deletions | additions       diff --git a/subsection_The_Gamma__rm_act__.tex b/subsection_The_Gamma__rm_act__.tex  index 598c613..daa13b0 100644  --- a/subsection_The_Gamma__rm_act__.tex  +++ b/subsection_The_Gamma__rm_act__.tex 
...
\end{equation}  Using this equation and the encounter geometry again, we find that the probability density for $\Gamma_{\rm obs}$ is   \begin{equation}\label{eqn:dpdGamma_obs}  \dfrac{dp}{d\Gamma_{\rm obs}} = 8\ 4\  \Gamma_{\rm act}^{-2}\ \Gamma_{\rm obs}. \end{equation}  Considering a fixed $\Gamma_{\rm act}$-value, this expression indicates we are more and more likely to observe such a devil with a wider and wider profile because more distant encounters are more likely and the more distant the encounter, the wider the observed pressure profile. Conversely, if we observe a $\Gamma_{\rm obs}$-value, the expression indicates a high likelihood it originated from a narrow (small $\Gamma_{\rm act}$) devil.  Again combining this expression with the recovery bias gives the probability density for a devil with a given $\Gamma_{\rm act}$-value to be observed with a $\Gamma_{\rm obs}$-value:  \begin{equation}\label{eqn:probability-density_Gammaobs-Gammaact}  f_{\rm \left( P_{\rm act} =\ {\rm const.} \right)}\ \dfrac{dp}{d\Gamma_{\rm obs}} = 8\ 4\  \Gamma_{\rm obs}/\Gamma_{\rm max}^2. \end{equation}  We see the dependence on $\Gamma_{\rm act}$ in each term cancels out, and the probability to observe a $\Gamma_{\rm obs}$-value in a narrow range depends only on $\Gamma_{\rm obs}$, meaning observing a wider profile is always more likely, no matter the profile's underlying width. 
...
\end{equation}  which, for a uniform $n(\Gamma_{\rm act}) = k$, gives  \begin{equation}\label{eqn:n-Gammaobs_from_uniform-n-Gammaact}  n(\Gamma_{\rm obs}) = 8k\ 4k\  \Gamma_{\rm max}^{-2}\ \left( \Gamma_{\rm obs} - \Gamma_{\rm min} \right) \Gamma_{\rm obs}. \end{equation}  Figure \ref{fig:n-Gammaobs_from_uniform-n-Gammaact} illustrates this result (using linear axes since the curves are not as steep as in the previous figures).