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Brian Jackson edited Whether_or_not_we_detect__.tex
almost 9 years ago
Commit id: e0d0d4f91579b7be4d2ec79c738bfe24762e92c7
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diff --git a/Whether_or_not_we_detect__.tex b/Whether_or_not_we_detect__.tex
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Whether or not we detect a devil with a given $P_{\rm act}$, the the fact that $b$ probably won't be zero means it 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
given by proportional to the infinitesimal area of the corresponding disk, giving $dp = 2 b\ db/b_{\rm max}$. Outside of $b_{\rm max}$, the probability 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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\label{eqn:probability-density_Pobs-Pact}
\end{equation}Figure \ref{fig:probabilty-density_Pobs-Pact} illustrates this density, assuming $P_{\rm min} = 0.1$ hPa and $P_{\rm max} = 10$ hPa.