deletions | additions       diff --git a/IGW_are_generated_by.tex b/IGW_are_generated_by.tex  index 46a3450..50435e5 100644  --- a/IGW_are_generated_by.tex  +++ b/IGW_are_generated_by.tex 
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For simplicity, we only focus on cases in which a convectively burning shell overlies the radiative core, irradiating it with IGW. A full understanding of the effects of IGW should also include the effects of IGW emitted during core convective phases, and the combined effects of IGW emitted by multiple convective shells. We focus on convective shell-generated IGWs because they are generated after core burning phases and thus have the final impact for a given burning phase. We use mixing length theory (MLT), as described in F14, to calculate IGW frequencies and fluxes. MLT produces convective velocities and Mach numbers similar to those seen in simulations (e.g., \citealt{meakinb:07}) and is likely adequate for our purposes.  Figure \ref{Massivestruc} \ref{fig:Massivestruc}  shows the density ($\rho$), mass [$M(r)$], and Brunt-V\"{a}is\"{a}l\"{a} frequency ($N$) profiles of our model during important convective shell phases. The first convective shell phase occurs during He-core burning, at which point the star has evolved into a red supergiant. At this stage, IGW are generated at the base of the surface convection zone and propagate toward the He-burning core. We have also shown profiles during shell C-burning, O-burning, and Si-burning, when the radiative core contains a mass of $M_c \sim 1 M_\odot$ and is being irradiated by IGW generated from the overlying convective burning shell. The basic features of each of these phases is quite similar, the main difference is that more advanced burning stages have smaller, higher density cores. We find that the characteristics of the convective burning shells (convective luminosities, turnover frequencies, mach numbers, and lifetimes) are similar to those listed in QS12, although the shell burning phases are generally more vigorous and shorter-lived than the core burning examined in Q12. Table 1 lists some of the parameters of our convective zones. \begin{table*}  \begin{center}  \caption{\label{tab:table} Properties of IGW AM transport during evolutionary stages corresponding to the stellar models shown in Figures \ref{Massivestruc} \ref{fig:Massivestruc}  and \ref{MassiveIGWtime}. Here, $r_c$ is the core radius that encloses the inner $\sim \! 1.2 M_\odot$, $T_{\rm shell}$ is the duration of the burning phase, and $\omega_{\rm con}$ is the convective turnover frequency. $\omega_*(r_c)$ is the wave frequency which dominates AM transport at $r_c$, while $P_{\rm min} = 2 \pi/\omega_*(r_c)$ is the minimum rotation period set by IGW during each phase. $P_{\rm min,Fe}$ is the minimum rotation period if AM is conserved until just before CC (when the iron core has a radius of 1500 km), and $P_{\rm min,NS}$ is the minimum rotation period if AM is conserved until NS birth. } \begin{tabular}{cccccccc}  \hline\  Burning Phase & $r_c$ (km) & $T_{\rm shell}$ (s) & $\omega_{\rm con}$ (Hz) & $\omega_*(r_c)$ (Hz) & $P_{\rm min}$ (s) & $P_{\rm min,Fe}$ (s) & $P_{\rm min,NS}$ (s) \\