deletions | additions       diff --git a/We_can_use_the_encounter__.tex b/We_can_use_the_encounter__.tex  index a40b398..cb8d4f0 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 $b_{\rm max} = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{\dfrac{P_{\rm act} - P_{\rm min}}{P_{\rm min}}}.$ This expression allows us to calculate the average miss distance $\langle b/\Gamma_{\rm act} \rangle = \int b/\Gamma_{\rm act}\ dp = 2/3\ b_{\rm max}/\Gamma_{\rm act} \approx 1/3 \sqrt{P_{\rm act}/P_{\rm min}}$, assuming $P_{\rm act} \gg P_{\rm min}$. Note that the average miss distance increases, albeit slowly, with increasing If, for example,  $P_{\rm act} \approx 10\ P_{\rm min}$, $\langle b \rangle \approx \Gamma_{\rm act}$, 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}