deletions | additions       diff --git a/We_can_use_the_encounter__.tex b/We_can_use_the_encounter__.tex  index 37e43cb..9281ef3 100644  --- a/We_can_use_the_encounter__.tex  +++ b/We_can_use_the_encounter__.tex 
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\label{eqn:dp_dGamma_obs}  \dfrac{dp}{d\Gamma_{\rm obs}} = \dfrac{\Gamma_{\rm obs}}{b_{\rm max}^2} = 4 \Gamma_{\rm act}^{-2} \left( \dfrac{P_{\rm min}}{P_{\rm act} - P_{\rm min}}\right) \Gamma_{\rm obs}.  \end{equation}  We require that $b \le b_{\rm max}$ in order for a devil to be detected, which limits the range of allowable values for $\Gamma_{\rm obs}$, given $P_{\rm act}$ and $\Gamma_{\rm act}$. We can use the relationship $\Gamma_{\rm obs}^2 = \Gamma_{\rm act}^2 + \left(2 b\right)^2$ and $b_{\rm max} = \left( \Gamma_{\rm act}/2 \right) \sqrt{\left( P_{\rm act} - P_{\rm min}\right)/P_{\rm min}}$ to solve for $\Gamma_{\rm obs}/\Gamma_{\rm act}$:  \begin{equation}  \label{eqn:Gamma_obs_limits}  1 \le \dfrac{\Gamma_{\rm obs}}{\Gamma_{\rm act}} \le \left[ 1 + \frac{1}{4} \left( \dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}} \right) \right]^{1/2}.  \begin{equation}  This range is actually quite narrow. For example, for $P_{\rm act} = 100\ P_{\rm min}$, we require $1 \le \Gamma_{\rm obs}/\Gamma_{\rm act} \lesssim 5$.  We can employ an analogous procedure involving Equation \ref{eqn:P_obs} to calculate the probability density for an encounter to give an observed profile depth between $P_{\rm obs}$ and $P_{\rm obs} + dP_{\rm obs}$:  \begin{equation}  \label{eqn:dp_dP_obs}  \dfrac{dp}{dP_{\rm obs}} = \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) \left(\dfrac{\Gamma_{\rm act}}{2 b}\right)^2 = \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) \left( \dfrac{P_{\rm min}}{P_{\rm act} - P_{\rm min}} \right).  \end{equation} We also require that $P_{\rm obs} \le P_{\rm act}$.