Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 314 / 4

The shock speed isApply the Strong-shock Rankine-Hugoniot conditions in the shock frame and transform the downstream velocity back to the ambient-medium frame. With the shock compression ratiothe post-shock values are

For $\gamma =5/3$, the shock data give ${\rho }_{\mathrm{ps}}=4{\rho }_0$ and $u_{\mathrm{ps}}=3V_s/4$. Since $V_s=R/(3t)$, the prescribed homologous spherical flow isPut $\xi =r/R(t)$ and $\rho ={\rho }_{0}D(\xi )$. The spherical continuity equation becomesso $D\propto {\xi }^9$. Matching the post-shock density givesThe material acceleration isThe radial Euler equations for an inviscid fluid thus give $dp/dr=3\rho r/(16t^2)$. Integrating from the centre to the shock,Since $p_{\mathrm{ps}}={\rho }_0{R}^2/(12t^2)$,

Resolve the uniform upstream field into components normal and tangential to the spherical shock:The ideal magnetohydrodynamic shock conditions leave the normal field continuous and multiply the tangential field by the gas compression ratio $\chi$. Therefore

The post-shock density is $\chi {\rho }_0$, while part c givesUsing the Alfvén speed $v_A=B/\sqrt{4\pi \rho }$ in Gaussian units,The ratio is largest in the equatorial plane, where the field is tangential to the shock: