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


  1. jfuller@caltech.edu

Abstract

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.

Introduction

\label{intro}

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.

Convectively Excited IGW

\label{igw}

IGW are generated by convective zones and propagate into neighboring stably stratified regions, carrying energy and AM. To estimate energy and AM fluxes carried by IGW, we use techniques similar to those of F14, QS12, and SQ14. We begin by constructing a sequence of stellar models using the MESA stellar evolution code (Paxton et al., 2011; Paxton et al., 2013). In what follows, we focus on a \(M=12 M_\odot\), \(Z=0.02\) model that has been evolved to CC. Details on the model can be found in Appendix \ref{model}. For our purposes, the most important model outputs are the local heat flux, convective mach numbers, and life time of convectively burning zones. As in SQ14, we find these quantities correlate most strongly with the helium core mass. Stellar models of larger zero-age main sequence (ZAMS) mass or with more mixing (due to overshoot or rotation) tend to have a higher He core mass and may exhibit different wave dynamics than our fiducial model. Our main goal here is simply to provide a rough estimate of IGW AM fluxes for a typical low-mass (\(M \lesssim 20 M_\odot\)) progenitor of a NS.

\label{fig:MassiveIGWhist} Kippenhahn diagram of our stellar model, with the x-axis showing the time until CC. Light blue shaded regions are convective. Solid colored lines show helium, carbon, oxygen, silicon, and iron core masses. We have labeled convective core burning phases with the element being burned in each phase. Dashed vertical lines show the locations (from left) of our helium core burning, carbon shell burning, oxygen shell burning, and silicon shell burning models used in Figures \ref{fig:Massivestruc} and \ref{fig:MassiveIGWtime}. The mass coordinate of the base of the convective shell, from which IGW propagate downwards, is labeled by a circle on each dashed line.

\label{fig:Massivestruc} Internal structure of our \(12 M_\odot\) stellar model, at the phases labeled in Figure \ref{fig:MassiveIGWhist}. At all stages shown, the star is a red supergiant with radius \(R \sim 10^3 R_\odot\). The He-burn stage is core helium burning, while the C-burn, O-burn, and Si-burn stages are shell carbon, oxygen, and silicon burning phases, respectively. Vertical shaded bars are drawn near the base of the surface convection zone for the He-burn stage, and near the base of the convective shell burning region of the other stages. Top: Density profiles \(\rho\) of the inner regions of our models. The envelope density profile is similar for each model, but central densities vary by orders of magnitude. Middle: Enclosed mass profiles. For C/O/Si-burning, the models are chosen such that the mass internal to the convective shell burning region is \(M_c \sim 1.2 M_\odot\). Bottom: The Brunt-Väisälä frequency, \(N\), of our models. We have also plotted horizontal dashed lines at the convective turnover frequencies \(\omega_c\) below the relevant convective zone. IGW propagate in the regions where \(\omega_c < N\), shown by the horizontal extent of the dashed lines.

A full understanding of AM transport by IGW should include the combined effects of waves emitted from each convective zone. For simplicity, we focus on cases in which a convective shell overlies the radiative core, irradiating it with IGW. These convective shell phases typically occur 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. Our MLT calculations yield convective velocities and Mach numbers that tend to be a factor of a few smaller than those seen in simulations (e.g., Meakin et al. 2006; Meakin et al. 2007; Meakin et al. 2007; Arnett et al. 2008). This could be due to the larger mass of their stellar model or the inadequacy of the MLT approximation. We proceed with our MLT results, but caution that realistic wave frequencies and fluxes may differ from those used here by a factor of a few.

Figure \ref{fig:MassiveIGWhist} shows a Kippenhahn diagram for our stellar model, and Figure \ref{fig:Massivestruc} shows the density (\(\rho\)), mass [\(M(r)\)], and Brunt-Väisälä 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 are more vigorous but shorter in duration. 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 and SQ13, although the shell burning phases are generally more vigorous and shorter-lived than the core burning examined in SQ13. Table 1 lists some of the parameters of our convective zones.

Burning Phase \(r_c\) (km) \(T_{\rm shell}\) (s) \(t_{\rm waves}\) (s) \(\mathcal{M}\) \(\omega_{\rm c}\) (rad/s) \(P_{\rm min}\) (s) \(P_{\rm min,Fe}\) (s) \(P_{\rm min,NS}\) (s)
He Core Burn \( 1.6 \times 10^7 \) \( 4 \times 10^{13}\) \( 2 \times 10^{5}\) \(0.06\) \( 3 \times 10^{-6}\) \( 2 \times 10^{5}\) \( 40\) \( 2 \times 10^{-3} \)
C Shell Burn \( 9.7 \times 10^3\) \( 3 \times 10^8\) \( 10^{6}\) \(0.002\) \( 4 \times 10^{-3}\) \( 2 \times 10^{3}\) \( 50\) \( 2.5 \times 10^{-3} \)
O Shell Burn \( 3.6 \times 10^3\) \( 4 \times 10^6\) \( 10^{5}\) \(0.004\) \( 2 \times 10^{-2}\) - - -
Si Shell Burn \( 1.7 \times 10^3\) \( 7 \times 10^3\) \( 2 \times 10^{3}\) \(0.02\) \( 4 \times 10^{-1}\) - - -

\label{tab:table}Properties of IGW AM transport during different evolutionary stages of the stellar model shown in Figures and \ref{fig:MassiveIGWhist} and \ref{fig:Massivestruc}. Here, \(r_c\) is the radius at the base of the convection zone in consideration, \(T_{\rm shell}\) is the duration of the burning phase, \(t_{\rm waves}\) is a wave spin-up timescale (Equation \ref{eqn:twave}), \(\mathcal{M}\) is the convective Mach number, and \(\omega_{\rm c}\) is the angular convective turnover frequency. For He and C burning phases, IGW may be able to spin down the core, and \(P_{\rm min}\) is the approximate 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. During O/Si burning, IGW are unlikely to extract sufficient AM from the core (see text), so we do not list \(P_{\rm min}\).

The total energy flux carried by waves emitted from the bottom of the convective zone is of order \[\label{eqn:Ewaves} \mathcal{M} L_{\rm c} \lesssim \dot{E} \lesssim \mathcal{M}^{5/8} L_{\rm c}\] (Goldreich et al., 1990; Kumar et al., 1999), where \(\mathcal{M}\) is the convective Mach number (defined as the ratio of MLT convective velocity to sound speed, \(v_{\rm c}/c_s\)), and \(L_{\rm c}\) is the luminosity carried by convection near the base of the convective zone. Many previous works have used the left-hand side of equation \ref{eqn:Ewaves} as an estimate for the IGW energy flux, although (Lecoanet 2013) argue that a more accurate estimate may be \(\dot{E} \sim \mathcal{M}^{5/8} L_{\rm c}\), which is larger by a factor of \(\mathcal{M}^{-3/8}\). We consider the left-hand side of \ref{eqn:Ewaves} to be a lower limit for the wave flux, and the right-hand side to be an upper limit.

For shell burning, this energy flux is dominated by waves with horizontal wave numbers and angular frequencies near \(\bar{m} \sim {\rm max}(r_c/H_c,1)\), and \(\bar{\omega} \sim \omega_{\rm c}\), respectively. Here, \(H_c\) and \(r_c\) are the pressure scale height and radial coordinate near the base of the convective zone, and we define the angular convective turnover frequency as \[\omega_{\rm c} = \frac{ \pi v_{\rm c}}{ \alpha H_c} \, ,\] where \(\alpha H_c\) is the mixing length. Our models use \(\alpha = 1.5\). For the convective shells we consider, \(H_c \sim r_c\) and we expect waves of low angular degree to be most efficiently excited. The characteristic AM flux carried by these waves is \[\label{eqn:Jwaves} \dot{J} \sim \frac{\bar{m}}{\bar{\omega}} \dot{E}.\] Turbulent convection generates waves with a spectrum of azimuthal numbers \(m\) and angular frequencies \(\omega\). The values given above are characteristic values which dominate the AM flux. The waves carrying the most AM flux sometimes damp before they reach the core, and might not be able to affect the spin of the core. Then the waves with \(\bar{m}\) and \(\bar{\omega}\) would not dominate the AM flux to the core; instead, other waves in the turbulent spectrum become important (see Appendix \ref{wavestar}).

As a first check to see if IGW can have any affect on the spin of the core of the star, we assume all waves can propagate to the core. We suppose that IGW could be important for the spin evolution if they are able to carry an amount of AM comparable to that contained in a young NS, which contains \(J_{NS} \approx 10^{48} \, g \, {\rm cm}^2 \, {\rm s}^{-1}\) for a rotation period of \(P_{\rm NS} = 10 \, {\rm ms}\). Then the characteristic timescale on which waves could affect the AM of the core is \[\label{eqn:twave} t_{\rm waves} = \frac{J_{\rm NS}}{\dot{J}} \, .\] Table 1 lists shell burning stage lifetimes \(T_{\rm shell}\) and wave spin-down time scales \(t_{\rm waves}\), evaluated using \(\dot{E} \propto \mathcal{M}\), \(\bar{m}=1\) and \(\bar{\omega}=\omega_c\). In all phases, \(t_{\rm waves} \ll T_{\rm shell}\) for our model, indicating that waves may be able to have a substantial impact on the spin rate of the core. We examine this impact in the following sections.

Core Spin-down by Internal Gravity Waves

\label{spindown}

The characteristic AM flux of equation \ref{eqn:Jwaves} is positive by definition, although we expect turbulent convection to generate prograde and retrograde waves in nearly equal quantities, so that the net AM flux is nearly zero. However, differential rotation can set up powerful wave filtration mechanisms (see F14), which filter out either prograde or retrograde waves. Consider a wave packet of angular frequency \(\omega\) and azimuthal number \(m\) that propagates across a radial region of thickness \(\Delta r\), whose endpoints have angular spin frequencies that differ by an amount \(\Delta \Omega\). If \(\Delta \Omega > \omega/m\), the waves will encounter a critical layer within the region, and will be absorbed. Only waves of opposing AM will penetrate through the layer; therefore, rapidly rotating regions will only permit influxes of negative AM, which will act to slow the rotation of the underlying layers.

IGW can therefore limit differential rotation to a maximum amplitude \(\Delta \Omega_{\rm max} \sim \omega/m\), provided the IGW AM flux is large enough to change the spin rate on time scales shorter than relevant stellar evolution times. In the case of a rapidly rotating core (which has contracted and spun-up) surrounded by a slowly rotating burning shell, we may expect a maximum core rotation rate of \(\omega\), provided waves of this frequency can propagate into the core. This maximum rotation rate assumes \(|m|=1\) waves dominate the AM flux, the actual rotation rate could be smaller if \(|m|>1\) waves have a substantial impact. Thus we define a maximum core rotation rate \[\label{eqn:Omegamax} \Omega_{\rm max} \sim \omega_* \,,\] where \(\omega_*\) is the characteristic frequency of waves able to penetrate into the core.

The maximum spin rate described above can only be established if waves of frequency \(\omega_*\) can change the spin rate of the core on short enough time scales. However, some of the waves generated by the convective zone may dissipate before reaching the core, and will therefore not be able to change its spin. Indeed, F14 showed that convectively excited IGW likely cannot change the spin rate of the cores of low mass red giants because most of the wave energy is damped out before the waves reach the core. Low frequency waves are particularly susceptible to damping because they have smaller radial wavelengths, causing them to damp out (via diffusive and/or non-linear processes) on shorter time scales.

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 \(v_{\rm rot} \sim 150 \,{\rm km \ s}^{-1}\) (Mink et al., 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, so \(J_0\) 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, \[J_{\rm ex} = \eta \dot{J}_*(r) T_{\rm shell} \, . \label{eqn:Jex}\] Here, \(\eta\) is an efficiency factor that accounts for the fact that only some of the wave flux is in low \(l\) retrograde waves capable of depositing negative AM in the core; we use \(\eta=0.1\) in our estimates.1 Figure \ref{fig:MassiveIGWtime} shows 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. During core He burning, the inner \(\sim \! 2 M_\odot\) is convective and IGW cannot propagate into it, hence, it will only be spun down if it is coupled (via its own IGW or via magnetic torques) to the radiative region above it. 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. This does not imply IGW have no effect, as the value of \(J_{\rm ex}\) for O/Si burning is larger than \(J_{\rm NS}\) (the typical AM content of a fairly rapidly rotating 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, the minimum rotation periods are 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) a combination of IGW and magnetic torques are responsible for spinning down the cores of massive stars.


  1. For waves with \(\omega \sim \omega_c\) launched from a thick convective zone with \(r \sim H\), the energy spectrum described in (Talon 2005) has an approximate spectrum \(d \dot{E}/dl \propto l e^{-l^2}\). The energy emitted in \(l=1\) waves is approximately one third of the total energy flux, and approximately one third of \(l=1\) waves are retrograde. Therefore we expect \(\eta=0.1\) to be a reasonable estimate for the energy flux emitted in low degree retrograde waves capable of propagating into the core.

\label{fig:MassiveIGWtime} Top: The thick black line shows the AM \(J_0\) contained within the mass coordinate \(M(r)\) of our massive star model rotating with a period of \(P_0 \approx 1.5 {\rm d}\) on the ZAMS, while the orange line shows the approximate AM \(J_{\rm NS}\) contained within \(M(r)\) for a NS rotating at \(P_{\rm NS} = 10 \, {\rm ms}\). The shaded regions indicate the AM \(J_{\rm ex}\) (equation \ref{eqn:Jex}) that can be extracted by IGW during the burning phases shown in Figure \ref{fig:Massivestruc}. The shaded regions are bounded by lines calculated with a pessimistic and optimistic estimate for IGW fluxes (see equation \ref{eqn:Ewaves}), and are shown for mass coordinates below the base of the convective zones (marked by the vertical dashed lines) from which IGW are launched. We have truncated the curves for He burning at the location of the convective core, into which the IGW cannot propagate. IGW can significantly slow the spin rate of the progenitor in regions where \(J_{\rm ex} > J_0\). In the event of prior core spin-down via waves/magnetic torques, IGW could be important for determining the spin of the newborn NS in regions where \(J_{\rm ex} > J_{\rm NS}\). Bottom: Spin periods \(P_0\) of our model in the absence of angular momentum transport. As the star evolves, the value of \(P_0\) decreases in the contracting core. We have also plotted the approximate minimum spin periods \(P_{\rm min}\) which may be enforced by IGW during the core He-burning and C-shell burning phases.

Stochastic Spin-up by Internal Gravity Waves

\label{spinup}

A stochastic influx of IGW into the core will also enforce a minimum expectation value of the core spin rate. Consider a non-spinning core being irradiated by IGW wave packets emitted from an overlying convective shell. The spherical symmetry of the background structure implies that the wave packets will contain randomly oriented AM vectors. Since we expect incoming waves to non-linearly dissipate before being reflected, the core will absorb their AM. At the end of the convective shell phase, the total core AM is the sum of the randomly oriented wave packet AM vectors emitted by the shell. Each wave packet will contain an AM vector of length \[\label{eqn:AMvec} J_w \sim \frac{2 \pi}{\bar{\omega}} \dot{J} \sim \frac{2 \bar{m} \pi}{\omega_{\rm c}^2} \mathcal{M} L_{\rm c} \, ,\] which we have calculated using equations \ref{eqn:Ewaves} and \ref{eqn:Jwaves}, and assuming typical wave packets have angular frequency \(\omega_{\rm c}\) and last for an eddy turnover time, \(2 \pi/\omega_{\rm c}\). The expected magnitudes of the \(x\), \(y\), and \(z\) components of the AM vector are \(J_w/\sqrt{3}\). Since our goal is to find a minimum rotation rate, we use \(\bar{m}=1\) to minimize the AM carried by each wave packet.

\label{fig:MassiveIGWspin} Normalized distribution of maximum spin periods in the pre-collapse iron core, \(P_{\rm Fe,max}\), due to stochastic spin-up via IGW. The left panel shows the distribution using the pessimistic wave flux (left hand side of equation \ref{eqn:Ewaves}), while the right panel shows the distribution using the optimistic wave flux (right hand side of equation \ref{eqn:Ewaves}). The top axis shows the corresponding maximum NS rotation period, \(P_{\rm max,NS}\), assuming conservation of angular momentum during the supernova. For CC SNe, \(P_{\rm max,NS}\) is most plausibly set during Si-burning (see text), which corresponds to the shaded red area. Stochastic IGW spin-up leads to values of \(P_{\rm max,NS}\) of tens to hundreds of milliseconds, similar to the initial rotation rates of most pulsars.

During each shell burning phase, the convective shell emits roughly \(N\) wave packets, with \[\label{eqn:Nwave} N \sim \frac{\omega_{\rm c} T_{\rm shell}}{2 \pi}\] where \(T_{\rm shell}\) is the length of the shell burning phase. For Si shell burning, \(N \sim 400\). This relatively small number demonstrates that the stochastic nature of the wave emission process can be important during late burning phases (in contrast to, e.g., He burning when \(N \sim 2 \times 10^7\)). The total core AM in each direction exhibits a random walk and thus has a Gaussian distribution centered around zero and with standard deviation \[\label{eqn:Jmean} \sigma_{J} = \sqrt{ \frac{N}{3} } J_w.\] The magnitude of the total AM vector then has a Maxwellian distribution, with standard deviation \(\sigma_J\). The corresponding spin frequency (assuming internal torques eventually restore rigid rotation in the core) also has a Maxwellian distribution, given by \[\label{eqn:jmaxwell} f(\Omega) = \sqrt{\frac{2}{\pi}} \frac{\Omega^2}{\sigma_\Omega^3} e^{- \Omega^2/(2 \sigma_\Omega^2)} \,.\] where \(\sigma_\Omega = \sigma_J/I_c\), and \(I_c\) is the moment of inertia of the radiative core. A little algebra yields the corresponding spin period distribution, \[\label{eqn:pdist} f(P) = \sqrt{\frac{2}{\pi}} \frac{\sigma_P^3}{P^4} e^{- \sigma_P^2/(2 P^2)} \,,\] with \(\sigma_P = 2 \pi/\sigma_\Omega\).

The expected angular spin frequency corresponding to equation \ref{eqn:jmaxwell} is \[\label{eqn:omex} \Omega_{\rm ex} = \sqrt{\frac{4 \omega_c T_{\rm shell} }{3 \pi^2}} \frac{J_w}{I_c} \, .\] The expected spin frequency scales as \(\Omega_{\rm ex} \propto \omega_c^{-1/2} L_c T_{\rm shell}^{1/2}\). Thus, more energetic and long-lived convection yields higher expected rotation rates. The short duration \(T_{\rm shell}\) of later burning phases largely counteracts their increased vigor, and we find that earlier burning phases are generally capable of depositing more AM. We consider the estimate of equation \ref{eqn:omex} to be robust against many uncertainties associated with IGW generation because it is not strongly dependent on the details of the IGW spectrum, and only depends weakly upon the convective turnover frequency \(\omega_c\). The main uncertainty is the value of the IGW flux, so we will consider both the optimistic and pessimistic limits of equation \ref{eqn:Ewaves}.

The stochastic spin-up process described above will only occur under certain conditions. First, as already mentioned, the core and burning shell must be slowly rotating, or else the stochastic spin-up will have a negligible effect. For example, stochastic spin-up is not important if the only prior core spin-down is provided by IGW as outlined in Section \ref{spindown}. Second, all waves (both prograde and retrograde) must be absorbed by the core. In the cores of massive stars, this is likely to occur because of non-linear breaking due to geometric focusing as waves approach the center of the star (see Appendix \ref{wavestar}). Third, stochastic spin-up can only proceed as long as \(\Omega_{\rm ex} < \omega_c\). If \(\Omega_{\rm ex}\) approaches \(\omega_c\), wave filtering processes as described in Section \ref{igw} will alter the subsequent dynamics. Most of our estimates below have \(\Omega_{\rm ex} \ll \omega_c\), so we believe they are valid estimates of minimum spin rates. For C-burning in the optimistic wave flux estimate (\(L_{\rm wave} \propto \mathcal{M}^{5/8}\)), however, \(\Omega_{\rm ex}\) approaches \(\omega_c\), so this value of \(\Omega_{\rm ex}\) lies near the maximum rotation rate achievable through stochastic spin-up for our stellar model.

Moreover, stochastic spin-up can only occur if other sources of AM transport (e.g., magnetic torques) operate on longer time scales. This could be the case during late burning phases when magnetic torques become ineffective (Heger et al. 2005; Wheeler et al. 2014). We can also estimate a minimum magnetic coupling time between core and envelope via the Alven wave crossing time \(t_A \approx r_c \sqrt{\rho_c}/B\), with \(B\) the approximate magnetic field strength. Typical neutron star field strengths of a few times \(10^{12} \, {\rm G}\) imply field strengths of \(\sim 10^8 \,{\rm G}\) in the iron core, which yields \(t_A \sim 5 \times 10^4 \,{\rm s}\), much longer than the Si shell burning time (see Table 1). Although magnetic torques may suppress stochastic spin-up during He/C/O burning phases, we expect them to have a negligible impact during Si burning.

It is also possible that IGWs will generate significant amounts of shear within the core, as IGW tend to amplify small amounts of shear (F14). In the absence of strong magnetic torques, such shear amplification may generate shear layer oscillations (Kumar et al., 1999; Talon et al., 2002) or differential rotation that encompasses much of the radiative core (Denissenkov et al., 2008). In our case, the breaking IGW could quickly develop shear layers near the center of the star due to its low moment of inertia. These shear layers can can only prevent the stochastic spin-up of the core if they can propagate to the core boundary, which occurs on the time scale \(t_{\rm shear} = I_c \omega_c/\dot{J}\). We find \(t_{\rm shear} \sim 2 \times 10^{4} \, {\rm s}\) for Si shell burning, using the conservative estimate from equation \ref{eqn:Ewaves}. We conclude that IGW may generate significant shear within the \(T_{\rm shell} \sim 7\times 10^{3} \, {\rm s}\) Si shell burning life time, although not enough to encompass the entire core and prevent further influx of IGWs. The stochastic spin-up process can still proceed as we have outlined, although the distribution of AM within the core at collapse is unclear, and significant differential rotation is possible.

Figure \ref{fig:MassiveIGWspin} shows the distribution in maximum spin period of the pre-collapse iron core, \(P_{\rm max,Fe}\), assuming the spin of the core is determined by stochastic IGW spin-up. We have plotted the values of \(P_{\rm max,Fe}\), if the core spin rate is set during C, O, or Si burning. We have also plotted the corresponding spin rate of the \(M_{\rm NS} \sim 1.4 M_\odot\), \(R_{\rm NS} \sim 12 \, {\rm km}\) NS, with \(I_{\rm NS} = 0.25 M_{\rm NS} R_{\rm NS}^2\), if its AM is conserved during the CC SN. We find that C, O, and Si burning generate maximum iron core rotation periods in the range of tens to thousands of seconds, depending on the burning phase and efficiency of IGW generation. Si burning most plausibly sets the pre-SN conditions because it is the last convective burning phase before CC and is least likely to be affected by magnetic torques. We find stochastic spin-up during Si burning leads to \(300 \, {\rm s} \lesssim P_{\rm max,Fe} \lesssim 10^4 \, {\rm s}\). The corresponding NS rotation rate is \(20\,{\rm ms} \lesssim P_{\rm NS} \lesssim 400 \, {\rm ms}\). Hence, we find that very slow core rotation rates, as suggested by (Spruit 1998), are unlikely. Nor do we expect that there is a population of NSs born with very long spin periods, \(P \gtrsim 2 \, {\rm s}\), at least from progenitors with ZAMS mass \(10 \, M_\odot < M < 20 \, M_\odot\).

The distribution of NS spin periods shown in Figure \ref{fig:MassiveIGWspin} appears broadly consistent with those inferred for young NSs (Faucher-Giguère et al., 2006; Popov et al., 2010; Gullón et al., 2014). We are therefore tempted to speculate that the stochastic wave flux scenario described above may be the dominant process setting the spin rates of newly born NSs. If so, this scenario predicts that the rotation rate and direction of the NS is uncorrelated with the rotation of the envelope of the progenitor star, in contrast to any sort of magnetic spindown mechanism. However, there are several caveats to keep in mind. First, the scenario presented above can only proceed if the core is initially very slowly rotating, which requires efficient magnetic/IGW core spin-down to occur before Si burning. Second, the NS rotation rate may be changed during the supernova, by fallback effects, or by the r-mode instability (Andersson 1998; Andersson et al. 1999, see Ott 2009 for a review). Finally, there is a considerable amount of uncertainty in the wave flux and frequency spectrum. Since the minimum core rotation rate set by stochastic waves is proportional to the wave energy flux (which is uncertain at an order of magnitude level), there is an equal amount of uncertainty in the induced rotation rates.

Discussion and Conclusions

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 (Heger et al., 2005; Wheeler et al., 2014) 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 (Cantiello et al., 2014). 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 (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 (Lai 1996; Gotthelf et al. 2013), and to the broad inferred birth spin period distribution of \(P_{\rm NS} \lesssim 500 \, {\rm ms}\) for ordinary pulsars (Faucher-Giguère et al. 2006; Popov et al. 2010; Gullón et al. 2014). 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 (Woosley, 1993; Yoon et al., 2006; Woosley et al., 2006; Metzger et al., 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 (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 (Cantiello et al., 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.

Acknowledgments

This paper was written collaboratively, on the web, using Authorea. JF acknowledges partial support from NSF under grant no. AST-1205732 and through a Lee DuBridge Fellowship at Caltech. DL is supported by a Hertz Foundation Fellowship and the National Science Foundation Graduate Research Fellowship under Grant No. DGE 1106400. EQ was supported in part by a Simons Investigator award from the Simons Foundation and the David and Lucile Packard Foundation. This research was supported by the National Science Foundation under grant No. NSF PHY11- 25915 and by NASA under TCAN grant No. NNX14AB53G.

Stellar Models

\label{model}

Our stellar models are constructed using the MESA stellar evolution code (Paxton et al., 2011; Paxton et al., 2013), version 6794. The models are run using the following inlist controls file.

aureplacedverbatimaa

The most important feature of this model is that it contains significant convective overshoot, especially above convective zones. It is non-rotating, thus there is no rotational mixing.

Just before core O-burning, we change to a 201-isotope reaction network: aureplacedverbatimaaa Although our choices affect details of the model (e.g., He core mass), the general features of our model are robust. It always explodes as a red supergiant. It always undergoes convective core C-burning, followed by shell C-burning, core O/Ne-burning, shell O-burning, core Si-burning, shell-Si burning, and then CC. The approximate convective properties (as described by MLT) are not strongly affected by model parameters. Since these properties are most important for IGW AM transport, we argue that the general features of IGWs described in this work are fairly robust against uncertain parameters in our massive star models.

Wave Frequencies, Fluxes, and Time Scales

\label{wavestar}

Here we describe our methods for estimating the wave frequency \(\omega_*(r)\) of waves that dominate the AM flux \(J_*(r)\) as IGW propagate through a star and are attenuated. For a more detailed discussion, see F14, whose methods are very similar to ours.

Consider a train of IGWs generated via shell convection which propagate inwards through underlying stable stratification. Upon generation, the IGW carry an energy flux \(\dot{E}_0\) and AM flux \(\dot{J}_0\). We assume the IGW have a frequency spectrum which is initially peaked around \(\omega_c\), with a power law fall off at higher frequencies such that \[\label{eqn:spectrum} \frac{d \dot{E}_0(\omega)}{d\omega} \sim \frac{\dot{E}_0}{\omega_c} \left(\frac{\omega}{\omega_c}\right)^{-a},\] where \(a\) is the slope of the frequency spectrum, which is somewhat uncertain. As in F14, we expect a spectrum slope in the range \(3 \lesssim a \lesssim 7\), and we use a value of \(a=4.5\) in our calculations. Lower frequency waves have shorter wavelengths and slower group velocities, making them more prone to both radiative and non-linear damping. Thus, as waves propagate inwards, low frequency waves may damp out, and increasing the wave frequency \(\omega_*\) of waves that dominate AM transport.

F14 show that radiative damping leads to \[\label{eqn:omstar2} \omega_*(r) = {\rm max} \bigg[ \omega_c \ , \ \bigg(\frac{4}{a} \int^{r_c}_r dr \frac{\lambda^{3/2} N_T^2 N K }{r^3} \bigg)^{1/4} \bigg] .\] and the corresponding AM flux \[\label{eqn:Jstar} \dot{J}_*(r) \sim \bigg[\frac{\omega_*(r)}{\omega_c}\bigg]^{-a} \dot{J}_0.\] In equation \ref{eqn:omstar2}, \(r_c\) is the radius of the inner edge of the convective zone, \(\lambda = l(l+1)\), \(l\) is the angular index of the wave (which corresponds to its spherical harmonic dependence, \(Y_{lm}\)), \(N_T\) is the thermal part of the Brunt-Väisälä frequency, and \(K\) is the thermal diffusivity. In what follows, we focus on \(l=1\) waves because they have the longest damping lengths and may dominate the AM flux when the waves are heavily damped. Moreover, focusing on \(l=1\) waves allows us to estimate maximum spin frequencies, although slower spin frequencies can be obtained when higher values of \(l\) and \(m\) contribute to the AM flux.

Non-linear Damping

IGW will overturn and break, leading to local energy/AM deposition, if they obtain sufficiently non-linear amplitudes. Here we estimate those amplitudes and the AM flux that can be carried toward the center of the star as waves are non-linearly attenuated. For traveling waves in the WKB limit, it is well known that the radial wave number is \[\label{eqn:kr} k_r = \frac{\sqrt{\lambda} N}{r \omega} \, .\] It is straightforward to show that the radial displacement \(\xi_r\) associated with IGW of frequency \(\omega\) carrying an energy flux \(\dot{E}\) is \[\label{eqn:eflux} | \xi_r | = \bigg[ \frac{\sqrt{\lambda} \dot{E} }{\rho N r^3 \omega^2} \bigg]^{1/2} \, .\] The waves become non-linear and break when \[\label{eqn:nl} | k_r \xi_r | = \bigg[ \frac{\lambda^{3/2} N \dot{E} }{\rho r^5 \omega^4} \bigg]^{1/2} \sim 1 \, .\] In the absence of damping, \(\dot{E}\) is a conserved quantity. Therefore, waves become more non-linear as they propagate into regions with larger \(N\), lower density, or smaller radius. In our problem, the geometrical focusing (i.e., the \(r\)-depdendence) is the most important feature of equation \ref{eqn:nl}, and causes waves to non-linearly break as they propagate inward. Note also the \(\omega^{-2}\) dependence of equation \ref{eqn:nl}, which causes low frequency waves to preferentially damp.

Equation \ref{eqn:nl} entails there is a maximum energy flux that can be carried by waves of frequency \(\omega\), \[\label{eqn:emax} \dot{E}_{\rm max} = \frac{ A^2 \rho r^5 \omega^4}{\lambda^{3/2} N} \, ,\] for waves that non-linearly break when \(|k_r \xi_r| = A \sim 1\). When the waves are highly non-linear, the waves which dominate the energy flux are those which are on the verge on breaking. To determine their frequency, we use the frequency spectrum \ref{eqn:spectrum} to find \[\label{eqn:efreq} \dot{E}_0 \bigg( \frac{\omega}{\omega_c} \bigg)^{1-a} \sim \dot{E}_{\rm max}.\] Solving equation \ref{eqn:efreq} yields the wave frequency which dominates energy transport, \[\label{eqn:omstarnl} \omega_* \sim {\rm max} \bigg[ \omega_c \ , \ \omega_c \bigg( \frac{ A^2 \rho r^5 \omega_c^4}{\lambda^{3/2} N \dot{E}_0} \bigg)^{-1/(a+3)} \bigg] \, .\] We expect frequency spectra with slopes somewhere near \(3 \lesssim a \lesssim 7\). Therefore the exponent in equation \ref{eqn:omstarnl} is quite small, and in most cases, \(\omega_*\) does not increase to values much larger than \(\omega_c\).

Substituting equation \ref{eqn:omstarnl} back into equation \ref{eqn:emax} allows us to solve for the energy and AM flux as a function of radius due to non-linear attenuation. The result is \[\label{eqn:jstarnl} \dot{J}_* \sim \bigg[ \frac{\omega_*(r)}{\omega_c} \bigg]^{-a} \dot{J}_0 \sim {\rm min}\bigg[ \dot{J}_0 \ , \ \bigg(\frac{A^2 \rho r^5 \omega_c^4}{\lambda^{3/2} N \dot{E}_0} \bigg)^{a/(a+3)} \dot{J}_0 \bigg] \, .\]

During C-shell burning, we find that radiative diffusion damps waves near the shell burning convective zone, while non-linear breaking damps waves near the center of the star. In this case, we first calculate \(\omega_*\) and its corresponding energy flux \(\dot{E}_*\) via equations \ref{eqn:omstar2} and \ref{eqn:spectrum}. We then substitute the value of \(\dot{E}_*\) for \(\dot{E}_0\) in equation \ref{eqn:omstarnl}. The appropriate value of \(\omega_*\) is then \(\omega_* = {\rm max} \big[ {\rm Eqn.}\) \ref{eqn:omstar2}\(, {\rm Eqn.}\)\ref{eqn:omstarnl}\( \big]\). The corresponding AM flux is \(\dot{J}_* \sim \Big[ \frac{\omega_*(r)}{\omega_c} \Big]^{-a} \dot{J}_0\).

The cores of massive stars nearing death cool primarily through neutrino emission, so it is not unreasonable to think that waves may be damped via neutrino emission. We calculate neutrino energy loss rates in the same manner as (Murphy 2004). We find that neutrino damping time scales are always longer than the wave crossing timescale \[\label{eqn:tcross} t_{\rm cross} = \int^{r_c}_0 \frac{dr}{v_g}\, ,\] where the IGW radial group velocity is \(v_g = r \omega^2/(\sqrt{\lambda} N)\). This is not surprising, as (Murphy 2004) found neutrino growth/damping rates were slower than stellar evolution time scales. Neutrino damping can therefore be safely ignored.

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