deletions | additions       diff --git a/IGW_are_generated_by.tex b/IGW_are_generated_by.tex  index 394fd00..98a31bf 100644  --- a/IGW_are_generated_by.tex  +++ b/IGW_are_generated_by.tex 
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In Appendix \ref{wavestar}, we calculate characteristic wave frequencies $\omega_*(r)$ and AM fluxes $\dot{J}_*(r)$ entering the core during different phases of massive star evolution. During core He and shell C-burning, waves of lower frequency are somewhat attenuated by radiative photon diffusion. Neutrino damping is likely irrelevant at all times. During shell burning phases, the waves become highly non-linear as they approach the inner $\sim 0.3 M_\odot$, and we expect them to be mostly dissipated via non-linearly breaking before reflecting at the center of the star.   The AM deposited by IGW is only significant if it is larger than the amount of AM contained within the core of the star. For a typical massive star that has a zero-age main sequence equatorial rotation velocity of $v_{\rot} \sim 150 \,{\rm km \ s}^{-1}$ (corresponding to a rotation period of $P_{\rm MS} \sim 1.5 \, {\rm d}$ for our stellar model), we caluclate the AM $J_0(M)$ contained within the mass coordinate $M_(r)$, given rigid rotation on the main sequence. In the absence of AM transport, this AM is conserved, although the core will spin-up as it contracts. Of course, magnetic torques may extract much of this AM and thus $J_0$ represents an upper limit to the AM contained within the mass coordinate $M(r)$. Both $J_0$ and the corresponding rotation profile are shown in the bottom panel of Figure \ref{fig:MassiveIGWtime}.  For each phase, we calculate the characteristic IGW spin down time scale, $t_*(r)$, on which the waves can change the spin rate by an amount $\omega_*(r)$,  \begin{equation}  \label{eqn:tstar}