deletions | additions       diff --git a/section_Formulating_the_Recovery_Biases__.tex b/section_Formulating_the_Recovery_Biases__.tex  index 894297c..87ac479 100644  --- a/section_Formulating_the_Recovery_Biases__.tex  +++ b/section_Formulating_the_Recovery_Biases__.tex 
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\begin{equation}\label{eqn:Pobs_from_Lorentz_profile}  P_{\rm obs} = \dfrac{P_{\rm act}}{1 + \left( \dfrac{2 b}{\Gamma_{\rm act}} \right)^2}.  \end{equation}  Clearly, unless $b = 0$, $P_{\rm obs} < P_{\rm act}$, and, it turns out, and  the profile width observed in the time series $\Gamma_{\rm obs}^\prime > \Gamma_{\rm act}^\prime$. act}^\prime$, as pointed out by \cite{Ellehoj_2010} and \cite{Lorenz_2014}.  These effects represent the signal distortion induced by the miss distance effect. Given $P_{\rm obs}$- and $P_{\rm act}$-values, we can solve Equation \ref{eqn:Pobs_from_Lorentz_profile} for $b$:  \begin{equation}\label{eqn:b_from_Lorentz_profile}  b = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm obs}}{P_{\rm obs}}}.  \end{equation}  There is a maximum closest approach distance $b_{\rm max}$ beyond which a devil will produce an undetectably small pressure signal, $P_{\rm obs} < P_{\rm min}$, and $b_{\rm max} = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{ \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}}}$. Devils with $b > b_{\rm max}$ will not be detected, which will bias our recovered population of devil parameters in ways that depend on the parameters themselves. It's worth noting that $\Gamma_{\rm act}$ may depend on $P_{\rm act}$, a point we will return to in Section. Again, this recovery bias results from the miss distance effect. Next, we use these equations to formulate the recovery biases and signal distortions resulting from the miss distance effect.