deletions | additions       diff --git a/Turning_to_Gamma_values_non__.tex b/Turning_to_Gamma_values_non__.tex  index b325b5f..fc63601 100644  --- a/Turning_to_Gamma_values_non__.tex  +++ b/Turning_to_Gamma_values_non__.tex 
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Likewise, non-central encounters with a dust devil will distort the profile full-width/half-max, giving a full-width/half-max $\Gamma_{\rm obs} > \Gamma_{\rm act}$. Having passed through its minimum at $b$, the observed pressure signal reaches half its value at a time $t = \frac{1}{2} \Gamma_{\rm obs}^\prime = \frac{1}{2} \Gamma_{\rm obs}/\upsilon$ by definition. At this time, the center of the devil is a radial distance from the barometer $r(\Gamma_{\rm obs}^\prime) = \sqrt{b^2 + \left( \Gamma_{\rm obs}/2 \right)^2}$ and $P(r) = \frac{1}{2} P_{\rm obs} = \dfrac{P_{\rm act}/2}{1 + \left( 2 b /\Gamma_{\rm act} \right)^2} = \dfrac{P_{\rm act}}{1 + \left( 2r(\Gamma_{\rm obs}^\prime)/\Gamma_{\rm act} \right)^2 }$. Solving for $\Gamma_{\rm obs}$ gives $\Gamma_{\rm obs} = \sqrt{\Gamma_{\rm act}^2 + \left( 2b \right)^2}$.  Turning to $\Gamma$-values, non-central ($b > 0$) encounters always widen and never narrow an observed profile, so we require $\Gamma_{\rm obs} > \Gamma_{\rm act}$. We also require that $b \le b_{\rm max}$, 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 $\Gamma_{\rm obs}/\Gamma_{\rm act}$:  $$  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}.