deletions | additions       diff --git a/IGW_are_generated_by.tex b/IGW_are_generated_by.tex  index a851b66..98c72ca 100644  --- a/IGW_are_generated_by.tex  +++ b/IGW_are_generated_by.tex 
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\begin{equation}  J_{\rm ex} = \dot{J}_*(r) T_{\rm shell}.  \end{equation}  We plot both a pessimistic and optimistic estimate, corresponding to the left and right-hand sides of equation \ref{eqn:Ewaves}, respectively. We find that the values of $J_{\rm ex}$ is comparable to $J_0$ for waves emitted during He core burning and C shell burning. This entails that IGW emitted during these phases may be able to significantly spin down the cores of massive stars. However, given the uncertainties, it is unclear whether IGW have a significant effect. During O and Si shell burning, we find that IGW most likely cannot remove the AM contained within the core. However, if the core has been spun down by IGW during previous burning phases, or via magnetic torques, IGW may still modify the core spin rate (see Section \ref{spinup}).  If IGW are able to spin down the core during He core burning and C shell burning, this entails a minimum possible core rotation period $P_{\rm min} = 2 \pi/\omega_*(r)$ at the end of these phases. The bottom panel of Figure \ref{fig:MassiveIGWtime} plots the value of $P_{\rm min}$, in addition to the rotation period $P_0$ corresponding to the AM $J_0$ that would occur in the absence of AM transport. If IGW are able to spin down the cores, they entail minimum rotation periods 10-100 times larger than those that would exist without AM transport. Thus, IGW may significantly spin down the cores of massive stars. Table 1 lists the values of $P_{\rm min}$ corresponding to He and C burning, as well as corresponding minimium spin periods for the pre-collapse iron core ($P_{\rm min,Fe}$) and for the neutron star remnant ($P_{\rm min,NS}$) given no subsequent AM transport. The minimum NS rotation period $P_{\rm min,NS}$ we calculate is on the order of milliseconds, shorter than those inferred for most newly born NSs. This entails that either IGW spin-down is significantly more effective than our conservative estimates, or (perhaps more likely) magnetic torques are the primary mechanism responsible for spinning down the cores of massive stars.