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The Rankine-Hugoniot jump conditions relate the mass density $\rho$, velocity $v$, and pressure $P$ on both sides of a shock. For ideal gases and non-oblique shocks the conservation of mass, momentum and energy across the shock can be written as follows \citep[][chap.~7]{1967pswh.book.....Z}, where the state before the front of the shock front is marked by the index 0, that behind the shock by index 1:  \begin{eqnarray}  \label{eqn:RH1}\rho_0 v_0 &=& & = &  \rho_1 v_1\\ \label{eqn:RH2}P_0+\rho_0 v_0^2 &=& & = &  P_1+\rho_1 v_1^2\\ \label{eqn:RH3}\frac{5 P_0}{2\rho_0}+\frac{v_0^2}{2}&=&\frac{5 P_0}{2\rho_0}+\frac{v_0^2}{2}& = &\frac{5  P_1}{2\rho_1}+\frac{v_1^2}{2} \ .\\ \end{eqnarray}  Initially, the stellar wind is relatively cold and thus the thermodynamic pressure can be neglected, setting $P_0=0$.