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 \(b_{\rm max} = \left( \dfrac{\Gamma_{\rm act}}{2} \right) \sqrt{\dfrac{P_{\rm act} - P_{\rm th}}{P_{\rm th}}}.\) This expression allows us to calculate the average miss distance \(\langle b/\Gamma_{\rm act} \rangle = \int b/\Gamma_{\rm act}\ dp = 2/3\ b_{\rm max}/\Gamma_{\rm act} \approx 1/3 \sqrt{P_{\rm act}/P_{\rm th}}\), assuming \(P_{\rm act} \gg P_{\rm th}\). If, for example, \(P_{\rm act} \approx 10\ P_{\rm th}\), \(\langle b \rangle \approx \Gamma_{\rm act}\), 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 \(\Gamma_{\rm obs} + d\Gamma_{\rm obs}\): \[\label{eqn:dp_dGamma_obs} \dfrac{dp}{d\Gamma_{\rm obs}} = \dfrac{\Gamma_{\rm obs}}{b_{\rm max}^2} = 4 \Gamma_{\rm act}^{-2} \left( \dfrac{P_{\rm th}}{P_{\rm act} - P_{\rm th}}\right) \Gamma_{\rm obs}.\] 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 \(\Gamma_{\rm obs}/\Gamma_{\rm act}\): \[\label{eqn:Gamma_obs_limits} 1 \le \dfrac{\Gamma_{\rm obs}}{\Gamma_{\rm act}} \le \sqrt{P_{\rm act}/P_{\rm th}}.\]

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 \(P_{\rm obs} + dP_{\rm obs}\): \[\label{eqn:dp_dP_obs} \dfrac{dp}{dP_{\rm obs}} = \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) \left(\dfrac{\Gamma_{\rm act}}{2 b_{\rm max}}\right)^2 = \left( \dfrac{P_{\rm act}}{P_{\rm obs}^2} \right) \left( \dfrac{P_{\rm th}}{P_{\rm act} - P_{\rm th}} \right).\] We also require that \(P_{\rm obs} \le P_{\rm act}\).

Figure \ref{fig:distortion_probabilities} shows the probability densities for \(\Gamma_{\rm obs}\) and \(P_{\rm obs}\), as well as the corresponding limits. To interpret the figure, consider \(\Gamma_{\rm act} = 10^{1.5}\) m in panel (a). Contours of \(\dfrac{dp}{d\Gamma_{\rm obs}}\) increase going up, indicating that the miss distance effect drives \(\Gamma_{\rm obs}\) toward larger values (within the limited range allowed). In panel (b), contours of \(\dfrac{dp}{dP_{\rm obs}}\) increase going down, showing \(P_{\rm obs}\) tends toward smaller values for a fixed \(P_{\rm act}\).