Past exam of the astrophysics course of the University of Oxford / 2018 / 1 / b / Solution

Use a steady, plane, nonradiative normal shock wave in a perfect gas, and measure the normal speeds $v_{\mathrm{u}},v_{\mathrm{d}}>0$ in the shock frame. The labels denote upstream and downstream. Conservation of mass, normal momentum and total energy givesandThe enthalpy term includes the pressure work. For very large upstream Mach number, neglect $p_{\mathrm{u}}$ compared with ${\rho }_{\mathrm{u}}{v}_{\mathrm{u}}^2$. If $r={\rho }_{\mathrm{d}}/{\rho }_{\mathrm{u}}$ is the shock compression ratio, the first two equations implySubstitution into the energy equation givesDiscard the no-shock solution $r=1$. For a nonrelativistic monatomic gas, $\gamma =5/3$, so the Strong-shock Rankine-Hugoniot conditions becomeThe factor four requires this adiabatic index and negligible radiative, magnetic and escaping-particle contributions to the flux budget. For example, the same strong, nonradiative calculation with $\gamma =4/3$ gives compression seven, not four.