The spin rate of pre-collapse stellar cores: wave driven angular momentum transport in massive stars

Jim Fuller1 California Institute of Technology & Kavli Institute for Theoretical Physics
Matteo Cantiello Kavli Institute for Theoretical Physics
Daniel Lecoanet University of California at Berkeley
Eliot Quataert University of California at Berkeley



The core rotation rates of massive stars have a substantial impact on the nature of core collapse supernovae and their compact remnants. We demonstrate that internal gravity waves (IGW), excited via envelope convection during a red supergiant phase or during vigorous late time burning phases, can have a significant impact on the rotation rate of the pre-SN core. In typical (\(10 \, M_\odot \lesssim M \lesssim 20 \, M_\odot\)) supernova progenitors, IGW may substantially spin down the core, leading to iron core rotation periods \(P_{\rm min,Fe} \gtrsim 50 \, {\rm s}\). Angular momentum (AM) conservation during the supernova would entail minimum NS rotation periods of \(P_{\rm min,NS} \gtrsim 3 \, {\rm ms}\). In most cases, the combined effects of magnetic torques and IGW AM transport likely lead to substantially longer rotation periods. However, the stochastic influx of AM delivered by IGW during shell burning phases inevitably spin up a slowly rotating stellar core, leading to a maximum possible core rotation period. We estimate maximum iron core rotation periods of \(P_{\rm max,Fe} \lesssim 10^4 \, {\rm s}\) in typical core collapse supernova progenitors, and a corresponding spin period of \(P_{\rm max, NS} \lesssim 400 \, {\rm ms}\) for newborn neutron stars. This is comparable to the typical birth spin periods of most radio pulsars. Stochastic spin-up via IGW during shell O/Si burning may thus determine the initial rotation rate of most neutron stars. For a given progenitor, this theory predicts a Maxwellian distribution in pre-collapse core rotation frequency that is uncorrelated with the spin of the overlying envelope.



Rotation is a key player in the drama that unfolds upon the death of a massive star. The angular momentum (AM) contained in the iron core and overlying layers determines the rotation rate at core collapse (CC), which could have a strong impact on the dynamics of CC and the subsequent supernova (see e.g MacFadyen et al., 1999; Woosley et al., 2002; Woosley et al., 2006; Yoon et al., 2006). Rotation may help determine the nature of the compact remnant, which could range from a slowly rotating neutron star (NS) to a millisecond magnetar or rapidly rotating black hole (see e.g. Heger et al., 2000; Heger et al., 2005). The former may evolve into an ordinary pulsar, while the latter two outcomes offer exciting prospects for the production of long gamma-ray bursts (GRB) and superluminous supernova. In each of these phenomena, rotating central engines are suspected to be the primary source of power (Woosley, 1993; Kasen et al., 2010; Metzger et al., 2011).

Despite rotation being recognized as an important parameter controlling the evolution of massive stars (Maeder et al., 2000), little is known about the rotation rates of the inner cores of massive stars nearing CC. The best observational constraints stem from measurements of the rotation rates of the compact remnants following CC. For instance, a few low-mass black hole X-ray binary systems have been measured to have large spins that can only be accounted for by high spins at birth (Axelsson et al., 2011; Miller et al., 2011; Wong et al., 2012). However, the rotation rates of young NSs show little evidence for rapid rotation (\(P \lesssim 10 \, {\rm ms}\)) at birth. The most rapidly rotating young pulsars include PSR J0537-6910 (\(P=16\,{\rm ms}\)) and the Crab pulsar (\(P=33\,{\rm ms}\)), whose birth periods have been estimated to be \(P_i \lesssim 10\,{\rm ms}\) (Marshall et al., 1998) and \(P_i \sim 19 \, {\rm ms}\) (Kaspi et al., 2002), respectively. Many young NSs appear to rotate much more slowly, with typical periods of hundreds of ms (Lai, 1996; Gotthelf et al., 2013; Dinçel et al., 2015). In general, pulsar observations seem to indicate a broad range of initial birth periods in the vicinity of tens to hundreds of milliseconds (Faucher-Giguère et al., 2006; Popov et al., 2010; Gullón et al., 2014). Hence, rapidly rotating young NSs appear to be the exception rather than the rule.

Theoretical efforts have struggled to produce slow rotation rates. In the absence of strong AM transport mechanisms within the massive star progenitor, NSs would invariably be born rotating near break-up (Heger et al., 2000). Heger et al. (2005) and Suijs et al. (2008) examined the effect of magnetic torques generated via the Tayler-Spruit (TS) dynamo (Spruit, 2002), and found typical NS spin periods at birth (assuming AM conservation during CC and the ensuing supernova) of several milliseconds. Wheeler et al. (2014) implemented magnetic torques due to MRI and the TS dynamo, and were able to reach iron core rotation rates of \(P_{c} \sim 500 \, {\rm s}\), corresponding to NS spin periods of \(P \sim 25 \, {\rm ms}\). These efforts are promising, but the operation of both mechanisms within stars has been debated (e.g. Zahn et al., 2007), and theoretical uncertainties abound.

Recent asteroseismic advances have allowed for the measurement of core rotation rates in low-mass red giant stars (Beck et al. 2012; Beck et al. 2014; Mosser et al. 2012; Deheuvels et al. 2012; Deheuvels et al. 2014). In these stars, the core rotates much faster than the surface and one cannot assume nearly rigid rotation as suggested in Spruit et al. (1998). However, the cores of low-mass red giants rotate much slower than can be explained via hydrodynamic AM transport mechanisms or magnetic torques via the TS dynamo (Cantiello et al., 2014). If similar angular momentum transport mechanisms operate in more massive objects, this suggests that the pre-collapse cores of massive stars may rotate slower than predicted by many previous theoretical investigations.

Internal gravity waves (IGW) constitute a powerful energy and AM transport mechanism in stellar interiors. Several studies (Kumar et al. 1997; Zahn et al. 1997; Kumar et al. 1999; Talon et al. 2002; Talon et al. 2003; Talon et al. 2005; Talon et al. 2008; Charbonnel et al. 2005; Denissenkov et al. 2008; Fuller et al. 2014, hereafter F14) have found that convectively generated IGW can redistribute large quantities of AM within low-mass stars. IGW may partially account for the rigid rotation of the Sun’s radiative interior and the slow rotation of red giant cores, although magnetic torques are also likely to be important (Denissenkov et al. 2008, F14). IGW may also be important in more massive stars, and (Lee 2014) found that convectively generated IGW in B[e]-type stars may instigate outbursts that expel mass into the decretion disk.

Convectively excited IGW may also have a strong influence on the evoution of massive stars nearing CC. Indeed, after core carbon exhaustion, waves are the most effective energy transport mechanism within radiative zones, as photons are essentially frozen in and neutrinos freely stream out. In two recent papers, Quataert et al. (2012) and Shiode et al. (2014) (hereafter QS12 and SQ14) showed that the prodigious power carried by convectively excited waves (on the order of \(10^{10} L_\odot\) during Si burning) can sometimes unbind a large amount of mass near the stellar surface, and may substantially alter the pre-collapse stellar structure. IGW are ubiquitous in simulations of late burning stages (Meakin et al., 2006; Meakin et al., 2007; Meakin et al., 2007), although existing simulations have not quantified their long-term impact.

In this paper, we examine AM transport due to convectively excited IGW within massive stars, focusing primarily on AM transport during late burning stages (He, C, O, and Si burning). We find that IGW are generally capable of redistributing large amounts of AM before CC despite the short stellar evolution time scales. IGW emitted from convective shells propagate into the radiative core and may be able to substantially slow its rotation. In the limit of very efficient core spin-down via magnetic torques, we show that the stochastic influx of AM via IGW set a minimum core rotation rate which is comparable to the broad distribution of low rotation rates (\(P \lesssim 500 \,{\rm ms}\)) observed for most young NSs.

Our paper is organized as follows. In Section 2, we describe our massive star models, the generation of IGW during various stages of stellar evolution, and the AM they transport. In Section 3, we investigate whether the IGW can spin down the cores of massive stars, attempting to determine a minimum core rotation period. In Section 4, we consider whether IGW can stochastically spin up a very slowly rotating core, attempting to determine a maximum core rotation period. In Section 5, we conclude with a discussion of our results and their implications for CC, supernovae, and the birth of compact objects.