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Brian Jackson edited Whether_or_not_we_detect__.tex
almost 9 years ago
Commit id: 0f67b0639bfa16f48a95bc8d912fbc9b0de0183a
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diff --git a/Whether_or_not_we_detect__.tex b/Whether_or_not_we_detect__.tex
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\end{equation}
$b_{\rm max}$ can also be written in terms of a devil's central pressure $P_{\rm act}$: $b_{\rm max} = \dfrac{\Gamma_{\rm act}}{2} \sqrt{ \dfrac{P_{\rm act}}{P_{\rm min}} - 1}$, giving the following expression for the probability density:
\begin{equation} \begin{equation}\label{eqn:dpdP_obs}
\dfrac{dp}{dP_{\rm obs}} = P_{\rm obs}^{-2} \left( \dfrac{P_{\rm min}}{P_{\rm act} - P_{\rm min}} \right) P_{\rm act}.
\label{eqn:dpdP_obs}
\end{equation}
Combining the recovery bias and distortion expression gives a probability density for a devil with a given $P_{\rm act}$ to be observed with a pressure between $P_{\rm obs}$ and $P_{\rm obs} + dP_{\rm obs}$:
\begin{equation} \begin{equation}\label{eqn:probability-density_Pobs-Pact}
f\ \dfrac{dp}{dP_{\rm obs}} = \left( \dfrac{P_{\rm min}}{P_{\rm max} - P_{\rm min}} \right) P_{\rm act}\ P_{\rm obs}^{-2}.
\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.