The gas-to-radiation pressure ratio gives
$$
\frac{\beta}{1-\beta}
=\frac{P_g}{P_{\rm rad}}
=\frac{3\rho k_B}{a_{\rm rad}\mu H\,T^3}.
$$
Hence
$$
\boxed{
T(\beta,\rho)=
\left[\frac{3k_B}{a_{\rm rad}\mu H}
\frac{1-\beta}{\beta}\rho\right]^{1/3}},
$$
and either pressure component gives
$$
\boxed{
P(\beta,\rho)=\frac{k_B}{\mu H\beta}
\left[\frac{3k_B}{a_{\rm rad}\mu H}
\frac{1-\beta}{\beta}\right]^{1/3}
\rho^{4/3}}.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 4 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-317/4/i/solution