deletions | additions       diff --git a/We_can_use_the_encounter__.tex b/We_can_use_the_encounter__.tex  index 8250610..4619887 100644  --- a/We_can_use_the_encounter__.tex  +++ b/We_can_use_the_encounter__.tex 
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We can use the encounter geometry to model the statistical probability for $P_{\rm obs}$ and $\Gamma_{\rm obs}$ to fall within a certain range of values, given a distribution of $P_{\rm act}$- and $\Gamma_{\rm act}$-values. The probability density for passing between $b$ and $b + db$ of a devil is $dp(b) = 2 b\ db / b_{\rm max}^2 $ for $b \le b_{\rm max}$. In this context, we take $b_{\rm max} = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{\dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}}}.$ th}}{P_{\rm th}}}.$  This expression allows us to calculate the average miss distance $\langle b/\Gamma_{\rm act} \rangle = \int b/\Gamma_{\rm act}\ dp = 2/3\ b_{\rm max}/\Gamma_{\rm act} \approx 1/3 \sqrt{P_{\rm act}/P_{\rm min}}$, th}}$,  assuming $P_{\rm act} \gg P_{\rm min}$. th}$.  If, for example, $P_{\rm act} \approx 10\ P_{\rm min}$, th}$,  $\langle b \rangle \approx \Gamma_{\rm act}$, meaning that, on average, $P_{\rm obs} \approx P_{\rm act}/5$ and $\Gamma_{\rm obs} \approx 5\ \Gamma_{\rm act}$. Holding $\Gamma_{\rm act}$ fixed, we can also use the probability density expression and Equation \ref{eqn:Gamma_obs} to calculate the probability density for an encounter to give an observed profile width between $\Gamma_{\rm obs}$ and $\Gamma_{\rm obs} + d\Gamma_{\rm obs}$:  \begin{equation}  \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 th}}{P_{\rm  act} - P_{\rm min}}\right) th}}\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 Equation \ref{eqn:Gamma_obs} 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 \sqrt{P_{\rm act}/P_{\rm min}}. th}}.  \end{equation}  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_{\rm max}}\right)^2 = \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) \left( \dfrac{P_{\rm min}}{P_{\rm th}}{P_{\rm  act} - P_{\rm min}} th}}  \right). \end{equation}  We also require that $P_{\rm obs} \le P_{\rm act}$.