deletions | additions       diff --git a/We_can_use_the_encounter__.tex b/We_can_use_the_encounter__.tex  index 04e811d..5941a5a 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 $b \le b_{\rm max}$. Holding $\Gamma_{\rm act}$ fixed, we can use the above 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 act} - P_{\rm min}}\right) \Gamma_{\rm obs}.  \end{equation}  We can employ an analogous procedure involving Equation {eqn:P_obs} to calculate the probability density for an encounter to give an observed profile width between $P_{\rm obs}$ and $P_{\rm obs} + dP_{\rm obs}$:  \begin{equation}  \label{eqn:dp_dP_obs}  \dfrac{dp}{dP_{\rm obs}} = \dfrac{P_{\rm act}}{P_{\rm obs}^2} \left(\dfrac{\Gamma_{\rm act}}{2 b}\right)^2.  \end{equation}