deletions | additions       diff --git a/During_each_shell_bu.tex b/During_each_shell_bu.tex  index 54ea397..3924316 100644  --- a/During_each_shell_bu.tex  +++ b/During_each_shell_bu.tex 
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\label{eqn:omex}  \Omega_{\rm ex} = \sqrt{\frac{4 \omega_c T_{\rm shell} }{3 \pi^2}} \frac{J_w}{I_c} \, .  \end{equation}  In MLT, the quantity $\mathcal{M} L_{\rm conv} \propto \omega_c^4$. Therefore, we find approximate scaling The expected spin frequency scales as  $\Omega_{\rm ex} \propto \omega_c^{5/2} \omega_c^{-1/2} L_c  T_{\rm shell}^{1/2}$. Thus, more vigorous burning phases with energetic and long-lived convection yields  higher convective turnover frequencies will lead to larger minimum core expected  rotation rates. However, the The short duration $T_{\rm shell}$ of  later burning phasestend to have a much shorter life $T_{\rm shell}$, which  largely counteracts their increased vigor. vigor, and we find that earlier burning phases are generally capable of depositing more AM.  The stochastic spin-up process described above will only occur under certain conditions. First, as already mentioned, the core and burning shell must be slowly rotating, or else the stochastic spin-up will have a negligible effect. Second, all waves (both prograde and retrograde) must be absorbed by the core. In the cores of massive stars, this is likely to occur because of non-linear breaking due to geometric focusing as waves approach the center of the star. Third, stochastic spin-up can only proceed as long as $\Omega_{\rm ex} \ll \omega_c$. If $\Omega_{\rm ex}$ approaches $\omega_c$, wave filtering processes as described in Section \ref{igw} will alter subsequent dynamics. Our estimates below have $\Omega_{\rm ex} \ll \omega_c$, therefore, we believe they are valid estimates of minimum spin rates. For C-burning in the optimistic wave flux estimate ($L_{\rm wave} \propto \mathcal{M}^{5/8}$), $\Omega_{\rm ex}$ approaches $\omega_c$, therefore this value of $\Omega_{\rm ex}$ lies near the maximum rotation rate achievable through stoachastic spin-up for our stellar model.