The Evolution of Massive Helium Stars, Including Mass Loss

Once revealed, a helium core in a close binary system continues to experience mass loss through winds. The uncertainty in the mass-loss rate and its dependence on metallicity dominate other factors, such as exactly when and how the helium star was formed. Here we use the mass-loss rate compilation of Yoon (2017), which is, in turn, an amalgamation of the rates of Tramper et al. (2016) for Wolf–Rayet stars of types WC and WO and Hainich et al. (2014) for WNE stars. In particular, as Yoon (2017) discussed, we take a mass-loss rate for WC and WO stars, defined as when the surface mass fraction of helium drops below Y_s = 0.9, of

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\,{\dot{M}}_{\mathrm{CO}} & = & -9.2\,+\,0.85\,\mathrm{log}\left(\displaystyle \frac{L}{{L}_{\odot }}\right)+0.44\,\mathrm{log}\,{Y}_{s}\\ & & +\ 0.25\,\mathrm{log}\left(\displaystyle \frac{{X}_{\mathrm{Fe}}}{{X}_{\mathrm{Fe}\odot }}\right).\end{array}\end{eqnarray} \tag{ 1 }$$

The metallicity in the form of iron is assumed to include all isotopes of the iron group and is X_{Fe ☉} = 1.46 × 10⁻³. All models calculated in this paper had Z = ${Z}_{\odot }$, so the metallicity term in these equations is zero. For further discussion of the metallicity dependence, see Eldridge & Vink (2006).

For WNE stars, nominally Y_s > 0.98,

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}\,{\dot{M}}_{\mathrm{WNE}} & = & -11.32\,+\,1.18\,\mathrm{log}\left(\displaystyle \frac{L}{{L}_{\odot }}\right)\\ & & +\ 0.6\,\mathrm{log}\left(\displaystyle \frac{{X}_{\mathrm{Fe}}}{{X}_{\mathrm{Fe}\odot }}\right).\end{array}\end{eqnarray} \tag{ 2 }$$

For surface helium mass fractions between 0.9 and 0.98, the mass-loss rate is linearly interpolated,

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{int}}=f\,{\dot{M}}_{\mathrm{CO}}\,+\,(1-f)\,{\dot{M}}_{\mathrm{WNE}},\end{eqnarray} \tag{ 3 }$$

with f = 12.5 (0.98 − Y_s).

It is assumed that the helium core is revealed at helium ignition and experiences this mass loss throughout its lifetime. Based on the models of Woosley & Heger (2007) and Sukhbold et al. (2018). A fit to the helium core mass in single massive stars at helium ignition as a function of ZAMS mass (i.e., the original hydrogenic star) is

$$\begin{eqnarray}&&{M}_{\mathrm{He},{\rm{i}}}\ \approx \ 0.0385\,{M}_{\mathrm{ZAMS}}^{1.603}\,{M}_{\odot }\ ({M}_{\mathrm{ZAMS}}\lt 30\,{M}_{\odot }).\end{eqnarray} \tag{ 4 }$$

This fit is valid for main-sequence masses in the range 6–30 ${M}_{\odot }$. For example, a 6 ${M}_{\odot }$ initial helium core mass corresponds to a 23.3 ${M}_{\odot }$ main-sequence star. This expression is plotted as a line through the green points in the upper panel of Figure 1. Above 30 ${M}_{\odot }$ and up to at least 140 ${M}_{\odot }$, a better fit for stars that do not lose their entire hydrogen envelope is

$$\begin{eqnarray}&&{M}_{\mathrm{He},{\rm{i}}}\ \approx \ 0.50\,{M}_{\mathrm{ZAMS}}\,-5.87\,{M}_{\odot }({M}_{\mathrm{ZAMS}}\geqslant 30\,{M}_{\odot }).\end{eqnarray} \tag{ 5 }$$

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Helium stars with initial masses between 1.6 and 3.2 ${M}_{\odot }$ develop degenerate cores of carbon and oxygen (<1.6–1.8 ${M}_{\odot }$); neon, oxygen, and magnesium (1.9–2.4 ${M}_{\odot }$); or silicon (2.5–3.2 ${M}_{\odot }$). Their final evolution is characterized by temperature inversions, flashes, off-center ignition, and CBFs. End states depend upon the mass loss during the final burning stages, which may be due to winds or a combination of winds and resumed transfer to the binary companion. Radii exceeding 10¹³ cm are common (Tables 1 and 2). Above about 3.2 ${M}_{\odot }$, for the present choice of mass-loss rates, this large expansion does not occur (Table 3). We first consider the case of mass loss by winds alone; then, we comment on how the results might be altered by late-stage mass loss to a companion (Section 3.1.3).

Table 1.  Carbon Ignition in Low-mass Models

Minit	Mign	MCO	MC−ign	L38	R13
[${M}_{\odot }$]	[${M}_{\odot }$]	[${M}_{\odot }$]	[${M}_{\odot }$]	[1038 erg s−1]	[1013 cm]
1.6	1.36	0.953	⋯	1.14	0.94
1.7	1.45	0.985	⋯	1.20	0.96
1.75	1.49	1.00	⋯	1.25	0.98
1.8	1.53	1.03	0.622	0.955	0.83
1.9	1.61	1.06	0.519	1.09	0.89
2.0	1.70	1.08	0.407	0.872	0.77
2.1	1.78	1.12	0.306	0.787	0.66
2.2	1.85	1.18	0.213	0.854	0.52
2.3	1.94	1.20	0.127	0.93	0.15
2.4	2.01	1.24	0.057	0.99	0.063
2.5	2.09	1.28	0.005	1.04	0.037
2.6	2.17	1.32	0	1.09	0.028
2.7	2.22	1.46	0	1.15	0.023

Note. For M_init > 1.75 ${M}_{\odot }$, M_ign and M_CO are the masses of the star and its CO core when carbon first ignites, and R₁₃ and L₃₈ are the star's radius and luminosity. Here M_C−ign is the mass shell where carbon ignites. For the three lighter models, approximate conditions are given when the star first develops a thin helium-burning shell. Carbon burning has not ignited.

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Table 2.  Late Evolution Below 2.5 ${M}_{\odot }$

Minit	Mcurrent	MCO	L38	R13
[${M}_{\odot }$]	[${M}_{\odot }$]	[${M}_{\odot }$]	[1038 erg s−1]	[1013 cm]	Outcome
1.6	1.21	1.21	⋯	⋯	CO WD
1.7	1.31	1.22	2.85	1.34	CO
1.75	1.33	1.26	3.27	1.39	CO
1.8	1.39	1.24	3.03	1.33	CO/NeO
1.9	1.48	1.26	3.18	1.41	NeO
2.0	1.59	1.26	3.25	1.48	NeO
2.1	1.73	1.23	3.19	1.53	NeO
2.2	1.80	1.28	3.71	1.63	NeO
2.3	1.87	1.32	4.16	1.74	NeO
2.4	1.96	1.32	4.18	1.77	NeO
2.5	2.07	1.37	0.96	0.72	Si flash

Note. For models from 1.7 to 2.4 ${M}_{\odot }$, conditions are given at the last model calculated and are not the terminal values. Here CO and NeO indicate the major constituents of the core at that time. Were the envelope not lost, continued growth of the core to the Chandrasekhar mass would lead to electron-capture supernovae in all cases from the 1.8 to 2.4 ${M}_{\odot }$ model.

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Table 3.  Silicon Ignition in Low-mass Models

Minit	Mfin	MCO	R13	ρc	Mign	η	Tign	Si-Ign
[${M}_{\odot }$]	[${M}_{\odot }$]	[${M}_{\odot }$]	[1013 cm]	[109 g cm−3]	[${M}_{\odot }$]		[109 K]
2.5	2.07	1.367	0.719	1.76	0.504	9.39	3.23	Deflagration
2.6	2.15	1.414	0.778	1.36	0.414	9.13	3.23	Weak pulse
2.7	2.22	1.459	0.760	1.17	0.38	9.09	3.18	Weak pulse
2.8	2.30	1.507	0.724	0.974	0.30	9.05	3.21	Weak pulse
2.9	2.37	1.556	0.647	0.772	0.18	9.04	3.18	Weak pulse
3.0	2.45	1.604	0.476	0.610	0.090	9.16	3.19	Deflagration
3.1	2.52	1.656	0.191	0.495	0.011	9.27	3.18	Deflagration
3.2	2.59	1.709	0.104	0.419	0	8.87	3.20	Deflagration
3.3	2.67	1.761	0.070	0.341	0	8.13	3.20	Normal
3.4	2.74	1.809	0.053	0.299	0	7.59	3.23	Normal

Note. Here M_init, M_fin, M_CO, and M_ign are the masses of the initial helium core, the final presupernova mass, the CO core at silicon ignition, and the shell where silicon burning ignites. In addition, T_ign and η are the temperature (in 10⁹ K) and electron degeneracy parameter at the location where silicon ignites, i.e., at ${M}_{\mathrm{ign}}$ and R₁₃ is the radius of the star then.

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3.1.1. M_He,i = 1.6–2.4 M_⊙; White Dwarfs and Electron-capture Supernovae

After central helium depletion, helium stars from the least massive considered here (1.6 ${M}_{\odot }$) to 2.4 ${M}_{\odot }$ develop degenerate CO or neon–oxygen–magnesium (NeO) cores surrounded by steep density gradients, thin helium-burning shells, and low-density convective helium envelopes. The radii of such stars can become quite large (Table 1) and grow with time (Table 2). The degenerate core also increases its mass as carbon and oxygen are added by helium burning. These attributes are similar to superasymptotic giant branch (SAGB) stars (e.g., Garcia-Berro & Iben 1994; Siess 2006) but differ in that the envelope here consists entirely of helium. Nevertheless, the term SAGB will be used to describe this structure. Another key difference is that the helium-burning shell here, despite being thin in mass, is not unstable. Perhaps a mild instability is suppressed by the implicit hydrodynamics of the code and large time steps, which are typically 10^6.5–10^7.5 s during the thin-shell epoch. Resolving the helium-burning shell requires very fine zoning though, down to 10⁻⁶ ${M}_{\odot }$. With this resolution, the burning breaks up into two regions: a nitrogen-burning shell, which lies just beneath the base of the convective envelope, and a broader helium-burning region, which is also radiative.

Provided these stars retain sufficient mass, they have two opportunities to ignite carbon burning: first, as the helium-burning ashes contract and heat up and second, as the helium-burning shell builds up a critical mass and ignites carbon closer to the surface. The critical mass for that second ignition is estimated here to be near 1.26 ${M}_{\odot }$. Consequently, the most massive white dwarf with a thick carbon shell should be 1.26 ${M}_{\odot }$.

Ignition of carbon within the helium-depleted core. Figure 2 and Table 1 show some characteristics of "first" carbon ignition in this mass range. Below 1.8 ${M}_{\odot }$, carbon burning does not ignite prior to the formation of an SAGB-like structure. The CO core is initially less than the 1.06 ${M}_{\odot }$ necessary for ignition (Nomoto & Hashimoto 1986, 1988). The 1.8 ${M}_{\odot }$ model itself is a transition case that ignites carbon burning in a flash 0.69 ${M}_{\odot }$ off-center (Figure 2) with a CO core mass of only 1.03 ${M}_{\odot }$. A second flash occurs 7500 yr later. Both flashes are weak, though, and fail to exhaust the carbon at the ignition sites or ignite a sustained CBF. The star at this point has not become a fully developed giant, and the pressure of the overlying helium is not negligible, hence the lower carbon ignition mass and incomplete burning. It remains mostly a CO white dwarf.

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It is not uncommon to see separate flashes preceding the formation of a CBF (Siess 2006; Farmer et al. 2015) or cases where flashes do not lead to flame formation. See, for example, the 7.0 ${M}_{\odot }$ model of Farmer et al. (2015), especially their Figure 3. This sort of evolution is thought to lead to "hybrid CO+NeO white dwarfs," and that is what is also found here for the 1.8 ${M}_{\odot }$ model. Most of the core remains CO, but there is a shell of partly burned carbon.

By comparing diffusion and burning timescales, Farmer et al. (2015) estimated a unique density for off-center carbon ignition, 2.1 × 10⁶ g cm⁻³. In good agreement, the 1.8 ${M}_{\odot }$ model here ignited carbon at 1.5 × 10⁶ g cm⁻³ when its central density was 1.30 × 10⁷ g cm⁻³.

The 1.9 ${M}_{\odot }$ model was the lowest mass to clearly ignite a sustained carbon CBF (Table 1) and did so with a core mass of 1.06 ${M}_{\odot }$ (Table 1). It did so after three flashes (Figure 2), the third of which developed into a flame that propagated to the center of the star. Even so, the composition remained contaminated by a substantial trace of residual, unburned carbon.

From 1.9 to 2.4 ${M}_{\odot }$, all stars ignite carbon CBFs (Nomoto & Iben 1985) that burn to the center of the star (Timmes et al. 1994) prior to becoming SAGB stars. This turns the composition of the core to neon, oxygen, and magnesium. As the mass of the core increases, a diminishing number of flashes precedes flame formation (Figure 2). The flash that bounds the eventual flame ignites systematically deeper in the star at a density that remains near 1.5 × 10⁶ g cm⁻³. The decreasing central density of more massive stars causes the ignition to move inward (Farmer et al. 2015). These flashes lead to an expansion of the core that temporarily weakens the helium-burning shell so that the base of the convective envelope moves out in mass. The flashes are less violent at higher mass, where the degeneracy is less. The 2.5 ${M}_{\odot }$ model was the last to ignite carbon off-center, doing so at an offset from the center of only 0.005 ${M}_{\odot }$.

In all models with a CBF, zoning was sufficiently fine that the flame moved by thermal diffusion. Rezoning was not allowed within the flame, but the calculation included standard overshooting as described in Sukhbold et al. (2018). Thermohaline mixing was not included, though the Ledoux criterion led to the mixing of regions with an inverted atomic mass when the temperature gradient was not strongly inverted. The flame speed is known to be sensitive to zoning, convective overshoot, and thermohaline mixing (Timmes et al. 1994; Siess 2006; Denissenkov et al. 2013). Sensitivity studies were not carried out, but the speeds here were about five times slower than Timmes et al. (1994), and their speeds should not be critical to the subsequent evolution.

Ignition of carbon as an SAGB star. The luminosity of models during their SAGB phase, which is due to helium shell burning, can be fitted to a power of the core mass; i.e., there is a "core luminosity relation" for helium SAGB stars (Jeffery 1988). Using the values in Tables 1 and 2 and earlier data points for core masses in the range 1.0–1.3 ${M}_{\odot }$,

$$\begin{eqnarray}&&L\ \approx 1.3\times {10}^{38}\,{M}_{c}^{4.1}\ \mathrm{erg}\ {{\rm{s}}}^{-1},\end{eqnarray} \tag{ 6 }$$

where M_c is the core mass (CO + NeO) in solar masses (see also Havazelet & Barkat 1979, their Equation (3.2)).

Burning 1 g of helium to 1 g of carbon and oxygen (50% by mass each) yields 7.3 × 10¹⁷ erg. The core luminosity relation thus implies that the compact core will increase in mass at a rate

$$\begin{eqnarray}&&{\dot{M}}_{c}\ \approx 2.8\times {10}^{-6}\ {M}_{c}^{4.1}\ \,{M}_{\odot }\ {\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 7 }$$

which is about (6–8) × 10⁻⁶ ${M}_{\odot }$ yr⁻¹ for core masses between 1.2 and 1.3 ${M}_{\odot }$. Growth rates in this range imply that carbon will ignite when the white dwarf reaches a mass of 1.25–1.3 ${M}_{\odot }$ (Table 2 of Saio & Nomoto 2004). Combining Equations (2) and (6), the mass loss to winds in the SAGB phase is

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{wind}}\ \approx 1.1\times {10}^{-6}\ {M}_{c}^{4.84}\ \,{M}_{\odot }\ {\mathrm{yr}}^{-1},\end{eqnarray} \tag{ 8 }$$

so for M_c ≈ 1.3 ${M}_{\odot }$, the core gains about 2 g for every 1 g lost to winds.

Carbon did not burn continuously in the shell that grew beneath the thin helium-burning shell in the SAGB stars. A critical mass of unburned carbon and oxygen accumulated before igniting and running away. Each runaway ignited a localized CBF, and the combination of convection and flame consumed the accumulated carbon layer, turning it to neon, oxygen, and magnesium. Similar to the hydrogen critical mass in classical novae, the critical mass of this carbon layer became smaller as the mass of the core and the gravitational potential at its edge increased. For models lighter than the 2.1 ${M}_{\odot }$ case, the critical mass, ∼0.1–0.2 ${M}_{\odot }$, was so large that only one layer was observed to form and run away during the time studied. For example, the 2.0 ${M}_{\odot }$ model ended carbon core burning with an NeO core mass of 1.15 ${M}_{\odot }$. Once an SAGB structure developed, helium burning in a thin shell increased the mass of the CO core to 1.26 before a runaway ignited at 1.22 ${M}_{\odot }$. Burning in this layer eventually converted the carbon and oxygen from 1.15 to 1.25 ${M}_{\odot }$ to NeO. Though the subsequent evolution was not followed, the critical mass for later runaways on this larger core will be smaller. That is, the critical mass depends on the current core mass, not the original helium star mass. For the 2.3 ${M}_{\odot }$ model, where the initial core mass was similar to the final mass in the 2.0 ${M}_{\odot }$ model, multiple carbon flashes were followed (Figure 3), and the critical mass was determined to be ∼0.01 ${M}_{\odot }$ for a core mass of 1.27 ${M}_{\odot }$. For the 2.4 ${M}_{\odot }$ model, carbon burning approached a steady state with helium burning, lagging just a few thousandths of a solar mass beneath, and no flashes were observed.

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No SAGB model was followed all the way to the Chandrasekhar mass. As the core grew more massive and compact, its surface gravity strengthened, making the helium- and nitrogen-burning shells thinner and requiring more zones for their resolution. Burning helium to carbon in zones of ever-decreasing mass required more time steps to maintain a given growth rate. The envelope expanded to large radii and partly recombined, causing additional numerical instability. The future evolution of the models can be estimated through assuming that (1) winds dominate the mass loss, (2) the core grows at twice the rate at which the total star loses mass (Equation (8)), (3) all cores over 1.26 ${M}_{\odot }$ ignite carbon burning in the accumulated layer, and (4) electron-capture supernovae happen when the core reaches 1.38 ${M}_{\odot }$. This growth rate is consistent with the numerical models evaluated during the last 0.1 ${M}_{\odot }$ of core growth (Table 2). It is estimated that the core of the 1.7 ${M}_{\odot }$ model will grow to 1.27 ${M}_{\odot }$ while the star loses 0.03 to end up as a white dwarf of 1.27 ${M}_{\odot }$, probably composed of NeO. Similarly, the 1.8 ${M}_{\odot }$ model will end up as a 1.34 ${M}_{\odot }$ NeO white dwarf. The 1.9 ${M}_{\odot }$ model will become an electron-capture supernova when its core mass reaches 1.38 ${M}_{\odot }$ and the envelope's mass is only 0.05 ${M}_{\odot }$; the 2.0 ${M}_{\odot }$ model will make a similar supernova with an envelope mass of 0.16 ${M}_{\odot }$; and so on. Whether any envelope can actually be retained to enable this final stage of core growth is very uncertain though.

3.1.2. M_He,i = 2.5–3.2 M_⊙; Silicon Flashes

Initial helium core masses of 2.5–3.2 ${M}_{\odot }$ correspond to stars with main-sequence masses 13.5–15.8 ${M}_{\odot }$ (Equation (4), Figure 1). At death, the mass ranges from 2.07 to 2.59 ${M}_{\odot }$, and the CO core mass ranges from 1.37 to 1.71 ${M}_{\odot }$ (Table 3). This range of final helium and CO core masses is known to be characterized by the lingering effects of off-center ignition and degenerate silicon flashes (e.g., Woosley & Heger 2015, Table 1).

As with carbon ignition, the displacement of oxygen ignition from the center shrinks with increasing mass. In the 3.5 ${M}_{\odot }$ model, both neon and oxygen burning ignite only 0.004 ${M}_{\odot }$ off-center. The oxygen-burning CBF propagates to the center, leaving behind composition and entropy profiles that set the stage for silicon ignition. How these flames progress may be different in a more realistic 3D simulation, and the assumption of a spherically symmetric flame propagation by conduction rather than turbulent undershoot mixing is questionable (Woosley & Heger 2015).

Silicon burning also ignites off-center for helium stars up to 3.2 ${M}_{\odot }$ (Table 3) with the displacement, again, a declining function of mass. Silicon ignition is defined as a region with exoergic energy generation capable of driving convection with a base temperature exceeding 3.2 × 10⁹ K. The core's temperature profile at silicon ignition is inverted by neutrino losses so that higher temperatures occur farther out in the star. Lower-mass stars retain their high degeneracy farther out, and this accounts for the violent runaway in the outer regions of the 2.5 ${M}_{\odot }$ model, but not the 2.6 ${M}_{\odot }$ model. At still greater masses, the higher temperature migrates deeper into the star, allowing another island of instability from 3.0 to 3.2 ${M}_{\odot }$.

The silicon flash in stars with violent runaways begins at a degeneracy parameter η ≳ 9 and reaches locally very high temperatures, around 6 × 10⁹ K, leading either to detonation or deflagration (Woosley & Heger 2015). If convection is left on until the energy transport exceeds 10⁴⁹ erg s⁻¹, the "sonic limit", a detonation forms that decays a short time later to a deflagration. If convection is more restricted, a hot, less dense layer of iron forms that is buoyant and also seeds a deflagration.

The strength of these explosions is poorly determined, especially in a 1D study, but the kinetic energy of any ejecta probably does not exceed a few × 10⁴⁹ erg. Typically, the runaways burn 0.1–0.5 ${M}_{\odot }$ of silicon to iron (Section 5.3). The fusion of a gram of silicon releases 1.7 × 10¹⁷ erg g⁻¹. This yield is evaluated from the existing composition, but typically, the fuel is rich in ³⁰Si and the ash in ⁵⁶Fe. The burning thus yields about 3.4 × 10⁵⁰ erg per solar mass, implying an explosive yield of order (0.3–1.7) × 10⁵⁰ erg, but the binding energy of the core at the time of ignition is (4–5) × 10⁵⁰ erg (e.g., 4.55 × 10⁵⁰ erg in the 3.0 ${M}_{\odot }$ model). It is only by transporting some fraction of the explosion energy to the envelope by a shock wave that any mass is ejected. The shock is produced by the large-amplitude oscillation of the core, and the more burning, the larger the amplitude and the stronger the shock.

Even stars where the runaway was not clearly supersonic, i.e., the 2.6–2.9 ${M}_{\odot }$ models, experienced mildly degenerate silicon flashes that gave weak shocks in the envelope. Typically, these shocks decayed by momentum conservation to less than 100 km s⁻¹ before reaching the star's surface. A tiny bit of matter is puffed off at low speeds, but that will be overtaken by the optically thick supernova before it brightens appreciably.

The radii of the star at silicon ignition—the same as "presupernova radius" in Table 4—ranges from 7 × 10¹² cm (2.5 ${M}_{\odot }$ model) to 1 × 10¹² cm (3.2 ${M}_{\odot }$ model). Given the low energy of the shock and small radius, the display resulting from the silicon flash will not be bright and will decline rapidly. A much brighter display could result when the remaining core completes silicon burning and collapses to a neutron star. Even a neutrino-powered wind impacting a solar-mass shell at roughly 1–10 au would give a luminous supernova (Woosley & Heger 2015). These possibilities are explored in Section 5.3.

Table 4.  Properties of the Model Sets

Minit	tHe/2	MHe/2	log LHe/2	log Teff,He/2	tfin	Mfin	log LC/Oign	log Teff,CO−ign	MCO	mHe	Ys
[${M}_{\odot }$]	[105 yr]	[${M}_{\odot }$]	[${L}_{\odot }$]	[K]	[105 yr]	[${M}_{\odot }$]	[${L}_{\odot }$]	[K]	[${M}_{\odot }$]	[${M}_{\odot }$]
Nominal Mass Loss
At helium burning and carbon ignition
1.8	19.66	1.73	3.27	4.82	57.03	1.53	4.37	3.82	1.03	0.49	0.99
1.9	17.79	1.82	3.35	4.83	51.32	1.61	4.45	3.82	1.06	0.55	0.99
2.0	16.04	1.92	3.41	4.84	46.59	1.70	4.35	3.83	1.08	0.59	0.99
2.1	14.68	2.01	3.48	4.85	42.53	1.78	4.31	4.31	1.09	0.63	0.99
2.2	13.56	2.10	3.53	4.86	39.10	1.86	4.31	4.59	1.10	0.67	0.99
2.3	12.50	2.20	3.59	4.87	36.19	1.94	4.33	4.67	1.14	0.70	0.99
2.4	11.76	2.29	3.64	4.87	33.63	2.02	4.35	4.71	1.18	0.73	0.99
2.5	10.86	2.38	3.69	4.88	31.43	2.10	4.37	4.73	1.21	0.76	0.99
At helium burning and oxygen ignition
2.5	10.86	2.38	3.69	4.88	31.73	2.07	4.37	3.85	1.37	0.67	0.99
2.6	10.28	2.47	3.74	4.89	29.76	2.15	4.47	3.86	1.41	0.70	0.99
2.7	9.69	2.56	3.79	4.89	28.02	2.22	4.51	3.87	1.46	0.72	0.99
2.8	9.08	2.65	3.83	4.90	26.48	2.30	4.55	3.89	1.51	0.75	0.99
2.9	8.64	2.74	3.87	4.91	25.10	2.37	4.59	3.92	1.56	0.77	0.99
3.0	8.21	2.83	3.91	4.91	23.88	2.45	4.61	3.99	1.61	0.78	0.99
3.2	7.54	3.01	3.99	4.92	21.74	2.59	4.65	4.34	1.70	0.82	0.99
3.5	6.65	3.28	4.09	4.93	19.18	2.81	4.69	4.55	1.85	0.86	0.99
4.0	5.60	3.72	4.24	4.95	16.10	3.15	4.78	4.69	2.12	0.91	0.99
4.5	4.89	4.16	4.36	4.97	13.97	3.49	4.85	4.75	2.38	0.93	0.99
5.0	4.75	4.55	4.47	4.98	12.37	3.82	4.91	4.86	2.65	0.94	0.99
6.0	3.93	5.39	4.64	5.01	10.24	4.45	5.02	4.95	3.15	0.93	0.99
7.0	3.40	6.22	4.78	5.02	8.87	5.05	5.10	4.99	3.63	0.88	0.99
8.0	3.05	7.03	4.90	5.04	7.91	5.64	5.17	5.00	4.11	0.82	0.99
9.0	2.79	7.83	5.00	5.05	7.22	6.20	5.21	5.01	4.59	0.73	0.99
10.0	2.58	8.61	5.08	5.06	6.70	6.75	5.27	5.02	5.07	0.61	0.99
11.0	2.43	9.39	5.16	5.05	6.30	7.05	5.27	5.24	5.45	0.35	0.64
12.0	2.30	10.17	5.22	5.05	5.99	7.27	5.28	5.22	5.55	0.20	0.38
14.0	2.10	11.70	5.34	5.06	5.52	8.04	5.36	5.24	6.11	0.21	0.19
16.0	1.95	13.20	5.43	5.06	5.18	8.84	5.39	5.25	6.79	0.24	0.22
18.0	1.84	14.69	5.51	5.07	4.92	9.62	5.42	5.25	7.42	0.25	0.23
20.0	1.70	16.23	5.58	5.12	4.72	10.39	5.47	5.30	8.01	0.26	0.25
25.0	1.59	19.68	5.71	5.12	4.35	12.53	5.61	5.34	9.76	0.25	0.22
30.0	1.49	22.85	5.81	5.12	4.08	14.81	5.72	5.37	11.63	0.25	0.21
40.0	1.37	29.48	5.98	5.13	3.74	19.55	5.89	5.40	15.67	0.27	0.19
50.0	1.29	36.32	6.11	5.13	3.52	24.48	6.02	5.42	20.00	0.32	0.18
60.0	1.24	43.25	6.21	5.14	3.36	29.53	6.13	5.42	24.31	0.38	0.17
70.0	1.20	50.26	6.30	5.14	3.25	34.66	6.21	5.42	28.81	0.46	0.16
80.0	1.17	57.32	6.37	5.14	3.16	39.83	6.29	5.40	33.38	0.53	0.16
90.0	1.15	64.25	6.43	5.14	3.09	45.04	6.35	5.34	38.04	0.60	0.16
100.0	1.13	71.37	6.49	5.14	3.04	50.27	6.40	5.29	42.70	0.66	0.16
110.0	1.12	78.41	6.54	5.15	2.99	55.51	6.45	5.22	47.45	0.71	0.16
120.0	1.11	85.45	6.59	5.15	2.95	60.78	6.49	5.16	52.05	0.76	0.16
1.5* Nominal Mass Loss
3.5	6.89	3.18	4.05	4.93	19.60	2.57	4.65	4.69	1.73	0.77	0.99
4.0	5.80	3.59	4.20	4.95	16.45	2.87	4.73	4.80	1.95	0.82	0.99
4.5	5.08	4.00	4.32	4.97	14.26	3.16	4.79	4.86	2.16	0.84	0.99
5.0	4.44	4.41	4.43	4.98	12.66	3.43	4.85	4.89	2.37	0.83	0.99
6.0	3.79	5.17	4.60	5.00	10.48	3.96	4.95	4.98	2.77	0.80	0.99
7.0	3.35	5.92	4.73	5.02	9.09	4.45	5.03	5.05	3.17	0.72	0.99
8.0	2.94	6.69	4.85	5.04	8.15	4.92	5.09	5.09	3.56	0.60	0.98
9.0	2.72	7.40	4.94	5.05	7.50	4.87	5.07	5.05	3.76	0.22	0.49
10.0	2.52	8.13	5.03	5.06	7.05	4.96	5.08	5.06	3.73	0.13	0.22
12.0	2.28	9.49	5.16	5.07	6.40	5.43	5.12	5.07	4.04	0.18	0.21
14.0	2.08	10.86	5.27	5.09	5.97	5.86	5.18	5.15	4.40	0.19	0.22
16.0	1.96	12.14	5.36	5.10	5.64	6.34	5.20	5.15	4.79	0.18	0.21
18.0	1.86	13.43	5.44	5.11	5.38	6.84	5.26	5.16	5.19	0.18	0.20
20.0	1.80	14.36	5.49	5.11	5.16	7.39	5.29	5.17	5.62	0.18	0.19
25.0	1.65	16.82	5.60	5.12	4.74	8.85	5.38	5.18	6.78	0.18	0.18
30.0	1.55	19.45	5.70	5.12	4.43	10.41	5.48	5.18	8.01	0.18	0.17
35.0	1.48	22.21	5.79	5.13	4.20	12.04	5.58	5.18	9.34	0.18	0.16

Note. Ages, masses, luminosities, and effective temperatures when helium has half burned ("He/2") and carbon (first eight rows) or oxygen (the rest of the table) ignites. The oxygen ignition values are also essentially the presupernova star properties. Here M_CO is the CO core mass, M_He is the remaining mass of helium in the star, and Y_s is the surface helium mass fraction.

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3.1.3. Radius Expansion and Late-stage Mass Transfer

Models with initial masses above 3.2 ignite all phases of nuclear burning, including silicon burning, at their centers (Table 3). Their radii also remain sufficiently small that a second stage of binary mass transfer is probably avoided (Figures 4 and 5).

The age, mass, luminosity, and effective temperature of the helium stars on the (helium) main sequence, when convective burning has reduced the central helium mass fraction to 0.50, are given in Table 4 and Figure 6. The luminosity as a function of mass is given, approximately, for the entire distribution by $L/{L}_{\odot }\approx 2.81\times {10}^{3}{M}^{1.71}$ (red curve), where M is the current mass in solar masses. A better fit can be obtained by breaking the fitting interval into two mass ranges: 3–15 ${M}_{\odot }$ (green line in Figure 6) and above 15 ${M}_{\odot }$ (blue line),

$$\begin{eqnarray}\begin{array}{rcl}L/{L}_{\odot } & \approx & 1.16\times {10}^{3}{M}^{2.13}\ \ (3\,{M}_{\odot }\leqslant M\leqslant 15\,{M}_{\odot })\\ & \approx & \ 4.92\times {10}^{4}\,M-4.57\times {10}^{5}\ \ (M\gt 15\,{M}_{\odot }).\end{array}\end{eqnarray} \tag{ 9 }$$

This second equation has the correct form for extrapolating to masses beyond the fit interval because the luminosity approaches the Eddington value at high masses and is thus proportional to M.

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Similarly, the lifetimes in Table 4 can be approximated but are not plotted,

$$\begin{eqnarray}\begin{array}{rcl}{t}_{f}/{10}^{5}\ \mathrm{yr} & \approx & 3.0+75.3{M}_{\mathrm{He},{\rm{i}}}^{-1.24}\ \ (3\,{M}_{\odot }\leqslant M\leqslant 75\,{M}_{\odot })\\ & \approx & \ 3.0\ \ (M\gt 75\,{M}_{\odot }),\end{array}\end{eqnarray} \tag{ 10 }$$

where again, at high masses, the lifetime approaches the correct (constant) Eddington value. Here M_He,i is the initial mass of the helium star. This fit is inaccurate for lower masses than indicated.

A fit for the final mass as a function of starting helium core mass (see bottom panel of Figure 1) that is good to better than a few percent for initial helium core masses up to 10 ${M}_{\odot }$ is

$$\begin{eqnarray}&&{M}_{\mathrm{fin}}\ \approx \ 0.939\ {M}_{\mathrm{He},i}^{0.872},\end{eqnarray} \tag{ 11 }$$

where all masses are in solar masses. For heavier helium cores,

$$\begin{eqnarray}&&{M}_{\mathrm{fin}}\ \approx \ 0.463\ {M}_{\mathrm{He},i}\,+1.49.\end{eqnarray} \tag{ 12 }$$

These equations can be combined with Equations (4) and (5) to give

$$\begin{eqnarray}&&{M}_{{\rm{f}}{\rm{i}}{\rm{n}}}\,\approx \,0.0548\,{M}_{{\rm{Z}}{\rm{A}}{\rm{M}}{\rm{S}}}^{1.40}\end{eqnarray} \tag{ 13 }$$

for main-sequence masses below 30 ${M}_{\odot }$ and

$$\begin{eqnarray}&&{M}_{{\rm{f}}{\rm{i}}{\rm{n}}}\,\approx \,0.232\,{M}_{{\rm{Z}}{\rm{A}}{\rm{M}}{\rm{S}}}\,-1.23\end{eqnarray} \tag{ 14 }$$

for main-sequence masses above 30 ${M}_{\odot }$. For very high masses, a good rule of thumb is that the final mass is one-fourth of the ZAMS mass. This is for the standard mass-loss rate and would need to be modified for other values.

For the standard choice of mass-loss rates (Equation (3)), the pair instability is first encountered for the 60 ${M}_{\odot }$ model. At the time of the instability (carbon depletion), the star has shrunk to 29.53 ${M}_{\odot }$ and consists mostly of oxygen and heavy elements. Helium has a small surface abundance, 0.16 by mass fraction, and the total mass of the star is small (Table 4). As with models having similar core mass in the study of Woosley (2017; see his Table 1), the instability in these lighter cores is a weak one, occurring at late times in the oxygen-burning shell and resulting in the low-energy ejection of only a few hundredths of a solar mass. The onset of the instability occurs at smaller masses here than in the earlier study because of the larger effective CO core mass (Figure 7). As Woosley (2017) pointed out, the strength of the pair instability is most sensitive to the CO core mass.

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The instability continues to be weak and brief for helium stars up to about 75 ${M}_{\odot }$ (final mass 37 ${M}_{\odot }$). The instability happens at late times in the oxygen-burning shell after a silicon core has already formed in hydrostatic equilibrium and the total energy of the ejecta is less than 10⁵⁰ erg. These low-energy, small-ejecta-mass, and small-radii models will not make luminous supernovae, but if the core collapses uneventfully to a black hole, the small amount of mass ejected may be the only optical display of their death. Their transients will be brief, faint, and very blue (see Woosley 2017 and Section 5). If, on the other hand, collapse leads to an explosive event, these small masses at radii 10¹²–10¹³ cm may enable a bright, collisionally powered supernova. The light curves for the heavier, more energetic models would be similar to those already published by Woosley (2017).

Moving on up in mass, one encounters PPISN of increasing duration and energy that resemble the helium stars of constant mass studied by Woosley (2017). The final mass is offset by about 5% due to the larger CO core. For example, Woosley (2017) found that the maximum helium core mass that made a PPISN rather than experiencing complete disruption in a single pulse pair-instability supernova (PISN) was about 63 ${M}_{\odot }$. Here the equivalent value is 60 ${M}_{\odot }$ (Table 5). The maximum remnant mass, 46 ${M}_{\odot }$, is also lighter. This would be the best current estimate for the onset of the black hole mass gap that might be detected by gravitational radiation experiments. It is considerably smaller than the 52 ${M}_{\odot }$ value given by Woosley (2017) and 50 ${M}_{\odot }$ value given by Belczynski et al. (2016). The larger values were for single stars. For helium stars, there is no hydrogen envelope to tamp the explosion and cause more material to fall back. The CO core is also larger in the new models.

Table 5.  Pulsational Pair-instability Supernovae

Minit	Mf	τ	E	Mej	Mrem	MFe
[${M}_{\odot }$]	[${M}_{\odot }$]	[s]	[1050 erg]	[${M}_{\odot }$]	[${M}_{\odot }$]	[${M}_{\odot }$]
60	29.53	6.4(4)	0.015	0.02	29.51	2.32
62	30.56	6.1(4)	0.020	0.05	30.51	2.36
64	31.57	5.8(4)	0.022	0.07	31.50	2.49
66	32.60	5.7(4)	0.030	0.10	32.50	2.55
68	33.63	5.8(4)	0.038	0.18	33.45	2.67
70	34.66	6.7(4)	0.060	0.43	34.23	2.84
75	36.83	8.0(4)	0.36	0.66	36.17	2.95
80	39.38	4.1(5)	1.5	1.95	37.43	3.22
85	41.95	9.4(5)	4.6	3.78	38.17	2.99
90	44.54	2.3(6)	5.6	5.78	38.76	2.62
95	47.13	1.1(7)	5.8	6.37	40.76	2.52
100	49.75	1.1(8)	5.7	7.15	42.60	2.73
100*	50.22	8.6(7)	4.1	7.29	42.93	2.22
105	52.24	9.1(9)	5.2	7.21	45.03	1.73
105*	52.82	7.6(9)	3.1	6.83	45.99	2.04
110	54.79	5.8(10)	5.2	10.57	44.22	2.58
110*	55.43	7.3(10)	5.9	10.63	44.80	2.14
115	57.42	1.5(11)	14.8	16.07	41.35	2.63
120	60.12	1.2(12)	35.6	56.61	3.51	1.76

Note. Here τ is the time between the onset of the first pulse and core collapse, and E₅₀ is the total kinetic energy of all mass ejected in units of 10⁵⁰ erg. Models 105*, 110*, and 115* were run using large networks coupled directly to the burning but are otherwise like models 100, 105, and 110. These and all other unstarred models used the approximation network and quasi-equilibrium assumption.

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For the standard mass-loss rate, the strong pulsational pair instability is first encountered for a final core mass of 37 ${M}_{\odot }$ (Table 5), which corresponds to an initial helium core mass near 75 ${M}_{\odot }$. Using Equation (5), this corresponds approximately to a star whose main-sequence mass was 160 ${M}_{\odot }$. Reductions in the mass-loss rate result in the instability being encountered at lower masses. The reduction may either be because the standard rate assumed for solar-metallicity stars here is too large (Vink 2017) or because of reduced metallicity. Though metallicity scaling is given in Equations (1) and (2), the exponent is uncertain (Eldridge & Vink 2006).

An additional 18 models were calculated with initial masses between 5 and 80 ${M}_{\odot }$. Half of these used a mass-loss rate of 50% of standard, and half used 20%. The results from this sparse grid are displayed along with the more densely sampled grids with normal and 1.5 times normal mass loss in Figure 8. Reducing the mass loss by a factor of 2 results in a strong pulsational pair instability (final mass over 37 ${M}_{\odot }$; Table 5) being encountered for an initial helium core mass of only 52 ${M}_{\odot }$. Reducing the standard mass loss by only a factor of 2 would allow a 110 ${M}_{\odot }$ star, either single or in a binary system, to become a PPISN.

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The fraction of mass lost for high-mass helium stars where the luminosity is almost linear in the mass and the lifetime approaches 3 × 10⁵ yr (i.e., the Eddington limit) is

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}M}{{M}_{\mathrm{init}}}\ \approx \ (1\,-\,\exp (-0.7F),\end{eqnarray} \tag{ 15 }$$

where F is the factor by which the overall mass loss is multiplied. For example, at F = 1, the star loses half its mass.

Winds remove the surface layers of the star, ultimately revealing what lies beneath. Initially, that material consists of almost pure helium, but as mass loss reveals the products of helium burning, a variable amount of carbon and oxygen appears. None of the stars here were so extreme as to lose all their helium. The mass loss occurs mostly during helium core burning, and the outer edge of the convective core recedes as mass is lost, leaving behind a gradient of helium. Table 4 and Figure 9 show that the surface consists of helium and nitrogen until some critical mass is reached, about 11 ${M}_{\odot }$ (final mass 7 ${M}_{\odot }$) for the nominal mass-loss case and 9 M_☉ (final mass 4.9 ${M}_{\odot }$) for a loss rate that is 1.5 times higher. Figure 9 is very similar to Figure 7 of Yoon (2017).

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As Yoon (2017) pointed out, the higher-mass limit is inconsistent with the observed luminosity of the faintest WC/WO stars, and this argues for mass-loss rates greater than the standard value used here. On the other hand, Yoon (2017) and McClelland & Eldridge (2016) argued that the temperatures of observed Wolf–Rayet stars are cooler than the models unless a low mass-loss rate is used. Vink (2017) also predicted mass-loss rates that are substantially smaller than even the standard one used here. He gave

$$\begin{eqnarray}&&\mathrm{log}{\dot{M}}_{\mathrm{Vink}}=13.3\ +\ 1.36\,\mathrm{log}\left(\displaystyle \frac{L}{{L}_{\odot }}\right)+0.61\mathrm{log}\left(\displaystyle \frac{Z}{{Z}_{\odot }}\right).\end{eqnarray} \tag{ 16 }$$

For solar metallicity and a 6 ${M}_{\odot }$ star, for example, with log L/${L}_{\odot }$ = 4.64 (Table 4), this expression gives 10⁻⁷ ${M}_{\odot }$ yr⁻¹, whereas Equation (2) gives a value 10 times larger.

In this paper, most models use the standard value of Yoon (2017). As we shall see (Section 5.1), however, a larger value may also be needed to produce SNe Ic if their progenitors must have a mass total less than 6 ${M}_{\odot }$ but lose their helium-rich layers before exploding. There is also an issue of whether mixing or mass loss is the essential ingredient in making a Type Ic.

Figure 10 shows the compactness parameter (O'Connor & Ott 2011),

$$\begin{eqnarray}&&{\xi }_{M}={\left.\displaystyle \frac{M/{M}_{\odot }}{R(M)/1000\mathrm{km}}\right|}_{{t}_{\mathrm{bounce}}},\end{eqnarray} \tag{ 17 }$$

as a function of final core mass for a variety of presupernova models. Customarily, ξ is evaluated at M = 2.5 ${M}_{\odot }$ and denoted ξ_2.5. In the figure, black points are for models with initial masses of 3.2–4.5 ${M}_{\odot }$ that have final masses of 2.59–3.49 ${M}_{\odot }$ (Table 4). These are, in turn, derived from stars whose original main-sequence masses were 13.8–15.6 ${M}_{\odot }$. For lighter stars, the final mass is less than 2.5 ${M}_{\odot }$, and ξ_2.5 is undefined, though it approaches zero. The lighter models (above 2.5 ${M}_{\odot }$) were evolved to the presupernova state using the large nuclear reaction network to calculate energy generation, neutrino losses, and neutronization. Normal mass loss was assumed. Fine surface zoning was employed, down to 5 × 10⁻⁹ ${M}_{\odot }$, and a relatively small boundary pressure, 10⁴ dyne cm⁻², was employed.

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The green points in Figure 10 also assumed the standard mass-loss rate but used a combination of the approximation and quasi-equilibrium networks. Surface zoning was coarser than for the black points, with the finest zones near 10⁻⁷ ${M}_{\odot }$. A surface boundary pressure of 10⁸ dyne cm⁻² was used. Blue points are for a more limited survey that used similar resolution but slightly reduced boundary pressure, 10⁷ dyne cm⁻². Red points came from a similar survey to the green points but with 1.5 times the mass-loss rate (see Table 4). Good agreement among different sets evolved with different physics; where they overlap in presupernova mass suggests the robustness of the pattern.

The overall behavior is less chaotic than that found by Sukhbold & Woosley (2014) and Sukhbold et al. (2018) for stars in this mass range (see Figure 8 of Sukhbold et al. 2018 for final helium cores from 3.5 to 6 ${M}_{\odot }$). Though clearer evidence might emerge with a greater number of models, there is less indication here of multiple solutions for a given final mass, especially branches with high compactness parameters. In general, the models, especially the lower-mass ones, look more likely to explode. Helium cores up to 6.0 ${M}_{\odot }$ in final mass have ${\xi }_{2.5}\lesssim 0.15$. A final helium core mass of 6.2 ${M}_{\odot }$ corresponds to an initial mass of 9.0 ${M}_{\odot }$ (Table 4), which, in turn, corresponds to a main-sequence mass of about 30 ${M}_{\odot }$. The helium core at death for a 30 ${M}_{\odot }$ star that did not lose its entire envelope would have been 10 ${M}_{\odot }$ (Figure 1) and would have had ξ_2.5 ≈ 0.40 (Sukhbold et al. 2018). Such a large difference will very likely affect whether the star explodes or not. Stars evolved in mass-exchanging binaries have very different deaths than single stars of the same initial mass.

Most striking is the absence of models with compactness ${\xi }_{2.5}\gtrsim 0.17$ for presupernova masses less than 6 ${M}_{\odot }$. In Sukhbold et al. (2018), there were many models with ξ_2.5 between 0.17 and 0.3 for helium core masses around 4.5–6.0 ${M}_{\odot }$ (see their Figure 6). These might have provided a population of low-mass black holes. While a single parameter characterization cannot substitute for a full study of core collapse, ξ_2.5 is known to correlate well with other, more physical measures of explodability (Ertl et al. 2016; Sukhbold et al. 2018), as well as with the results of simple 1D neutrino transport calculations (Sukhbold et al. 2016). It thus seems likely that the distribution of black hole masses will be different for binaries and single stars. Since all mass measurements of black hole masses are made in binaries, usually with a history of interaction, this has important implications. The peak in compactness where a large number of black holes are likely to be born is also shifted slightly upward. In Sukhbold et al. (2018), the peak was centered at helium core masses of 7 ${M}_{\odot }$. Here it is at 8 ${M}_{\odot }$.

Why are the results so different, even when compared for the same final helium core masses? Several factors contribute. First, a helium core inside a massive star has different boundary conditions, since the former is bounded by a dense, hot hydrogen-burning shell, and the latter is not. The helium core in a star that still has its hydrogen envelope is arbitrarily defined as where the hydrogen mass fraction first goes below 0.01 moving inward. Typical temperatures for that shell during helium burning inside a massive star are near 4 × 10⁷, and the pressure is a few × 10¹⁶ dyne cm⁻². In bare helium stars, such conditions are usually achieved a few tenths ${M}_{\odot }$ beneath the surface. Helium cores in massive stars, so defined, evolve like slightly heavier bare helium cores.

Second, and more importantly, the carbon mass fraction at carbon ignition is substantially higher in the mass-losing helium stars than in the mass-gaining helium cores of massive hydrogenic stars (Figure 11). Inside a massive star, the helium core grows because of mass processed through the hydrogen-burning shell. As it grows, the extent of the central convective core also increases (Figure 3 of Woosley et al. 2002). Thus, at late times, extra helium is mixed into the burning region, turning more carbon to oxygen. In a mass-losing star, the opposite occurs. The convective core recedes (second panel of Figure 12), leaving behind a gradient in the carbon abundance with a larger abundance resulting from less destruction by ¹²C(α, γ)¹⁶O. This larger carbon abundance has a substantial impact on the presupernova core structure. As previously noted (Barkat 1994; Timmes et al. 1996; Sukhbold & Woosley 2014), a major change in core structure happens when the star transitions from burning carbon in its center convectively to burning it radiatively. For single stars, this occurs near 20 ${M}_{\odot }$, or for a final helium core mass of 6.2 ${M}_{\odot }$. For the mass-losing helium stars, the transition is pushed up to 7.2 ${M}_{\odot }$. A similar shift would occur for a smaller rate for the ¹²C(α, γ)¹⁶O reaction (see Figure 24 of Sukhbold & Woosley 2014). Here we use an S-factor at 300 keV of 165 keV barns, which is within the current experimental error bar but could be slightly high (deBoer et al. 2017).

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Finally, the outer structure of the helium core, and thus the strength of its helium-burning shell, are different when the core loses mass. A set of 22 helium stars from 3.5 to 14 ${M}_{\odot }$ was evolved without mass loss. The compactness plot (not shown) in Figure 10 would be about halfway between the Sukhbold et al. (2018) set and the present study. In Table 6, a subset of these is compared with their counterparts in the mass-losing set that ended up with the same final helium core mass. The final mass of the carbon-depleted core (M_ONe) and compactness parameter, ξ_2.5, are given. For comparison, the ONe core masses and compactness parameters are given for seven equivalent mass-losing stars from the standard set. These stars began their lives as much heavier helium stars, M_init, but ended with final masses, M_fin, very close to those of the stars evolved at constant mass. The compactness parameters are systematically lower and less variable than for the constant-mass stars.

Table 6.  Compactness in Models without Mass Loss

Minit,0	MONe	ξ2.5	Minit	Mfin	MONe	ξ2.5
[${M}_{\odot }$]	[${M}_{\odot }$]	No $\dot{M}$	[${M}_{\odot }$]	[${M}_{\odot }$]	[${M}_{\odot }$]	$\dot{M}$
4.0	1.78	0.132	5.25	3.98	1.76	0.097
4.5	1.86	0.162	6.125	4.52	1.71	0.109
5	1.81	0.129	7.00	5.04	1.83	0.159
5.5	2.00	0.200	7.875	5.56	1.87	0.167
6	2.35	0.265	8.625	5.98	1.75	0.102
6.5	2.18	0.180	9.625	6.54	2.02	0.127
7	2.51	0.285	10.75	6.95	2.31	0.194

Note. The first three columns give the constant mass of the models evolved without mass loss, the location of the combined neon- and carbon-burning shells in the presupernova star, and the compactness parameter. The next four columns give similar information for the models evolved with mass loss. Here M_fin is the presupernova mass, chosen to be close to that of the models with no mass loss.

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Figure 12 shows the convective history and presupernova structure for two of the models in Table 6, both of which ended their lives with a mass very near 6.0 ${M}_{\odot }$. Neither star changed its total mass appreciably after carbon ignition. In the mass-losing 8.63 ${M}_{\odot }$ model, a strong, coupled carbon-, oxygen-, and neon-burning shell ignites at 1.75 ${M}_{\odot }$ about a year before the explosion. This burning expands the overlying material making the radius at 2.5 ${M}_{\odot }$ larger, i.e., the compactness parameter smaller. In the constant-mass star, a similar shell ignites at 2.36 ${M}_{\odot }$, but only about a week before the explosion. This burning is too late and too far out to greatly affect the radius at 2.5 ${M}_{\odot }$, and the star dies with a large compactness parameter.

As to why this difference exists, it may have to do with the modulation of the helium-burning shell, which sets the boundary condition for the core. Figure 12 shows that while the mass-losing cores initially produce larger CO cores, the helium stars evolved without mass loss develop deeper, stronger helium-burning shells in their later evolution that modulate the presupernova density structure.

Given the broad range of masses explored, a great variety of supernova light curves is possible. These include essentially all SNe I that do not have rotationally powered light curves and are Type Ia. Some models closely resemble those already in the literature and will have similar observable properties. Based upon Figure 10, it is expected that most models with final masses up to 6 ${M}_{\odot }$ will explode and leave neutron star remnants. This includes stars with initial helium core masses up to 9 ${M}_{\odot }$ (Table 4) and thus main-sequence masses up to 30 ${M}_{\odot }$. It is noteworthy that Type Ib and Ic supernovae will come from a main-sequence mass range that is typically much greater than Type IIp. For final masses below about 2.0 ${M}_{\odot }$, the core is so compact that both the explosion energy and ⁵⁶Ni production probably decline below what is needed for typical SNe Ib and Ic (Sukhbold et al. 2016). This leaves 2–6 ${M}_{\odot }$ as the likely presupernova mass for common SNe Ibc. Previous studies of models in this mass range have shown good agreement with observations (Dessart et al. 2012, 2016). The transition from Ib to Ic for progenitor masses around 5 ${M}_{\odot }$ with residual helium masses of ∼0.3 ${M}_{\odot }$ found by Dessart et al. (2015) is in reasonable accord with Table 4, which shows the CO core being uncovered for final masses above 5 ${M}_{\odot }$ for the 1.5 times standard mass-loss cases. Residual helium in the present models is ≳0.2 ${M}_{\odot }$. Pending further study, this might be supportive of the higher mass-loss rate.

The progenitor of SNe Ib iPTF13bvn was inferred to have a mass at death near 3.5 ${M}_{\odot }$ (Bersten et al. 2014; Eldridge et al. 2015). A final mass of 3.5 ${M}_{\odot }$ corresponds to present models with initial helium cores masses of 4.5–5 ${M}_{\odot }$ (Table 4). These would have come from main-sequence stars of about 20 ${M}_{\odot }$ and would have a log luminosity and temperature of 4.85 and 4.75, respectively. This is within the allowed observational range (Figure 1 of Eldridge et al. 2015).

An island of possible SNe Ib or Ic with presupernova masses near 10–12 ${M}_{\odot }$ is also noted, but these will be rarer and probably have broader, fainter light curves due to their slower expansion speed. Most of the stars with final masses between 7 and 30 ${M}_{\odot }$ will produce black holes.

For final masses above 30 ${M}_{\odot }$, and especially above 37 ${M}_{\odot }$, PPISN result. For a given final mass, the light curves should be similar to those calculated for helium stars evolved without mass loss by Woosley (2017).

Some models are sufficiently different from those already published to be worthy of immediate attention. Chief among these are the low-mass stars with extended envelopes and the stars that experience silicon flashes (see also Kleiser et al. 2018 for more realistic transport studies of similar parameterized models).

Typical of stars that have large radii as presupernovae but have not experienced a strong silicon flash is the 2.7 ${M}_{\odot }$ model (Table 3), which has a presupernova mass of only 2.21 ${M}_{\odot }$. Assuming that its envelope was not lost to the binary companion during the second phase of expansion, the presupernova star had a radius of 6.5 × 10¹² cm. Two explosions were introduced in this model using pistons at 1.29 and 1.38 ${M}_{\odot }$. The smaller piston mass was at the edge of the iron core; the larger was where the entropy per baryon was S/N_Ak = 4.0. Explosion energies of 2.6 and 4.7 × 10⁵⁰ erg, as measured by the kinetic energy of ejecta at infinity, were generated for the mass cut at 1.38 ${M}_{\odot }$, and one with explosion energy 5.1 × 10⁵⁰ erg was generated for the mass cut at 1.29. These limits on piston mass were chosen to approximate the minimum and maximum ⁵⁶Ni production. The explosions with the piston farther out produced very little ⁵⁶Ni, 0.0025 and 0.0029 ${M}_{\odot }$ for the low and high explosion energy, respectively. The deeper piston produced 0.071 ${M}_{\odot }$. This is an upper bound, since ejecting neutronized iron from deeper in would not increase ⁵⁶Ni production. More realistic explosion models usually produce less (Sukhbold et al. 2016). A version of the high nickel yield model was also calculated in which the ejected composition was moderately but artificially mixed.

Figure 13 shows the results. All explosions of this mass star exhibit an initial peak lasting about 4 days. More energetic explosions are brighter, as expected from scaling laws for SNe IIP (Sukhbold et al. 2016). A larger radius would also give a brighter initial explosion, but larger radii were not found for models that did not experience silicon flashes (Figure 5). The envelope would also be in greater danger of being lost to the companion. Effective temperatures on days 0.5, 1, 3, and 5 for the higher-energy model were 33,000, 23,000, 13,000, and 10,500 K, so this would be a very blue transient. Velocities are typically around 5000 km s⁻¹ for the less energetic model and 7000 km s⁻¹ for the more energetic one (Figure 13). Most of the mass exterior to 1.46 ${M}_{\odot }$ is helium, with about 10% by mass of carbon and oxygen from a prior helium-burning shell extending to 1.82 ${M}_{\odot }$. Matter interior to 1.46 ${M}_{\odot }$ is mostly oxygen with traces of heavier freshly synthesized elements, notably silicon, magnesium, and ⁵⁶Ni. There is also, of course, a primordial abundance of all elements characteristic of solar metallicity in the ejected helium.

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After the first week, the display, in those models that make appreciable ⁵⁶Ni, is dominated by its decay. In models without mixing, this decay produces a pronounced secondary maximum with luminosity roughly proportional to nickel mass. The effective temperature at these later times is not well determined in the KEPLER code but is estimated to be about 7000 K after the first week. With a moderate amount of mixing (helium mixed to a mass fraction of 0.3 at the bottom of the ejecta and ⁵⁶Ni exceeding 0.01 out to 1.95 ${M}_{\odot }$), the minimum between the two peaks is eroded.

Recently, De et al. (2018) reported observations of SN 2014ft, a SN Ic that they attributed to an "ultra-stripped" helium star in a binary system. This will be discussed further in Section 5.3, but the models here also bear a superficial resemblance to that event. The observed light curve is initially very blue, evolves rapidly, and has two peaks. The initial peak was brighter than 10⁴³ erg with a temperature greater than 32,000 K but declined within a day to 10,000 K and a luminosity of approximately 1 × 10⁴² erg s⁻¹. Over the next 6 days, the supernova rebrightened to over 2 × 10⁴² erg s⁻¹, while the temperature remained near 10,000 K. Velocities in the second peak were ∼10,000 km s⁻¹, and the spectrum showed lines of helium and carbon.

Compared with the 2.7 ${M}_{\odot }$ model (Figure 13), temperatures, velocities, composition, and even the bolometric light curve agree qualitatively. The initial peak in SN 2014ft was much briefer, though, and brighter than the model. The secondary peak also occurred earlier. A model with a larger radius and greater explosion energy would have had a brighter first peak, and one with a smaller ejected mass would have evolved more quickly. For the opacities, mass loss, and zoning employed here, though, a radius much greater than 100 ${R}_{\odot }$ is unlikely for models that avoid becoming SAGB stars (Figure 5). Explosion energies much greater than 5 × 10⁵⁰ erg are also not expected for stars in this mass range (Sukhbold et al. 2016). A larger explosion energy would also give greater velocities than observed. The observed initial luminosity is thus problematic, though perhaps only by a factor of 2. A smaller ejected mass is certainly possible and could be obtained by modestly altering the uncertain mass-loss rate or removing only part of the envelope in a final binary interaction. All in all, the model seems promising and worth further exploration. It is similar in properties to the one suggested by De et al. (2018) but somewhat simpler in that it is the first supernova in the binary that makes SN 2014ft, not the second (see their Figure 6), and all that has been assumed is that a main-sequence star of about 14 ${M}_{\odot }$ lost its envelope to a companion when it ignited helium burning.

Some low-mass models may have experienced explosive mass ejection due to silicon deflagration or detonation a few weeks prior to their final explosion (Section 3.1.2). Here the major uncertainty is how much silicon burns promptly (on a sonic crossing time for the dense core) to iron in the runaway. The greater the speed of the mass ejection, the farther this matter travels before the iron core finally collapses and launches a second, more powerful shock. Ejected envelopes that expand to 10¹⁴–10¹⁵ cm before core collapse give very luminous, long-lasting secondary explosions, even for supernovae with low kinetic energies. Those that eject less mass and spread to ≲10¹⁴ cm give fainter, briefer ones. If a very small mass of silicon burns, less than ∼0.1 ${M}_{\odot }$, too little mass is ejected too slowly to greatly modify the light curve of the terminal explosion. The small mass falls back or is overtaken by the terminal shock long before the supernova becomes bright.

Table 7 shows some results for specific cases. Four presupernova models based on the 2.5 ${M}_{\odot }$ progenitor each had a total mass of 2.07 ${M}_{\odot }$ at the time of their initial instability. Since a remnant of ∼1.4 ${M}_{\odot }$ is left by the explosion and additional matter is ejected by the silicon flash, the amount left to be ejected in the final explosion is small. Silicon deflagrations are parameterized by the amount of silicon that burns to iron in the runaway, an adjustable parameter in this study. Since the deflagration ignites far off-center in this model (at 0.49 ${M}_{\odot }$), and only matter external to the ignition point burns in the initial flash, the amount of silicon that can burn is limited. The most energetic model considered burned 0.294 ${M}_{\odot }$. On the other hand, about 0.1 ${M}_{\odot }$ of silicon must burn to cause a large-amplitude oscillation of the bound core and launch an appreciable shock. This determined the least energetic case considered. Typical ejection speeds were 100–800 km s⁻¹ (model 2.5D), though higher speeds were present in both (Figure 14). In model 2.5A, the ejecta had not reached terminal speed, and some would have fallen back had the supernova not exploded first. For models 2.5C and 2.5D, the entire envelope was ejected.

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Table 7.  Low-mass Supernovae

Minit	Mfin	ΔMSi	Mej	tdelay	MFe	Mpist	KESN	M(56Ni)
[${M}_{\odot }$]	[${M}_{\odot }$]	[${M}_{\odot }$]	[${M}_{\odot }$]	[day]	[${M}_{\odot }$]	[${M}_{\odot }$]	[1051 erg]	[${M}_{\odot }$]
2.5A	2.07	0.102	0.11	27	1.29	1.29	0.34	0.04
2.5B	2.07	0.124	0.25	19	1.29	1.29	0.32	0.04
2.5C	2.07	0.171	0.70	20	1.29	1.34	0.13	<0.01
2.5D	2.07	0.294	0.70	50	1.22	1.30	0.22	<0.01
2.7A	2.21	⋯	⋯	⋯	1.29	1.38	0.26	0.0027
2.7B	2.21	⋯	⋯	⋯	1.29	1.38	0.47	0.0024
2.7C	2.21	⋯	⋯	⋯	1.29	1.29	0.52	0.071
2.7Cm	2.21	⋯	⋯	⋯	1.29	1.29	0.52	0.071
3.0A	2.45	0.446	0.73	62	1.32	1.40	0.25	<0.01
3.0B	2.45	0.446	0.73	62	1.32	1.40	0.42	<0.01
3.0C	2.45	0.446	0.73	62	1.32	1.32	0.86	0.06
3.2A	2.59	0.138	0.016	15	1.36	1.36	0.97	0.10
3.2B	2.59	0.138	0.016	15	1.36	1.36	0.97	0.05
3.2Bm	2.59	0.138	0.016	15	1.36	1.36	0.97	0.05

Note. A suffix "m" indicates a model that was artificially mixed. All models except 2.7 experienced a silicon deflagration with mass ejection as indicated.

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Table 7 also shows that, usually, the more powerful the silicon flash, the longer the wait until iron core collapse. Times varied from 19 to 62 days. During this time, the ejected or partly ejected envelope expanded to a few × 10¹³ cm (model 2.5A) to 10¹⁵ cm (model 2.5D).

Following iron core collapse, terminal explosions were introduced in each model using a piston, M_pist, to generate a final kinetic energy of KE_SN (Table 7). In those cases with low-energy flashes, where radioactivity might contribute substantially to the light curve, the piston was situated as deeply as possible, at the edge of the iron core, and a moderately high explosion energy was invoked. This resulted in the synthesis of 0.05–0.1 ${M}_{\odot }$ of ⁵⁶Ni, a rather standard amount of ⁵⁶Ni for SNe Ib and Ic. For other models (2.5C, 2.5D, 3.0A, and 3.0B), the piston was placed where the entropy rose to S/N_Ak = 4, a standard value often used to describe the mass cut in core-collapse supernovae. This was well outside the iron core, where the density was lower. As a result, only a small amount of ⁵⁶Ni was synthesized (Table 7).

The impact of this piston-ejected mass with the previously ejected envelope had dramatic consequences for the light curve that were most sensitive to the mass of the previously ejected shell and its radius. For the 2.5 ${M}_{\odot }$ models (Figure 14), the time the light curve stayed near its peak value varied from days to weeks. While the approximate nature of the effective temperature is to be emphasized for a single-temperature code, these are all very blue transients. The brightest events that peaked later had the cooler temperatures, 43,000 and 23,000 K for models 2.5C and 2.5D, respectively. Still cooler temperatures might have been obtained had the ejected shell been at a larger radius.

The sharp falloff in luminosity around 25 days in models 2.5C and 2.5D occurs as the shock reaches the edge of the shell ejected by the silicon flash and the supernova becomes optically thin. In a sense, these very luminous displays are simply an extension of supernova shock breakout as the high-velocity ejecta interacts with the circumstellar medium. The photosphere remains outside of the shock until the shock exits the shell, and light diffuses ahead of it. The main source of energy is circumstellar interaction, and the radiating region piles up, in the 1D code, in a very thin shell. These light curves are thus also similar to what has been seen for circumstellar medium interaction before, e.g., in PPISN (Woosley 2017).

The faster-evolving models 2.5A and 2.5B, on the other hand, have light curves that, even by day 1, are dominated by diffusion from the shock-heated layers well inside the shell ejected in the flash. The decline rate thus reflects their large initial radius, which, even before the silicon flash, was 7.3 × 10¹² cm, and not so much the mass or energy of the previously ejected shell. A much higher energy explosion would be required for models 2.5A or 2.5B to decline more quickly. Even before the silicon flash, these models had large radii.

Two other models explored the consequences of silicon deflagration in more massive stars (Table 3). A low-energy deflagration in the 3.2 ${M}_{\odot }$ model ejected only 0.02 ${M}_{\odot }$ with energy 5 × 10⁴⁶ erg. When the iron core collapsed 15 days later, this matter, with characteristic speeds less than 1000 km s⁻¹, was still within 10¹⁴ cm. A simulated explosion with the piston located just outside the iron core at 1.36 ${M}_{\odot }$ with kinetic energy 0.97 × 10⁵¹ erg gave the light curve in Figure 15. The initial transient is again essentially shock breakout with circumstellar interaction. Matter piles up in a thin shell beneath ionized matter that is optically thick. Radiation diffuses ahead. The bright emission ceases when the shock reaches the edge of the shell. Substantial radioactivity was produced in model 3.2A (Table 7), and this powers a bright secondary peak at about 10 days. Due to the uncertainty in where to put the mass cut, the nickel mass was varied. Model 3.2B (red line in the figure) shows the effect of moderately mixing the ejecta and dividing the ⁵⁶Ni mass by 2.

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The similarity of this model to the observed properties of SN 2014ft (iPTF14gqr; De et al. 2018) is striking and better than for the 2.7 ${M}_{\odot }$ models. Indeed, its discoverers proposed a silicon flash as a possible explanation in their paper. At the time of their first observation, the supernova had cooled to 32,000 K with a luminosity of ∼2 × 10⁴³ erg s⁻¹. Here that luminosity occurs on the fading tail of a much brighter, hotter event when the age of the supernova is 1.6 days and the temperature is 26,000 K. The temperature at the second peak, 10,000 K, and its luminosity, about 2 × 10⁴² erg s⁻¹, also agree with SN 2014ft, but the maximum occurs substantially later, at about 11 days post-explosion versus 7 days in the observations. This is because of the larger mass of ejected helium and heavy elements in the present model, 1.21 ${M}_{\odot }$, as opposed to the 0.3 ${M}_{\odot }$ estimated by De et al. (2018). One could vary the parameters in order to obtain a better match, but until an actual explosion model is available, this would just duplicate what De et al. (2018) have already done.

In the 3.0 ${M}_{\odot }$ models, the effect of more energetic silicon flashes was explored. More mass is ejected to larger radii, and the final light curves are brighter. This is perhaps more natural than in the 2.5 ${M}_{\odot }$ models, since the flash occurs closer to the center of the star and has the potential of burning more silicon to iron. Explosions with even mild kinetic energies produce very bright light curves in these models (Figure 16). Varying the explosion energy and the location of the piston did not alter the light curve qualitatively, though larger explosion energies did modestly increase the peak luminosity. Because of the very strong circumstellar interaction, the production of even 0.06 ${M}_{\odot }$ of ⁵⁶Ni in model 3.0C had no discernible effect on the light curve. A broader light curve, more consistent with common superluminous SNe I (GalYam 2019), would have been produced had the matter ejected by the silicon flash expanded farther. The amount of silicon burned in the flash is already maximal, but if the mass of the ejected envelope had been smaller, it would have acquired a larger velocity and expanded farther before core collapse. The timescale for silicon shell burning and flame propagation might possibly be lengthened by a factor of 2. Models like 3.2 that burn a lot of silicon in the flash are thus worth further exploration as superluminous (SLSN-I) prototypes.

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Stars in this mass range develop SAGB-like structures with low-density helium envelopes surrounding dense cores that slowly grow as mass accretes through a stable, thin, helium-burning shell. Helium stars with initial masses below 2.5 ${M}_{\odot }$ also fail to ignite silicon burning (Table 2) and thus avoid iron core collapse. Below 1.8 ${M}_{\odot }$, even carbon burning does not ignite promptly following central helium burning. From 1.9 to 2.4 ${M}_{\odot }$, stars ignite carbon burning off-center, forming flames that eventually burn to the center of the star (Figure 2). The subsequent evolution depends upon how much low-density envelope is retained and burns though the thin shell.

If the envelope is lost to a companion when the star first expands and develops an SAGB-like structure, helium stars with initial masses below 1.9 ${M}_{\odot }$ end up as CO white dwarfs with masses up to ∼1.05 ${M}_{\odot }$ (Table 1). Stars from 1.9 to 2.4 ${M}_{\odot }$ become NeO white dwarfs but can retain up to ∼0.1 ${M}_{\odot }$ of unburned carbon on their outsides (less for higher masses). This might be important if the star ever becomes a classical nova.

If the envelope is lost only to winds and not the companion, the core continues to grow until the envelope is gone or the star explodes. Carbon eventually ignites when the core mass approaches 1.26 ${M}_{\odot }$, i.e., for all models with initial masses greater than 1.7 ${M}_{\odot }$ (Table 2). If this is the first time carbon has ignited, a flame might move to the center of the star producing an NeO white dwarf, or it may stall (Denissenkov et al. 2013; Farmer et al. 2015; Brooks et al. 2016). If carbon has already burned in the interior, it ignites again, sometimes repeatedly, as thick layers of carbon-rich ash accumulate beneath the thin helium shell. The critical mass for these carbon flashes decreases as the mass of the core grows, and eventually, for the highest masses in this range, carbon burns steadily in a thin shell.

For this wind-dominated case, the 1.6 ${M}_{\odot }$ model is the heaviest to unambiguously make a CO white dwarf. Higher mass-loss rates or late-stage mass transfer would raise this limit. In principle, a CO white dwarf as massive as 1.26 ${M}_{\odot }$ could be created in a model that had not yet ignited carbon (e.g., the 1.7 ${M}_{\odot }$ model) and lost its envelope just as the core reached 1.26 ${M}_{\odot }$. If the envelope is not lost in heavier models, the core ignites carbon at 1.26 ${M}_{\odot }$ and eventually grows to the Chandrasekhar mass, about 1.38 ${M}_{\odot }$, producing an electron-capture supernova. This explosion would transpire in an extended red giant–like structure composed entirely of helium, and these SNe I would have different properties from both Type IIP or Ib events (Section 5; Figure 14). With no mass loss except winds, this would happen for all models between 1.9 and 2.5 ${M}_{\odot }$, a broad range. In practice, the wind-dominated case may only be appropriate for the more massive stars in this range. The radii of the 2.4 and 2.5 ${M}_{\odot }$ models at carbon ignition are relatively small (Table 1) and might not readily interact with a companion.

In Table 8, it is assumed that all models below 2.4 ${M}_{\odot }$ lose their envelopes to their companions and become white dwarfs. There may be a narrow range where the radius is not large enough for this second stage of mass transfer, and electron-capture supernovae result (see also Tauris et al. 2015).

Off-center silicon ignition characterizes all models in this mass range. The 2.5 ${M}_{\odot }$ model is the lightest to experience iron core collapse. It ignites silicon 0.504 ${M}_{\odot }$ off-center (Table 3) under very degenerate circumstances. Depending upon the degeneracy and offset of the ignition, the silicon flash develops into a violent runaway with properties that cannot be accurately represented in a 1D study. Similar behavior was observed for the 3.1, 3.2, and 3.3 ${M}_{\odot }$ models, where the temperature ran away to a peak value near 6 × 10⁹ K and initiated a deflagration or detonation (Section 3.1.2).

The explosive silicon burning does not unbind the core but initiates a large-amplitude oscillation that launches a shock into the helium envelope. Depending upon how much silicon burns, a parameter in these calculations, either a small fraction of the envelope may be ejected at a few hundred km s⁻¹ or the entire envelope may be ejected with a speed of about 1000–2000 km s⁻¹. After the oscillations damp, the remaining core evolves as in the lower-mass stars. A silicon flame is kindled that burns to the center of the star. After a bit more silicon burning in a shell, the core collapses to a neutron star. Even the neutrino-powered wind from the proto-neutron star has the potential of creating a bright display when it runs into the ejected envelope in some of these models (Section 5.3; Figure 16).

Common SNe Ib and Ic should come from these stars. Presupernova masses range from 2.6 to 6 ${M}_{\odot }$ (Table 4). This is also the region where the greatest number of successful explosions might make appreciable ⁵⁶Ni. Even should helium stars below 2.5 ${M}_{\odot }$ keep their envelopes until they die, the steep density gradient at their edges (Figure 5) would preclude making much nickel. Stars in this mass range should be comparatively easy to explode (Figure 10). Thus, it is reassuring that this is also the mass range that produces explosions whose light curves and spectra agree with observations (Dessart et al. 2012, 2015, 2016). A scarcity of SNe Ib and Ic with final masses above 6 or 7 ${M}_{\odot }$ would be consistent with the lack of Type IIP progenitors with helium cores above ∼6 ${M}_{\odot }$, i.e., main-sequence masses above ∼20 ${M}_{\odot }$ (Smartt 2009). Perhaps both have a common cause: difficulty exploding more massive cores (Figure 10).

Of course, the compactness parameter, ξ_2.5 is, by itself, an overly simple representation of core structure, but it is known to correlate with other measures of explodability (e.g., Ertl et al. 2016; Sukhbold et al. 2018). Somewhat surprisingly, in addition to being smaller for presupernova masses less than 6 ${M}_{\odot }$, the compactness parameter is less chaotic for stars in binaries where the hydrogen envelope is lost than in single stars (Figure 10; Section 4.2). Pending further study, fewer low-mass black holes are expected in this range.

The small mass of the exploding star compared to its companion, which now has its original mass plus, perhaps, the hydrogen envelope of the primary, suggests that, absent strong kicks, the system will remain bound. These are thus the likely progenitor systems of massive X-ray sources that contain neutron stars.

Most of the helium stars in this range seem likely to make black holes, with a pronounced peak between 7 and 9 ${M}_{\odot }$ (Figure 10). There is also a narrow range of models with final masses around 10–12 ${M}_{\odot }$ that might explode. Given their large masses, these (neutrino-powered) explosions might produce broader, fainter light curves (Ensman & Woosley 1988), inconsistent with common SNe Ic. Because of their high mass, they may also be relatively rare. The black hole peak at 8 ${M}_{\odot }$ is derived from helium stars with initial masses near 14 ${M}_{\odot }$, which, in turn, come from main-sequence masses near 40 ${M}_{\odot }$. The 11 ${M}_{\odot }$ supernova would be derived from a star with an original helium core mass of 22 ${M}_{\odot }$ and a main-sequence mass near 55 ${M}_{\odot }$. If these models do explode, then a new potentially detectable mass gap in black hole masses would exist around 10–12 ${M}_{\odot }$. This is a less robust gap than the one above 46 ${M}_{\odot }$ (see below) and more sensitive to uncertain physics, like the rates for ¹²C(α, γ)¹⁶O and mass loss.

The new results for compactness differ substantially from those for single stars (Figure 10; Sukhbold et al. 2018) and suggest that the distribution of black hole masses will be different in mass-exchanging binaries and single stars.

The relation between helium core mass and main-sequence mass used here applies to nonrotating stars only, which is probably an unrealistic approximation. With substantial rotation, gamma-ray bursts and magnetars might come from these sorts of stars. With rotation, the main-sequence mass required to produce a given helium core mass would be reduced, especially in the case of chemically homogeneous evolution.

A new limit is derived for the maximum black hole mass implied by the pulsational pair instability. In a binary system, pulses will eject all mass external to 46 ${M}_{\odot }$ (Table 5). This is smaller than the previous limit, 52 ${M}_{\odot }$, given by Woosley (2017) because (a) the limit is smaller for helium cores that lack a hydrogen envelope to tamp the explosion (see also Woosley 2017) and (b) the CO core is larger in presupernova stars for helium stars that lose their hydrogen envelope early on. This limit is consistent with all LIGO detections thus far (Abbott et al. 2018)

Helium cores with solar metallicity and initial masses greater than 60 ${M}_{\odot }$ encounter a weak instability in the oxygen-burning shell (Table 5) that results in the low-energy ejection of a small amount of mass. The stronger instability, with ejected kinetic energies greater that 10⁴⁹ erg, begins for helium core masses greater than 75 ${M}_{\odot }$. These corresponds to main-sequence masses of roughly 130 and 160 ${M}_{\odot }$ (Equation (5)) for the weak and strong pulsational pair instabilities, respectively. If the mass-loss rate is reduced by a factor of 2, as might occur for lower metallicity, these numbers are about 70% as great (Figure 8); i.e., a main-sequence mass of only 110 ${M}_{\odot }$ could produce violent PPISN. Of course, these numbers can be reduced if the core is a product of chemically homogeneous evolution. Then, a 75 ${M}_{\odot }$ "helium star" could be produced by a rapidly rotating main-sequence star with a mass that is just slightly larger.

While the mass-loss history of the first star in the binary to fill its Roche lobe is uncertain, the evolution of the second, the mass acceptor, is even more so. If the mass exchange is conservative, the "effective" main-sequence mass for the secondary in Equations (4) and (5) is its initial mass plus the mass lost by the primary, which is the main-sequence mass minus the final mass (Equation (13)). This is an upper bound, since some mass will be lost, especially if there is a common envelope phase.

Several caveats should be noted. First, the 46 ${M}_{\odot }$ limit derived here for black holes remaining following pulsational pair-instability supernovae (Section 3.3) strictly applies only to the mass of the primary immediately after the first explosion. An uncertain amount of matter could accrete onto the black hole, especially if it experiences "hypercritical" accretion (e.g., Belczynski et al. 2002; Bethe et al. 2007) during a second common-envelope stage. O'Shaughnessy et al. (2005) estimated the fractional increase in black hole mass to be 5%–25%. When the second explosion happens, assuming the envelope has been lost, that helium star also will be subject to the 46 ${M}_{\odot }$ limit, but the original primary could have grown to over 46 ${M}_{\odot }$.

Second, in addition to increasing the mass of the secondary, accretion may also increase its spin. If the star rotates rapidly enough, it will evolve chemically homogeneously. Very massive helium stars could result, even at solar metallicity, and a second common envelope might be avoided, though the orbit would remain wide.

Finally, because the more massive stars have a constant hydrogen-burning lifetime given by the Eddington limit, about 3 million yr, and a constant helium-burning lifetime of ∼10% of that, it may not be uncommon for very massive stars born at the same time to die at nearly the same time, even if they have different masses. The delay time between supernovae could be short, and the first mass exchange might begin when both stars have already developed helium cores.

While this paper is about stars that lose their hydrogen envelopes near helium ignition, binary mass transfer is certainly capable of producing supernovae that retain some of their hydrogen. Interacting binaries are the probable origin of SNe IIb like SN 1993J. Mergers can also produce supernovae like 1987A. For a recent discussion including many light curves, see Eldridge et al. (2018). In general, a presupernova star that retains even a small hydrogen envelope in an interacting binary will much more closely resemble the outcome of single-star evolution. The small mass of the envelope will affect the optical display, but the helium core, explosion physics, remnant masses, and nucleosynthesis will be more like those of single stars of the same initial mass.