deletions | additions       diff --git a/During_each_shell_bu.tex b/During_each_shell_bu.tex  index 5295110..a464f08 100644  --- a/During_each_shell_bu.tex  +++ b/During_each_shell_bu.tex 
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Moreover, 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 a few times $10^{12} \, {\rm G}$ 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.  Figure \ref{fig:MassiveIGWspin} shows the distribution in maximum spin period of the pre-collapse iron core, $P_{\rm max,Fe}$, assuming the spin of the core is determined by stochastic IGW spin-up. We have plotted the values of $P_{\rm max,Fe}$, if the core 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 $P_{\rm max,Fe}  \lesssim {\rm several} few}  \times 10^3 \, {\rm s}$. 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, $P \gtrsim 2 \, {\rm s}$. The distribution of NS spin periods shown in Figure \ref{fig:MassiveIGWspin} may be broadly consistent with those inferred for young NSs \citealt{faucher:06,popov:10,gullon:14}. We are therefore tempted to speculate that the stochastic wave flux scenario described above may be the dominant process setting the spin rates of newly born NSs. If so, the scenario predicts that the spin rate/direction of the NS is uncorrelated with the spin rate/direction of the progenitor star, in contrast to any sort of magnetic spindown mechanism. However, there are several caveats to keep in mind. First, the scenario presented above can only proceed if the core is initially very slowly rotating, and requires efficient magnetic/wave core spin down to occur before Si burning. Second, the NS rotation rate may be changed during the supernova, by fallback effects, or by the r-mode instability \citep[See e.g.][for a review]{ott:2009}. Finally, there is a considerable amount of uncertainty in the wave flux and frequency spectrum. Since the minimum core rotation rate set by stochastic waves is proportional to the wave energy flux (which is uncertain at an order of magnitude level), there is an equal amount of uncertainty in the induced rotation rate.