deletions | additions       diff --git a/subsection_bf_Wave_Leakage_Time__.tex b/subsection_bf_Wave_Leakage_Time__.tex  index 387049e..1088f52 100644  --- a/subsection_bf_Wave_Leakage_Time__.tex  +++ b/subsection_bf_Wave_Leakage_Time__.tex 
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t_{\rm leak} = \frac{ E_{\rm ac} }{\dot{E}_{\rm leak}} \, .  \end{equation}    We have calculated the leakage timescales for waves at frequencies $\omega_{\rm max} = 2 \pi \nu_{\rm max}$ for stellar models on the lower RGB. Figure \ref{fig:DipoleTime} shows the exact  value of $t_{\rm leak}$ calculated from equation \ref{eqn:tleak2}, and $t_{\rm leak}$  approximated from equation \ref{eqn:tleak}, with $T$ calculated via equations \ref{eqn:integral2} and \ref{eqn:integral}. Clearly, evaluating $t_{\rm leak}$ via equation \ref{eqn:tleak} with $T$ calculated from equation \ref{eqn:integral} is a very good approximation, accurate to within $\sim \! 10 \%$ for our stellar models. However, using the approximation of equation \ref{eqn:integral2} is not very accurate, and generally produces a value of $t_{\rm leak}$ too large by a factor of $\sim \! 2$.     
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