deletions | additions       diff --git a/subsection_Converting_Between_the_Observed__.tex b/subsection_Converting_Between_the_Observed__.tex  index c8ec156..d093e1e 100644  --- a/subsection_Converting_Between_the_Observed__.tex  +++ b/subsection_Converting_Between_the_Observed__.tex 
...
Consider a distribution of observed values $\rho(\Gamma_{\rm obs}, P_{\rm obs}) = \dfrac{d^2N}{d\Gamma_{\rm obs}\ dP_{\rm obs}}$. The small number of devils $dN = f\ \rho(\Gamma_{\rm obs}, P_{\rm obs})\ d\Gamma_{\rm act}\ dP_{\rm act}$ contributing are those that had closest approach distances between $b$ and $b + db$ of the detector. Thus, we can convert $f\ \rho(\Gamma_{\rm obs}, P_{\rm obs})$ to $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ by integrating from $b = 0$ to $b_{\rm max}$ and setting $\Gamma_{\rm act}(b) = \sqrt{\Gamma_{\rm obs}^2 - \left( 2b \right)^2}$ and  $P_{\rm act}(b) = P_{\rm obs}\left( \Gamma_{\rm obs}/\Gamma_{\rm act} \right)^2$. To calculate the integral, we also need to re-cast the upper limit $b_{\rm max}$ to express the maximum possible radial distance, $b_{\rm max} = \left( \dfrac{\Gamma_{\rm max}}{2} \right) \sqrt{\dfrac{P_{\rm max} - P_{\rm min}}{P_{\rm min}}}$. The integral to convert from $\rho(P_{\rm act}, \Gamma_{\rm act})$ to $\rho(\Gamma_{\rm obs}, P_{\rm obs})$ extends from $b = 0$ to $\left( \Gamma_{\rm obs}/2 \right) \sqrt{\left( P_{\rm obs} - P_{\rm min} \right)/P_{\rm min}}$ and is given by  \begin{comment}  \begin{eqnarray}  \label{eqn:convert_from_actual_to_observed_density}  \rho(\Gamma_{\rm %\begin{eqnarray}  %\label{eqn:convert_from_actual_to_observed_density}  %\rho(\Gamma_{\rm  obs}, P_{\rm obs}) &=& &\int_{b = 0}^{b(\Gamma_{\rm obs}, P_{\rm obs})} & &f&\ \rho(\Gamma_{\rm act}(b), P_{\rm act}(b))\ \dfrac{2b\ db}{b_{\rm max}^2}\\ &=& %&=&  &\int_{b = 0}^{\left( \Gamma_{\rm obs}/2 \right) \sqrt{\left( P_{\rm obs} - P_{\rm min} \right)/P_{\rm min}}}& &A_{\rm max}^{-1}\ \Gamma_{\rm act} \sqrt{\dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}}} \upsilon\ \kappa\ \left( \Gamma_{\rm act}/{\rm m} \right)^{2/3}& \\ && %&&  && &\rho(\sqrt{\Gamma_{\rm obs}^2 - \left( 2b \right)^2}, P_{\rm obs}\left( \Gamma_{\rm obs}/\Gamma_{\rm act} \right)^2)&\ \dfrac{2b\ db}{\left( \dfrac{\Gamma_{\rm %\dfrac{\Gamma_{\rm  max}}{2} \right)^2 \left( \dfrac{P_{\rm max} - P_{\rm min}}{P_{\rm min}} \right)}. \end{eqnarray}  \end{comment} %\end{eqnarray}  %\end{comment}  \begin{equation}  \label{eqn:convert_from_actual_to_observed_density}