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\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. numerically\footnote{It is possible to remove all trigonometric functions from eqn.~\ref{eqn:ode} by means of addition formulae, but that introduces singularities into the solution. Thus, we numerically solve the ODE in the form of eqn.~\ref{eqn:ode}.}.