deletions | additions       diff --git a/develop_ode.tex b/develop_ode.tex  index 19753c4..d657300 100644  --- a/develop_ode.tex  +++ b/develop_ode.tex 
...
For general $P(z)$ this ODE needs to be solved 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}.}.   The solution to the ODE determines the locations of the shock front. Thus allows us to calculate the pre-shock velocity perpendicular to the shock front using eqn.~\ref{eqn:v0} and the post-shock temperature $T$. $T_{\mathrm{post-shock}}$.  From eqn.~\ref{eqn:RH3} with negligible pre-shock presure and $v_0=4\;v_1$ we derive: \begin{equation}  T(z) T_{\mathrm{post-shock}}(z)  = \frac{3}{20} \frac{\mu m_{\textrm{H}}}{k} v_0(z)^2,\label{eqn:T} \end{equation}  where $\mu=1.4$ is the mean molecular weight, $m_{\textrm{H}}$ the mass of the hydrogen atom and $k$ the Boltzman constant.