The Pair-instability Mass Gap for Black Holes

Substantial adjustments to the carbon abundance after helium burning are possible because of uncertainties in the critical reaction rates for 3α and ¹²C ${\left(\alpha ,\gamma \right)}^{16}{\rm{O}}$ and the treatment of convective mixing (Farmer et al. 2019, 2020; Renzo et al. 2020). Historically, the rate for ¹²C ${\left(\alpha ,\gamma \right)}^{16}{\rm{O}}$ during helium burning has been a major source of uncertainty for studies of stellar evolution and nucleosynthesis (Weaver & Woosley 1993; Tur et al. 2010; West et al. 2013). This rate is characterized by an "S-factor" evaluated at a typical Gamow energy for reactions during helium burning, which is taken to be 300 keV. DeBoer et al. (2017) have reviewed the experimental situation and offered their own analysis, concluding that this critical S-factor is most likely $140\pm {21}_{-11}^{+18}$ keV b. This is smaller than some other recent publications (see their Table 4) that gave a value near 165 keV b, but usually with larger error bars. In particular, Buchmann (1996), which serves as a reference point in the present study, gave a best value of S₃₀₀ = 146 keV b with a possible range from 62 to 270 kev b (see also Buchmann & Barnes 2006). In the same time frame Buchmann et al. (1996) found a best value of S₃₀₀=165 ± 75 keV b. Here we shall consider a possible range of S-factors from 110 to 205 keV b with a standard value of 146 keV b. The lower bound is that of deBoer et al. (2017). The upper bound is a bit larger than theirs and more consistent with Buchmann et al. (1996). Past studies of nucleosynthesis (Section 2.2) have favored such large values. Computationally these variations were affected by multiplying the standard fitted rate of Buchmann (1996) by factors, f_Buch, ranging from 0.75 to 1.4 (Table 1). As will be shown, M_low is more sensitive to the rate for ¹²C ${\left(\alpha ,\gamma \right)}^{16}{\rm{O}}$ when it is small than when it is large, which justifies, for now, not exploring larger multipliers than 1.4.

Table 1. Nonrotating Helium Stars of Constant Mass

MHe	X(12C)	fBuch	MBH	${{KE}}_{\exp }$
(M⊙)			(M⊙)	(1051 erg)
50	0.0677	1.4	42.5	1.10
51	0.0666	1.4	42.7	1.03
52	0.0656	1.4	45.4	0.90
53	0.0646	1,4	45.3	1.43
54	0.0636	1.4	46.1	0.77
56	0.0617	1.4	44.7	1.23
58	0.0600	1.4	43.5	2.25
60	0.0583	1.4	40.6	2.41
62	0.0568	1.4	20.1	2.97
64	0.0554	1.4	0	4.58
50	0.0986	1.2	42.9	0.93
51	0.0972	1.2	43.8	0.81
52	0.0959	1.2	43.8	0.86
53	0.0947	1.2	45.0	0.76
54	0.0935	1.2	45.6	0.74
56	0.0909	1.2	45.4	1.05
58	0.0887	1.2	46.0	1.78
60	0.0869	1.2	42.5	2.27
62	0.0850	1.2	32.0	2.53
64	9.0831	1.2	0	3.77
50	0.140	1	44.9	0.41
51	0.138	1	45.5	0.48
52	0.137	1	46.1	0.47
53	0.135	1	45.8	0.53
54	0.134	1	46.5	0.46
56	0.131	1	47.8	0.67
58	0.129	1	46.5	1.31
60	0.126	1	46.2	1.75
62	0.124	1	41.9	1.96
64	0.122	1	24.1	2.70
66	0.120	1	0	4.07
50	0.187	0.8	49.0	0.05
51	0.186	0.8	49.5	0.096
52	0.184	0.8	49.9	0.14
53	0.183	0.8	50.4	0.14
54	0.181	0.8	51.0	0.15
56	0.179	0.8	53.1	0.19
58	0.176	0.8	56.5	0.13
60	0.173	0.8	53.4	0.90
62	0.171	0.8	54.2	1.11
64	0.169	0.8	47.0	1.74
66	0.166	0.8	45.7	2.19
68	0.164	0.8	31.0	2.71
70	0.164	0.8	0	4.51
54	0.190	1/1.35	50.9	0.21
56	0.187	1/1.35	52.7	0.24
58	0.184	1/1.35	56.6	0.11
60	0.181	1/1.35	52.9	0.67
62	0.178	1/1.35	52.1	1.10
64	0.176	1/1.35	52.3	2.16
66	0.173	1/1.35	41.7	1.97
68	0.171	1/1.35	13.0	3.37
70	0.169	1/1.35	0	4.91
56	0.261	0.75/1.35	55.9	0.001
58	0.258	0.75/1.35	57.8	0.007
60	0.254	0.75/1.35	59.6	0.020
62	0.252	0.75/1.35	60.7	0.10
64	0.249	0.75/1.35	63.1	0.042
66	0.246	0.75/1.35	63.5	0.17
68	0.244	0.75/1.35	64.1	0.32
70	0.241	0.75/1.35	51.4	1.39
72	0.239	0.75/1.35	25.5	2.85
74	0.236	0.75/1.35	11.6	3.95
76	0.234	0.75/1.35	0	4.53

Note. The S-factor used for the ¹²C ${\left(\alpha ,\gamma \right)}^{16}{\rm{O}}$ reaction rate in these calculations was 146 keV b multiplied by the indicated multiplication factor, f_Buch. In some runs the 3α rate was also multiplied by 1.35. X(¹²C) is the central carbon mass fraction just before central carbon burning ignites. Numbers in bold type are maximum masses for a given choice of reaction rates.

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Shen et al. (2020) have given a recent reevaluation of the electric quadrupole part of the reaction rate S_E2 = 80 ± 40 keV b, which is larger than the 45 keV b adopted by deBoer et al. (2017) and more consistent with Buchmann et al. (1996). They recommend that the error bar on the deBoer et al. rate be raised from +18 to +39 keV b, i.e., the total S-factor is more likely to be larger than 146 keV b than smaller.

The rate for the 3α reaction depends on the radiative width for the "Hoyle state," an 0⁺ resonance at an excitation energy of 7.65 MeV in ¹²C. A small correction is generally added to the radiative width for pair emission. Summarizing previous work, Freer & Fynbo (2014) gave what might be considered the previous standard value for this width, Γ_rad = 3.78 meV (i.e., 3.78 × 10⁻³ eV). This is close to the value used in Caughlan & Fowler (1988), Γ_rad+pair = 3.70 meV (Hale in Wallerstein et al. 1997). New recent measurements (Eriksen et al. 2020; Kibédi et al. 2020), though by no means the final word, give Γ_rad = 5.1(6) meV, implying an upward revision of about 35%. This large increase is substantially outside the error bar of previous measurements and has major implications for stellar nucleosynthesis (Section 2.2), but it is considered here, for now, as a limiting case. Traditionally, this reaction rate was assumed to be known to 15% accuracy (e.g., Rolfs & Rodney 1988). In the present study, we define f_3α as the multiplier on the Caughlan & Fowler (1988) rate for 3α. That is, f_3α = 1.35 corresponds approximately to the new rate of Kibédi et al. (2020).

In our past works, including Woosley (2017), the multiplying factor in Kepler for the Buchmann (1996) recommended value (146 keV b) was f_Buch = 1.2. Within the context of a then very large range of possible values for the S-factor, Weaver & Woosley (1993) carried out a survey of nucleosynthesis in massive stars to determine what value worked best for nucleosynthesis, finding S₃₀₀ = 170 ± 20 keV b. That value was used in Kepler prior to 1996. After 1996, this was refined to be 1.2 times the rate given by Buchmann (1996), or S₃₀₀ = 175 keV b. This value has remained constant ever since, in part, to facilitate comparison among various calculations carried out with the Kepler code, but it has always remained within the experimental error bar and close to its centroid.

Tur et al. (2007, 2010) studied the influence of uncertainties in the 3α and ¹²C ${\left(\alpha ,\gamma \right)}^{16}{\rm{O}}$ reactions on nucleosynthesis and found that an increase in the 3α rate of 10% had about the same effect as an 8% decrease in ¹²C ${\left(\alpha ,\gamma \right)}^{16}$ O. Hence, for small variations, it is the the ratio of the reaction rates, λ_{α γ} (¹² C)/λ_3α (or f_3α /f_Buch), that matters most. West et al. (2013) and Austin et al. (2014) considered constraints on the ratio of these two rates required to fit nucleosynthesis in massive stars. They found that multipliers that gave a best fit obeyed the relation f_Buch = 1.0 f_3α + (0.35 ± 0.2), including the effect on the s-process (their R_α,12 and R_3α are equivalent to our f_Buch and f_3α ). The best fit using only the intermediate-mass isotopes was f_Buch = 1.0f_3α + (0.25 ± 0.3). That is, nucleosynthesis favored a relatively large rate for ¹²C ${\left(\alpha ,\gamma \right)}^{16}$ O, f_Buch ∼ 1.3, and any increase in 3α required a commensurate increase in the rate for ¹²C ${\left(\alpha ,\gamma \right)}^{16}$ O.

We will be exploring values outside this recommended range, including values of f_Buch as low as 0.75 and f_3α as large as 1.35. This large decrease in the ratio of the rates is necessary to appreciably raise the black hole mass but may have negative implications for nucleosynthesis that will need to be explored in the future. See Figure 4 of West et al. (2013).

In a very close binary, where little or no extended hydrogen envelope can be tolerated, the evolution of pure helium stars can be used to approximate the final presupernova state (e.g., Woosley 2017). Helium stars or carbon–oxygen stars will also be the end state of evolution where the metallicity is sufficiently high that the envelope is lost to a wind or the rotation is so rapid as to induce chemically homogeneous evolution (Maeder 1987) on the main sequence. Once revealed, the helium star will also experience mass loss by winds, especially if the metallicity is large. During the post-helium-burning evolution, the evolution is so fast that mass loss by winds is negligible. As will be shown (Section 2.4), helium cores evolved at constant mass are a good approximation to the evolution of heavier helium cores that include mass loss. It is the final mass that matters.

This simplification motivates a study of reaction-rate sensitivity in helium stars of constant mass. Similar studies have been carried out by Farmer et al. (2019, 2020). We consider here the variation of just the rates for ¹²C ${\left(\alpha ,\gamma \right)}^{16}{\rm{O}}$ and 3α because of their key role in determining the carbon abundance when the star becomes unstable. It is assumed that once an iron core is formed, typically around 2 M_⊙, that iron core and all the remaining star collapses to a black hole. This will not necessarily be the case if the star is very rapidly rotating (Section 4).

Models are calculated using the Kepler code and the same physics and zoning as in Woosley (2017). Convection is left on during interpulse periods, but only in the core, where hydrostatic equilibrium prevails. Convection speeds are always limited to 20% sonic, and convective velocities do not adjust instantaneously, but with a growth rate that is limited to the (inverse of the) convective timescale. Interpulse convection can sometimes play an important role in mixing combustible fuel, i.e., helium and carbon, to deeper depths in the star where it affects the strength of the next pulse. All models contained in excess of 2400 mass shells, with finer zoning near the center and surface. Typical simulations took about 20,000 time steps, though sometimes much more, to reach core collapse. Nuclear energy generation was generally treated using the 19-isotope approximation network in Kepler and the 128-isotope quasi-equilibrium network once the central oxygen mass fraction fell below 0.05. About 20 models were also calculated using an adaptable network (Rauscher et al. 2002) with typically 250 isotopes from hydrogen to germanium coupled directly to the structure calculation. Changing networks caused final remnant masses to vary by at most 1 M_⊙, except near the critical mass, where the star fully exploded after the first pulse (Section 2.3.2). Small differences made a large difference in outcome for these cases, but the value of M_low for a given choice of nuclear physics was not greatly affected.

2.3.1. Lower Boundary to the Pair-instability Mass Gap

2.3.2. Upper Boundary to the Pair-instability Mass Gap

Uncertainties in the reaction rates can also have an important effect on the upper boundary of the pair-instability gap (Table 2). The least massive black hole above the gap, M_high, now ranges from 136 to 161 M_⊙ for the same choices of reaction rates as in Table 1. There is a notable shift for f_Buch = 1.2 from the value given by Heger & Woosley (2002), M_high = 133.3 M_⊙, to the new value, M_high = 139 M_⊙, a change of 4.3% resulting solely from different code physics.

Table 2. Upper Bound on Pair-instability Supernovae

MHe	fBuch	X(12C)	MBH	${{KE}}_{\exp }$
(M⊙)			(M⊙)	(1051 erg)
135	1.4	0.0311	0	99.4
136	1.4	0.0309	136	0
138	1.2	0.0487	0	103
139	1.2	0.0485	139	0
143	1.0	0.0758	0	109
144	1.0	0.0755	144	0
150	0.8	0.116	0	116
151	0.8	0.116	151	0
151	1.0/1.35	0.115	0	120
152	1.0/1.35	0.114	152	0
160	0.75/1.35	0.176	0	131
161	0.75/1.35	0.176	161	0

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About half of this change comes from directly coupling a large reaction network of about 200 isotopes to the stellar structure calculation, taking smaller time steps, and other minor changes to the Kepler code in the past 20 yr. The calculations of Heger & Woosley (2002) only used a 19-isotope approximation network for energy generation and "coprocessed" with a much larger network to obtain nucleosynthesis. The quasi-equilibrium network in Kepler was not used owing to problems at that time with convectively coupled zones containing a varying amount of oxygen at explosive temperatures.

The other half comes from different assumptions regarding convection during the implosion. In Heger & Woosley (2002), it was assumed that convection did not occur during the implosion once carbon was depleted in the central zone. Here, for the standard cases, time-dependent mixing-length convection is allowed to proceed in zones that are not supersonically collapsing, i.e., v_col < c_s, but the convective speed used in the mixing-length calculation is limited, in all zones, to v_conv < 0.2 c_s. Here c_s is the local sound speed. Additional studies in which the zonal collapse speed limiter was reduced to v_col < 0.3 c_s while retaining v_conv < 0.2 c_s showed little deviation from Table 2. If those limiters were further decreased, though, to v_col < 0.1 c_s and v_conv < 0.1 c_s, M_high was reduced. More efficient mixing of oxygen downward during the implosion leads to greater burning and increases M_high. The values in Table 2 are thus probably upper limits for nonrotating models, and the actual values could be 2% smaller. As noted previously, small changes in physics can have a large effect near the cusp of a transition from full collapse to full explosion. These changes thus had a much greater effect on M_high than M_low.

Rotation could also substantially decrease M_high by weakening the pair instability, but that was not explored here. See Section 4 for the effect of rotation on M_low.

Stellar winds reduce the mass of the star and alter the thickness of the helium-burning shell relative to the carbon–oxygen core. Being farther out in the star, the influence of the residual helium is reduced relative to carbon, but helium burns at a lower temperature and generates more energy, and so it can still have a minor effect. Stars with the large masses considered here have convective helium-burning cores that extend through more than 90% of their mass when helium is half burned, so most of the post-helium-burning star is carbon and oxygen. Some helium always remains in the outer layers of the star, though, no matter what the mass-loss rate (see, e.g., Table 4 of Woosley 2019). The convective core recedes as mass is lost from the surface, leaving a gradient of helium outside. Typically at carbon ignition helium still has an appreciable abundance in the outer 15% of the star. Burning at the base of this shell during implosion drives convection that mixes additional helium down, increasing the strength of the shell burning.

To illustrate the effect, we consider here a single set of reaction rates, f_Buch = 1.2 and f_3α = 1.0, and explore some initial mass and mass-loss combinations that produce presupernova stars in the same mass range as Table 1. The mass-loss rate is taken from Yoon (2017) for 10% solar metallicity with f_WR = 1. Vink (2017) and Sander et al. (2020) predict a smaller mass-loss rate for such stars. Table 3 shows some results. For a typical case, a helium core with an initial mass of 76 M_⊙ ends up with a final mass of 55.9 M_⊙, mostly composed of carbon and oxygen, with a central carbon fraction of 0.0862. This is very similar to the 56 M_⊙ helium star, which, when evolved at constant mass in Table 1, had a carbon mass fraction of 0.0909. The small difference is because the shrinking helium convective core mixes in less helium to the center in the mass-losing star toward the end of helium burning. A more important difference, though, is the amount of helium in the outer regions of the star. The model with mass loss had 0.94 M_⊙ of ⁴He prior to pair pulsations, while the constant-mass model had 1.94 M_⊙. Both helium shells had their bases at about 47 M_⊙, though, i.e., the outer 9 M_⊙ of both presupernova stars contained appreciable helium.

Table 3. Effect of Mass Loss

Minit	Mfin	X(12C)	MBH	${{KE}}_{\exp }$
(M⊙)	(M⊙)		(M⊙)	(1051 erg)
68	50.2	0.0931	42.7	1.27
70	51.6	0.0913	45.2	0.66
72	53.0	0.0895	44.8	0.62
74	54.4	0.0878	45.7	0.58
76	55.9	0.0862	45.5	0.86
78	57.2	0.0846	45.0	1.41
80	58.7	0.0831	41.0	1.47

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Table 3 shows that the effect of this residual helium and mass loss on the remnant mass is generally small for the reaction rates considered. Comparing the results including mass loss to the corresponding stars with f_Buch = 1.2 and f_3α = 1 in Table 1, M_low is reduced by less than 1 M_⊙. This is comparable to variations introduced by zoning, network, and time step criteria. The small effect of the helium shell also suggests that the essential conclusions of Table 1 might be obtained ignoring the helium altogether and just modeling stars of just carbon and oxygen.

Many factors affect the carbon abundance in the core of a massive star following helium burning: the reaction rates for 3α and ¹²C ${\left(\alpha ,\gamma \right)}^{16}$ O, the treatment of semiconvection and convective overshoot mixingv(Imbriani et al. 2001), rotationally induced mixing, radiative mass loss (Section 2.4), and whether the star is in a mass-exchanging binary (Woosley 2019). Fortunately, the structure of a very massive star during its final stages of helium burning is quite simple. Because the luminosity is nearly Eddington and the structure near that of an n = 3 polytrope, helium stars of the mass considered are almost fully convective throughout most of their helium-burning evolution. Their compositions and entropies, except for a small mass near the surface, are nearly constant. The helium-exhausted material includes most of the matter that participates in burning during a PPISN or PISN. The remainder, near the surface, while composed of a mixture of helium and carbon, has an electron mole number near Y_e = 0.50. The star's structure, which is most sensitive to the electron density, is thus not very much affected by the ratio of helium to carbon.

These characteristics and a desire to better understand how the star's composition affects M_low motivate an exploration of stars consisting initially of just carbon and oxygen in a ratio that does not vary with location in the star. The composition reflects what exists at the end of helium burning and condenses many of the uncertainties mentioned above into a single parameter, the carbon mass fraction. Once a grid of explosions has been calculated for a suitable range of carbon mass fractions, the future properties of the supernova can be determined for any model by comparing its actual carbon mass fraction at helium depletion to the grid. Since the carbon mass fraction is bounded, so too is the maximum-mass black hole that can be produced following the pulsational pair instability.

Unfortunately, such a simple approach suffers from two complications. First, as discussed in Section 2.4, helium shell burning can weaken the pair instability. Whether the outer layers of the star are carbon or helium makes little difference to their structure, but the burning of helium during the implosion can weaken the first pulse. As a consequence, the remnant masses coming out of pure carbon–oxygen cores could underestimate the actual value of M_low. The other complication is that helium burning in such massive stars produces not just ¹²C and ¹⁶O but appreciable ²⁰Ne and even some ²⁴Mg as well. The abundances of these heavier nuclei can become comparable to the carbon abundance itself for those cases where the ¹²C ${\left(\alpha ,\gamma \right)}^{16}{\rm{O}}$ reaction rate is large and the 3α rate small. For the converse cases, perhaps of greater interest here, where the carbon abundance after helium burning is large, the carbon shell dominates and both the burning of helium and the abundance of neon have little effect. The effects of neon are also mitigated by the fact that neon burning never becomes an exoergic phase in such massive stars. Neutrino losses dominate, and the products of neon burning are oxygen and magnesium, so half the ²⁰Ne ends up in ¹⁶O anyway. The bulk energy yields of the neon-rich and neon-poor compositions also differ very little. Burning the "0.05/Ne" composition in Table 4 to ²⁸Si gives 4.63 × 10¹⁷ erg g⁻¹, whereas burning the "0.05" composition to ²⁸Si gives 4.69 × 10¹⁷ erg g⁻¹. Nevertheless, having a substantial abundance of neon can sometimes affect the strength of carbon shell burning, and it will be necessary to demonstrate the sensitivity of the answer to the neglect of neon and magnesium in the initial composition of helium in the outer layers.

Table 4 gives the results. Carbon–oxygen stars of constant mass with the initial carbon mass fractions indicated were evolved until the star either completely disperses as a PISN or its iron core collapses. Except for the three columns noted, the remainder of the initial star was all ¹⁶O. The range of carbon mass fractions span what we consider to be reasonable values for current reaction rates and convective treatments. No helium star in Section 2.3 gave a central carbon mass fraction less than 0.05 or greater than 0.25. In two of the exceptional series of models, those marked "0.05/Ne" and "0.1/Ne," more realistic neon-rich compositions were considered. Helium star models in Table 1 show that when the carbon mass fraction after helium burning is 0.05, the mass fractions of ²⁰Ne and ²⁴Mg are 0.07 and 0.01, respectively. When carbon is 0.1, ²⁰Ne and ²⁴Mg are 0.05 and 0.005. Surveys using these "neon-rich" compositions are also shown for comparison in Table 4.

The effects of a helium shell were also considered in a series of models labeled "0.05/He." In the inner 85% of their mass, these stars had compositions of 5% carbon and 95% oxygen by mass. In their outer 15%, though, the composition was 5% carbon, 90% oxygen, and 5% helium. This typically amounted to about 0.4 M_⊙ of helium, small even for mass-losing stars (Section 2.4). As expected, the presence of helium generally led to larger black hole masses. The effects of both neon and helium in the "0.05" case were an increase of about 2 M_⊙ (5%) each in M_low. The "0.05" case should give an upper bound on these effects because the carbon plays an increasingly dominant role in the others.

Comparing Table 4 for carbon–oxygen stars with Table 1 for helium stars that have similar carbon mass fractions at helium depletion, one sees that discrepancy is never great, but it is least when the carbon mass fraction is large. Not only is the neon abundance smaller in such cases, but the effects of carbon shell burning dominate those of helium shell burning. Conversely, when the carbon abundance is small, neon and helium have a bigger effect. For X(¹²C) = 0.05, M_low is 41 M_⊙ for carbon–oxygen stars but 46 M_⊙ for helium stars with f_Buch = 1.4. About half the difference is due to the large neon mass fraction, and the other half is due to the helium shell (columns "0.05/He" and "0.05/Ne"). For X(¹²C) = 0.125, on the other hand, M_low is 47 M_⊙ for carbon–oxygen stars and 48 M_⊙ for helium stars with f_Buch = 1. For X(¹²C) = 0.25, M_low is 64 M_⊙ for carbon–oxygen stars and 64 M_⊙ for helium stars with f_Buch = 0.75 and a 3α rate 1.35 times standard. Variations of 1–2 M_⊙ are expected in any survey owing to zoning, time step, network, number of pulses, etc. Thus, unless the carbon abundance is less than about 0.1, the simpler carbon–oxygen cores can be substituted for helium stars in our search for M_low.

As with the helium stars, the maximum black hole mass in Table 4 is not very sensitive to the exact value of the initial fraction of carbon so long as that fraction is small. For carbon fractions from 0.05 to 0.15, M_low varies only between 41 and 48 M_⊙ (and 41 M_⊙ is an underestimate as we have just discussed). For larger values, though, as would occur for small values of the ¹²C ${\left(\alpha ,\gamma \right)}^{16}{\rm{O}}$ reaction cross section or a large value for triple-alpha, M_low increases dramatically.

To summarize, for an S-factor greater than 110 keV b (deBoer et al. 2017) and a triple-alpha rate no greater than 1.35 times the traditional value, the carbon mass fraction in our models is no greater than 0.25 at helium depletion. This robustly implies a most massive black hole below the pair-instability gap of M_low = 64 M_⊙ with an expected error of about 2 M_⊙. Pending further experiments, we regard this as the upper limit to the black hole mass one can obtain in a close mass-exchanging binary by adjusting only the nuclear reaction rates. Additional corrections for mass loss are small (Table 3 versus Table 1).

Rotation weakens the pair instability (Fowler & Hoyle 1964), but the effect on the dynamics of the explosion is not great unless the star maintains a large degree of differential rotation. Marchant & Moriya (2020) give the modification to the structural adiabatic index, Γ₁, required for instability,

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1}\lt \displaystyle \frac{4}{3}\,-\,\displaystyle \frac{2}{9}\,{\omega }_{{\rm{k}}}^{2}+{ \mathcal O }({\omega }_{{\rm{k}}}^{4}),\end{eqnarray} \tag{ 1 }$$

where ${\omega }_{{\rm{k}}}={\rm{\Omega }}/\sqrt{{Gm}/{r}_{{\rm{e}}}^{3}}$ is the ratio of the actual angular rotation rate, Ω, to the Keplerian rotation rate for a given mass shell; r_e is its equatorial radius, here just taken to be the radius; and m is the mass inside that radius. Since a real star has neither constant Γ₁ nor ω_k, this criterion may or may not be satisfied in different regions of the star at different times. Whether there is a global instability depends on the integrated average (Marchant & Moriya 2020).

The strength of the pair instability also depends sensitively on the temperature and is weaker when the star first begins to contract and gain momentum and greater at the end when oxygen burns explosively. The accumulated momentum during the collapse plays an important dynamical role. Typical maximum decrements in Γ₁ from the pair instability for the masses being studied, once the central temperature exceeds 2 × 10⁹ K, are a few percent locally (relative to a fiducial 1.33) and ∼1% globally (Figure 1). This implies that differential rotation in the deep interior with ω_k ≳ 10% will be required to alleviate the instability. Smaller amounts of rotation can still affect the outcome, though, by leveraging the evolution early on when the instability is weak. Marchant & Moriya (2020) found a typical increase in black hole mass of ∼4% when magnetic torques (Spruit 2002) were included in the models and ∼15% when they were not. Since ignoring magnetic torques would give a very rapid rotation rate for pulsars and possibly overproduce gamma-ray bursts (GRBs; Woosley & Heger 2006), the inclusion of magnetic torques is probably more realistic.

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Unfortunately, rapid rotation affects the evolution of massive stars in many ways, and disentangling them complicates any conclusion regarding the black hole birth function. Rotation leads to mixing that results, at a minimum, in larger helium cores for a given main-sequence mass. At a maximum, the star may become fully mixed on the main sequence and experience chemically homogeneous evolution (Maeder 1987). Rotationally induced mixing can transport additional helium into the convective helium-burning core at late times, thus decreasing the carbon yield (Heger et al. 2000; Limongi & Chieffi 2018, their Fig. 19) and lowering the remnant mass (Table 4). Rotation reduces the central density, making the star evolve like one of lower mass (Marchant & Moriya 2020). Rapid rotation can lead to mass shedding at the equator, especially in stars that are already near the Eddington limit—as these are. Rotation can mix helium or carbon down during the implosion, so that shell burning has a greater effect during the first pulse. Rotation can alter the way the collapsing iron core behaves and allow explosions to develop where the star would have collapsed if neutrino transport were the only mechanism. If the rotation of the remnant is too rapid, not all of the star can immediately collapse to a black hole without making a disk. The formation of a "collapsar" may produce a jet that explodes the remaining star or, at the least, inhibits its full collapse.

The amount and distribution of angular momentum also depend critically on uncertain mechanisms for its transport and the rates for mass loss. Transport concentrates angular momentum in the outer layers of the star where mass loss removes it. To neglect mass loss is to overestimate the role of rotation in the evolution, but including it adds another uncertain parameter to the outcome, a parameter that will itself vary in an uncertain way with metallicity.

For all these reasons, the results in this section are less precise than those in the previous ones. A one-dimensional code is used to calculate the effects of a two-dimensional phenomenon—rotation. Angle averaging and a semiempirical formulation are necessary. Mass loss by winds is neglected, though see Section 4.3. What is most important is the angular momentum distribution the star has at carbon ignition since radiation-driven mass loss can presumably be neglected after that. Though we follow all stars through helium burning, they approach carbon ignition with essentially the same angular momentum distribution they started with—nearly rigid rotation characterized by a single parameter, J_init, the initial total angular momentum of the star.

Rotational mixing and angular momentum transport by the usual dynamical processes (Heger et al. 2000) were included in all stages of the evolution, including the interpulse periods. Time-dependent convection is also left on during interpulse periods in that portion of the star that remains in hydrostatic equilibrium, i.e., is not moving supersonically. Except where specified, angular momentum transport by magnetic torques is included (Spruit 2002; Heger et al. 2005).

Unlike our past studies of pair instability, an approximate centrifugal support term has been added to the momentum equation as

$$\begin{eqnarray}&&f\frac{9}{4}\,\frac{{j}^{2}}{{r}^{3}}\approx f\,\frac{{{\rm{\Omega }}}^{2}}{r},\end{eqnarray} \tag{ 2 }$$

where j is the average specific angular momentum of the shell, r is the radius of the shell, and Ω is the angular velocity of the shell. Shells are assumed to be spherical and rigidly rotating (solid-body rotation for each shells). Different shells may rotate at different angular velocity. The factor 9/4 is because, for a spherical shell in solid rotation, the average angular momentum at the equator is 3/2 times the average. The factor f accounts for averaging the centrifugal force over the surface and should be 2/3. Our approximation does not take into account any deformation of the shell or that the centrifugal forces would be larger at the equator than at the pole.

The helium stars studied here are assumed to initially rotate rigidly and are characterized by a total angular momentum J_init = (0.15, 0.3, 0.6, 1.2) × 10⁵³ erg s. Several choices of reaction rates are explored. Except for rotation, the physics and computational procedure here are the same as for nonrotating helium stars in Section 2.3.

Rotational mass shedding is included and is important for many models with J_init ≳ 6 × 10⁵² erg s. A surface zone is assumed to be lost when its angular velocity exceeds 50% of Keplerian (i.e., ω_k > 0.5). This value, corresponding to a ratio of centrifugal force to gravity of 25%, is an upper bound since the luminosity of the stars considered here is typically ∼ 90% Eddington. In any case, a larger value would make the approximation of spherical stars highly questionable. Values of ω_k this large only exist in layers near the surface. The central density rises from around 1000 g cm⁻³ prior to carbon ignition (T_9,c = 0.5) to 10⁷ g cm⁻³ during silicon burning. As the star attempts to maintain rigid rotation, the surface of the star is spun up, resulting in mass loss. This loss continues all the way through the evolution.

In addition to their angular momentum and mass, the models in Table 6 are characterized by several choices of nuclear reaction rates chosen to span most of the range for the nonrotating stars in Section 2.3 and Table 1. Three choices typify a central carbon mass fraction at carbon ignition, X(¹²C), that is either "small," f_Buch = 1.2, f_3α = 1; "medium," f_Buch = 1.0, f_3α = 1.35; or "large," f_Buch = 0.75, f_3α = 1.35. Magnetic torques are included all the time in all models except 64b, 66b, 68b, and 80b.

Table 6. Helium Stars with Constant Mass and Rotation

MHe	fBuch	Jinit	X(12C)	Mshed	Mpulse 1	KEpulse 1	MBH	${{KE}}_{\exp }$	Jfinal
(M⊙)		(1052 erg s)		(M⊙)	(M⊙)	(1051 erg)	(M⊙)	(1051 erg)	(1052 erg s)
54	1.2	1.5	0.0936	0	53.3	0.12	45.9	0.89	0.79
56	1.2	1.5	0.0913	0	51.6	0.44	46.8	0.72	0.43
58	1.2	1.5	0.0891	0	49.2	0.96	46.6	1.41	0.72
60	1.2	1.5	0.0871	0	49.9	1.56	45.4	2.34	0.32
62	1.2	1.5	0.0851	0	43.6	2.26	41.1	2.64	0.35
54	1.2	3	0.0932	0	53.3	0.041	46.2	1.33	1.30
56	1.2	3	0.0910	0	53.6	0.19	48.5	0.82	1.18
58	1.2	3	0.0889	0	50.6	0.62	48.1	0.92	0.41
60	1.2	3	0.0868	0	50.5	1.13	47.2	1.78	0.67
62	1.2	3	0.0849	0	49.4	1.77	44.2	2.68	0.59
56	1.2	6	0.0892	0.42	54.0	0.25	46.8	1.17	1.73
58	1.2	6	0.0874	0.37	57.0	0.036	48.6	1.12	1.51
60	1.2	6	0.0856	0.26	57.7	0.18	52.4	0.87	0.96
62	1.2	6	0.0839	0.02	54.2	0.59	49.1	1.04	0.38
64	1.2	6	0.0797	0.01	53.9	0.11	49.5	1.64	0.27
64b	1.2	6	0.0822	0	54.1	1.01	50.2	1.49	2.85
66	1.2	6	0.0786	0	50.4	1.75	43.6	3.05	0.15
66b	1.2	6	0.0805	0	50.2	1.78	44.5	2.81	1.75
68	1.2	6	0.0770	0	25.1	3.08	25.1	3.08	0.47
68b	1.2	6	0.0790	0	30.5	2.90	30.5	2.90	0.87
58	1.2	12	0.0844	2.11	55.4	0.018	47.2	0.81	0.92
60	1.2	12	0.0817	1.95	57.2	0.019	48.5	0.84	0.44
62	1.2	12	0.0805	1.73	58.2	0.42	51.4	1.69	1.84
64	1.2	12	0.0797	1.67	59.1	0.19	46.7	1.11	0.46
64b	1.2	12	0.0792	0.28	63.5	0.015	54.8	1.24	6.74
66	1.2	12	0.0786	1.56	60.7	0.28	55.5	0.97	0.53
66b	1.2	12	0.0778	0.23	64.5	0.14	56.5	1.17	6.86
68	1.2	12	0.0770	1.50	60.4	0.45	53.8	1.20	0.42
68b	1.2	12	0.0766	0.21	67.0	0.043	59.1	1.40	7.20
70	1.2	12	0.0759	1.32	51.7	1.54	45.1	3.12	0.14
72	1.2	12	0.0745	1.21	48.6	1.87	43.4	3.10	0.16
74	1.2	12	0.0730	1.04	28.6	2.83	28.6	2.83	0.89
62	1.0/1.35	6	0.177	0	60.6	0.058	58.2	0.28	2.93
64	1.0/1.35	6	0.175	0	63.1	0.054	59.2	1.09	1.85
66	1.0/1.35	6	0.173	0	58.6	0.56	54.5	1.04	0.82
68	1.0/1.35	6	0.170	0	58.3	0.95	55.0	1.48	1.60
70	1.0/1.35	6	0.168	0	54.5	1.35	50.7	1.99	1.47
72	1.0/1.35	6	0.166	0	46.6	1.98	43.8	2.47	1.09
74	1.0/1.35	6	0.164	0	0	4.84	0	4.84	0
66	1.0/1.35	12	0.169	1.56	61.7	0.14	61.7	0.14	4.32
68	1.0/1.35	12	0.167	1.66	64.2	0.074	61.3	0.98	2.19
70	1.0/1.35	12	0.165	1.43	65.2	0.14	61.8	0.48	1.44
72	1.0/1.35	12	0.163	1.36	64.8	0.29	61.2	0.87	0.74
74	1.0/1.35	12	0.161	1.35	66.7	0.27	61.3	1.03	0.72
76	1.0/1.35	12	0.159	0.97	64.5	0.93	58.0	2.33	1.68
78	1.0/1.35	12	0.158	0.94	59.5	1.44	56.4	2.02	2.58
68	0.75/1.35	3	0.244	0	68.0	0	67.9	0.001	2.97
70	0.75/1.35	3	0.241	0	67.2	0.10	67.2	0.10	2.41
72	0.75/1.35	3	0.239	0	61.6	0.88	61.6	0.88	1.68
74	0.75/1.35	3	0.237	0	33.7	2.89	33.7	2.89	0.36
68	0.75/1.35	6	0.243	0	68.0	0	67.9	0.004	5.87
70	0.75/1.35	6	0.240	0	70.0	0	69.9	0.007	5.92
72	0.75/1.35	6	0.238	0	69.0	0.096	69.0	0.096	4.74
74	0.75/1.35	6	0.236	0	66.4	0.63	66.4	0.63	3.80
76	0.75/1.35	6	0.234	0	44.1	2.35	44.1	2.35	1.19
78	0.75/1.35	6	0.232	0	0	5.12	0	5.12	0
68	0.75/1.35	12	0.239	1.39	66.6	0	66.3	0	8.32
70	0.75/1.35	12	0.237	1.43	68.6	0	68.3	0	8.92
72	0.75/1.35	12	0.235	1.16	70.8	0	69.9	0.033	7.90
74	0.75/1.35	12	0.233	1.19	70.4	0.12	70.4	0.12	6.40
76	0.75/1.35	12	0.231	0.87	73.4	0.051	73.3	0.051	8.62
78	0.75/1.35	12	0.229	0.75	73.0	0.015	73.0	0.015	7.60
80	0.75/1.35	12	0.227	0.28	69.9	0.85	69.9	0.85	6.33
80b	0.75/1.35	12	0.227	0.03	74.9	0.29	74.9	0.29	9.11

Note. See Table 1 for the definition of f_Buch. J_init is the initial total angular of the helium star. It can only decrease owing to mass loss, though angular momentum can be transported within the star. X(¹²C) is the central carbon mass fraction just before central carbon burning ignites. M_shed is the mass shed by rotation up to the point when oxygen burning ignites. M_{pulse 1} and KE_{pulse 1} are the mass following the first pulse and the kinetic energy of the initial ejecta. A zero value for KE_{pulse 1} indicates stable oxygen ignition with no explosive mass loss. M_BH and KE_fin are the mass of the black hole remnant and the total kinetic energy of all ejecta including the first pulse. J_fin is the angular momentum of the black hole remnant. Models 64b, 66b, 68b, and 80b were calculated without including the transport of angular momentum by magnetic torques and are of questionable validity. Otherwise, the most massive black hole for each choice of reaction rates and angular momentum is given in boldface. Black hole masses for models with ${J}_{\mathrm{final}}\gtrsim {\left({M}_{\mathrm{BH}}/33.5{M}_{\odot }\right)}^{2}$ are upper bounds. The Kerr parameter is the ratio of these two quantities.

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The rotating models in Table 6 will be denoted by their mass and the choice of reaction rates and angular momentum used in their calculation. Model 60_1.2,12 thus refers to the 60 M_⊙ model that used f_Buch = 1.2, f_3α = 1, and an initial angular momentum J_init = 12 × 10⁵² erg s.

Model stars (Table 6) were generated by allowing an extended configuration of pure helium with an initial central density of 0.1 g cm⁻³ to relax to thermal and hydrostatic equilibrium with a central density of about 200 g cm⁻³ and a central temperature of ∼ 2.2 × 10⁸ K where helium burning ignited. The initial angular momentum, J_init, characterizes that state. Helium burning lasts about 300,000 yr in all models and produces the central carbon mass fractions given in the table. Typical radii at central helium depletion (helium mass fraction is 1%) are near 1.5 × 10¹¹ cm, and the luminosities are ∼ 1.05 × 10⁴⁰ erg s⁻¹ (M/60 M_⊙).

After helium depletion, carbon burns away at the centers of all models without the nuclear energy generation ever exceeding neutrino losses and generating convection. Due to efficient magnetic torques, the star rotates nearly rigidly up until carbon ignition except for a small mass at the surface. The angular velocity for most of the interior is ${\rm{\Omega }}\approx 5\times {10}^{-4}{\left(60{M}_{\odot }/M\right)}^{2}({J}_{\mathrm{init},52}/3)$ rad s⁻¹ when the central temperature is 5 × 10⁸ K and the density is 2500 g cm⁻³. The ratio of angular velocity to its Keplerian value, ω_k, varies with radius, but for a point midway in mass in the star it is typically ${\omega }_{{\rm{k}}}\approx 0.029{\left(60{M}_{\odot }/M\right)}^{2}({J}_{\mathrm{52,0}}/3)$ at that time (Figure 1). These values are reduced in cases where there has been appreciable rotational mass shedding.

Prior to any pulsational mass loss, all models with J_init = 1.2 × 10⁵³ erg s and many with J_init = 6 × 10⁵² erg s experience rotational mass shedding. This begins late in helium burning, but it mostly happens near and just after carbon ignition. The mass lost is ≲ 2 M_⊙, which is a small fraction of the total mass but often contains a substantial fraction of the total angular momentum. For example, Model 60_1.2,12 shrank to 59.04 M_⊙ at carbon ignition, and in the process, its angular momentum, J₅₂, declined from 12 to 7.84. By oxygen ignition, these values had further decreased to 58.05 M_⊙ and 6.35, respectively. Even before pulsations began, the initial angular momentum for Model 60_1.2,12 had almost halved, making Model 58_1.2,6 and Model 60_1.2,12 similar in outcome. The two stars had masses of 57.6 and 58.0 M_⊙ after rotational mass shedding and produced black holes of 48.6 and 48.5 M_⊙. Other pairs, like 62_1.2,6 and 64_1.2,12, did not yield such similar results because of the stochastic nature of multiple pulses and sensitivity to slight changes in the angular momentum distribution and composition. The agreement was especially poor for models near the threshold of full explosion, like 66_1.2,6 and 68_1.2,12, and for models with little rotational mass shedding.

Still, the implication is that it would not be fruitful to consider models with much larger rotation rates than J_init,52 = 12. The excess momentum would simply be shed early in the evolution, and the maximum black hole mass would not be greatly increased. This is particularly true since the criterion for rotational mass loss, ω_k = 0.5 at the surface, is already too large for stars so near the Eddington limit.

Also shown in Table 6 is the energy of the first pulse and the mass of the star after that pulse (including the effects of rotational mass shedding prior to the pulse). These characteristics follow expectations more consistently than final properties. For a given angular momentum, the mass ejected and the kinetic energy of that mass, KE_pulse1, increase with initial mass. For a given initial mass, M_pulse1 increases with initial angular momentum, while KE_pulse1 decreases. In cases where the central carbon mass fraction and initial angular momentum are high, the stars often ignite central oxygen burning stably. Such stars may also experience a long phase of numerous, weak pulsations due to the burning of carbon in a shell. The mass lost prior to oxygen burning in such models is small, though.

A strong first pulse, on the other hand, implies a long wait before the next pulse while the star relaxes from a highly extended, weakly bound state. During this time, angular momentum can be transported out of the core by shear, magnetic torques, and convection. For these cases, subsequent pulses are not as influenced by rotation as the first one and can also be quite strong.

Indeed, the transport of angular momentum by magnetic torques prior to oxygen ignition and during the long interpulse periods, when they exist, is a major uncertainty in these calculations. Eliminating them reduces rotational mass shedding. Angular momentum that might have been transported to the surface and lost remains concentrated in the central core, where it influences the subsequent evolution. Figure 2 shows the evolution of the angular velocity and the ratio of the angular velocity to Keplerian, ω_k, for Model 66_1.2,12 with and without magnetic torques (models in Table 6 that end in b, like "66b," did not include magnetic torques). Early on, near carbon ignition at T₉ = 0.5, the rotational profiles are similar for the two models, though rotational mass shedding has already reduced the angular momentum in the model without magnetic fields. As time passes, the divergence increases, and by the time the iron core collapses, there is more than an order-of-magnitude difference in the angular momenta of the remnants.

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The chief cause of divergence in this case is magnetic braking during the thousand-year-long interval following the second pulse in the model that includes magnetic fields. During that time, the angular momentum in the inner 55.5 M_⊙ of the star (the part destined to be a black hole) declines from 4.2 × 10⁵² erg s to 5.4 × 10⁵¹ erg s. The mass of the star is constant during this interval, so the excess angular momentum piles up in a mass of just a few M_⊙ near the surface. This is later ejected in a final third pulse, so that the star that included magnetic fields ends up making a black hole with spin J₅₂ = 0.53, whereas the one without magnetic fields makes a black hole of nearly the same mass, but with J₅₂ = 6.86.

It would not be realistic to ignore magnetic torques during these long Kelvin–Helmholtz episodes while cores recover from prior explosive expansion. The compositions, densities, temperatures, and rotation rates are similar to those in stable carbon burning. On the other hand, experimental validation of the simple prescription used is lacking. The correct solution might lie between the two extremes. This means that any estimate of the Kerr parameter for the black holes coming from these rapidly rotating models, especially those with large carbon abundance, is not reliable.

The dimensionless Kerr parameter is $a/M={cJ}/{{GM}}^{2}={J}_{52}{\left(33.5{M}_{\odot }/M\right)}^{2}$. For a 70 M_⊙ black hole with a/M = 1, the critical angular momentum is 4.40 × 10⁵² erg s. Compact remnants in Table 6 with much more angular momentum than this cannot form black holes without losing mass and shedding the excess. This includes many of the more massive models with large carbon mass fractions in Table 6. Only a small fraction of this angular momentum is in the collapsing iron core itself. Most is in the outer layers. The most likely fate of such cores is to form a black hole surrounded by an accretion disk. This probably leads to the production of jets (Woosley 1993; MacFadyen & Woosley 1999) and mass ejection. That is, these are excellent candidates for making long soft GRBs, and GRBs may be intimately related to the formation of the most massive LIGO sources (see also Marchant & Moriya 2020). The bursts, of course, happened long ago when the black holes formed, not when they merged. They are probably not the most common form of GRBs since they involve quite massive black holes and the surface layers of the star. Given the large angular momentum, the disk might form at large radius and thus have a long characteristic timescale. A timescale of minutes is implied by the freefall time for the outer layers and is not an unreasonable estimate for the burst duration.

How much mass is ejected? Most of the final angular momentum is in the outer part of the remnant (Figure 2). Consider the case of 76_0.75/1.35,12. This model leaves a 73.3 M_⊙ remnant with J_final,52 = 8.62, too much angular momentum for a black hole with this mass. But J₅₂ = 4.2 exists at 69 M_⊙, corresponding to a/M = 1. A probable outcome, then, is that a black hole with an initial mass of ∼69 M_⊙ forms with an accretion disk of about 4 M_⊙. Jet formation will eject some, but not all, of this 4 M_⊙. Based on calculations by Zhang et al. (2004), the fraction ejected might not be large. It depends on the efficiency with which lateral shocks induced by the jet eject matter in the equatorial plane. The implication is that some of the large black hole masses in Table 6 should be treated as upper bounds but are perhaps not too far from correct. The final black hole of all models with ${J}_{52}\gtrsim {\left(M/33.5{M}_{\odot }\right)}^{2}$ would have a dimensionless Kerr parameter near a/M = 1.

The maximum-mass black hole for a given choice of reaction rates and angular momentum is given in boldface in Table 6. Models without magnetic torques are ignored. These limits range from little modification to those derived for nonrotating stars in Table 1 if J_init,52 ≲ 3 to increases as great as 11% for maximal rotation. Our heaviest black hole is 73 M_⊙ (though, as previously noted, that model has too much spin to fully collapse to a black hole). This is a larger increase than estimated by Marchant & Moriya (2020) for stars with magnetic torques included. The difference probably involves the consideration here of more rapid rotation and, especially, choices of reaction rates that produce more carbon. Without magnetic fields the increase could potentially be greater, though probably not much larger as Model 80b illustrates.

The rotation rates assumed here are similar to those used to model GRB progenitors (Yoon & Langer 2005; Woosley & Heger 2006; Yoon et al. 2006; Cantiello et al. 2007). Chemically homogeneous evolution is common for these stars and is consistent with the need to keep LIGO progenitors compact inside close binaries (Mandel & de Mink 2016).

Consider a 70 M_⊙ main-sequence star with initial angular momentum J = 4 × 10⁵³ erg s. This corresponds to an equatorial rotational speed at the surface of 390 km s⁻¹ and an angular velocity there that is 32% Keplerian. With this rotation rate, the star evolves chemically homogeneously (Mandel & de Mink 2016). It never becomes a giant and consumes all the hydrogen within its mass, except for a small fraction near the surface, while still on the main sequence. It thus evolves directly into a helium star similar to the ones considered in this section.

If mass loss were negligible, this initial angular momentum would be preserved and the star would die with values in excess of the largest ones in Table 6, but that is not realistic. On the other hand, with solar metallicity and conventional mass-loss rates, the star would end up with too small a mass to encounter the pair instability and would have a negligible amount of angular momentum.

Consider, then, the results of substantially decreasing the star's mass-loss rate, perhaps due to a small initial metallicity. Following Szécsi et al. (2015), we consider a metallicity of 2% solar, more specifically an iron mass fraction of 2.92 × 10⁻⁵. Their 67 M_⊙ model had a rotational velocity of 400 km s⁻¹, comparable to the 390 km s⁻¹ for our 70 M_⊙ model. Their model also evolved chemically homogeneously. For the mass-loss prescription they employed, their star had a mass at the end of hydrogen burning of 62.2 M_⊙. Most of their mass loss occurred after the star became a Wolf-Rayet star with a surface hydrogen mass fraction less than 0.3.

Szécsi et al. (2015) used a complicated prescription for mass loss involving multiple sources and interpolation. Here we just use the rates of Nieuwenhuijzen & de Jager (1990) and Yoon (2017) with a multiplicative factor less than 1 for the Yoon rate. Given the uncertain weak dependence of Yoon's mass-loss rate on metallicity, these reductions are reasonable. With this prescription, our 70 M_⊙ metal-poor star reaches a central hydrogen mass fraction of 0.3 with a mass of 69.4 M_⊙. Up to that point mass loss was taken from Nieuwenhuijzen & de Jager (1990) and was small. Continued evolution used the Yoon rates with a multiplicative factor of 0.5. The star finished hydrogen burning with a mass of 63.7 M_⊙ and a residual angular momentum of 2.3 × 10⁵³ erg s. The model of Szécsi et al. (2015) had a luminosity at hydrogen depletion of 10^6.34 L_⊙ and an effective temperature of 80,600 K. Our values were 10^6.37 L_⊙ and 86,800 K. The slight mismatch in starting mass and rotation rate is inconsequential compared with uncertainties in the mass-loss rate.

It is necessary to follow the continued evolution through helium burning. Published mass-loss rates for metal-poor helium stars span a wide range (Vink 2017; Yoon 2017; Sander et al. 2020; Sander & Vink 2020; Woosley et al. 2020). Here we continued to use the rate of Yoon (2017) scaled, for the given metallicity (2% solar), by factors of 1, 0.5, and 0.25. All runs were started from the model with a central hydrogen mass fraction of 0.3. These multipliers gave a range of average mass-loss rates during helium burning from 4 × 10⁻⁵ M_⊙ yr⁻¹ to 1 × 10⁻⁵ M_⊙ yr⁻¹. The instantaneous rate was about a factor of two smaller early in helium burning when the star was still a WNE star and larger later on when it is a WC star. These rates are consistent with what Sander & Vink (2020) have recently published for a metallicity of 2% to 5% solar. With these three rates, the final masses of the WC stars at carbon ignition were 47.2, 56.7, and 62.7 M_⊙, and their angular momenta were (2.7, 9.7, 19) × 10⁵² erg s. This spans most of the range of angular momenta sampled in Table 6. Other choices of initial mass and angular momentum could have filled the grid.