deletions | additions       diff --git a/During_each_shell_bu.tex b/During_each_shell_bu.tex  index 86cd466..2c30f12 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. For C-burning in the optimistic wave flux estimate ($L_{\rm wave} \propto \mathcal{M}^{5/8}$), $\Omega_{\rm ex}$ approaches $\omega_c$, and therefore  this value of $\Omega_{\rm ex}$ lies near the upper limit of maximum rotation rate  achievable through  stoachastic spin-up. spin-up for our stellar model.  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_2005,wheeler:14}). We can also estimate a minimum magnetic coupling time between core and envelope via the Alven wave crossing time $t_A \approx r_c \sqrt{\rho_c}/B$, with $B$ the approximate magnetic field strength. Typical neutron star field strengths of $10^{12} \, {\rm G}$ imply field strengths of $\sim 10^8 \,{\rm G}$ in the iron core, which yields $t_A \sim 5 \times 10^4 \,{\rm s}$, much longer than the Si shell burning time (see Table 1). Although magnetic torques may reduce stochastic spin-up during C/O burning phases, we expect them to have a negligible impact during Si burning.