deletions | additions       diff --git a/During_each_shell_bu.tex b/During_each_shell_bu.tex  index 1d2dfdf..169a379 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 described above for long as  $\Omega_{\rm ex} \ll \omega_{c}$. \omega_c$.  If $\Omega_{\rm ex}$ approaches $\omga_c$, $\omega_c$,  wave filtering processes as described in Section \ref{spin} 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. 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, $P \gtrsim 2 \, {\rm s}$.