deletions | additions       diff --git a/develop_ode.tex b/develop_ode.tex  index 63fc482..3bfe60b 100644  --- a/develop_ode.tex  +++ b/develop_ode.tex 
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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}  \rho(r) = \frac{\dot M}{4 \pi r^2 v_{\inf}}\ v_{\infty}}\  . \end{equation}  Figure~\ref{fig:sketch} shows that $v_0$ depends on the position of the shock:  \begin{equation}  \label{eqn:v0}v_0 = v_{\inf} v_{\infty}  \sin \psi \end{equation}  with   \begin{equation}\label{eqn:angle} 
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\end{equation}  Inserting equation~\ref{eqn:rho} to \ref{eqn:deriv} into eqn.~\ref{eqn:4}, we arrive at an ordinary differential equation, that describes the functional form of the shape of the shock front:  \begin{equation}  P(z) = \frac{3}{4}\rho_0v_0^2 = \frac{3}{4} \frac{\dot{M}}{4\pi v_{\inf}(z^2+\omega^2)} v_{\inf}^2 v_{\infty}(z^2+\omega^2)} v_{\infty}^2  \sin^2(\psi) \end{equation}  This equation can be simplified to  \begin{equation}\label{eqn:ode} 
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\end{equation}  with  \begin{equation}  c = \frac{16\pi P(z)}{3 \dot{M} v_{\inf}}\ v_{\infty}}\  . \end{equation}