right)_s.}
$$
The <chain rule> gives $\Gamma_1(\Gamma_2-1)/\Gamma_2=\Gamma_3-1$. Define the <pressure> derivatives $\chi_\rho=(\partial\log P/\partial\log\rho)_T$ and $\chi_T=(\partial\log P/\partial\log T)_\rho$. The <first law of thermodynamics>, with $du=Tds+P\,d\rho/\rho^2$, gives
$$
\Gamma_3-1=\frac{P\chi_T}{\rho T c_V},\qquad c_P-c_V=\frac{P}{\rho T}\frac{\chi_T^2}{\chi_\rho}.
$$
Combining this with $d\log P=\chi_\rho d\log\rho+\chi_Td\log T$ on an adiabat gives
$$
\boxed{\Gamma_1=\chi_\rho+\chi_T(\Gamma_3-1)=\chi_\rho\frac{c_P}{c_V},\qquad\gamma\equiv\frac{c_P}{c_V}=\frac{\Gamma_1}{\chi_\rho}.}
$$
Thus the <specific-heat ratio> is not generally equal to the three <stellar adiabatic exponents>.
For the mixture, $\chi_\rho=\beta$ and $\chi_T=4-3\beta$. The <specific heat capacity at constant volume>, obtained by differentiating $u$ at fixed <mass density>, is
$$
c_V=\mathcal R\frac{24-21\beta}{2\beta}.
$$
Therefore the <adiabatic exponents of a monatomic gas-radiation mixture> and its <specific-heat ratio> are
$$
\boxed{\begin{aligned}
\Gamma_3-1&=\frac{8-6\beta}{24-21\beta},\\
\Gamma_1&=\frac{32-24\beta-3\beta^2}{24-21\beta},\\
\Gamma_2&=\frac{32-24\beta-3\beta^2}{24-18\beta-3\beta^2},\\
\gamma&=\frac{32-24\beta-3\beta^2}{\beta(24-21\beta)}=\frac{\Gamma_1}{\beta}.
\end{aligned}}
$$
If a relation involving only the exponents is wanted, eliminate $\beta$ from $\Gamma_1=\beta+(4-3\beta)(\Gamma_3-1)$:
$$
\gamma=\frac{\Gamma_1(4-3\Gamma_3)}{\Gamma_1-4\Gamma_3+4}\qquad(0<\beta\le1).
$$
For a pure monatomic <perfect gas>, $\beta=1$ and $\gamma=\Gamma_1=\Gamma_2=\Gamma_3=5/3$. For any calorically perfect gas with constant heat capacities, the same equality holds with its own $\gamma_g$. In a genuine gas-radiation mixture, $0<\beta<1$, $\gamma=\Gamma_1/\beta$ while the exponents are given separately above.
In the radiation limit, $\Gamma_1=\Gamma_2=\Gamma_3=4/3$ and the <adiabatic temperature gradient> is $1/4$. This follows independently from photon <entropy>: a comoving volume $V$ has $S\propto VT^3$, so an adiabat obeys $T\propto V^{-1/3}$ and $P\propto V^{-4/3}$. However, \b[$c_P/c_V$ is singular in the pure-radiation limit], not $4/3$. This is the <pure-radiation constant-pressure heat-capacity singularity>. The equation $P=aT^4/3$ fixes <temperature> whenever <pressure> is fixed, so an ordinary constant-pressure <temperature> derivative is not available; along the mixture limit $\gamma\to\infty$ and $\beta\gamma\to4/3$. For photons alone, mass-specific quantities additionally require a material mass label. The often quoted radiation index $4/3$ is its pressure-density adiabatic exponent, not a finite constant-pressure/constant-volume heat-capacity ratio.
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