deletions | additions       diff --git a/A_quartic_function_achieves_a__.tex b/A_quartic_function_achieves_a__.tex  index d29bb92..a3e615c 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, an issue all issues  we discuss further in Section \ref{sec:dicussion_and_conclusions}.