= Solution
Assume the material gas is monatomic. Its specific enthalpy together with the radiation enthalpy is
$$
h=\frac52\mathcal RT+\frac{4a_{\rm rad}T^4}{3\rho}
=\mathcal RT\left(\frac4\beta-\frac32\right).
$$
At constant total pressure, logarithmic differentiation of
$P=\mathcal R\rho T+a_{\rm rad}T^4/3$ gives
$$
\frac{d\beta}{dT}\bigg|_P=-\frac{4(1-\beta)}T.
$$
Consequently the <specific heat capacity at constant pressure> is
$$
\boxed{
c_P=\left(\frac{\partial h}{\partial T}\right)_P
=\frac{\mathcal R}{2\beta^2}
(32-24\beta-3\beta^2)}.
$$
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