= Solution
In a <radiative shock>, a narrow dissipative jump first converts upstream <kinetic energy> into internal <energy>. The immediate strong adiabatic compression is $(\gamma+1)/(\gamma-1)$, and the immediate <temperature> is the result of part 1(c). Downstream gas then radiates, loses <specific enthalpy> and compresses further. Its <shock cooling length> is of order $u_2t_{\rm cool}$, with the immediate downstream speed used only as an estimate because the gas slows as it cools.
For the idealization of equal far-upstream and cooled downstream <temperatures>, unchanged composition and negligible magnetic stress, write $p=c_T^2\rho$. Conserved <mass> and normal <momentum flux> across the whole layer give
$$
\rho_1u_1=\rho_fu_f,\qquad
c_T^2\rho_1+\rho_1u_1^2=c_T^2\rho_f+\rho_fu_f^2.
$$
With $r=\rho_f/\rho_1=u_1/u_f$, the nontrivial <isothermal shock> root is
$$
\boxed{r=\frac{u_1^2}{c_T^2}=\mathcal M_{T,1}^2.}
$$
It can greatly exceed the adiabatic strong-shock compression. An upstream <adiabatic sound speed> instead defines $\mathcal M_{{\rm ad},1}=u_1/(\sqrt\gamma c_T)$, so the same result is $r=\gamma\mathcal M_{{\rm ad},1}^2$; the two <Mach numbers> must not be confused. The radiated <energy> flux is $\rho_1u_1(u_1^2-u_f^2)/2$ when the two endpoint <specific enthalpies> agree. The gas's <specific entropy> rises at the initial jump and falls during cooling, while the escaping radiation carries <entropy> away. A transverse frozen-in <magnetic field>, or an imposed <temperature> floor, can limit the compression. Strongly cooling shocks can therefore collect thin dense shells without violating total <conservation of energy>.
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