deletions | additions       diff --git a/subsection_Converting_Between_the_Observed__.tex b/subsection_Converting_Between_the_Observed__.tex  index e11f953..a0a16f8 100644  --- a/subsection_Converting_Between_the_Observed__.tex  +++ b/subsection_Converting_Between_the_Observed__.tex 
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$P_{\rm act}(b) = P_{\rm obs}\left[ 1 + \left( \dfrac{2b}{\Gamma_{\rm act}} \right)^2 \right]$. To perform the integral, we also need to re-cast the upper limit $b_{\rm max}$ in terms of the observed parameters, $b_{\rm max} = \frac{1}{2} \sqrt{\Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2}$. This expression shows, for example, that for $\Gamma_{\rm obs} = \Gamma_{\rm min}$, we should only consider $b = 0$, i.e. only a direct encounter. The integral to convert from $\rho(P_{\rm act}, \Gamma_{\rm act})$ to $\rho(P_{\rm obs}, \Gamma_{\rm obs})$ is  \begin{equation}  \label{eqn:convert_from_actual_to_observed_density}  \rho(P_{\rm obs}, \Gamma_{\rm obs}) = \int_{b = 0}^{\frac{1}{2} \sqrt{\Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2}} f\ \rho(P_{\rm act}(b), \Gamma_{\rm act}(b))\ \left(\frac{1}{2}  \Gamma_{\rm obs}^2 - \Gamma_{\rm min}^2 \right)^{-1}\ 2b\ 4b\  db. \end{equation}