deletions | additions       diff --git a/sectionSpin_Evolutio.tex b/sectionSpin_Evolutio.tex  index 8202ebf..0997149 100644  --- a/sectionSpin_Evolutio.tex  +++ b/sectionSpin_Evolutio.tex 
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Our results (see Figure \ref{fig:MassiveIGWtime}) indicate that the entire helium core ($M \lesssim 4 M_\odot$) is likely to be substantially spun down by IGW excited at the base of the surface convection zone. The IGW may not penetrate into the inner $\sim 1.5 M_\odot$ because this region contains the convective He-burning core. However, we expect the convective central core to couple relatively strongly to the overlying He outer core, either through IGW emitted by the convective core, through magnetic torques, or during subsequent burning phases. Figure \ref{fig:MassiveIGWtime} indicates that a plausible upper limit to the core angular frequency during the He-burning stage is $\Omega_{\rm max} \sim \omega_* \sim 4 \times 10^{-5}{\rm Hz}$. The corresponding minimum rotation period is $P_{\rm min} \sim 2 \, {\rm days}$. In the absence of AM transport, using a typical main sequence equatorial rotation velocity of $v = 100 \, {\rm km}\,{\rm s}^{-1}$ \citep{demink:13}, the He-burning core would rotate at $P \sim 3 \,{\rm hr}$. IGW will therefore substantially spin down the core during the He core burning phase.  IGW will continue to spin down the core during subsequent burning phases. Table 1 lists convective turnover frequencies $\omega_{\rm con}$, dominant wave frequencies $\omega_*$, and minimum core rotation periods enforced during different stages of evolution. IGW launched by the C-burning shell limit the core angular spin frequency to $\Omega \lesssim 2 \times 10^{-3} \, {\rm Hz}$ and spin period to $P \gtrsim 3 \times 10^3 \,{\rm s}$. In contrast, the core would be spinning at $P \sim 40\,{\rm s}$ given no internal AM transport. Similarly, the O-burning shell will provide limits of $\Omega \lesssim 10^{-2} \, {\rm Hz}$ and $P \gtrsim 600 \,{\rm s}$, in contrast to $P \sim 5 \, {\rm s}$ without AM transport. In our model, it is unclear whether waves launched during Si burning can significantly spin down the core. We find that core spin-down during C-burning is our most robust result because $t_* \ll T_{\rm shell}$ everywhere in the inner core  during this phase. The results above indicate that IGW can likely spin down the core to rotation rates at least one hundred times slower than in the absence of AM transport. Magnetic torques may supplement IGWs and result in even slower rotation rates. Either way, the effects of IGW through O-shell burning imply iron core rotation rates (when the iron core radius is $R_c \approx 1500 \,{\rm km}$) of $P_{\rm Fe} \gtrsim 130 \,{\rm s}$. Such a slow rotation rate is unlikely to strongly affect the dynamics of the subsequent SN (***REF***), and would result in an NS rotation period of $P_{\rm NS} \gtrsim 7 \,{\rm ms}$ in the absence of AM transport during the SN.