deletions | additions       diff --git a/Results.tex b/Results.tex  index b458b2e..04e267a 100644  --- a/Results.tex  +++ b/Results.tex 
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\begin{equation}  d_{\mathrm{cool}} \approx 20.9 \mathrm{ AU}  \left(\frac{10^5\mathrm{ cm}^{-3}}{n_0}\right)  \left(\frac{v_{\mathrm{shock}}}{500\textnormal{ \left(\frac{v_{\mathrm{shock}}}{500\mathrm{  km s}^{-1}}\right)^{4.5}\ . \end{equation}  The derivation for this formula assumes a cylindrical cooling flow. In contrast, the external pressure will continue to compress the gas, as it starts cooling. Since denser gas emits more radiation and thus cools faster, $d_{\mathrm{cool}}$ is only an upper limit. With this in mind, figure~\ref{fig:rhocool} (lower panel) indicates that the cooling lenghts for our fiducial model is consistent with the X-ray observations that do not resolve the wind shock \citep{2008A&A...488L..13S}. On the other hand, a model with a wind mass loss rate of only $10^{-10}$~M$_{\odot}$~yr$^{-1}$, has a much larger $d_{\mathrm{cool}}$. Since only a very small fraction of the stellar mass loss is heated to X-ray emitting temperatures (Fig.\ref{fig:result}, rightmost panel) this scenario does not provide enough X-ray luminosity to explain the observations (paper~I). This model shows that the external presure that confines the wind must be within an order of magnitude or so form the values we assumed for our fiducial model. Significantly higher pressures require unrealistically fast outflows to push the shock front out to 40~AU and lower presures do not allow a mass flux high enough to power the X-ray luminosity.