deletions | additions       diff --git a/sectionSpin_Evolutio.tex b/sectionSpin_Evolutio.tex  index 2e2b6d3..ee2aad6 100644  --- a/sectionSpin_Evolutio.tex  +++ b/sectionSpin_Evolutio.tex 
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Using our spin-down time scales derived above, we can begin to speculate on the spin evolution of the cores of typical NS progenitors. We have focused on a low mass ($12 M_\odot$), solar metallicity model at red supergiant stages of evolution. These stars invariably contain an extended, slowly spinning convective zone that is a nearly unlimited AM sink. The core spin rate can therefore be changed without any significant change in envelope rotation rate. In what follows, we ignore magnetic torques, which may substantially slow core rotation rates \citep{Heger_2005,wheeler:14}. We therefore consider our results to represent maximum expected spin rates, which will be enforced regardless of the efficacy of magnetic torques.  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 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.