deletions | additions       diff --git a/subsection_Converting_Between_the_Observed__.tex b/subsection_Converting_Between_the_Observed__.tex  index 87f276c..b7c727e 100644  --- a/subsection_Converting_Between_the_Observed__.tex  +++ b/subsection_Converting_Between_the_Observed__.tex 
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\subsection{Converting Between the Observed and Actual Parameter Distributions}  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 the former density over $b$ 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, i.e. the distance at which $P_{\rm act} = P_{\rm max}$: $b_{\rm max} $b(\Gamma_{\rm obs}, P_{\rm obs})  = \left( \Gamma_{\rm act}/2 \right) \left[ \left( P_{\rm act} - P_{\rm obs} \right)/P_{\rm obs} \right]^{1/2}$. Making the replacement $\Gamma_{\rm act} = \left( P_{\rm obs}/P_{\rm max} \right)^{1/2} \Gamma_{\rm obs}$ from Equation \ref{eqn:P_obs_Gamma_obs} gives $b(\Gamma_{\rm obs}, P_{\rm obs}) = \left(\Gamma_{\rm obs}/2\right) \left[ \left(P_{\rm max} - P_{\rm act}\right)/P_{\rm max} \right]^{1/2}$. The integral to convert from $\rho({\rm act}) \equiv \rho(P_{\rm act}, \Gamma_{\rm act})$ to $\rho({\rm obs}) \equiv \rho(\Gamma_{\rm obs}, P_{\rm obs})$ is then \begin{equation}  \label{eqn:convert_from_actual_to_observed_density}