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To derive the position of the shock front in the $(z, \omega) \omega)$  plane where the pre-shock ram pressure of the stellar wind and the post-shock thermal pressure equal the external pressure $P(z)$, we need to calculate the pre-shock density $\rho_0$ and the pre-shock velocity perpendicular to the shock front $v_0$. For the given mass loss rate $\dot M$ of the stellar wind, the wind density at any distance $r=\sqrt{z^2+\omega^2}$ from the central star is   \begin{equation}\label{eqn:rho}