= Solution
The three <stellar adiabatic exponents> are defined by
$$
\Gamma_1=\left(\frac{\partial\log P}{\partial\log\rho}\right)_s,
\qquad
\Gamma_3-1=\left(\frac{\partial\log T}{\partial\log\rho}\right)_s,
$$
$$
\frac{\Gamma_2}{\Gamma_2-1}
=\left(\frac{\partial\log P}{\partial\log T}\right)_s.
$$
For the gas-radiation mixture,
$$
\chi_\rho=\left(\frac{\partial\log P}{\partial\log\rho}\right)_T=\beta,
\qquad
\chi_T=\left(\frac{\partial\log P}{\partial\log T}\right)_\rho=4-3\beta,
$$
and
$$
c_V=\frac{\mathcal R}{2\beta}(24-21\beta).
$$
It follows that
$$
\boxed{\Gamma_3-1=\frac{8-6\beta}{24-21\beta}},
$$
$$
\boxed{\Gamma_1=\frac{32-24\beta-3\beta^2}{24-21\beta}},
\qquad
\boxed{\Gamma_2=\frac{32-24\beta-3\beta^2}
{24-18\beta-3\beta^2}}.
$$
Their defining derivatives imply the identity
$$
\boxed{\frac{\Gamma_1}{\Gamma_3-1}
=\frac{\Gamma_2}{\Gamma_2-1}}.
$$
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