deletions | additions       diff --git a/12 vs 13 [3].tex b/12 vs 13 [3].tex  index 8600835..356a021 100644  --- a/12 vs 13 [3].tex  +++ b/12 vs 13 [3].tex 
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^{12}CO gives very inaccurate data when we assume that it is optically thin.  \subsection{Correction}  To correct for the fact that \textsuperscript{12}CO is opticaly thick, we We can  use the following equation: optically thin ^{13}CO data to correct the ^{12}CO.  \begin{equation} %\label{eqn:mass-energy}   \frac{ T_{\rm mb 12}}{T_{\rm mb 13}} = \frac{[^{12}{\rm CO}]}{[ ^{13}{\rm CO}]} \frac{1 - e^{-\tau_{12}}}{\tau_{12}} \\  \frac{1 - e^{-\tau_{12}}}{\tau_{12}} = \frac{ T_{\rm mb 12}}{T_{\rm mb 13}} \frac{[ ^{13}{\rm CO}]}{[^{12}{\rm CO}]}  from the Dunhan paper  \end{equation}  taken from the Dunhan paper. This provides a correction for corrects the ^{12}CO data, by calculating  the optical depth of each\textsuperscript{12}CO  voxel based on the \textsuperscript{13}CO equivalent voxel. and multiplying by this factor.  The results of after  this correction  are shown below.