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\end{equation}  Inserting eqn.~\ref{eqn:psi} this can be simplified to  \begin{equation}\label{eqn:ode}  \frac{\rm{d}\omega}{\rm{d}z} = \frac{\omega z \pm (\omega^2+z^2)^{3/2}  \sqrt{c (\omega^2+z^2)^2 (\omega^2+z^2-c(\omega^2+z^2)^2)}}{z^2 (1-c(\omega^2+z^2))}}{z^2  - c(\omega^2+z^2)^2} \end{equation}  with  \begin{equation}  c(z) = \frac{4\pi P(z)}{\dot{M} v_{\infty}}\ . = \frac{1}{R_0^2},  \end{equation}  where $R_0(z)$ is the zylindrical radius of the shock front if no further conditions are placed. In the case of young stars, however, there is another constraint. Due to the circum-stellar disk the stellar wind cannot expand freely in the $z=0$ plane; it is constrained to $R_{\textrm{shock}}(0) = R_{\textrm{inner}}$. Thus, we start the integration at $R_{\textrm{inner}} = 0.05$~AU.  For general $P(z)$ this ODE needs to be solved numerically. The negative root describes the compression shock that heats up the gas.