LONG GAMMA-RAY TRANSIENTS FROM COLLAPSARS

A paucity of metals in a star affects its evolution in several ways—reduced energy generation by the CNO cycle in the hydrogen burning shell, reduced opacity, and reduced mass loss. Of these effects, the reduction in mass loss is least well understood, but the lack of atomic lines in the atmosphere on the main sequence and of grains as a red giant seems certain to lead to some reduction. A common assumption is that the mass loss for massive main-sequence stars depends upon some fractional power of the metallicity (Kudritzki 2002). More important and less certain is the dependence of mass lost as a red or blue supergiant on metallicity. It is thought that mass loss from cool giants is more dependent upon pulsations and grain formation (Reimers 1977; Smith et al. 2011). One might expect therefore a rapid falloff in mass loss below some value necessary for significant grain production. It has been estimated that red giant mass loss will be significantly less below 0.1 Z_☉ (Bowen & Willson 1991; Zijlstra 2004).

If stars do not lose mass then they conserve angular momentum. Since the natural course of evolution leads to the contraction and spin up of the inner star, shear instabilities and magnetic torques will concentrate an increasing amount of angular momentum in the outer regions of the star. While the inner part does spin faster due to contraction, it actually loses most of its initial angular momentum by the time it reaches central carbon burning. Heger & Woosley (2010) surveyed the evolution of non-rotating zero-metal massive stars from 10 M_☉ to 100 M_☉. In a parallel study currently underway (A. Heger & S. E. Woosley 2012, in preparation), we are surveying the evolution of rotating stars in the same mass range with metallicity 0, 10⁻³, and 10⁻¹ that of the Sun. The two stars discussed here have been extracted from that survey and are typical in the way that they accumulate large amounts of angular momentum in their outer layers before dying as supergiants. All calculations of stellar evolution presented here used the KEPLER code (Weaver et al. 1978; Woosley et al. 2002; Heger et al. 2000) and included angular momentum transport by magnetic torques (Spruit 2002). Models V24 and V36 are 24 M_☉ and 36 M_☉ main-sequence stars with metallicity 0.1% solar. Both stars rotated rigidly on the main sequence with a moderate surface equatorial speed of about 200 km s⁻¹ which is about 20% of the Keplerian value. Both pre-supernova stars had hydrogenic envelopes that were, throughout most of their mass, radiative. Model V36, however, had a low-mass surface convection zone that included 0.11 M_☉. Model V36 was consequently a yellow supergiant at death (L = 2.2 × 10³⁹ erg s⁻¹; R = 5.4 × 10¹³ cm; T_eff = 5700 K). Model V24 was a rather large blue supergiant (L = 1.1 × 10³⁹ erg s⁻¹, R = 1.0 × 10¹³ cm; T_eff = 11, 300 K).

These two stars ended their lives with cores of helium and heavy elements of 8.3 M_☉ and 15.0 M_☉, respectively, and angular momentum distributions as shown in Figure 1. While the lack of sufficient angular momentum within the helium core precludes making a disk around a black hole within that mass, there is ample rotation in the outer part of the hydrogen envelope to do so. This could be a very common occurrence. From the survey of A. Heger & S. E. Woosley (2012, in preparation), the total angular momentum of a massive star at birth that has an equatorial rotation speed of about 20% Keplerian on the main sequence is

$$\begin{equation} J_{\mathrm{tot}}= I \omega \approx 2 \times 10^{52} \left(\frac{M}{M_\odot }\right)^{\!1.8}\,\mathrm{erg}\,\mathrm{s}. \end{equation} \tag{ 1 }$$

If the star does not lose mass, most of this angular momentum becomes concentrated, in the pre-supernova star, in a nearly rigidly rotating hydrogenic envelope with radius either ≲ 10¹³ cm (blue supergiant) or ∼10¹⁴ cm (red supergiant). Though the density declines with radius in the actual envelope, one can obtain some interesting scaling relations by assuming constant density. For a moment of inertia, I ≈ 0.4MR², the angular velocity at the surface, R = 10¹⁴ R₁₄ cm, is

$$\begin{equation} \omega \approx 1.2 \times 10^{-10} \, R_{14}^{-2} \left(\frac{ M}{20\,M_\odot }\right)^{\!0.8} \,\mathrm{rad}\,\mathrm{s}^{-1}. \end{equation} \tag{ 2 }$$

The specific angular momentum of this rigidly rotating envelope is j = (2/3)ωr², or about 10¹⁸ cm² s⁻¹ at the surface of a red supergiant and 10²⁰ cm² s⁻¹ for a blue supergiant, if no mass is lost.

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These can be compared with the angular momentum required to make a disk around a non-rotating black hole, $j = 2 \sqrt{3} GM/c = 3.1 \times 10^{17} (M_{\mathrm{BH}}/20\,M_\odot)\,\mathrm{cm}^2\,\mathrm{s}^{-1}$ or $j = 2/\sqrt{3} GM/c = 1.0 \times 10^{17} (M_{\mathrm{BH}}/20\,M_\odot)\,\mathrm{cm}^2\,\mathrm{s}^{-1}$ for a maximally rotating hole (Kerr parameter a = 1). All massive stars that do not lose mass have sufficient angular momentum in their outermost layers to make a disk around any black hole formed in their collapse.

Of course, stars do lose mass and the fact that, for constant ω, j∝r² means that the first mass to be lost contains most of the angular momentum. In practice, we find that a red supergiant that loses more than a few percent of its total mass before dying will probably not make a black-hole–disk system.

This sensitivity to mass loss is illustrated by the evolution of a 25 M_☉ model, O25, with 10% solar metallicity in a calculation that included mass loss at one-half the standard value for that metallicity. The star lost 1.17 M_☉ (5%) of its mass and died as a red supergiant with a radius of 1.1 × 10¹⁴ cm. The value of j in its outermost layers was 2.8 × 10¹⁷ cm² s⁻¹, too little to form a disk around a Schwarzschild black hole, but possibly just enough to form a disk around a mildly rotating one. The point where j declined to one-half its surface value in the pre-supernova star was located 5.5 M_☉ interior. This is borderline case. Stars with less mass loss, i.e., lower metallicity, and stars that never become red giants could make long gamma-ray transients (Table 1).

Table 1. Summary of Model Characteristics

Type	Model	Mdisk	R	R/vesc	$\dot{M}$	1% $\dot{M}c^2$
		(M☉)	(cm)	(s)	(10−4 M☉ s−1)	(1048 erg s−1)
BSG	V24	10.0	(0.4–10) × 1012	20,000	5	9
	V36	10.2	(1–50) × 1012	60,000	1	2
RSG-loZ	O25	0.	1.2 × 1014	107	⋅⋅⋅	⋅⋅⋅
RSG-bin	S25	2.8	(0.1–9) × 1013	3 × 106	0.01	0.02
Pair SN	Z250A	19.6	(0.9–8) × 1013	3 × 105	0.7	1
	Z250B	20.6	(0.4–6) × 1013	105	2	4
	Z250C	12.1	(0.2–2) × 1013	3 × 104	4	8
WR-bin	8A	0.18	(2.5–5) × 1010	100	20	40
	8B	1.19	(1–5) × 1010	100	100	200
	16A	0.01	(3–4) × 1010	100	1	2
	16B	0.83	(2–4) × 1010	40	200	300

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Based upon theoretical models for single massive stars, the red supergiant progenitors of common Type IIp supernovae are not expected to have rapidly rotating envelopes. Expansion to a red giant slows the surface layers. Torques transfer the angular momentum of the more rapidly rotating inner regions outwards concentrating it in the outermost layers. For stars of sufficient metallicity, stellar winds are efficient at removing these layers and, along with them, most of the star's angular momentum. For example, a 25 M_☉ solar metallicity star rotating rigidly with an equatorial speed of 200 km s⁻¹ on the main sequence ends up a 12.1 M_☉ supernova progenitor if commonly used estimates for mass loss are employed (Woosley & Heger 2007). The surface rotation rate of that red supergiant (R = 9 × 10¹³ cm) is 2 × 10⁻¹² rad s⁻¹ and its angular momentum at the surface is 1.6 × 10¹⁶ cm² s⁻¹. Nowhere in the star does any matter have enough angular momentum to form a disk around a black hole. If a black hole forms without making an outgoing shock, the star abruptly disappears. This sort of "un-nova" might be a common occurrence in the modern universe (Kochanek et al. 2008), though see Smith et al. (2011).

The outcome can be quite different in a binary system, however. A stellar merger could lead to a rapidly rotating envelope (Ivanova & Podsiadlowski 2003), but such mergers probably lead to ejection of a common envelope. An interesting alternative is a detached binary system with a small enough separation for tidal interaction, but too distant to merge or exchange mass (Figure 2).

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The circularization and synchronization timescales for binary stars, one of which is a red giant, have been considered by Zahn (1966, 1989) and Verbunt & Phinney (1995). For a primary with mass, M, envelope mass M_env, luminosity L, and radius R, in an orbit having radius d, with a companion of mass M₂, the circularization time τ_circ is (Verbunt & Phinney 1995; Phinney 1992)

$$\begin{equation} \frac{1}{\tau _{\mathrm{circ}}} = f \left(\frac{L}{M_{\mathrm{env}}R^2}\right)^{\!1/3} \, \frac{M_{\mathrm{env}}}{M} \frac{M_2}{M} \frac{M+M_2}{M} \left(\frac{R}{d}\right)^{\!8}\!, \end{equation} \tag{ 3 }$$

where f is a number close to unity. The lifetime of a 25 M_☉ main-sequence star as a red supergiant is approximately 700,000 yr, depending upon uncertain convection parameters. The mass of the envelope that is convective at any one time, however, is only about 3 M_☉ and the star loses more than this in its lifetime. A more appropriate timescale might be the time the star spends losing its last 3 M_☉, which is about 250,000 yr. The mass of the star when it becomes a supernova is 12.1 M_☉ and we assume a companion mass of, e.g., 6 M_☉. The luminosity of the star as a red supergiant is 1.0 × 10³⁹ erg s⁻¹ and its moment of inertia is approximately 4 × 10⁶¹ g cm⁻². Equation (3) then implies that during its last 250,000 yr, stars in orbits with separations d  ≲  3.8R ≈ 18 AU will circularize.

The synchronization timescale for co-rotation is somewhat different (Zahn 1977, 1989) and, in the present case, is easier to achieve

$$\begin{equation} \frac{1}{\tau _{\mathrm{sync}}} = 6 q^2 \lambda _{\mathrm{sync}}\left(\frac{L}{M_{\mathrm{env}}R^2}\right)^{\!1/3} \, \frac{MR^2}{I} \, \left(\frac{R}{d}\right)^{\!6}. \end{equation} \tag{ 4 }$$

Most of the angular momentum is in the hydrogen envelope and, from the same 25 M_☉ model, MR²/I = 2.9. With q ≈ 1 and λ_sync ≈ 0.02 (Zahn 1989),

$$\begin{equation} \frac{1}{\tau _{\mathrm{sync}}} \approx 0.1 \left(\frac{L}{M_{\mathrm{env}}R^2}\right)^{\!1/3} \, \left(\frac{R}{d}\right)^{\!6}, \end{equation} \tag{ 5 }$$

which only requires d  ≲  6.7R for synchronization in 250,000 yr. In such a system, the period will be shorter than about 30 years (ω = 10⁻⁸ rad s⁻¹). Systems much closer will merge or experience mass exchange, so we take this to be a typical rotation period for the envelope in a detached binary system that might achieve corotation. Longer period systems will still transfer significant rotation to the envelope, however.

This is about four orders of magnitude more angular momentum in the outer envelope than in the case of the single star. Since the outer several solar masses of the pre-supernova star are convective, this rotation rate will persist to some depth. All of that mass would have sufficient angular momentum to form an accretion disk if its helium core collapsed to a black hole. Even then though, the total angular momentum of the envelope, 2.0 × 10⁵³ erg s, would be a small fraction of the orbital angular momentum of the binary pair of stars. The addition of only 1% of the angular momentum estimated here would still result in large quantities of material forming a disk, so the range of binary separations allowed is actually larger, perhaps by a factor ∼(100)^1/6, or about two.

Since the timescale for material to fall in from 10¹³ to 10¹⁴ cm is longer than any realistic viscous timescale for the disk, the accretion rate will be given by the collapse timescale of the envelope. For matter at the base of the hydrogen shell where a disk would first form, the enclosed mass is 9.3 M_☉ and the radius about 1 × 10¹³ cm. At the outer edge of the star the mass is 12.1 M_☉ and the radius, 9.8 × 10¹³ cm. The corresponding free fall times ($\tau _{\mathrm{ff}} = (24 \pi G \bar{\rho })^{-1/2} = 446\, (\bar{\rho }/1\,\mathrm{g}\,\mathrm{cm}^{-3})^{-1/2}\,\mathrm{s}$) are 2.1 × 10⁵ s and 5.7 × 10⁶ s suggesting that the event would last for months. Accreting 3 M_☉ in 10⁶ s with 10% efficiency for converting accreted mass to outgoing energy would give a jet power of 10⁴⁸ erg s⁻¹. Conversion of only 10% of the jet power (1% of the accreted mass energy) into gamma-rays would explain the recently discovered transient Sw-1644+57 (Levan et al. 2011; Burrows et al. 2011), especially if moderate beaming were invoked.

Though the focus here is on red supergiants because they give the longest transients; tidal locking would also occur in blue supergiants with smaller radii. The key point is that forced co-rotation with a body in Keplerian orbit gives, for a large range of orbital separations, envelope material that is rotating too fast to accrete, without hindrance, into a stellar mass black hole. Since the timescale for accretion is set by the collapse time for the outer part of the star, not the viscous time of the disk, a transient from a tidally locked blue supergiant would be similar in duration, within a factor of a few, to those studied for single blue supergiants in Section 3.1. Blue supergiants, however, will have less mass in the outer layers of the envelope and hence would have less powerful outbursts (the Appendix and Quataert & Kasen 2011).

Following helium burning, non-rotating helium cores with masses greater than 133 M_☉ will collapse directly into black holes with no outgoing shock (Heger & Woosley 2002). The collapse is caused by the pair instability and, above this mass, nuclear burning is unable to reverse the implosion. Rotation, unless it is very rapid and highly differential, does not change this mass limit greatly since the instability exists in the deep interior where the ratio of centrifugal force to gravity is small.

The collapse of a rotating pair-unstable star to a black hole was followed in two dimensions by Fryer et al. (2001). First, a 300 M_☉ main-sequence star was evolved to the point of instability in the KEPLER (1D) code. This yielded a helium core of 180 M_☉. The initial star was assumed to rotate rigidly on the main sequence with a ratio of equatorial speed to Keplerian of 20%. Angular momentum transport, especially by shear mixing and Eddington Sweet circulation, was followed throughout the evolution, but magnetic torques were neglected. As a result, the collapsing helium core rotated sufficiently rapidly to produce a disk outside of a massive black hole once about 140 M_☉ of the 180 M_☉ core of helium and heavy elements had collapsed. Fryer et al. (2001) speculated that the accretion of the remaining 30–40 M_☉ might produce a very bright, lengthy GRB.

We have recalculated the evolution of similar stars, but with magnetic torques (Spruit 2002) included in the simulation. Models Z250A, B, and C are stars with zero metallicity and a mass at the time they formed of 250 M_☉. They differ in the amount of angular momentum each had on the main sequence. Z250A had 0.75 × 10⁵⁴ erg s corresponding to a rotation rate of 170 km s⁻¹, or 12% Keplerian. Models Z250B and Z250C had 1.0 × 10⁵⁴ erg s and 1.5 × 10⁵⁴ erg s of angular momentum and rotational speeds of 220 km s⁻¹ (15% Keplerian) and 310 km s⁻¹ (19% Keplerian). These made helium cores of 143 M_☉, 166 M_☉, and 222 M_☉ respectively, all of which collapsed directly to black holes following helium depletion. Unlike in Fryer et al. (2001) however, none of these helium cores had sufficient angular momentum to make a disk around a black hole anywhere in their interior. Because mass loss was neglected in these models, the angular momentum lost from the helium cores of these stars due to magnetic coupling instead ended up concentrated in the outer part of their hydrogen envelopes.

Figure 3 shows the evolution of the angular momentum in Model Z250B. The evolution of the other two models was similar. Inadequate angular momentum exists in the helium core to form a disk but, for Model Z250B, the outer 16 M_☉ can still do so. As in the tidally locked red supergiant case, the collapse timescale for this material is days to months, so a high energy transient of around 10⁴⁷ erg s⁻¹, possibly boosted by beaming, could exist for that time.

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It should be noted, however, that mass loss is even more uncertain in these models than in the others. Due to rotationally induced mixing, the envelopes of all three models were appreciably enhanced in C, N, and O and thus were red supergiants when they died. Neglecting mass loss in such a situation is probably not justified. Also there may be other mechanisms for losing the envelopes of such massive stars besides line- and grain-driven mass loss (Smith & Owocki 2006).

Helium stars in close tidally locked binaries have been frequently invoked as progenitors for common GRBs (Tutukov & Cherepashchuk 2003; Podsiadlowski et al. 2004; Izzard et al. 2004; Bogomazov et al. 2007; van den Heuvel & Yoon 2007). On the positive side, such events should occur frequently in nature and, if the core of a massive Wolf–Rayet star collapses to a black hole while the surface is forced to rotate at a fraction of its Keplerian speed, disk formation around a black hole is assured. On the negative side, however, the production of a GRB with typical duration ∼10 s requires that the inner part of the helium star rotate rapidly, not just its surface. The inclusion of magnetic torques in tidally locked stars leads to too little rotation to make a disk in that part of the star which might accrete rapidly (Woosley & Heger 2006; van den Heuvel & Yoon 2007). On the other hand, leaving out magnetic torques makes it difficult to produce the spin rates of ordinary pulsars (Heger et al. 2005) and could lead to an overabundance of GRB progenitors.

This is not to say that close binary systems do not produce some form of gamma-ray transient though. The angular momentum in their cores may be adequate to produce a millisecond magnetar, and thus a common GRB, just not a (Type 1) collapsar. If a black hole forms, the accretion of the outer layers of the star may lead to the formation of a disk and GRB much longer than typical. It is this latter possibility that we explore here.

Following van den Heuvel & Yoon (2007), we consider the evolution of helium cores of 8 M_☉ and 16 M_☉ whose surfaces have become tidally locked with a closely orbiting companion. Van den Heuvel & Yoon (2007) found that the shortest possible periods were 2.05 hr (ω = 8.5 × 10⁻⁴ rad s⁻¹) and 2.47 hr (ω = 7.1 × 10⁻⁴ rad s⁻¹) for 8 M_☉ and 16 M_☉ helium stars, respectively, and they gave examples of systems containing a compact object and a helium core that might approach these limiting values. An 8 M_☉ helium core in a tidally locked binary with a 0.8 M_☉ companion filling its Roche lobe had a longer period of 7.17 hr (ω = 2.4 × 10⁻⁴ rad s⁻¹). Here we calculate two models for each helium core mass. Model 8A has an enforced rotation rate in its outer 0.5 M_☉ of 2 × 10⁻⁴ rad s⁻¹ throughout its evolution. Models 8B, 16A, and 16B have rotation rates of 8.5 × 10⁻⁴ rad s⁻¹, 2 × 10⁻⁴ rad s⁻¹, and 7 × 10⁻⁴ rad s⁻¹, respectively.

Figure 4 shows the evolution of the angular momentum distribution as a function of mass for these four models. During helium burning, magnetic coupling maintains a state of nearly rigid rotation with an angular speed given by the surface boundary condition. For both Model 8A and 16A, this corresponds to a ratio of rotational speed to Keplerian at the surface of ∼8%. For Models 8B and 16B, the ratio is ∼32%. Roughly the inner 75% (8 M_☉) or 85% (16 M_☉) of the helium star is convective during helium burning and rigid rotation is enforced in that region. The outer part of the star is radiative, but magnetic torques there suppress differential rotation. During carbon burning and more advanced stages, the inner core contracts and rotates more rapidly. Magnetic torques are too inefficient to enforce rigid rotation. Later, during carbon burning, the stars ignite a convective helium burning shell which persists until the star dies. This convective shell does not extend all the way to the surface though, but does extend beyond the 0.5 M_☉ at the surface where corotation is enforced. Thus, all stars die with an angular velocity in their helium convective shell given by the surface boundary condition, but have more rapid rotation rates in their cores, due to contraction during advanced burning stages.

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Figure 5 shows the corresponding angular momentum in the evolving models. Shear instabilities and magnetic torques transport angular momentum out of the inner core and to the surface. In the absence of a binary companion, the surface would have actually spun up, but would have been removed by mass loss. In the present parameterized calculations though, there is no mass loss and the set rotation rate of the surface absorbs any excess angular momentum delivered to it. Consequently, the total angular momentum of the star decreases with time. For Model 8A at helium ignition, it was 9.0 × 10⁵¹ erg s, but for the pre-supernova star it was 1.6 × 10⁵⁰ erg s.

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Figure 6 shows the final distribution of angular momentum in the stars compared with that required to make a disk around a black hole, once the mass interior to the given sample point has collapsed. Four curves are given. Three are for the minimum angular momentum required to form a disk at the last stable orbit of (1) a non-rotating (Schwarzschild) black hole, (2) a Kerr black hole, and (3) a black hole with rotation given by the actual angular momentum distribution of the pre-supernova star. The fourth curve is the angular momentum distribution in the pre-supernova star. Because the inner parts of the star are not rotating very rapidly, curves (1) and (3) are similar except near the surface. Only where the actual angular momentum lies above curve (3) can a disk form. Material with less angular momentum plunges directly into the hole.

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None of the models has enough angular momentum to form a disk in its inner core. Thus a Type 1 collapsar will not occur. The angular momentum in the inner 1.7 M_☉ of the stars is still appreciable, however. Models 8A and 16A would form 1.4 M_☉ (gravitational mass) pulsars with periods of 7.4 ms and 2.6 ms, respectively. Models 8B and 16B would form pulsars of 3.2 ms and 1.5 ms. Model 16B thus qualifies as a "millisecond magnetar" candidate and might power a common GRB if a pulsar is able to form before accretion turns it into a black hole. Models 8B, 16A, and 16B could also make powerful pulsar-powered supernovae. Model 8A and other more slowly rotating systems probably would not.

Of greater interest for the present paper, however, is the large angular momentum in the outer layers of three of the models. These would be relevant if the center of the star collapsed to a black hole rather than a millisecond pulsar. Model 8A has angular momentum sufficient to form a disk in its outer 0.074 M_☉. Two other models (Table 1) have similar large rotation in their surface layers. These results are consistent with those of van den Heuvel & Yoon (2007) who found that their most rapidly rotating 8 M_☉ and 16 M_☉ helium cores ended their lives with j in outer layers of 3.7 × 10¹⁷ cm² s⁻¹ and 6.0 × 10¹⁷ cm² s⁻¹. The radius and gravitational potential of the high angular momentum surface layers allows an order of magnitude estimate of the free-fall timescale (Table 1), about 100 s.

The tidal interaction actually acts to brake the rotation of the surface layers compared to what they would have had in a star without mass loss and no companion star. To illustrate this, Model 8C was calculated, starting from the rigidly rotating helium burning stage of Model 8B (total angular momentum 3.6 × 10⁵¹ erg s), but with no surface boundary condition and no mass loss. That is, the rotation rate of the surface layers was allowed to adjust to be consistent with whatever angular momentum was transported to them. Without mass loss, the star was forced to conserve angular momentum overall and the very outer layers ended up rotating very rapidly (Figure 6). The angular speed, rather than being ω = 7.5 × 10⁻⁴ rad s⁻¹ was 3.7 × 10⁻³ rad s⁻¹. In fact, the outer tenth of a solar mass rotated so rapidly that it would be centrifugally ejected. In a realistic calculation, this matter and more underlying matter would probably have been ejected as a disk or a centrifugally boosted wind. The deep interior still lacked sufficient angular momentum to form a disk around a black hole, but the amount of surface material that could form a disk (if mass loss did not remove it) was significant, about 2 M_☉, comparable to that in Model 8B. Two other models like 8C were calculated that included mass loss appropriate for a Wolf–Rayet star at solar metallicity and one-tenth that value. The first ended up with a final mass of 3.95 M_☉ and no surface layers with sufficient angular momentum to form a disk. The second ended up with a mass of 7.09 M_☉ and still had enough angular momentum in its outer solar mass to make a disk.

So the major role of the close companion in the calculations done here is to preserve the angular momentum in the outer layers that might otherwise be removed by mass loss. A companion may also be essential so that the helium core rotates rapidly in the first place, following the loss of the star's hydrogen envelope. When magnetic torques are included in the calculation, the helium cores of supernova progenitors that are red and blue supergiants rotate too slowly for any portion of that core to make a disk. A close companion could either preserve angular momentum in the core by removing the envelope very early in the evolution or add angular momentum by tidal locking later on.