deletions | additions       diff --git a/A_quartic_function_achieves_a__.tex b/A_quartic_function_achieves_a__.tex  index 0957755..cb2fe08 100644  --- a/A_quartic_function_achieves_a__.tex  +++ b/A_quartic_function_achieves_a__.tex 
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A quartic function achieves a reasonable (but informal) fit to the $\Gamma_{\rm obs}^\prime$-histogram, while a power-law with the same form as Equation \ref{eqn:n-Pact_from_uniform_n-Pobs} and an index of -1 does the same for the $P_{\rm obs}$-histogram. To work back to the underlying distribution of profile widths $n(\Gamma_{\rm act})$, we apply Equation \ref{eqn:n-Gammaact_from_n-Gammaobs}, which suggests a function quadratic in $\Gamma_{\rm act}$ for $n(\Gamma_{\rm act})$. Applying Equation \ref{eqn:n-Pact_from_n-Pobs} to the $P_{\rm obs}$-histogram fit suggests the underlying distribution $n(P_{\rm act}) \propto P_{\rm act}^{-1}$. \citet{Ellehoj_2010} suggest a form for $n(\Gamma_{\rm obs}) \propto \exp\left(-P_{\rm obs}/P_0\right)$, from which Equation \ref{eqn:n-Gammaact_from_n-Gammaobs} suggests $n(\Gamma_{\rm obs}) \propto \left( P_{\rm act}/P_0 - 2 \right) \exp\left(-P_{\rm act}/P_0\right)$.   A more formal application of our formulation to these data would likely involve a formal least-squares minimization for all the function parameters, including the normalization constants for each histogram and the $\Gamma_{\rm max}$ and $P_{\rm max}$-values. It would also involve consideration of the binning procedure used to make the histograms in the first place, all issues place and the counting statistics, particularly in the bins at either end of the distribution. For example, how can  we discuss further use the largest $P_{\rm obs}$-value to infer $P_{\rm max}$. The role of the detection scheme itself  in Section \ref{sec:dicussion_and_conclusions}. distorting the recovered population of devils is another important issue that must be considered.