Jim Fuller edited spindown.tex  about 9 years ago

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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 burning, waves of lower frequency are significantly attenuated by radiative photon diffusion. Neutrino damping is likely irrelevant at all times. During C/O/Si 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-linear breaking instead of 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. A typical massive star has a zero-age main sequence equatorial rotation velocity of $v_{\rm rot} \sim 150 \,{\rm km \ s}^{-1}$ \citep{de_Mink_2013}, corresponding to a rotation period of $P_{\rm MS} \sim 1.5 \, {\rm d}$ for our stellar model. Using this rotation rate, we calculate 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, causing the core to spin-up as it contracts. Of course, magnetic torques may extract much of this AM and in the contracting core $J_0$ represents an upper limit to the AM contained within the mass coordinate $M(r)$. Both $J_0$ and the corresponding evolving rotation profiles are shown in Figure \ref{fig:MassiveIGWtime}. We also plot the approximate AM $J_{\rm NS}$ contained within a NS rotating at $P_{\rm NS} = 10 \, {\rm ms}$, which is more than two orders of magnitude smaller than the value of $J_0$ within the inner $1.4 M_\odot$.  In the top panel of Figure \ref{fig:MassiveIGWtime}, we plot the amount of AM capable of being extracted by IGW during each burning phase,   \begin{equation}  J_{\rm ex} = \dot{J}_*(r) T_{\rm shell} \, .  \label{eqn:Jex}  \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 $J_{\rm ex}$ are comparable to $J_0$ for waves emitted during He core burning and C shell burning. This implies 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, if the core retains its full AM from birth. If, however, This does not imply IGW have no effect, as the value of $J_{\rm ex}$ for O/Si burning is larger than $J_{\rm NS}$. Therefore,  if the core has been spun down by IGW or magnetic torques during previous burning phases, IGW during late burning phases may be critical in modifying 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 profile $P_0$ corresponding to the AM profile $J_0$ 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, which is shorter than that inferred for most newly born NSs. Therefore 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.