We have demonstrated that convectively generated internal gravity waves (IGW) in massive stars are capable of redistributing angular momentum (AM) on short time scales. In particular, we have focused on the effects of IGW generated during late stages of massive star evolution (He burning and later) for typical neutron star (NS) progenitors (\(10 M_\odot \lesssim M \lesssim 20 M_\odot\)). It may seem surprising that AM transport via IGW can act on the short stellar evolution timescales of massive stars nearing core collapse (CC). However, the huge convective luminosities inside evolved massive stars ensure large fluxes of IGW (QS12, SQ14) that can transport energy and AM on short timescales. We therefore encourage efforts to incorporate the effects of IGW in stellar evolution codes focusing on the final stages of massive star evolution.
We have shown that inwardly propagating IGW launched from convective shells may be able to slow down the core to much slower spin rates than would be obtained in the absence of other AM transport mechanisms. For our \(12 M_\odot\) model, IGW generated at the base of the convective zone during the core He burning red supergiant phase may slow the core to minimum spin periods of \(P_{\rm min,He} \sim 2 \, {\rm days}\). IGW launched during C shell burning may also be able to substantially slow the spin of the core. These convective phases lead to pre-collapse iron cores with rotation periods \(P_{\rm min,Fe} \gtrsim 50\,{\rm s}\), corresponding to initial NS rotation periods of \(P_{\rm min,NS} \gtrsim 2.5 \, {\rm ms}\). The rotation periods listed above are minimum periods for our stellar model. Calculations of rotation rates including magnetic torques \citep{Heger_2005,wheeler:14} typically yield rotation periods several times larger. Magnetic torques may therefore be the dominant AM transport mechanism responsible for extracting AM from massive stellar cores, although it is likely that both mechanisms play a significant role.
Stochastic influxes of IGW during late burning phases can also lead to the spin-up of an otherwise very slowly rotating core. This occurs in the case of very efficient prior core spin-down via IGW/magnetic torques. Such efficient core spin-down is not unreasonable, especially given that the cores of low mass red giant stars rotate slower than can be accounted for using existing prescriptions for hydrodynamic mechanisms or magnetic torques via the Tayler-Spruit dynamo \citep{cantiello:14}. It is thus quite plausible that massive star cores are efficiently spun down via waves/magnetic torques, after which they are stochasticly spun up via waves launched during O/Si burning. If this mechanism determines the core spin rate before death, it predicts a Maxwellian distribution in spin frequency, with typical iron core spin periods of \(300 \, {\rm s} \lesssim P_{\rm Fe} \lesssim 10^4 \, {\rm s}\). We thus find it extremely unlikely that magnetic torques can enforce very large pre-collapse spin periods as claimed by \cite{Spruit_1998}. Additionally, we speculate that the stochastic spin-up process is relatively insensitive to binary interactions or winds that have stripped the stellar envelope, as long as these processes do not strongly modify the core structure and late burning phases. We also express a word of caution, as Si burning is notoriously difficult for stellar evolution codes to handle, and the properties of Si burning produced by our MESA evolutions have large associated uncertainties. The rough energy and AM fluxes in convectively excited waves are, however, reasonable at the order of magnitude level.
If AM is conserved during the supernova, stochastic IGW spin-up entails NS birth periods of \(20 \, {\rm ms} \lesssim P_{\rm NS} \lesssim 400 \, {\rm ms}\), albeit with significant uncertainty. These estimates are comparable to spin periods of typical young NSs (\citealt{lai:96,gotthelf:13}), and to the broad inferred birth spin period distribution of \(P_{\rm NS} \lesssim 500 \, {\rm ms}\) for ordinary pulsars (\citealt{faucher:06,popov:10,gullon:14}). Therefore, stochastic wave spin-up could be the dominant mechanism in determining the rotation periods of pre-collapse SN cores and newborn NSs. In this scenario, there is little or no correlation between the spin of the progenitor and the spin of the NS it spawns. Although torques during the supernova may modify the spin rate of the NS, they would have to be very finely tuned to erase the stochastic spin-up occurring during shell burning. Any sort of purely frictional spin-down processes would likely slow the NS to rotation periods larger than typically inferred for young NSs.
We have quantitatively considered the implications of IGW AM transport in two distinct limits. In the first, we neglect AM transport by magnetic fields and consider the limit in which the core is rotating much faster than the convective shell because of AM conservation during core contraction. In this case, IGW emitted from convective shells propagate into the radiative core and may be able to substantially slow its rotation, enforcing a maximum rotation rate. The second limit we consider is when the stellar core has been efficiently spun-down via magnetic coupling to the envelope and/or IGW in earlier phases of stellar evolution. In this case, we have shown that the stochastic influx of AM via IGW from shell burning leads to a spin-up of the stellar core and a minimum core rotation rate. Taken together, these limits enforce iron core rotation periods \(50 \, {\rm s} \lesssim P_{\rm Fe} \lesssim 10^4 \, {\rm s}\) and initial NS rotation periods of \(2.5 \, {\rm ms} \lesssim P_{\rm NS} \lesssim 400 \, {\rm ms}\). We expect these limits to be robust against many uncertain factors in massive star evolution, e.g., birth spin rate, mass loss, mixing, and the effects of magnetic fields.
There is ample evidence that some CC events occur with rapidly rotating cores. In particular, long GRBs almost certainly require a rapidly rotating central engine \citep{1993ApJ...405..273W,Yoon_2006,Woosley_2006,Metzger_2011}, and the picture advanced above must break down in certain (although somewhat rare) circumstances. It is not immediately clear what factors contribute to the high spin rate in GRB progenitors, as our analysis was restricted to “typical" NS progenitors with \(10 M_\odot \lesssim M \lesssim 20 M_\odot\), which explode to produce type-IIp supernovae during a red supergiant phase \citep[see e.g.][]{Smartt_2009}. We speculate that GRB progenitors (if occurring in effectively single star systems) have never undergone a red supergiant phase, as torques via magnetic fields and/or waves are likely to spin down the helium core by coupling it with the huge AM reservoir contained in the slowly rotating convective envelope. Alternatively, it may be possible that stars with very massive He cores, which exhibit more vigorous pre-SN burning phases, can generate stochastic wave spin-up strong enough to produce a GRB. Our preliminary examination of more massive models indicates that stochastic spin-up may lead to slightly larger spin rates, but is unlikely to generate very rapid rotation. A third possibility is that spin-up via mass transfer/tidal torques in binary systems is required for GRB production \citep{Cantiello_2007}. A merger or common envelope event after the main sequence could also remove the extended convective envelope and prevent it from spinning down the core.
The population of massive stars approaching death is complex, and factors such as initial mass, rotation, metallicity, binarity, magnetic fields, overshoot, mixing, winds, etc., will all contribute to the anatomy of aging massive stars. We have argued that AM transport via convectively driven IGW is likely to be an important factor in most massive stars. But it is not immediately obvious how this picture will change in different scenarios, e.g., electron capture supernovae, very massive \((M_i \gtrsim 40 M_\odot)\) stars, interacting binaries, etc. We hope to explore these issues in subsequent works.