CORE-COLLAPSE SUPERNOVAE FROM 9 TO 120 SOLAR MASSES BASED ON NEUTRINO-POWERED EXPLOSIONS

The pre-SN evolution of Sk −69° 202 and its explosion as SN 1987A have been studied extensively for almost 30 yr (e.g., Arnett et al. 1989; Utrobin et al. 2014, and references therein), yet a comprehensive, self-consistent model for the progenitor star, its ring structure, and the explosion is still lacking. The observational constraints are tighter, though, than for any other SN, so it is worthwhile to try to use SN 1987A as a calibration point.

It is known that the progenitor star was a Type B3-I blue supergiant, Sk −69° 202 (Walborn et al. 1987), with an effective temperature of 14,000–17,000 K (Humphreys & McElroy 1984; Crowther et al. 2006; Smith 2007), a luminosity of $3\mbox{--}6\times {10}^{38}$ erg s⁻¹ (Humphreys & McElroy 1984; Arnett et al. 1989), and thus a radius of $2\mbox{--}4\times {10}^{12}$ cm. Since the pre-SN luminosity is determined chiefly by the helium core mass, a core mass of 6 ± 1 ${M}_{\odot }$ has been inferred theoretically (e.g., Woosley 1988). Based on analysis of the SN light curve, this core was surrounded by about 10 ${M}_{\odot }$ of envelope in which helium and nitrogen were enriched but hydrogen was still a major constituent (Shigeyama et al. 1988; Woosley 1988). The star had been a red supergiant before dying as a blue one in order to produce the colliding wind structure seen shortly after the explosion. Spectral analysis, as well as the light curve, gives an explosion energy near $1.5\times {10}^{51}$ erg (Arnett et al. 1989; Utrobin et al. 2014), and the late-time light curve shows unambiguously that the explosion produced 0.07 ${M}_{\odot }$ of ⁵⁶Ni (Bouchet et al. 1991; Suntzeff et al. 1992). The explosion also ejected approximately 1.4 ± 0.7 ${M}_{\odot }$ of oxygen (Fransson & Kozma 2002).

2.2.1.1. Models for Sk −69° 202

Numerous attempts have been made to model the pre-SN evolution of Sk −69° 202. The final stages are complicated by the widespread belief that SN 1987A may have been a merger event (Podsiadlowski 1992; Morris & Podsiadlowski 2007), though see Smith (2007). In any case, the star that exploded was single at the time, in hydrostatic and, given the necessary timescale for the red-to-blue transition, probably in approximate thermal equilibrium. Its helium core mass is thus constrained by the pre-SN luminosity, and so single-star models are relevant.

Five principal models for SN 1987A are considered here, the ones above the blank line in Table 1, and span the space of possible helium and heavy-element core structures (Figure 2). The 3D explosion and observable properties of four of them have been explored previously (Utrobin et al. 2014) and agree well with experimental data. These models are N20, W15, W18, and W20.

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Table 1.  SN 1987A Models

Model	${M}_{{\rm{preSN}}}$/${M}_{\odot }$	${M}_{{\rm{He}}}$/${M}_{\odot }$	${M}_{{\rm{CO}}}$/${M}_{\odot }$	L/1038 erg s−1	${T}_{{\rm{eff}}}$	${\zeta }_{2.5}$	Z/${Z}_{\odot }$	Rotation
W18	16.93	7.39	3.06	8.04	18,000	0.10	1/3	Yes
N20	16.3	6	3.76	5.0	15,500	0.12	low	No
S19.8	15.85	6.09	4.49	5.65	3520	0.13	1	No
W15	15	4.15	2.02	2.0	15,300	...	1/4	No
W20	19.38	5.78	2.32	5.16	13,800	0.059	1/3	No
W16	15.37	6.55	2.57	6.35	21,700	0.11	1/3	Yes
W17	16.27	7.04	2.82	7.31	20,900	0.11	1/3	Yes
W18x	17.56	5.12	2.12	4.11	19,000	0.10	1/3	Yes
S18	14.82	5.39	3.87	4.83	3520	0.19	1	No

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Model N20 (Shigeyama & Nomoto 1990; Nomoto & Hashimoto 1988; Saio et al. 1988), a historical model, was "prepared" by artificially combining the pre-SN structure of an evolved 6 ${M}_{\odot }$ helium core from one calculation with the hydrogen envelope from another. This model was thus defined to satisfy observational constraints on helium core mass, hydrogen envelope mass, radius, and luminosity, but lacks the physical basis of a pre-SN model evolved intact from the main sequence onward. Since the helium core of model N20 was evolved using the Schwarzschild criterion for convection (Nomoto & Hashimoto 1988), its carbon–oxygen (CO) core is large compared with some of the other models (Table 1), which used restricted semiconvection in order to produce a blue supergiant progenitor. It is not clear that, had the 6 ${M}_{\odot }$ core been included in a self-consistent calculation with an envelope always in place, a blue supergiant would have resulted rather than a red one.

Model W15 (Woosley et al. 1988), another early model calculated shortly after SN 1987A occurred, was produced by evolving a single 15 ${M}_{\odot }$ star from beginning to end, but relied on opacities that are now outdated, reduced metallicity, and greatly restricted semiconvection to produce a blue supergiant. Mass loss was not included. The small ratio (less than about 0.6) of the CO core mass to helium core mass in W15 is characteristic of all models in Table 1 that produce blue supergiant progenitors. As far as we know, this is a hallmark of all solutions that use Ledoux convection or restricted semiconvection to achieve this goal. The helium core mass and luminosity of W15 are too small compared with more accurate observations of Sk −69° 202. Still, W15 is included here because it has been studied in the literature and it brackets, on the low side, the observed luminosity and helium core mass of Sk −69° 202 (Figure 2).

Model W20 (Woosley et al. 1997) used more modern opacities (Rogers & Iglesias 1992; Iglesias & Rogers 1996) and included mass loss. Its luminosity, radius, and helium core mass agree well with observations of Sk −69° 202 (Figure 2). Its pre-SN oxygen mass, 0.68 ${M}_{\odot }$, is on the small side compared with observations (Fransson & Kozma 2002), especially when allowance is made for the processing of some oxygen into heavier elements during the explosion. The envelope is also not enriched in helium. The surface mass fraction of helium is 0.26.

Model W18 is a more recent unpublished model that will be presented in greater detail later in this section. It achieves the desired goals of making a blue supergiant progenitor, producing a relatively large mass of oxygen, and displaying large enhancements in surface helium and nitrogen abundances, by invoking reduced metallicity, restricted semiconvection, and a substantial amount of rotation. Rotation makes the helium core larger and also stirs more helium into the envelope, raising the mean atomic weight there, and reducing the opacity. A larger helium core implies a larger luminosity, somewhat above the observed limit for SN 1987A (Figure 2), but the increased helium fraction in the envelope facilitates a blue solution (Saio et al. 1988; Langer 1992).

Three other models similar to W18 are also given in Table 1 and Figure 2 to illustrate some sensitivities, but were not used in the remainder of the paper. W16 and W17 are identical to W18 except for their different masses and rotation rates. The angular momentum on the ZAMS for models W16, W17, and W18 was ${J}_{{\rm{ZAMS}}}\;=\;$ (2.7, 2.9, 3.1) $\times {10}^{52}$ erg s, respectively. The pre-SN core properties for W16 and W17 are similar to W18, and one would expect similar results were they to be employed as standard central engines. W18x, on the other hand, is slightly different in input physics, yet quite different in outcome. Like W18, W18x includes rotation, but the total angular momentum on the main sequence was reduced by ∼25% to ${J}_{{\rm{ZAMS}}}\;=\;2.3\times {10}^{52}$ erg s. This small reduction in J resulted in a dramatic decrease in the helium core mass and pre-SN luminosity (Table 1; Figure 2). Another model, not shown, with an angular momentum 50% of W18, gave almost identical results to W18x, showing that W18 itself was on the verge of becoming a chemically homogeneous model (Maeder 1987; Langer 1992; Woosley & Heger 2006). Nevertheless, we chose to use model W18 itself as one of our standard models because it gave reasonable agreement with the observed properties of Sk −69° 202; had a core structure similar to the nonrotating, solar-metallicity stars in the present survey; was available a few years ago when this study began; and had been previously studied in the literature (Utrobin et al. 2014; Ertl et al. 2015).

In addition to the above models with restricted semiconvection and low metallicity, a nonrotating, solar-metallicity red supergiant progenitor was also examined. "S19.8" (Woosley et al. 2002) was not considered by Utrobin et al., but has a similar helium core structure to W18 and N20 (Table 1). Because the surface boundary pressure is small in both cases, the structure of the helium core is insensitive to whether the envelope is that of a blue or red supergiant. Despite its vintage, model S19.8 was calculated using essentially the same initial composition and stellar physics as in the present survey—including opacities, mass loss, semiconvection, and metallicity. The luminosity, helium core mass, and hydrogen envelope mass of pre-SN model S19.8 are all what one would want for SN 1987A, but the pre-SN star was a red supergiant. Also given in Table 1 is an 18 ${M}_{\odot }$ pre-SN model from the present survey. The luminosity is very similar to Sk −69° 202 and the structure similar to S19.8.

The point of these calculations was not to generate the best possible model for SN 1987A itself, but to use its well-studied properties to bracket and calibrate a central engine capable of exploding helium cores with similar characteristics. It was important to study a range of models to test the sensitivity of the results to the choice of a model for SN 1987A (see Section 4 for results).

2.2.1.2. The Pre-SN Evolution of Model W18

While the Crab SN was not studied spectroscopically, modern observations of its remnant suggest that the kinetic energy was much lower than that of SN 1987A and other common SNe from more massive stars (Arnett et al. 1989; Kasen & Woosley 2009; Yang & Chevalier 2015). The event has been successfully and repeatedly modeled as the explosion of a star near 10 ${M}_{\odot }$ (Nomoto et al. 1982; Nomoto 1987; Smith 2013; Tominaga et al. 2013). Such stars are known to have compact structures that explode easily by the neutrino-heating mechanism, even in spherical symmetry (Kitaura et al. 2006; Fischer et al. 2010; Melson et al. 2015). Use of an 87A-like central engine in stars of this sort thus overestimates the power of the SN and gives an unphysically large kinetic energy (Section 3.1.3). Systematics of the pre-SN core structure, to be discussed later, suggest that, for KEPLER models, this low-energy branch of SNe should characterize events from 9.0 (or less) to 12.0 ${M}_{\odot }$.

The fiducial model employed to represent the core physics in this low-mass range was a 9.6 ${M}_{\odot }$ zero-metallicity model (Z9.6) provided by A. Heger (2012, private communication). This star was also calculated using the KEPLER code using physics described in Heger & Woosley (2010), and, while a star of zero metallicity, it has a very similar core structure to the lightest solar-metallicity models employed here (Figure 3). The model has the merit of having been successfully exploded in 1D and multidimensional calculations (Janka et al. 2012; Melson et al. 2015), developing an explosion dynamically similar to collapsing O–Ne–Mg core progenitors (Kitaura et al. 2006; Janka et al. 2008). It also gives similar nucleosynthesis to the electron-capture SNe (S. Wanajo et al. 2015, in preparation) often mentioned as candidates for the Crab (Wanajo et al. 2011c; Jerkstrand et al. 2015), but has the advantage of being an iron-core-collapse SN similar to the ones we are currently modeling.

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The 17 models in this mass range, 9.0–13.0 ${M}_{\odot }$, in intervals of 0.25 ${M}_{\odot }$, are based on the recent study of Woosley & Heger (2015). They use the same stellar physics, especially for the propagation of convectively bounded oxygen- and silicon-burning flames, as in that work (see also Timmes et al. 1994), but differ by carrying a much larger network throughout the whole evolution. They are new models.

Stars from 9.0 to 10.25 ${M}_{\odot }$ ignite silicon in a strong flash that, for the 10.0 and 10.25 ${M}_{\odot }$ models, could be violent enough to eject the hydrogen envelope and helium shell prior to iron-core collapse. What actually happens is uncertain, though, and depends on how much silicon burns in the flash (Woosley & Heger 2015). For the present study, silicon burning in the flash was constrained to be small by using a small convective transport coefficient, and envelope ejection did not occur. For models 10.5 ${M}_{\odot }$ and heavier, all fuels ignited stably in the center of the star.

The nuclear reaction network used was complete up to Bi (Z = 93) and contained roughly 1000 isotopes during helium burning and over 1200 during oxygen and silicon burning. Pre-SN stars contained approximately 1900 isotopes. Woosley & Heger (2015), who were interested chiefly in stellar structure and not nucleosynthesis, truncated their network at Ge. The extra nuclei included here allowed a better representation of the s-process in the pre-SN star and its modification during the explosion, but did not greatly alter the structure.

Models in this mass range were calculated using the same nuclear and stellar physics as in Woosley & Heger (2007). The survey includes 151 models from 15 to 30 ${M}_{\odot }$ taken from the "S-series" in Sukhbold & Woosley (2014) and 19 additional models from 13.1 to 14.9 ${M}_{\odot }$ calculated the same way. In order to demonstrate the agreement among models from two different surveys, several additional models from 12 to 13 ${M}_{\odot }$ were calculated for comparison with models from Section 2.3.1, but were not used in the survey. The pre-SN structures of stars calculated both ways were in good agreement, having, e.g., nearly identical values of the mass enclosed by the inner 3000 km (Figure 4).

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The mesh of stellar masses was relatively fine, 0.1 ${M}_{\odot }$, as is necessary to calculate the rich variation of stellar structure in this mass range.

A total of 13 models with varying mass increments were studied in this range—31, 32, 33, 35, 40, 45, 50, 55, 60, 70, 80, 100, 120 ${M}_{\odot }$—all coming from Woosley & Heger (2007). Starting at about 40 ${M}_{\odot }$, these models lost their entire envelopes and ended their lives as Wolf–Rayet stars. The final masses are thus sensitive to the treatment of mass loss for both red supergiant stars and stripped cores. The final pre-SN masses for the 60, 70, 80, 100, and 120 ${M}_{\odot }$ stars were 7.29, 6.41, 6.37, 6.04, and 6.16 ${M}_{\odot }$, respectively. Some additional properties of all the pre-SN models are given in Table 2.

Table 2.  Progenitor Data

${M}_{{\rm{i}}}\ ({M}_{\odot })$	${M}_{{\rm{f}}}\ ({M}_{\odot })$	${M}_{\alpha }\ ({M}_{\odot })$	${M}_{{\rm{CO}}}\ ({M}_{\odot })$	log(${R}_{{\rm{f}}}$) (cm)
9.0	8.746	1.571	1.402	13.46
9.25	8.978	1.783	1.452	13.45
9.5	9.208	2.041	1.501	13.46
9.75	9.445	2.305	1.555	13.49
10.0	9.675	2.476	1.608	13.55
10.25	9.927	2.590	1.648	13.58
10.5	10.194	2.672	1.689	13.58
10.75	10.440	2.752	1.741	13.59
11.0	10.688	2.819	1.781	13.60
11.25	10.889	2.842	1.793	13.60
11.5	10.754	2.806	1.781	13.60
11.75	10.775	2.999	1.910	13.63
12.0	10.902	3.116	2.086	13.65
12.25	11.083	3.220	2.166	13.66
12.5	11.188	3.342	2.254	13.68
12.75	11.404	3.418	2.332	13.68
13.0	11.567	3.516	2.416	13.69
13.1	11.469	3.636	2.496	13.71
13.2	11.538	3.660	2.515	13.71
13.3	11.703	3.656	2.509	13.70
13.4	11.733	3.708	2.566	13.71
13.5	11.777	3.762	2.610	13.72
13.6	11.766	3.824	2.654	13.73
13.7	11.885	3.835	2.680	13.73
13.8	11.864	3.905	2.726	13.73
13.9	12.124	3.876	2.707	13.73
14.0	12.079	3.950	2.765	13.74
14.1	12.084	4.012	2.831	13.74
14.2	12.315	3.994	2.805	13.74
14.3	12.347	4.038	2.864	13.74
14.4	12.384	4.082	2.904	13.75
14.5	12.201	4.196	3.006	13.76
14.6	12.599	4.131	2.952	13.75
14.7	12.531	4.214	2.999	13.76
14.8	12.780	4.200	3.013	13.75
14.9	12.894	4.223	3.040	13.76
15.0	12.603	4.352	3.144	13.77
15.1	12.933	4.328	3.130	13.76
15.2	12.584	4.459	3.257	13.78
15.3	13.121	4.378	3.193	13.76
15.4	13.226	4.419	3.221	13.77
15.5	13.279	4.451	3.227	13.77
15.6	13.230	4.518	3.292	13.77
15.7	13.187	4.579	3.342	13.78
15.8	13.414	4.579	3.378	13.78
15.9	13.563	4.600	3.370	13.78
16.0	13.145	4.749	3.510	13.79
16.1	13.741	4.672	3.435	13.78
16.2	13.551	4.765	3.524	13.79
16.3	13.861	4.756	3.538	13.78
16.4	13.974	4.787	3.552	13.79
16.5	13.992	4.827	3.581	13.79
16.6	13.863	4.904	3.657	13.79
16.7	14.067	4.915	3.679	13.79
16.8	14.190	4.942	3.701	13.79
16.9	14.244	4.983	3.739	13.80
17.0	14.301	5.018	3.801	13.80
17.1	14.362	5.058	3.831	13.80
17.2	14.348	5.107	3.853	13.80
17.3	14.514	5.141	3.900	13.81
17.4	14.297	5.216	3.955	13.82
17.5	14.573	5.220	3.978	13.81
17.6	14.689	5.254	4.000	13.82
17.7	14.654	5.308	4.056	13.83
17.8	14.794	5.338	4.087	13.83
17.9	14.882	5.372	4.135	13.83
18.0	14.936	5.401	4.176	13.83
18.1	15.065	5.441	4.200	13.83
18.2	15.024	5.485	4.223	13.84
18.3	15.156	5.527	4.301	13.84
18.4	15.151	5.566	4.315	13.84
18.5	14.943	5.651	4.385	13.85
18.6	15.247	5.654	4.409	13.84
18.7	15.289	5.684	4.414	13.84
18.8	14.977	5.799	4.522	13.85
18.9	15.042	5.838	4.555	13.86
19.0	15.503	5.804	4.541	13.85
19.1	15.576	5.838	4.584	13.85
19.2	15.545	5.884	4.608	13.86
19.3	15.536	5.929	4.642	13.86
19.4	15.537	5.975	4.715	13.86
19.5	15.670	6.002	4.710	13.86
19.6	15.650	6.051	4.773	13.87
19.7	15.757	6.087	4.817	13.87
19.8	15.851	6.123	4.851	13.87
19.9	15.857	6.169	4.886	13.87
20.0	15.848	6.207	4.970	13.87
20.1	15.900	6.256	4.955	13.87
20.2	15.835	6.292	5.020	13.87
20.3	16.057	6.316	5.035	13.87
20.4	15.935	6.376	5.101	13.88
20.5	16.093	6.402	5.126	13.87
20.6	15.660	6.533	5.243	13.89
20.7	15.941	6.505	5.217	13.89
20.8	16.017	6.539	5.232	13.89
20.9	16.232	6.558	5.288	13.89
21.0	16.119	6.617	5.335	13.89
21.1	16.183	6.651	5.360	13.89
21.2	16.372	6.682	5.373	13.90
21.3	16.306	6.720	5.436	13.90
21.4	16.163	6.772	5.475	13.90
21.5	16.044	6.842	5.537	13.91
21.6	16.331	6.848	5.565	13.90
21.7	16.390	6.880	5.630	13.91
21.8	16.336	6.924	5.629	13.91
21.9	16.514	6.969	5.708	13.91
22.0	16.218	7.026	5.776	13.91
22.1	16.297	7.045	5.786	13.92
22.2	16.520	7.091	5.839	13.92
22.3	16.368	7.142	5.892	13.92
22.4	16.385	7.174	5.932	13.92
22.5	16.364	7.201	5.959	13.93
22.6	16.489	7.252	6.012	13.93
22.7	16.386	7.307	6.066	13.93
22.8	16.260	7.330	6.093	13.94
22.9	16.433	7.385	6.147	13.94
23.0	16.468	7.419	6.174	13.94
23.1	16.464	7.450	6.215	13.94
23.2	16.444	7.497	6.269	13.95
23.3	16.533	7.525	6.282	13.95
23.4	16.475	7.594	6.351	13.95
23.5	16.564	7.626	6.379	13.95
23.6	16.131	7.668	6.420	13.96
23.7	16.274	7.720	6.475	13.96
23.8	16.053	7.788	6.546	13.96
23.9	16.460	7.799	6.573	13.96
24.0	16.062	7.848	6.601	13.96
24.1	16.366	7.853	6.614	13.96
24.2	15.236	8.077	6.830	13.98
24.3	16.357	7.951	6.712	13.97
24.4	15.981	8.018	6.769	13.97
24.5	15.988	8.038	6.782	13.97
24.6	15.967	8.101	6.869	13.98
24.7	15.743	8.152	6.897	13.98
24.8	16.215	8.161	6.910	13.98
24.9	16.121	8.165	6.908	13.98
25.0	15.849	8.251	7.011	13.99
25.1	15.827	8.302	7.039	13.99
25.2	15.076	8.414	7.127	13.99
25.3	16.264	8.355	7.080	13.99
25.4	15.621	8.432	7.169	13.99
25.5	15.445	8.460	7.197	13.99
25.6	15.599	8.525	7.256	14.00
25.7	15.867	8.569	7.315	14.00
25.8	15.954	8.572	7.312	14.00
25.9	15.461	8.649	7.387	14.00
26.0	15.462	8.688	7.385	14.00
26.1	15.473	8.723	7.460	14.00
26.2	15.400	8.775	7.489	14.01
26.3	15.276	8.826	7.549	14.00
26.4	15.147	8.886	7.593	14.01
26.5	15.616	8.896	7.638	14.01
26.6	15.317	8.959	7.683	14.01
26.7	15.073	8.988	7.712	14.01
26.8	15.268	9.027	7.757	14.02
26.9	15.449	9.031	7.770	14.01
27.0	15.397	9.159	7.896	14.02
27.1	15.133	9.170	7.909	14.02
27.2	15.281	9.184	7.921	14.02
27.3	15.068	9.238	7.983	14.02
27.4	15.019	9.316	8.029	14.02
27.5	15.008	9.306	8.058	14.02
27.6	14.862	9.392	8.137	14.02
27.7	14.831	9.443	8.183	14.03
27.8	14.846	9.493	8.263	14.03
27.9	15.169	9.441	8.226	14.03
28.0	15.120	9.521	8.289	14.03
28.1	14.747	9.599	8.352	14.03
28.2	14.561	9.655	8.399	14.03
28.3	14.532	9.668	8.412	14.03
28.4	14.235	9.776	8.527	14.03
28.5	14.578	9.697	8.456	14.03
28.6	14.310	9.794	8.535	14.03
28.7	14.363	9.838	8.582	14.03
28.8	14.442	9.908	8.664	14.03
28.9	14.292	9.941	8.694	14.03
29.0	15.087	9.867	8.662	14.03
29.1	14.264	10.013	8.807	14.04
29.2	14.609	10.021	8.802	14.03
29.3	14.181	10.080	8.849	14.03
29.4	14.030	10.138	8.897	14.03
29.5	13.674	10.195	8.963	14.03
29.6	14.094	10.215	8.976	14.03
29.7	14.113	10.264	9.024	14.04
29.8	13.803	10.278	9.036	14.03
29.9	13.882	10.379	8.277	14.03
30.0	13.805	10.359	9.101	14.03
31	13.634	10.821	9.611	14.01
32	13.414	11.270	10.070	13.99
33	13.239	11.599	10.396	13.96
35	13.663	12.561	11.426	12.16
40	15.336	14.682	13.581	11.91
45	13.018	13.018	13.018	10.83
50	9.817	9.817	9.817	11.58
55	9.380	9.380	9.380	11.72
60	7.289	7.289	7.289	10.49
70	6.408	6.408	6.408	10.63
80	6.368	6.368	6.368	10.62
100	6.036	6.036	6.036	10.58
120	6.160	6.160	6.160	11.55

Note. ${M}_{{\rm{i}}}$ is the ZAMS mass, ${M}_{{\rm{f}}}$ is the final pre-SN mass, and ${R}_{{\rm{f}}}$ is the corresponding final radius. He and CO core masses have been measured using 20% mass fraction point X(H)$\;\leqslant \;0.2$ for ${M}_{\alpha }$ and X(He)$\;\leqslant \;0.2$ for ${M}_{{\rm{CO}}}$.

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A detailed discussion of our methodology for neutrino-driven explosions for large sets of progenitor models can be found in Ertl et al. (2015), and more technical information is also given in Ugliano et al. (2012). Here we summarize the essential characteristics.

The calculations with P-HOTB assume spherical symmetry and are 1D. They follow the evolution from the onset of stellar collapse as determined by the KEPLER code, when the collapse speed anywhere in the core first exceeds 1000 km s⁻¹, through core bounce and post-bounce accretion to either BH formation or successful explosion. The fallback of matter that does not become gravitationally unbound in the explosion is determined by running the calculation for days to weeks. PNS cooling and neutrino emission are followed for typically 15 s, at which time the shock has crossed ∼10¹⁰ cm and is outside of the CO core. By that time the power of the neutrino-driven wind, which pushes the dense ejecta shell enclosed by the expanding SN shock and the wind-termination shock, has decayed to a dynamically insignificant level. The models are then mapped to a grid extending to larger radii, and the inner boundary is moved from below the neutrinosphere to 10⁹ cm in order to ease the numerical time-step constraint.

Spherically symmetric, first-principles simulations do not yield explosions by the neutrino-driven mechanism except for low-mass progenitors with oxygen–neon–magnesium cores (Kitaura et al. 2006; Janka et al. 2008) or small iron cores (Melson et al. 2015). These stars are characterized by an extremely dilute circum-core density and small compactness values, ${\xi }_{M}$, for $M\geqslant 1.5$. Ignoring the beneficial effects of hydrodynamic instabilities in initiating the revival of the stalled bounce shock, explosions require a suitable tuning of the neutrino source to revive the shock by neutrino heating.

In order to do so, we exclude the inner 1.1 ${M}_{\odot }$ of the high-density core of the PNS at densities significantly above the neutrinospheric region from the hydrodynamical modeling, and we replace them by an inner grid boundary that is contracted to mimic the shrinking of the core. At this domain boundary, time-dependent neutrino luminosities and mean energies (Scheck et al. 2006; Arcones et al. 2007; Ugliano et al. 2012) are imposed. Outside of this boundary (between neutrino optical depths below ∼10, initially, and up to several thousand after seconds of PNS cooling), the lepton-number and energy transport of all species of neutrinos is solved using a fast, analytical, gray transport approximation (Scheck et al. 2006). The time-dependent neutrino emission from PNS accretion, which is determined by the progenitor structure of the collapsing star and by the dynamics of the shock front, is thus included in our models and ensures that corresponding feedback effects are taken into account, at least as far as they can be represented in spherical symmetry.

The excised PNS core is represented as a time-dependent neutrino source using a simple one-zone model whose behavior is determined by applying fundamental physics relations (Ugliano et al. 2012). The free parameters of this core model are calibrated by comparing the results of explosion simulations for suitable progenitor models to observations of well-studied SNe. For this purpose we consider important diagnostic parameters of the explosion, i.e., the explosion energy and the mass of ⁵⁶Ni produced, as well as important properties of the PNS as a neutrino source, i.e., the total release of neutrino energy and the timescale of this neutrino emission. For the neutrino properties, the detected neutrino signal from SN 1987A is used as a benchmark, but we also make sure that the neutrino-energy loss of our PNS models is consistent with binding energies of neutron stars based on microphysical EOSs.

Excluding the high-density core of the PNS from direct modeling can be justified by the still incomplete knowledge of the EOS in the supernuclear regime and significant uncertainties in the neutrino opacities in dense, correlated nuclear matter, which overshadow all fully self-consistent modeling. Of course, modeling SN explosions in spherical symmetry can only be an intermediate step, and ultimately three-dimensional (3D) simulations will be needed. Below we will argue that 1D simulations in the case of tuned explosions exhibit some aspects that are more similar to current 3D models than two-dimensional (2D) simulations. According to current multidimensional calculations, the onset of the explosion differs between 2D and 3D, with the former favoring earlier explosions because of differences in the behavior of turbulence in the post-shock region (e.g., Hanke et al. 2012; Couch 2013). Moreover, the imposed constraint of axisymmetry in 2D leads to artificial explosion geometries with pronounced axis features. Also the long-time phase of simultaneous accretion and ejection of neutrino-heated matter seems to differ considerably between 2D and 3D (Müller 2015), which can lead to more energetic explosions in 3D (Melson et al. 2015; Müller 2015).

In the following, we present essential information on how the PNS is described as a neutrino source in our simulations and how the calibration of this neutrino source is done by referring to observational data. Since results of nearly the same set of pre-SN models using P-HOTB have been recently discussed by Ertl et al. (2015), our description here can be terse and focused mainly on newly introduced aspects.

The central 1.1 ${M}_{\odot }$ core of the PNS resulting from the collapse of each pre-SN star is excised from the computational volume and replaced by a contracting inner grid boundary a few milliseconds after core bounce, shortly after the expanding shock has converted to a stalled accretion shock at an enclosed mass of more than 1.1 ${M}_{\odot }$. For the hydrodynamics we apply the conditions that are needed to maintain hydrostatic equilibrium at this Lagrangian grid boundary (Janka & Müller 1996). The shrinking of the cooling and deleptonizing PNS is mimicked by the contraction of the inner grid boundary, R_ib, which is described by an exponential function (Equation (1) in Arcones et al. 2007) with the parameter values given in Section 2 of Ugliano et al. (2012). In order to avoid too severe time-step restrictions that could hamper the tracking of the combined evolution of PNS and ejecta over more than 20 s, we choose the case of "standard" instead of "rapid" contraction shown in Figure 1 of Scheck et al. (2006). This may lead to an underestimation of the accretion luminosity and of the gradual hardening of the radiated neutrino spectra compared to self-consistent SN simulations, but in our approach this can be well compensated by a corresponding adjustment of the calibration of the inner core as a neutrino source. Since the accretion luminosity computed by the transport adds to core luminosity imposed at the inner grid boundary, the calibration sets constraints to the combined value of these two contributions.

As detailed in Ugliano et al. (2012), we describe the high-density PNS core of mass ${M}_{{\rm{c}}}\;=\;1.1\;{M}_{\odot }$ by a simple one-zone model under the constraints of energy conservation and the virial theorem including the effects associated with the growing pressure of the accretion layer, whose accumulation around the PNS core is followed by our hydrodynamic simulations. The one-zone core model provides a time-dependent total core-neutrino luminosity, ${L}_{\nu ,{\rm{c}}}(t)$, which determines the boundary luminosity imposed at the inner grid boundary (at R_ib) of the computational grid, suitably split between ${\nu }_{e}$, ${\bar{\nu }}_{e}$, and heavy-lepton neutrinos (see Ugliano et al. 2012; Scheck et al. 2006). Besides the luminosities of the different neutrino species, also the mean energies (or spectral temperatures) of the inflowing neutrinos are prescribed at R_ib (see Ugliano et al. 2012).

The total core-neutrino luminosity ${L}_{\nu ,{\rm{c}}}(t)$ is determined by the core mass, M_c, the core radius, ${R}_{{\rm{c}}}(t)$, the rate of contraction of this radius, ${\dot{R}}_{{\rm{c}}}(t)$, the mass of the PNS accretion mantle around the core, m_acc (taken to be the mass between the inner grid boundary and a lower density of ${10}^{10}$ g cm⁻³), and the mass accretion rate of the PNS, ${\dot{m}}_{\mathrm{acc}}$, as computed in our hydrodynamical runs. Equation (4) of Ugliano et al. (2012) for the core luminosity can be rewritten as

$$\begin{eqnarray}&&{L}_{\nu ,{\rm{c}}}(t)=\displaystyle \frac{1}{3({\rm{\Gamma }}-1)}\left[(3{\rm{\Gamma }}-4)({E}_{{\rm{g}}}+S)\displaystyle \frac{{\dot{R}}_{{\rm{c}}}}{{R}_{{\rm{c}}}}+S\displaystyle \frac{{\dot{m}}_{\mathrm{acc}}}{{m}_{\mathrm{acc}}}\right],\end{eqnarray} \tag{ 2 }$$

with the factors

$$\begin{eqnarray}&&{E}_{{\rm{g}}}+S=-\displaystyle \frac{2}{5}\;\displaystyle \frac{{{GM}}_{{\rm{c}}}}{{R}_{{\rm{c}}}}\left({M}_{{\rm{c}}}+\displaystyle \frac{5}{2}\;\zeta \;{m}_{\mathrm{acc}}\right),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&S=-\zeta \;\displaystyle \frac{{{GM}}_{{\rm{c}}}{m}_{\mathrm{acc}}}{{R}_{{\rm{c}}}},\end{eqnarray} \tag{ 4 }$$

and with the adiabatic index Γ and coefficient ζ ($={ \mathcal O }(1)$) being free parameters of the model. We note that through the S-dependent terms the PNS core luminosity depends on the structure (e.g., compactness) of the progenitor star, which determines the mass accretion rate ${\dot{m}}_{\mathrm{acc}}$ and thus the mass m_acc of the accretion mantle of the PNS. The core radius as a function of time is prescribed by

$$\begin{eqnarray}&&{R}_{{\rm{c}}}(t)={R}_{{\rm{c}},{\rm{f}}}+\displaystyle \frac{{R}_{{\rm{c}},{\rm{i}}}-{R}_{{\rm{c}},{\rm{f}}}}{{(1+t)}^{n}},\end{eqnarray} \tag{ 5 }$$

where the time t is measured in seconds. The initial PNS core radius, ${R}_{{\rm{c}},{\rm{i}}}$, is set equal to the initial radius of the inner boundary of the hydrodynamic grid, ${R}_{\mathrm{ib}}^{{\rm{i}}}$, and the final PNS core radius, ${R}_{{\rm{c}},{\rm{f}}}$, and the exponent n are also free parameters of our model. Note that R_c should be considered as a representative radius of the high-density core. Since we grossly simplify the radial structure of the PNS core by a one-zone description, the numerical value of R_c has no exact physical meaning except that is represents a rough measure of the size of the PNS core. However, neither its time evolution nor its final value need to be equal to the inner boundary radius of the hydrodynamic grid or to the radius of the 1.1 ${M}_{\odot }$ core of a self-consistently computed PNS model using a nuclear EOS. Nevertheless, with a suitably chosen (calibrated) time dependence of R_c, our model is able to reproduce basic features of the neutrino emission of cooling PNSs, and the choice of n and ${R}_{{\rm{c}},{\rm{f}}}$ can be guided by the temporal behavior of ${R}_{{\rm{c}}}(t)$ in sophisticated PNS cooling calculations (see below).

Since both of the expressions of Equations (3) and (4) are negative and ${\dot{R}}_{{\rm{c}}}\lt 0$ and ${\dot{m}}_{\mathrm{acc}}\gt 0$ for the contracting and accreting PNS, respectively, the first term in the brackets on the right-hand side of Equation (2) is positive, while the second term is negative. The former term represents the gravitational binding energy release as a consequence of the PNS core settling in the gravitational potential, including the compressional work delivered by the surrounding accretion layer. In contrast, the latter (negative) term accounts for the additional internal energy (and pressure) of the core that is needed for gravitational stability when the overlying mantle grows in mass. This second term thus reduces the energy that can be radiated in neutrinos.

The four free parameters of the PNS core model, Γ, ζ, n, and ${R}_{{\rm{c}},{\rm{f}}}$, determine the core-neutrino emission as given by Equation (2) and thus the solution to a given SN explosion. The observational cases that serve our needs for calibrating these free parameters are SN 1987A for high-mass stars and the Crab SN, SN 1054, for low-mass stars. The progenitors and explosion properties of these SNe are distinctly different and require different values for our model parameters, as we shall argue below. For both SN 1987A and SN 1054 the explosion energy and ⁵⁶Ni mass are fairly well determined and reasonably good guesses for the progenitors exist.

For the SN 1987A progenitor, Ertl et al. (2015) considered five different 15–20 ${M}_{\odot }$ models, W18, N20, S19.8, W15, and W20. The parameter values that were found to approximately reproduce the known explosion energy and ⁵⁶Ni production of SN 1987A, (1.3–$1.5)\times {10}^{51}$ erg and 0.0723–0.0772 ${M}_{\odot }$, respectively (Utrobin 2005; Utrobin & Chugai 2011; Utrobin et al. 2014), are given in Table 3. In the table, E₅₁ is the explosion energy in units of 10⁵¹ erg = 1 B = 1 bethe, and M(⁵⁶Ni + 1/2 Tr) is the mass of shock-produced ⁵⁶Ni plus one-half of the iron-group species in the neutrino-powered wind. Because Y_e in the wind is not precisely determined, the actual ⁵⁶Ni synthesis is between M(⁵⁶Ni) and M(⁵⁶Ni + Tr) (Ertl et al. 2015). In practice, Γ and for most cases ${R}_{{\rm{c}},{\rm{f}}}$ are held constant (${\rm{\Gamma }}\;=\;3$ and ${R}_{{\rm{c}},{\rm{f}}}\;=\;6.0$ km), while n and ζ are adjusted in an iterative process.

Table 3.  PNS Core Model Parameters in P-HOTB

Model	${R}_{{\rm{c}},{\rm{f}}}$ (km)	Γ	ζ	n	E51	M(56Ni+ 1/2 Tr)
Z9.6	7.0	3.0	0.65	1.55	0.16	0.0087
S19.8	6.5	3.0	0.90	2.96	1.30	0.089
W15	6.0	3.0	0.60	3.10	1.41	0.068
W18	6.0	3.0	0.65	3.06	1.25	0.074
W20	6.0	3.0	0.70	2.84	1.24	0.076
N20	6.0	3.0	0.60	3.23	1.49	0.062

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Once the parameter values of the PNS core model are determined for a given SN 1987A progenitor, that same core history is used in all the other pre-SN stars of different masses. In this way a set of SNe calibrated to a given SN 1987A progenitor is generated. This set of explosions is named according to the calibration model, so we have the "W18 series," the "N20 series," etc. In this paper we mainly focus on the sets of explosions for the W18 and N20 calibrations.

A problematic point in the results of Ugliano et al. (2012) was an overestimation of the explosion energies for lower-mass SNe. Because of the calibration of the PNS core parameters with SN 1987A models, a high core-neutrino luminosity caused a strong neutrino-driven wind for low-mass neutron stars. This led to explosion energies in excess of 1.5 B for most stars between 10 and 15 ${M}_{\odot }$. In particular, for progenitors below ∼12 ${M}_{\odot }$ such high energies are, on the one hand, not compatible with recent self-consistent 2D and 3D models of the explosion, which obtain much lower explosion energies in the range of ≲0.1 B to at most a few 0.1 B for progenitors from ∼8.8 ${M}_{\odot }$ to ∼12 ${M}_{\odot }$ with oxygen–neon–magnesium or iron cores (Kitaura et al. 2006; Janka et al. 2008, 2012; Melson et al. 2015; Müller 2015). Given the relatively dilute layers and steep density gradients that surround the cores of these progenitors, the hydrodynamic explosion models also suggest that it will be very difficult to reach higher energies by the neutrino-driven mechanism and our present knowledge of the involved physics. On the other hand, high explosion energies for low-mass progenitors are also disfavored by observations, which seem to suggest some correlation between the progenitor (or ejecta) mass and the explosion energy (e.g., Poznanski 2013; Chugai & Utrobin 2014), although the correlation appears weakened in light of a more detailed statistical analysis (Pejcha & Prieto 2015).

Ertl et al. (2015) applied a "quick fix" to cure the problem of overenergetic low-mass explosions by using a linear scaling of the compression parameter ζ of the PNS core model, coupled to the decreasing compactness ${\xi }_{1.75}$ for progenitors with $M\leqslant 13.5$ ${M}_{\odot }$. Here we elaborate on the background of this measure and introduce an alternative, similarly effective but physics-wise more transparent approach than the ζ modification.

Figure 5 displays the mass inside of a radius of 3000 km for all considered progenitor stars up to 30 ${M}_{\odot }$ at the onset of core collapse. For most appropriate comparison we evolved all models until they had contracted to the same central density of $3\times {10}^{10}$ g cm⁻³. The low-mass stars with $M\leqslant 12.0$ ${M}_{\odot }$ are clearly separated from the more massive progenitors by smaller enclosed masses ${M}_{3000}\;=\;m(r\leqslant 3000\;\mathrm{km})$. Since the Crab remnant is thought to have originated from the explosion of a star in the ∼9–10 ${M}_{\odot }$ range (e.g., Nomoto et al. 1982; Nomoto 1987; Smith 2013; Tominaga et al. 2013), we call the models with small M₃₀₀₀ "Crab-like," whereas we consider the progenitors with $M\gt 12.0$ ${M}_{\odot }$ and considerably larger values of M₃₀₀₀ as "SN 1987A-like." Our calibration models for SN 1987A possess M₃₀₀₀ cores that join with the SN 1987A-like cases (see in Figure 5 the purple star and blue diamond for the W18 and N20 progenitors, respectively, and the colored symbols for the other calibration models).

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Since the value of M₃₀₀₀ provides the mass that is located within 3000 km from the center, it correlates with the mass of the PNS that forms in the collapse of the stellar core. It is important to realize that the contraction behavior and final radius of the innermost 1.1 ${M}_{\odot }$ (which we describe here by our analytic one-zone model) depend in a microphysical model on the mass of the newly formed NS. This is shown in Figure 6, where we plot the time evolution of the 1.1 ${M}_{\odot }$ cores for two self-consistent PNS cooling simulations with all relevant microphysics (Mirizzi et al. 2015), carried out with the nuclear EOS of Lattimer & Swesty (1991). The slower and less extreme contraction in the case of the low-mass PNS compared to the high-mass PNS suggests that less neutrino energy is more gradually released in the low-mass case. This difference needs to be taken into account when modeling the explosions of the low-mass progenitor stars of our "Crab-like" sample in order to avoid the overestimation of the explosion energy in this subset of the progenitors.

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Because the simple one-zone core model does not allow us to just adopt the core-radius evolution from a self-consistent simulation, we calibrate the "Crab-like" explosions by reproducing the observationally inferred explosion energy of SN 1054 (∼0.05–0.15 B; Smith 2013; Tominaga et al. 2013; Yang & Chevalier 2015) and ⁵⁶Ni mass ($\lesssim {10}^{-2}$ ${M}_{\odot }$; Smith 2013; Tominaga et al. 2013) with a suitable progenitor model.

Table 3 lists the parameter values obtained in our Crab calibration of the Z9.6 model. We again use ${\rm{\Gamma }}\;=\;3.0$ and $\zeta \;=\;0.65$ as in the W18 series, but now employ a reduced value of n = 1.55 and a larger value of ${R}_{{\rm{c}},{\rm{f}}}\;=\;7$ km, because these values describe a slower contraction of the PNS core to a larger final radius (see Equation (5)) as found for a low-mass PNS in the self-consistent PNS cooling simulation (Figure 6).

Since model Z9.6 for the Crab progenitor is in the extreme corner of the Crab-like sample in terms of mass and M₃₀₀₀ (see the blue circle in Figure 5) and the PNS masses in this sample vary considerably (see Figures 8 and 9), we determine the values of n and ${R}_{{\rm{c}},{\rm{f}}}$ for the other Crab-like progenitors (with masses M) by interpolating as function of M₃₀₀₀ between the Crab model and the SN1987-like progenitor with the smallest value of M₃₀₀₀, which is the case at 15.2 ${M}_{\odot }$ in Figure 5, model SW14-15.2. On the SN 1987A-like side we use here the SN 1987A calibration values of n and ${R}_{{\rm{c}},{\rm{f}}}$ for the W18 series. Our interpolation functions for $y\equiv n,{R}_{{\rm{c}},{\rm{f}}}$ therefore are

$$\begin{eqnarray}&&y(x)={y}_{0}+\displaystyle \frac{{y}_{1}-{y}_{0}}{{x}_{1}-{x}_{0}}(x-{x}_{0}),\end{eqnarray} \tag{ 6 }$$

with $x\;=\;{M}_{3000}(M)$, ${x}_{0}\;=\;{M}_{3000}({\rm{Z}}9.6)$, ${x}_{1}\;=\;{M}_{3000}$(SW14-15.2), $y(x)\;=\;X[{M}_{3000}(M)]$, ${y}_{0}\;=\;X[{M}_{3000}({\rm{Z}}9.6)]$, and y₁ = X[M₃₀₀₀ (SW14-15.2)]. For all SN 1987A-like progenitors, we still apply the same values of the model parameters as obtained for the SN 1987A calibration of each series and listed in Table 3.

For the 9.0 ${M}_{\odot }$ progenitor, which has a lower mass M₃₀₀₀ than Z9.6, we use the same values of all core model parameters as for the 9.6 ${M}_{\odot }$ case. Although we employ Z9.6 as our template case for Crab and can also refer to a recent 3D explosion simulation of this model (Melson et al. 2015) for its parameter calibration, our 9.0 ${M}_{\odot }$ explosion is equally well compatible with Crab. The computed properties of this explosion (energy around 0.1 B and ⁵⁶Ni mass below 0.01 ${M}_{\odot }$; see following section) turn out to be fully consistent with the observational limits for Crab.

The basic explosion properties of all SN 1987A calibration models considered here can be found in Table 1 of Ertl et al. (2015). Figure 7 shows the evolution toward explosion for our calibration models Z9.6, W18, and N20. The shock radius and characteristic mass trajectories, as well as neutrino luminosities and mean neutrino energies, are given in the upper panels, and the different contributions to the ejecta energy are given in the lower panels.

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The expansion of the shock radius in model Z9.6 reproduces the results of multidimensional simulations (Melson et al. 2015) fairly well, and the diagnostic explosion energy also reaches ∼0.1 B after roughly 0.5 s to asymptote to a final energy of about 0.16 B after several seconds. This energy is in the upper range of estimates for SN 1054 but still compatible with them (Smith 2013; Tominaga et al. 2013; Yang & Chevalier 2015). Also the luminosities and mean energies of electron neutrinos (${\nu }_{e}$) and antineutrinos (${\bar{\nu }}_{e}$), which are the crucial agents for driving and powering the explosions, are in the ballpark of results of sophisticated transport calculations for this stellar model (see Figure 12 of Mirizzi et al. 2015). In all displayed runs the escaping luminosities (solid lines in the second panel from top) for ${\nu }_{e}$ and ${\bar{\nu }}_{e}$ are enhanced compared to the core luminosities (dashed lines) owing to the neutrino emission produced by the accretion and mantle cooling of the PNS.

While the Z9.6 progenitor explodes relatively quickly (at ${t}_{\mathrm{exp}}\;\sim \;$0.15 s after core bounce, defined as the time when the outgoing shock passes 500 km), it takes more than 0.5 s for the shocks in W18 and N20 to expand to 500 km and to accelerate outward (vertical dashed lines in Figure 7). In all three cases the explosion sets in after the mass shell ${M}_{4}\;=\;m(s\;=\;4)$, where the progenitor entropy reaches a value of 4k_B per baryon, has fallen through the shock (see also Ertl et al. 2015).

Figures 8 and 9 provide an overview of the explosion and remnant properties of all simulated models for our Z9.6, W18, and N20 series. The general features are similar to the results obtained by Ugliano et al. (2012) for a different progenitor set, and they are identical to those discussed by Ertl et al. (2015) for stars with ZAMS masses M greater than 13 ${M}_{\odot }$. In the mass range M ⩽ 13 ${M}_{\odot }$ our present work not only employs updated progenitor models, but in contrast to the (seemingly ad hoc) ζ reduction applied by Ertl et al. (2015), we also introduced the more elaborate treatment of the Crab-like models (i.e., of cases $M\leqslant 12.0$ ${M}_{\odot }$) described in Section 3.1.3 to cure the problem of overenergetic low-mass explosions in Ugliano et al. (2012).

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Figures 8 and 9 demonstrate that the ζ modification used by Ertl et al. (2015) (see Figure 3 there) leads to basically the same overall explosion results as our present Crab-like calibration for the low-mass progenitors. This can be understood by the mathematical structure of the core luminosity as given by Equation (2), where a similar effect on ${L}_{\nu ,{\rm{c}}}$ can be achieved by either a reduction of the absolute value of S (by decreasing ζ) or an increase of R_c through a larger value of ${R}_{{\rm{c}},{\rm{f}}}$. As discussed in detail by Ertl et al. (2015), the explosions of these stars set in fairly late after an extended phase of accretion, and the explosion is powered by a relatively massive and energetic neutrino-driven wind. This seems compatible with the recent result of multidimensional simulations by Müller (2015), who found a long-lasting phase of simultaneous accretion and mass ejection after the onset of the explosion for stars in the mass range between 11 and 12 ${M}_{\odot }$.

While SNe near the low-mass end of the investigated mass range tend to explode with the lowest energies and smallest production of ⁵⁶Ni, our results exhibit a moderately strong correlation of these explosion parameters with ZAMS mass. Overall, our results appear compatible with the observational data collected by Poznanski (2013), in particular in view of the weakness of the M–E_exp correlation concluded from a critical assessment of the observational analysis by Pejcha & Prieto (2015).

Although the remainder of the paper focuses on models based on the W18 and N20 calibrations, some results from P-HOTB are also given for the other three series, W15, W20, and S19.8. Among the SN 1987A models, W15 has both an unusually low compactness parameter (Table 1 in Ertl et al. 2015) and an extremely low value of M₃₀₀₀ (Figure 5), more Crab-like than SN 1987A-like. Nevertheless, we found that W15, calibrated as an engine as described in Section 3.1.3, was almost as successful a central engine as W18, N20, and S19.8. In contrast, W20 is fairly inconspicuous with respect to compactness and M₃₀₀₀, but turned out to be the clearly weakest engine with the largest number of failed explosions.

The behavior of W15 can be understood in terms of the two parameters M₄ and ${\mu }_{4}$ that determine neutrino-driven explosions, as discussed by Ertl et al. (2015). M₄ is the stellar mass enclosed by the radius where the entropy per baryon is $s\;=\;4\;{k}_{{\rm{B}}}$ and ${\mu }_{4}$, the gradient, dM/dr, just outside of this location. W15 has much lower values of both M₄ and ${\mu }_{4}$ than all the other engine models. For this reason W15 has a low accretion component of the neutrino luminosity. Exploding SN 1987A with sufficiently high energy using W15 therefore requires that the parameters of the high-density core model provide a sufficiently powerful explosion, on their own, without much assistance from accretion. With this strong core component of the neutrino emission W15 also acts as a strong engine for other progenitor stars.

W20 is weak for other reasons. Its values of M₄ and ${\mu }_{4}$ are actually quite similar to those of N20, which is a considerably stronger engine. The main reason for the different strengths of these two cases is the fact that we tuned the N20 parameters such that SN 1987A exploded with about 1.5 B (Table 3), whereas for W20 we accepted an SN 1987A explosion energy of only 1.24 B (see Table 1 in Ertl et al. 2015). The higher explosion energy for N20 was necessary to obtain enough ⁵⁶Ni from this explosion and is responsible for N20 being the more successful engine.

While perhaps not widely recognized, massive stars can make an important, though not dominant, contribution to the abundance of ⁷Li (though not ⁶Li). The ⁷Li is made mostly by the neutrino process, which spalls helium in the deeper layers to make ³He, which combines with ⁴He to make ⁷Li (after a decay). Some ³He is also made in the hydrogenic envelope. The total contribution to each is about 20% of their solar value. ⁶Li, ⁹Be, and ¹⁰B must be made elsewhere, perhaps by cosmic-ray spallation.

¹¹B, on the other hand, is abundantly produced by the neutrino process. Spallation of carbon by μ- and τ-neutrinos produces the isotope in the carbon shell. Without neutrinos, ¹¹B would be greatly underproduced. Its abundance in the Sun thus provides a useful thermometer for measuring the typical temperature of the neutrinos (though their spectrum varies significantly from thermal). The assumed temperature of the neutrinos here was 6 MeV. A slightly smaller value would have sufficed, but larger values are ruled out.

Carbon, perhaps surprisingly, is adequately produced in massive stars, but the yield is very sensitive to the treatment of mass loss and, to a lesser extent, the IMF (Figure 27). Too much carbon is made for the mass-loss rates assumed here (see also Maeder 1992), but a mass-loss rate half as large is certainly allowed and would produce carbon nicely. In the unrealistic case of no mass loss, only about one-third of carbon would be made in massive stars. Care should be taken though because folding the yields from a population of solar-metallicity stars with an IMF is very approximate compared with a full model for galactic chemical evolution. Lower-metallicity massive stars would presumably have diminished winds and would make less carbon. The isotope ¹³C is always underproduced in massive stars and is presumably made in the winds and planetary nebulae of lower-mass stars.

Nitrogen too is underproduced and needs to be mostly made elsewhere. Presumably ¹⁴N is made by the CNO cycle in lower-mass stars and ¹⁵N is made in classical novae by the hot CNO cycle.

Oxygen is the normalization point, and both ¹⁶O and ¹⁸O are copiously produced, both by helium burning. The yield of ¹⁸O is sensitive to the metallicity, but that of ¹⁶O is not directly, though it is sensitive by way of the mass loss. ¹⁷O is not made owing to its efficient destruction by ¹⁷O(p, α)¹⁴N. Perhaps it too is made in classical novae.

Fluorine is somewhat underproduced in the overall ensemble, despite having a very substantial production by the neutrino process and helium burning in low-mass stars.

The abundance of neon is uncertain in the Sun, but ²⁰Ne and ²¹Ne are well produced and ²²Ne is overproduced. The synthesis of ²²Ne depends on the initial metallicity of the star, however, since CNO is converted into ²²Ne during helium burning. Lower-metallicity stars would make less, and the abundance of ²²Ne might come down in a full study of galactic chemical evolution. The yield of ²²Ne is also sensitive to mass loss, more so than ²⁰Ne and ²¹Ne, and lower metallicity would imply lower mass loss.

All isotopes of sodium, magnesium, and aluminum are produced reasonably well by carbon and neon burning. Magnesium and aluminum are a little deficient, however, and this is the beginning of a trend that persists through the intermediate-mass elements. It could reflect a systematic underestimate of the CO core size due to the neglect of rotation or inadequate overshoot mixing, since these isotopes will be produced in greater abundances in larger CO cores, but increasing the core size might make the stars harder to blow up. Or it may indicate an incompleteness in the present approach. Many of the stars that made BHs here would have contributed to the intermediate-mass elements. Perhaps rotation or other multidimensional effects on the explosion play a role?

The major isotopes of the intermediate-mass elements from silicon through calcium are consistently co-produced in solar proportions. The total mass of these, however, is about 12% of that of ¹⁶O, which is only half the solar value of 22%. A similar underproduction was seen by Woosley & Heger (2007) (their Figure 8), so it is not solely a consequence of the new approach. Part of the difference might be picked up by SNe Ia, which, aside from being prolific sources of iron, can also produce a significant amount of intermediate-mass elements (Iwamoto et al. 1999, and Section 6.5). Still, the systematic underproduction of so many species generally attributed to massive stars is troubling. Better agreement between intermediate-mass element and oxygen productions existed in earlier studies (Timmes et al. 1995; Woosley et al. 2002), which used a larger solar abundance for ¹⁶O.

Several isotopes with particularly anomalous production in this mass range warrant mention. ⁴⁰K is greatly overproduced compared even with the abundance in the zero-age sun. The difference presumably reflects the lengthy time in which decay occurred between the last typical SN and the Sun's birth. ⁴⁴Ca is underproduced in massive stars. Presumably it is made by sub-Chandrasekhar mass models for SNe Ia (Section 6.5). ⁴⁸Ca, along with several neutron-rich iron-group nuclei like ⁵⁰Ti and ⁵⁴Cr, is presumably made in a neutron-rich nuclear statistic equilibrium as might exist in a rare variety of SNe Ia igniting at high density. ⁴⁵Sc is due to the s-process, and its underproduction is a portent of problems to come (Section 6.4.4).

The iron yields here were calibrated to be the maximum calculated in P-HOTB (Tables 7 and 8), where it was assumed that the neutrino wind makes an appreciable contribution. This was one of the agreements forced on the KEPLER recalculation (Figure 12).

As expected, even taking this upper bound, the iron group is severely underproduced in massive stars, since most of the iron in the Sun has been made by SNe Ia. In the next section, we shall consider the consequences of combining both varieties of SNe. The ratio of the mass of new iron made here as ⁵⁶Ni to new oxygen (neglecting the initial iron and oxygen in the star because it had solar metallicity) is proportional to the ratio of (P – 1) for the two species, where P here is the unnormalized production factor (Section 6). When normalized to the solar mass fractions, the fraction of solar iron made in core-collapse SNe is

$$\begin{eqnarray}&&{F}_{{\rm{Fe}}}=\displaystyle \frac{{P}_{{\rm{Fe}}}-1}{{P}_{{\rm{Ox}}}-1}.\end{eqnarray} \tag{ 12 }$$

For the series W18 ${P}_{{\rm{Fe}}}$ = 2.17 and ${P}_{{\rm{Ox}}}$ = 7.16; for the N20 series ${P}_{{\rm{Fe}}}$ = 3.09 and ${P}_{{\rm{Ox}}}$ = 8.38. Both sets of numbers include the low-mass contributions from the Z9.6 series. In both cases the implied iron production is 28%. Given the way this was calculated, using yields normalized to the maximum production in Table 4, this is probably an upper bound, though not by much.

While SNe Ia make most of the iron, it is noteworthy that massive stars do contribute appreciably to many species in the iron group. ⁵⁰V is well produced by carbon burning and ⁵⁸Fe by the s-process. Cobalt, copper, and the nickel isotopes are well produced and in the case of ⁶²Ni actually overproduced. All three elements are made mostly by the α-rich freeze-out (as ⁵⁸Ni, ⁵⁹Cu, and ${}^{\mathrm{60,61,62}}$Zn) and are thus most sensitive to the treatment of the inner boundary in KEPLER and the explosion itself of P-HOTB. The abundances of ⁵⁸Ni and ⁶²Zn, which makes ⁶²Ni, are sensitive to the neutron excess in the innermost zones. In KEPLER, appreciable electron capture occurs as these zones fall to very high density and are heated to high temperature by the shock. In P-HOTB the neutron excess in these deepest zones can be changed by the neutrinos flowing through, but this has been treated approximately. As a result, the composition of the neutrino-powered wind has not been accurately determined in either code. A higher value of Y_e in those zones would reduce the synthesis of both ⁵⁸Ni and ⁶²Ni, so their production here must be regarded as uncertain. An appreciable, though lesser, contribution to ⁶²Ni also comes from the s-process in the helium shell.

Also noteworthy is the underproduction of ⁵⁵Mn in massive stars. This species is made mostly as ⁵⁵Co in a normal (α-deficient) freeze-out from nuclear statistical equilibrium, and its synthesis is not so uncertain as that of nickel, which is made deeper in.

In contrast to previous studies (e.g., Woosley & Heger 2007), the production of species above the iron group falls off rapidly. This is problematic given that the synthesis of s-process isotopes up to about A = 90 is often attributed to massive stars (Käppeler et al. 2011). The problem is exacerbated by the fact that these are secondary elements and so are expected to be even more underproduced in stars of lower metallicity. If anything, one would like to produce an excess of isotopes in this metallicity range compared with solar values.

Figure 24 shows that production in the mass range A = 65–90 is 20%–50% of what is needed. Previous studies (Woosley et al. 2002; Woosley & Heger 2007) using the same code and stellar and nuclear physics showed a slight overproduction in the same mass range. The change is a direct consequence of fewer massive stars exploding (Brown & Woosley 2013). A larger value for the ²²Ne(α, n)²⁵Mg reaction rate might help here. For calibration, the present study used reaction rates at 3 × 10⁸ K of $2.58\times {10}^{-11}$ mole⁻¹ s⁻¹ for ²²Ne(α, n)²⁵Mg and $8.13\times {10}^{-12}$ mole⁻¹ s⁻¹ for ²²Ne ${(\alpha ,\gamma )}^{26}$ Mg, the same as used for the past 15 yr. Longland et al. (2012) recently suggested a best value of $3.36\times {10}^{-11}$ mole⁻¹ s⁻¹ (range (2.74–4.15) $\times \;{10}^{-11}$) for ²²Ne(α, n)²⁵Mg and $1.13\times {10}^{-11}$ mole⁻¹ s⁻¹ (range ($0.932\mbox{--}1.38)\times {10}^{-11}$) for ²²Ne ${(\alpha ,\gamma )}^{26}$ Mg. Use of more modern rates would thus improve the agreement slightly, but would probably not eliminate the discrepancy. An improved treatment of overshoot mixing might also make a difference (Costa et al. 2006). Future studies to further study this discrepancy are planned. For the time being, we take the deficient production of the light s-process as the strongest indicator yet that something may be awry in our present 1D modeling. The possibility that a substantial fraction of heavy-element production occurs in stars that blow up because of rotation or other multidimensional effects must be considered.

Accompanying the underproduction of light s-process elements is a notable underproduction of the heavy p-process, again compared with past results. Some have suggested that the light p-process could also be made in the neutrino-powered wind (Hoffman et al. 1996; Fröhlich et al. 2006; Wanajo et al. 2011a, 2011b).

The synthesis processes and sites of the long-lived radioactivities ²⁶Al and ⁶⁰Fe and their importance to gamma-line astronomy have been extensively discussed in the literature (Timmes et al. 1995; Limongi & Chieffi 2006a, 2006b). Here, using the W18 calibration, the IMF-averaged ejection masses for ²⁶Al and ⁶⁰Fe are, respectively, $2.80\times {10}^{-5}$ and $2.70\times {10}^{-5}$ solar masses per typical massive star. Using the N20 calibration instead gives $3.63\times {10}^{-5}$ and $3.20\times {10}^{-5}$ solar masses per star. Gamma-ray line observations (Wang et al. 2007) imply a number ratio of ${}^{60}\mathrm{Fe}$ to ${}^{26}\mathrm{Al}$ in the interstellar medium of 0.148, which implies a mass ratio of 0.34. The ratios of 0.97 and 0.88 from our two calibrations thus exceed observations almost by a factor of three. Still, this is an improvement from Woosley & Heger (2007), where the ratio, using the same nuclear physics as here, was 1.8. This factor-of-two decrease reflects both the trapping of substantial ⁶⁰Fe in stars that no longer explode and weaker typical explosion energies. Limongi & Chieffi (2006a) have discussed other uncertainties that might bring the models and observations into better agreement, including a modified IMF, mass loss, modified cross sections, and overshoot mixing.

As in Woosley & Heger (2007), we also note that alternate, equally justifiable choices for key nuclear reaction rates, especially for ²⁶Al(n, p)²⁶Mg, ²⁶Al ${(n,\alpha )}^{23}$Na, and ${}^{\mathrm{59,60}}$Fe(n, γ)^60,61Fe, might bring this ratio in our models down by an additional factor of two, hence within observational errors. This possibility will be explored in a future work. Interestingly, the production ratio for ${}^{60}\mathrm{Fe}$ to ${}^{26}\mathrm{Al}$ is sensitive to stellar mass. Averaging over just the explosions from 9 to 12 ${M}_{\odot }$, the ratio of masses ejected is close to 2.6 for both the W18 and N20 calibrations, while for heavier stars it is 0.65 (Table 6).

Table 6.  Integrated Statistics (see Section 6.4 for Descriptions; All Masses in ${M}_{\odot }$)

Cal.	$\bar{E}$ (erg)	$\bar{{M}_{b}}$	$\bar{{M}_{g}}$	Lower $\bar{{M}_{\mathrm{BH}}}$	Upper $\bar{{M}_{\mathrm{BH}}}$	$\bar{{M}_{{\rm{Ni}}},l}$	$\bar{{M}_{{\rm{Ni}}},u}$	SN%	($\gt 12$)	($\gt 20$)	($\gt 30$)
W15.0	0.68	1.55	1.40	8.40	13.3	0.040	0.049	66	47	8	2
W18.0	0.72	1.56	1.40	9.05	13.6	0.043	0.053	67	48	9	2
W20.0	0.65	1.54	1.38	7.69	13.2	0.036	0.044	55	37	3	0
N20.0	0.81	1.56	1.41	9.23	13.8	0.047	0.062	74	52	13	5

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We have also examined the potential effect of ejecting the ²⁶Al that resides in the envelopes of the stars that implode to become BHs. While not lost to winds, even weak explosions may eject these loosely bound envelopes while allowing the helium core to collapse to a singularity (Nadezhin 1980; Lovegrove & Woosley 2013). For these, we summed the total mass of ²⁶Al still in the hydrogen envelopes of the stars that collapsed and added these to the yields of the stars prior to the integration. We find only a very minor improvement to the total integrated yield of aluminum, of about 7%. This helps to reduce the underproduction of ²⁶Al, but not appreciably.

Additional information on ⁶⁰Fe production is available from the ⁶⁰Fe/⁵⁶Fe ratio observed in cosmic rays (Israel et al. 2015). There the ratio is thought to reflect chiefly the synthesis of both species in massive stars, diluted with some fraction of unprocessed material prior to acceleration. Observations of cosmic rays near the earth give a mass ratio that, when propagated back to the source, is ⁶⁰Fe/⁵⁶Fe = (0.80 ± 0.30) $\times {10}^{-4}$ by mass. Our IMF average of the mass ratio for ⁶⁰Fe/⁵⁶Fe ejected in both sets of explosions and winds is near $7\times {10}^{-4}$. Assuming a standard dilution factor of 4 to one for normal ISM to massive star ejecta (Higdon & Lingenfelter 2003; Israel et al. 2015) gives an expected ratio in the local cosmic rays of $1.4\times {10}^{-4}$, in reasonable agreement with the observations. Again, though, a reduction of a factor of about 2 in ⁶⁰Fe would be desirable. This ratio too is sensitive to the mass range of SNe that are sampled (Table 6) and is larger in lower-mass stars and somewhat smaller in heavier ones. Note that the iron in Table 6 is expressed in solar masses per massive star death, while that in Table 4 is per SN.

The correct model or models for SNe Ia are currently under much debate. In terms of nucleosynthesis, however, the models segregate into two general categories: explosions near the Chandrasekhar mass by a combination of deflagration and detonation, and prompt detonations in sub-Chandrasekhar mass white dwarfs. The requirement that any successful model makes ∼0.6 ${M}_{\odot }$ of ⁵⁶Ni in order to explain the light curve and ∼0.2 ${M}_{\odot }$ of intermediate-mass elements to explain the spectrum determines the acceptable bulk nucleosynthesis, so that the models differ only in detail. Here, as a representative example of the Chandrasekhar mass model, we use the nucleosynthesis from Nomoto's highly successful "W7" model (Iwamoto et al. 1999). For the sub-Chandrasekhar mass model we use model 10HC of Woosley & Kasen (2011). This detonation of a 1.0 ${M}_{\odot }$ carbon–oxygen white dwarf capped by 0.0445 ${M}_{\odot }$ of accreted helium produced 0.636 ${M}_{\odot }$ of ⁵⁶Ni in an explosion with final kinetic energy $1.2\times {10}^{51}$ erg. The contribution from each SN Ia model was normalized so as to produce a solar proportion of ⁵⁶Fe/¹⁶O when combined with the integrated yield of massive stars from 9 to 120 ${M}_{\odot }$.

The combined nucleosynthesis is given in Figure 28 for a sub-Chandrasekhar white dwarf and in Figure 29 for the Chandrasekhar mass model. Most of the intermediate-mass elements are unchanged, though some of these isotopes do change. Not surprisingly, the introduction of SNe Ia improves the overall fit especially for the iron group. The sub-Chandrasekhar mass model greatly improves the production of ⁴⁴Ca (made by helium detonation as ⁴⁴Ti) and seems to be required if only because of that unique contribution. The Chandrasekhar mass model, on the other hand, does a much better job of producing manganese, suggesting that this component may also be needed (see also Seitenzahl et al. 2013a; Yamaguchi et al. 2015). The Chandrasekhar mass model overproduces nickel, however. This is a known deficiency of model W7 having to do with excess electron capture at high density. It might be circumvented in more modern multidimensional models (e.g., Seitenzahl et al. 2013b) especially if the white dwarf expands appreciably during the deflagration stage prior to detonating.

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As noted previously (Section 6.4.3), our massive star models are responsible for producing about 28% of the iron in the Sun. The average SN Ia makes very nearly 0.6 ${M}_{\odot }$. That being the case, and assuming that the typical successful core-collapse SN (mostly SNe IIp in the present study) makes 0.04–0.05 ${M}_{\odot }$ of new iron (Table 4), the implied ratio of SN Ia event rates to SN IIp is 1/6 to 1/5. If core-collapse SNe made a little less iron, as seems likely, then the SN Ia rate would need to be a bit bigger. This ratio is consistent with observations for spiral galaxies similar to the Milky Way (Li et al. 2011).

We can also use these abundances, particularly the fact that each massive star above 9 ${M}_{\odot }$ produces an average of 0.57 ${M}_{\odot }$ of ${}^{16}{\rm{O}}$, to infer a rate of SNe II (assuming that stars below 9 ${M}_{\odot }$ produce little oxygen in their evolution). From the solar abundance of ¹⁶O of $6.6\times {10}^{-3}$ (Lodders 2003), we find that there must have been 0.012 massive stars per solar mass of Population I material. Assuming that 66% of these stars become SNe, as we found in our models, we find that integrated over galactic time, there must be 0.0076 core-collapse SNe (visible or invisible) per solar mass of Population I material in order to explain the present solar abundance. This seems a reasonable, albeit very approximate, number, especially if the SN rate was different in the past.