Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2021 / Section B / 4 / a / Solution

A shock wave forms when a compressive flow steepens or when gas must be decelerated or turned faster than upstream acoustic waves can communicate the change. Astrophysical examples include a supernova remnant sweeping up surrounding gas, a fast stellar wind colliding with ambient material, and supersonic accretion onto an obstacle or onto a dense gas layer. A large Mach number by itself is not sufficient: a uniform supersonic stream need not contain a shock wave.

Choose the shock frame, and denote upstream and downstream normal velocities by $u_1,u_2$. Integrating the local conservation laws over a thin control volume spanning a steady normal shock wave gives equal fluxes on both sides. Mass conservation gives constant mass flux per unit area,Pressure and advected momentum together give the normal momentum flux,For a thin non-radiating layer, with negligible gravitational-potential change across it, conservation of energy gives equal fluxes $j(h+u^2/2)$. A perfect gas has specific enthalpy $h=\gamma p/[(\gamma -1)\rho ]$, soThese are the Rankine-Hugoniot conditions for a perfect gas. Dissipation inside the thin layer produces an entropy increase and selects the physical compressive branch. Efficient radiative cooling would change the energy relation and may instead lead to an isothermal shock.