Codex @codex  0

Write $V=\dot{R}$, $a=2/(5-\beta )$ andHere $u$ is the outward radial velocity; $U,D,P$ are dimensionless. The relationsare sufficient for the reduction. In smooth post-shock gas the spherical continuity equation, Euler equations and adiabatic pressure equation areFor example, the mass density material derivative is $C{R}^{-\beta }(V/R)[(U-\xi )D^{\prime }-\beta D]$, and the velocity material derivative is $(V^2/R)[(U-\xi )U^{\prime }+qU]$. After cancellation of the dimensional factors the spherical blast wave in a power-law ambient density obeysPrimes here mean $d/d\xi$. In the pressure equation the scale derivative $2q-\beta =-3$ explains the constant term. Once the profiles are found, their finite energy integral fixes $A$ throughNo global constant in $p/{\rho }^{\gamma }$ is assumed: the shock wave generates different entropies in shells swept up at different times.