deletions | additions       diff --git a/Whether_a_devil_with_a__.tex b/Whether_a_devil_with_a__.tex  index 66b1943..fa62be8 100644  --- a/Whether_a_devil_with_a__.tex  +++ b/Whether_a_devil_with_a__.tex 
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Whether a devil with a given $P_{\rm act}$ is detected, the the fact that $b$ probably isn't zero means the devil will usually be detected with $P_{\rm obs} < P_{\rm act}$. From the encounter geometry, we can see that the infinitesimal probability $dp$ for the center of a devil to pass within a certain range of radial distances, between $b$ and $b + db$, is proportional to the infinitesimal area of the corresponding disk, giving $dp = 2 b\ db/b_{\rm max}^2$. Outside of $b_{\rm max}$, the probability to detect a devil with a given $P_{\rm act}$ and $\Gamma_{\rm act}$  is assumed zero. For the Lorentz profile, we can relate the differential range of distances $db$ to $dP_{\rm obs}$: \begin{equation}  db = \frac{1}{2}\Gamma_{\rm act}\left[ \dfrac{P_{\rm act}}{P_{\rm obs}} - 1 \right]^{-1/2} \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) dP_{\rm obs} = \frac{1}{2}\left( \dfrac{\Gamma_{\rm act}}{2} \right)^2\dfrac{P_{\rm act}}{P_{\rm obs}^2} \dfrac{ dP_{\rm obs} }{b},  \end{equation} 
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\begin{equation}\label{eqn:probability-density_Pobs-Pact}  f_{\left( \Gamma_{\rm act} = {\rm const.} \right)}\ \dfrac{dp}{dP_{\rm obs}} = \left( \dfrac{P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right) P_{\rm act}\ P_{\rm obs}^{-2}.  \end{equation}Figure \ref{fig:probabilty-density_Pobs-Pact} illustrates this density, assuming $P_{\rm min} = 0.1$ and $P_{\rm max} = 10$ in arbitrary units.