deletions | additions       diff --git a/subsection_The_P__rm_act__.tex b/subsection_The_P__rm_act__.tex  index 25668bd..33f2def 100644  --- a/subsection_The_P__rm_act__.tex  +++ b/subsection_The_P__rm_act__.tex 
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\subsection*{The $P_{\rm act}$ Recovery Bias}  Not surprisingly, the devils with the smallest $P_{\rm act}$-values are the most difficult to detect since relatively close (small $b$) encounters are required for $P_{\rm obs} > P_{\rm min}$ and such encounters are more rare. For example, $b = 0$ would be required for a devil with $P_{\rm act} = P_{\rm min}$ to be detected, an encounter geometry with a infinitesimally small probability.     The probability for a devil to be detected is proportional to the area $A$ over its pressure profile for which $P(r) \ge P_{\rm min}$, which is $A = \pi r^2(P_{\rm act}) = \pi \left( \dfrac{\Gamma_{\rm act}(P_{\rm act})}{2} \right)^2 \left( \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}} \right)$. We're interested in the way in which this bias shifts the recovered distribution $n(P_{\rm obs})$ toward larger $P_{\rm obs}$, so we can calculate the relative recovery probability by taking the ratio $A(P_{\rm \begin{equation}  f \equiv A(P_{\rm  act})/A(P_{\rm max}) = \left( \dfrac{\Gamma_{\rm act}(P_{\rm act})}{\Gamma_{\rm act}(P_{\rm max})} \right)^2 \left( \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right)$. \right).   \end{equation}  For now, we'll assume $\Gamma_{\rm act}$ is independent of $P_{\rm act})$, so $A(P_{\rm act})/A(P_{\rm max}) = \left( \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right)$.