deletions | additions       diff --git a/During_each_shell_bu.tex b/During_each_shell_bu.tex  index 0264cb8..e4a3e2d 100644  --- a/During_each_shell_bu.tex  +++ b/During_each_shell_bu.tex 
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\end{equation}  In MLT, the quantity $\mathcal{M} L_{\rm conv} \propto \omega_c^4$. Therefore, we find approximate scaling $\Omega_{\rm ex} \propto \omega_c^{5/2} T_{\rm shell}^{1/2}$. Thus, more vigorous burning phases with higher convective turnover frequencies will lead to larger minimum core rotation rates. However, the later burning phases tend to have a much shorter life $T_{\rm shell}$, which largely counteracts their increased vigor.  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. Finally, stochastic spin-up can only occur if other sources of AM transport (e.g., magnetic torques) operate on longer time scales. This could be the case during late burning phases when magnetic torques become ineffective (\citealt{heger:05,wheeler:14}). (\citealt{Heger_2005,wheeler:14}).  Figure \ref{fig:MassiveIGWspin} shows a plot of the distribution in angular spin frequency $\Omega$ and spin period $P$, assuming the spin of the core is determined by stochastic wave fluxes. We have plotted the spin rate of the $M_{\rm Fe} \sim 1.4 M_\odot$, $R_{\rm Fe} \sim 1500 \, {\rm km}$ iron core before CC, if its spin rate is set during C, O, or Si burning. We have also plotted the corresponding spin rate of the $M_{\rm NS} \sim 1.4 M_\odot$, $R_{\rm NS} \sim 12 \, {\rm km}$ NS, with $I_{\rm NS} = 0.25 M_{\rm NS} R_{\rm NS}^2$, if its AM is conserved during the CC SN. We find that C, O, and Si burning all generate maximum iron core rotation periods on the order of $P \lesssim {\rm several} \times 10^3 \, {\rm s}$. Si burning most plausibly sets the pre-SN conditions, since it is the last convective burning phase before CC. The corresponding NS rotation rate is $P_{\rm NS} \lesssim 300 \, {\rm ms}$. Hence, we find that very slow core rotation rates, as speculated by Spruit \& Phinney, are unlikely. Nor do we expect that that there is likely to exist a population of NSs born with very long spin periods, $P \gtrsim 2 \, {\rm s}$.