Parameterizing the Supernova Engine and Its Effect on Remnants and Basic Yields

A core-collapse supernova is produced when the core of a massive star collapses under its own weight, forming a neutron star. The gravitational potential energy released in this collapse is (2–3) × 10⁵³ erg. Extracting 1% of this energy has been the focus of core-collapse engine astronomers for the past 80 years (Burrows 2013). The current favored engine behind supernova invokes convection above the neutron star formed in the collapse. When the collapse of the core is stopped by nuclear forces and neutron degeneracy pressure, it sends a bounce shock through the star that quickly stalls. The region between the proto-neutron star and the stalled shock is convectively unstable (Herant et al. 1994). This convection is able to revive the shock, driving an explosion. An increasing number of simulations have produced explosions under this paradigm, and it has slowly gained traction and become the leading theory model for the supernova engine (Fryer & Young 2007; Takiwaki et al. 2014; Lentz et al. 2015; Melson et al. 2015; Burrows et al. 2016).

Additionally, the observational support for the convective supernova engine continues to grow (for a review, see Fryer 2016). One of the strongest pieces of evidence demonstrating that at least some supernovae are powered by this convective engine is the recent observations of the ⁴⁴Ti distribution in the Cassiopeia A remnant (Grefenstette et al. 2014, 2017). Unlike other elements observed in supernova remnants that emit only when they are shock-heated, decay lines from ⁴⁴Ti can be detected in hard X-rays, allowing astronomers to determine the full mass distribution of this isotope. Since ⁴⁴Ti is produced in the innermost ejecta, it provides an ideal probe of the engine asymmetries. NuSTAR has been able to map the distribution of ⁴⁴Ti, demonstrating the multi-modal but not bimodal structure of this isotope that not only requires engine asymmetries, but rules out the jet/magnetar engines for the supernova that produced the Cassiopeia A remnant (Grefenstette et al. 2014). With these observations, the convective engine is now the clear leading theory for supernova explosions.

Many details of this convective engine remain unknown. For example, the time it takes to drive an explosion varies with simulations. Similarly, the explosion energy is sensitive to the simulation details. The uncertainties lead to a range of predictions for the remnant masses and yields of a given progenitor. Accurate remnant masses and yield measurements may provide clues to the nature of this convective engine, but to understand these clues, we must first understand the range of results from this engine paradigm.

In this paper, we develop a broadly parameterized model to study more fully the possible solutions from the convection-enhanced supernova engine. To capture the convective engine more accurately, we implement a three-part parameterization for the energy injection: power, duration, and the extent of the energy injection region. With these models, we produce a range of explosion energies, compact remnant masses, and nucleosynthetic yields. In Section 2, we describe details of past remnant mass calculations, including aspects of the explosion that set the mass of the compact remnant. Section 3 describes our progenitors and our method to implement explosions. In Section 4, we review the results of our simulation suite, studying both the remnant mass and key yields typically observed in supernova remnants. Without constraints, a wide range of results are possible. By limiting the allowed explosion energy, we place constraints on the remnant mass and yields, and we conclude by comparing these constrained results to current remnant mass and remnant yield observations.

The supernova explosion directly determines the fate of the compact remnant. In one extreme, if an explosion is unable to launch, material accretes onto the proto-neutron star, ultimately causing it to collapse to form a black hole. Without an explosion or further ejection, the entire star accretes onto the black hole, forming a black hole mass equal to the mass of the star at collapse. Instead, if a strong explosion is produced, the material above the proto-neutron star is ejected leaving behind a neutron star with a mass set to that of the proto-neutron star. In between these two extremes in the explosion energy, a range of remnant masses can be produced.

Initial remnant mass distributions were based solely on the structure of the progenitor star (Timmes et al. 1996) guided by nucleosynthetic yield requirements. Such estimates did not include an understanding of the explosion mechanism itself. More recent remnant-mass estimates have included constraints based on the supernova engine. For most of these, the method has focused on increasing the energy deposition due to neutrinos (Fröhlich et al. 2006; Fischer et al. 2010; Ugliano et al. 2012; Ertl et al. 2016; Sukhbold et al. 2016). For example, Perego et al. (2015) tap energy from the μ and τ neutrinos, calibrating their model by fitting SN 1987A. With these calibrated engines, they can produce a "best-set" of explosion models versus progenitor mass. These models typically still deposit the energy at the neutrino gain region. In the supernova engine, convection can redistribute the energy across the entire convective engine. Here we want to probe the broader range of explosion possibilities where we include this redistribution of the energy.

Initial models considering these effects were done analytically. The energy of the convective engine model can be estimated by determining the energy stored in the convection region when pressure of this region overcomes the pressure of the infalling star. The corresponding explosion energies from this analysis are a few times 10⁵¹ erg, explaining why most supernova energies also lie in this range (even though 10⁵³ erg of gravitational potential energy is released in the collapse). With estimates based on this convective engine, Fryer & Kalogera (2001) and Fryer et al. (2012) predicted the remnant mass as a function of progenitor mass for a range of different stellar evolution models. Another approach has been to induce explosions in one-dimensional models and calculate the yields and remnant masses from these explosions. The problem with this latter technique is that the results vary wildly upon how the explosion is induced. In this work, we strive to better understand how much the results can vary within the convective paradigm and our parameterized models are designed to produce a full range of results. From this range, we can use observational constraints to limit our explosion models.

The processes that set the remnant mass can be separated into three basic phases: the core mass post-bounce, the timing of the explosion and the fallback post-explosion. As electron capture produces a runaway collapse, the core collapses. When the core approaches nuclear densities, strong nuclear forces and neutron degeneracy pressure halt the infall, producing a bounce. When the bounce shock moves outward, it leaves behind a dense core, the seed of the proto-neutron star. The bounce of the core depends upon the entropy and for rapidly spinning stars, the rotation in the core. The masses of this initial proto-neutron star are typically ∼0.9 ± 0.2 M_⊙ (Fryer et al. 2012).

After the stall of the bounce shock, the region between the dense core and the stalled shock is typically unstable to Rayleigh–Taylor instabilities. Other instabilities can also exist, most notably the standing accretion shock instability (Houck & Chevalier 1992; Blondin et al. 2003). During this phase, material from the collapsing star is transported down to the proto-neutron star and is assimilated onto this core. This phase continues until either the energy in this convective region is sufficient to drive an explosion or the proto-neutron star collapses to form a black hole. The longer it takes to explode, the larger the proto-neutron star core becomes. By modifying the onset of the energy deposition and its power, our simulations produce a range of remnant masses at the launch of the explosion. If the mass exceeds the maximum neutron star mass, the core will collapse to form a black hole with no explosion. The most massive stellar-mass black holes are formed in these failed explosions.

If the energy in the convective region is sufficent to drive an explosion, the infalling material is pushed outward. This explosion shock decelerates as it pushes outward, causing its velocity to drop below the escape velocity. This material will ultimately fall back onto the proto-neutron star (Colgate 1971; Fryer 2006). This fallback accretes onto the neutron star and is responsible for making high-mass neutron stars and low-mass black holes. We follow our explosions out to 4000 s at which time we can determine which material will fall back (based on the escape velocity) and accurately calculate the final remnant mass.

In this paper, we focus on the fate of stellar collapse as a function of the energy injection in the supernova engine. By studying a wide range of injection engines, we can study trends in the remnant mass and final supernova yields as a function of explosion energy. For this study, we use three different progenitor masses (15, 20, and 25 M_⊙) computed using the KEPLER code (Weaver et al. 1978; Woosley et al. 2002; Heger & Woosley 2010). We chose these as representative models for exploding systems where the convective engine can produce a range of results. For low-mass core-collapse systems (roughly 8–10 M_⊙), the envelope is so weakly bound (Ibeling & Heger 2013; Woosley & Heger 2015) that an explosion typically occurs quickly with little fallback (Kitaura et al. 2006). It is unlikely that there is a lot of variation in the engine for these progenitors. Likewise, for stars without extensive mass loss above 30 M_⊙ (e.g., low metallicity stars), most simulations predict that the convective engine fails to drive an explosion (Fryer 1999; Heger et al. 2003; O'Connor & Ott 2013; Sukhbold et al. 2016), although see Müller et al. (2016b).

For these calculations, a 19-species nuclear network (Weaver et al. 1978) is used at low temperatures (until the end of oxygen burning); for silicon burning and beyond, KEPLER switches to a quasi-equilibrium approach that provides an efficient and accurate means to treat silicon burning, including convection and then transitions to a nuclear statistical equilibrium network after silicon depletion. For convection, we use the Ledoux criterion and mixing length theory. For semiconvection, we use a diffusion coefficient that is 10% that of thermal diffusion, which roughly corresponds to the efficiency resulting from the formulation by Langer et al. (1983) with an α value of 0.04. We also include thermohaline mixing according to Heger et al. (2005), but the process has no major impact on the models. The models use the corrected thermal neutrino rates described in Sukhbold et al. (2017). All processes are formulated for use with a general equation of state (see the Appendix in Heger et al. 2005) and mixing is implemented as a diffusive process. For consistency with prior work, the initial composition is taken from Grevesse & Noels (1993). The models are the same as those presented in Jones et al. (2015) but evolved to the presupernova stage.

Figure 1 shows the density profiles of our three progenitor stars. Although the inner density mostly shows the extent of the collapse (the core of the 25 M_⊙ has not collapsed as deeply into its potential well), the outer densities show the more compact structure of the 25 M_⊙ star. The sharp decrease in the density of the 20 M_⊙ above 10⁸ cm reflects the entropy jump in the star at this radius. This is easier to see in the entropy profiles shown in Figure 2. At roughly 1.5 M_⊙, the entropy in the 20 M_⊙ rises sharply. This corresponds to the lower-edge of the oxygen-burning shell, which may produce the seeds to convection within the supernova engine (Müller et al. 2016b).

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Our simulations are set up to mimic the basic convective-engine paradigm with our one-dimensional models. In fast rotating models, centrifugal support can slow the collapse before the material reaches nuclear densities, causing it to bounce at a lower density (Mönchmeyer & Müller 1989; Fryer & Heger 2000) and our one-dimensional models would not capture the collapse/bounce phase. At still higher rotation speeds, disks can form around the collapsed core. In both cases, the angular momentum profiles from these collapses produce submillisecond pulsars and it is likely that such explosions are rare. Unless the core is rapidly rotating, the collapse and bounce phase can be modeled reasonably accurately in one-dimension. Our one-dimensional collapse code utilizes a Lagrangian hydrodynamics scheme coupled to gray neutrino transport with three neutrino species: electron, anti-electron, $\mu +\tau$ neutrinos (Herant et al. 1994; Fryer et al. 1999). This code includes general relativistic effects (spherically symmetric), an equation of state for dense nuclear matter combining the Swesty–Lattimer equation of state at high densities (Lattimer & Douglas Swesty 1991) and the Blinnikov equation of state at low densities (Blinnikov et al. 1996), and an 18-isotope nuclear network (Fryer et al. 1999). With this code, we follow the collapse and bounce of the core of our core-collapse progenitors.

Within the convection-enhanced neutrino-driven supernova paradigm, energy from the hot proto-neutron star and the continually accreting material drives convection. If the convection is rapid, this energy is redistributed across the entire convective region. To mimic this explosion process, we have introduced a series of parameters including the region into which the energy is deposited (representing the size of the convective region) as well as the energy deposition rate. In the simplest models, once the explosion is launched, energy deposition halts. Material can continue to accrete even after the launch of the shock. This fallback material can drive further outflows, depositing additional energy (Fryer 2009) and are seen in many multi-dimensional core-collapse calculations (e.g., Lentz et al. 2015). To include this effect, we allow a broad range of durations, including calculations that allow a time-dependent energy deposition rate. Although this allows us to include more properties of the convective engine than past studies, the reader should bear in mind that these are still one-dimensional simulations and do not include the full effects of multi-dimensional calculations.

Finally, we have included a small number of models for our most massive progenitor that have extremely late-time energy depositions. Late-time energy deposition can either be through late-time fallback or some magnetically driven engine, e.g., a magnetar.

We vary many of the parameters independently. However, in an actual engine, some parameters depend upon each other. For example, a higher-power drive will produce more vigorous convection and can produce a larger convective region. Similarly, a weak drive likely ultimately produces a smaller convective region. But other factors (like the initial seeds and rotation) can also affect the size of the convective region. Hence, for this paper, we vary these parameters independently. Likewise, a strong explosion is likely to have a shorter deposition time. Our parameter space allows for a strong explosion with a long deposition time. In this manner, we produce stronger explosions than one would expect with the convective engine. In such cases, our models do not represent the classic convective engine alone, rather mimicking magentically driven or fallback energy sources. If we limited the results to the classic convective engine, we could constrain our parameters somewhat.

4.1. Compact Remnant Masses

Ideally, for a given progenitor star, there would be a one-to-one correspondence between explosion energy and remnant mass. The convective engine depends upon the growth of the convective instabilities that, in turn, depend upon a variety of features of the progenitor, including the progenitor's rotation and the magnitude of the turbulence in the progenitor's burning layers. The asphericities in the burning layers provide the seeds for the convection in the engine. These seemingly small differences can cause large differences in the growth of the convective motions above the proto-neutron star, leading to a wide range of explosion energy for roughly similar progenitors: see, for example, Couch & Ott (2015). Therefore, it is possible that progenitors of equal mass can produce very different explosion energies.

The explosion energy is set to the kinetic energy minus the absolute value of the gravitational potential energy 4000 s into the explosion. But because the nature of the explosion can vary wildly, this may not be the case. Table 1 shows the different choices for the energy deposition and the resultant compact masses. There are multiple injection parameters that produce roughly the same explosion energy. In this manner, we can determine whether the yields only depend upon the explosion energy and progenitor, or whether the nature of the explosion can alter the yield.

Table 1.  Remnant Masses

Model	Mprog	Mbounce	Minja	tinj	Einj	Eexp	Mremnant
	(M⊙)	(M⊙)	(M⊙)	s	1051 erg	1051 erg	(M⊙)
M15aE0.34	15	1.30	0.3	0.1	3	0.34	1.94
M15aE0.54	15	1.30	0.3	0.1	4	0.54	1.91
M15aE0.82	15	1.30	0.3	0.1	5	0.82	1.88
M15aE2.47	15	1.30	0.3	0.1	9	2.47	1.52
M15aE3.63	15	1.30	0.3	0.3	10	3.63	1.51
M15aE4.79	15	1.30	0.3	0.4	20	4.79	1.50
M15bE0.74	15	1.30	0.02	0.4	3	0.74	1.73
M15bE0.92	15	1.30	0.02	0.3	4	0.92	1.75
M15bE1.37	15	1.30	0.02	0.2	5	1.37	1.80
M15bE1.43	15	1.30	0.02	0.1	6	1.43	1.75
M15bE1.48	15	1.30	0.02	0.2	7	1.48	1.79
M15bE1.69	15	1.30	0.02	0.2	10	1.69	1.52
M15bE2.63	15	1.30	0.02	0.2	20	2.63	1.53
M15bE5.08	15	1.30	0.02	0.2	40	5.08	1.50
M15bE10.7	15	1.30	0.02	0.2	80	>10.7	1.53
M15cE0.49	15	1.30	0.1	0.4	3	0.49	1.89
M15cE0.98	15	1.30	0.1	0.4	5	0.98	1.89
M15cE1.79	15	1.30	0.1	0.4	7	1.79	1.65
M15cE1.81	15	1.30	0.1	0.4	8	1.81	1.63
M15cE1.86	15	1.30	0.1	0.3	9	1.86	1.63
M15cE1.90	15	1.30	0.1	0.3	10	1.90	1.62
M15cE1.94	15	1.30	0.1	0.3	12	1.94	1.61
M15cE2.06	15	1.30	0.1	0.3	15	2.06	1.59
M15cE2.24	15	1.30	0.1	0.3	25	2.24	1.56
M15cE2.60	15	1.30	0.1	0.3	45	2.60	1.52
M15cE3.43	15	1.30	0.1	0.3	90	3.43	1.51
M20aE0.53	20	1.56	0.1	0.50	4	0.53	3.40
M20aE0.65	20	1.56	0.1	0.12	4	0.65	3.03
M20aE0.81	20	1.56	0.1	0.12	7	0.81	2.70
M20aE0.85	20	1.56	0.1	0.50	7	0.85	2.62
M20aE1.39	20	1.56	0.1	0.12	10	1.39	1.93
M20aE1.47	20	1.56	0.1	0.50	10	1.47	2.23
M20aE2.43	20	1.56	0.1	0.12	20	2.43	1.86
M20aE2.50	20	1.56	0.1	0.50	20	2.50	1.93
M20aE4.15	20	1.56	0.1	0.12	50	4.15	1.85
M20bE0.78	20	1.56	0.2	0.12	5	0.78	2.85
M20bE1.04	20	1.56	0.2	0.12	6	1.04	2.47
M20bE1.19	20	1.56	0.2	0.12	8	1.19	2.28
M20bE1.52	20	1.56	0.2	0.12	10	1.52	1.97
M20bE2.60	20	1.56	0.2	0.12	25	2.60	1.90
M20bE4.33	20	1.56	0.2	0.12	50	4.33	1.87
M20cE0.75	20	1.47	0.1	0.5	6	0.75	2.76
M20cE0.84	20	1.47	0.1	0.5	7	0.84	2.62
M20cE1.00	20	1.47	0.1	0.5	8	1.00	2.35
M20cE1.65	20	1.47	0.1	0.5	10	1.65	1.78
M20cE2.76	20	1.47	0.1	0.5	15	2.76	1.76
M20cE2.85	20	1.47	0.1	0.5	20	2.85	1.74
M20cE5.03	20	1.47	0.1	0.5	50	5.03	1.74
M20cE8.86	20	1.47	0.1	0.5	100	8.86	1.74
M25aE0.99	25	1.83	0.1	0.1	5.0	0.99	4.89
M25aE1.57	25	1.83	0.1	0.1	10	1.57	3.73
M25aE4.73	25	1.83	0.1	0.1	20	4.73	2.38
M25aE6.17	25	1.83	0.1	0.1	35	6.17	2.38
M25aE7.42	25	1.83	0.1	0.1	50	7.42	2.37
M25aE14.8	25	1.83	0.1	0.1	100	14.8	2.35
M25bE0.76	25	1.83	0.02	0.3	4.0	0.76	5.54
M25bE1.86	25	1.83	0.02	0.5	6.0	1.86	3.52
M25bE1.92	25	1.83	0.02	1	8.0	1.92	3.13
M25bE8.40	25	1.83	0.02	0.28	50.0	8.40	2.38
M25bE9.73	25	1.83	0.02	0.69	100	9.73	2.35
M25bE18.4	25	1.83	0.02	0.69	200	18.4	2.35
M25d1E3.30	25	1.83	0.02	0.7, 10–30b	25	3.30	2.35
M25d1E4.72	25	1.83	0.02	0.7, 10–30b	50	4.72	2.35
M25d1E7.08	25	1.83	0.02	0.7, 10–30b	100	7.08	2.35
M25d2E2.53	25	1.83	0.02	0.7, 100–300b	20	2.53	2.35
M25d2E2.64	25	1.83	0.02	0.7, 100–300b	35	2.64	2.35
M25d2E2.78	25	1.83	0.02	0.7, 100–300b	50	2.78	2.35
M25d2E3.07	25	1.83	0.02	0.7, 100–300b	100	3.07	1.83
M25d3E0.89	25	1.83	0.02	0.7, 1000–3000b	7	0.89	4.66
M25d3E0.92	25	1.83	0.02	0.7, 1000–3000b	8	0.92	1.84
M25d3E1.04	25	1.83	0.02	0.7, 1000–3000b	10	1.04	1.84
M25d3E1.20	25	1.83	0.02	0.7, 1000–3000b	50	1.20	1.84
M25d3E1.52	25	1.83	0.02	0.7, 1000–3000b	100	1.52	1.83

Notes.

^aM_inj is the mass of the region above the proto-neutron star where energy is injected, t_inj is the time of this injection and E_inj is the total energy injected. Much of this energy is lost and the final explosion energy is E_exp. ^bThe latter injection timescale corresponds to a fallback or magnetar drive.

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Figure 3 plots remnant mass versus explosion energy for these models. These masses are given in bayronic mass of the collapsing core. The gravitational mass of the remnant is likely to be ∼10%–15% lower. Hence, our 15 M_⊙ progenitor produces remnants with gravitational masses in the 1.2–1.7 M_⊙ range. More massive progenitors generally produce more massive remnants (Fryer 1999): for explosions between (1–2) × 10⁵¹ erg, remnant masses can be as high as 2 M_⊙ for a 20 M_⊙ progenitor and over 4 M_⊙ for a 25 M_⊙ progenitor.⁹

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Because fallback is less in stronger explosions (Fryer et al. 2012), there is a clear trend where more energetic explosions produce lower-mass remnants. Similarly, the high binding energy of more massive stars means that, for a fixed explosion energy, the more massive progenitors generally produce more massive remnants. Indeed, with our normal explosion methods, the remnant mass for our 25 M_⊙ progenitor with "standard" supernova explosion energies of $(1\mbox{--}2)\times {10}^{51}$ erg produces remnants with gravitational masses ranging from 2 to 4.25 M_⊙. For a few times 10⁵⁰ erg explosion, most of the 25 M_⊙ star remains bound, making a massive black hole.

Table 1 shows the wide range of total injection energies in our models. The naming convention of each run is "M" followed by the mass of the progenitor (15, 20, 25) with the letters a, b, or c corresponding to the series, and "E" followed by the explosion energy (kinetic and thermal minus the binding) at the end of the simulation. As energy is injected into the layers above the proto-neutron star, this region begins to emit neutrinos copiously. Hence, the injection energy can be much higher than the final explosion energy. Explosion energies above a few times 10⁵¹ erg are difficult to achieve without extremely high injection energies and it is unlikely that these high energies occur without a different engine.

Recent results from Sukhbold et al. (2016) argue that they can make <2 M_⊙ remnants with progenitor systems with initial masses above 20 M_⊙ with 10⁵¹ erg explosions. These explosions are driven by enhanced neutrino energy deposition. The neutrino deposition region is typically very close to the surface of the proto-neutron star and these models closely mimic our small injection-region models. However, our standard explosions, with energy injections less than 1 s were unable to reproduce these results. Either these progenitors are very different than ours (have much less compact cores) or our models are missing some aspect of the explosion. One solution is that late-time energy injection prevents the fallback.¹⁰ We were able to reproduce more modest remnant masses by constructing explosions with long-term energy injection. In a scenario where continued accretion through fallback or magnetar dipole radiation injects energy at late times, we can minimize the fallback and the final remnant mass. In such a scenario, we can produce gravitational masses below 1.6 M_⊙ even for <10⁵¹ erg explosions.

Our progenitors are all modeled as single stars. Wind mass loss ejects half of the mass in our 25 M_⊙ star, 7 M_⊙ of our 20 M_⊙ star, and 4.5 M_⊙ in our 15 M_⊙ star. Binary mass transfer can remove the hydrogen and even helium envelopes allowing further mass loss through winds. If this mass loss affects the core, it will alter the fate of the collapse. If it does not alter the inner ∼3 M_⊙, the explosion (and unless there is considerable fallback, the remnant mass) is not affected by the mass loss. If mass loss does alter the core structure (e.g., it occurs before helium depletion in the core) or fallback is extensive, binary mass transfer can dictate the final remnant mass.

Our predictions include a few additional uncertainties. We choose when to start driving the energy after the bounce—the timing is determined by the instability growth time. This reflects the time for the convection to develop between the proto-neutron star and the stalled shock. Altering this can change the final remnant mass by 0.1–0.2 M_⊙. In addition, after the launch of the shock, we "accrete" onto the proto-neutron star after the material exceeds a density of between 10¹⁰–10¹² g cm⁻³, arguing that above this density, neutrino cooling is rapid, allowing it to accrete quickly. For models with over a solar mass of fallback and late-time drives, this density limit can also change the mass by 0.1–0.2 M_⊙ (the mass increases with a larger density limit).

4.2. Basic Yields

The yields from a given progenitor are also sensitive to the details of the explosion. For this paper, we use the publicly available TORCH code (Timmes et al. 2000) (http://cococubed.asu.edu) to post-process our ejecta trajectories and calculate a few basic yields from the supernova explosion: oxygen, neon, magnesium, silicon, sulfur, argon, calcium, and iron peak elements using a 489 isotope network. The initial abundances are taken from the network in our stellar models, limited to 19 isotopes (mostly alpha-chain, including the 8 isotopes considered here). For these nuclear network calculations, we use the first 2000 (equally spaced, 10 ms) timesteps, following the ejecta for 20 s, well after nuclear burning is complete. Harris et al. (2017) argued that the yields do not change dramatically with better sampling and 10 ms sampling is sufficient for most yields. The Torch code subsamples these time dumps if more resolution is needed. Both the choice of the initial abundances and this time sampling will be discussed in more detail in a later paper (S. Andrews et al. 2018, in preparation). These calculations are intended to demonstrate the range in yields. The exact yields will vary with a more complete isotope distribution in the initial conditions and understanding the nucleosynthetic yield errors must also include errors in the initial composition, time sampling, and network conditions (see S. Andrews et al. 2018, in preparation).

As the supernova explosion plows through the star, the shock hitting the stellar material accelerates it while compressing and heating it. The shock-heated material then adiabatically cools as the star expands. The final yields are set by the peak temperature and density at this peak temperature as well as the timescale of the cooling. Typically, astronomers use one of two profiles, both assuming adiabatic expansion, but with different expansion evolutions (Magkotsios et al. 2010; Harris et al. 2017). Originally, an exponential decay was assumed (Fowler & Hoyle 1964; Hoyle et al. 1964):

$$\begin{eqnarray}&&T={T}_{0}{e}^{-t/\tau }\end{eqnarray} \tag{ 1 }$$

and

$$\begin{eqnarray}&&\rho ={\rho }_{0}{e}^{-t/3\tau },\end{eqnarray} \tag{ 2 }$$

where the pre-shock temperature and density are T₀ and ρ₀ and a decay time τ = 446/ρ₀^0.5. Comparisons to explosion calculations suggests an alternative, constant-velocity expansion that produces a power-law evolution (Magkotsios et al. 2010) similar to some wind calculations (Panov & Janka 2009):

$$\begin{eqnarray}&&T={T}_{0}/(2t+1)\end{eqnarray} \tag{ 3 }$$

and

$$\begin{eqnarray}&&\rho ={\rho }_{0}/{(2t+1)}^{3}.\end{eqnarray} \tag{ 4 }$$

Figure 4 shows temperature and density profiles for a set of mass points for two different explosions (strong explosions of a 15 M_⊙ and 25 M_⊙ progenitor). Although the power-law evolution is the better fit to the data, especially after the ejecta drops out of nuclear statistical equilibrium where the time evolution is most critical, neither analytic model captures all aspects of the expansion. In particular, a succession of shocks can actually cause the temperature and density to increase as the material expands and the evolution is not necessarily monotonic.

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The yields for our models are listed in Table 2. The star itself produces many of the yields studied here during its lifetime. One of the effects of the supernova explosion is to determine what material is ejected and what remains part of the compact remnant. But the shocks in the supernovae can drive nuclear burning, further altering the abundances in the star, both destroying and producing the elements in this study. For example, when the supernova shock hits the silicon layer, it drives fusion, producing iron and destroying silicon. Silicon, on the other hand, can be produced when the shock heats the oxygen shell.

Table 2.  Explosive Yields

Model	MO (M⊙)	MNe (M⊙)	MMg (M⊙)	MSi (M⊙)	MS (M⊙)	MAr (M⊙)	MCa (M⊙)	MFe (M⊙)
M15aE0.34	0.29	0.064	0.013	0.022	0.0051	0.00021	4.4 × 10−5	5.5 × 10−8
M15aE0.58	0.25	0.11	0.027	0.043	0.017	0.00081	6.4 × 10−5	2.9 × 10−6
M15aE0.82	0.25	0.11	0.041	0.073	0.040	0.0033	6.4 × 10−5	0.0010
M15aE2.47	0.25	0.098	0.057	0.083	0.045	0.0037	0.00015	0.16
M15aE3.63	0.27	0.086	0.083	0.089	0.046	0.0037	0.00032	0.29
M15aE4.79	0.27	0.0085	0.087	0.093	0.046	0.0037	0.00029	0.30
M15bE0.74	0.28	0.11	0.054	0.072	0.040	0.0033	0.00012	0.12
M15bE0.92	0.27	0.11	0.054	0.071	0.040	0.0033	0.00012	0.10
M15bE1.37	0.25	0.11	0.041	0.073	0.040	0.0034	0.00011	0.054
M15bE1.43	0.25	0.11	0.043	0.074	0.041	0.0034	0.00013	0.098
M15bE1.48	0.25	0.10	0.045	0.073	0.041	0.0034	0.00011	0.065
M15bE1.69	0.26	0.010	0.0052	0.072	0.040	0.0034	0.00039	0.27
M15bE2.63	0.26	0.0088	0.075	0.087	0.046	0.0037	0.00034	0.27
M15bE5.08	0.24	0.081	0.077	0.13	0.076	0.0069	0.00043	0.35
M15vE10.7	0.26	0.10	0.052	0.11	0.051	0.0037	0.00040	0.27
M15cE0.49	0.26	0.11	0.023	0.047	0.023	0.0016	8.0 × 10−5	6.4 × 10−5
M15cE0.98	0.25	0.11	0.034	0.054	0.030	0.0023	8.9 × 10−5	0.00040
M15cE1.79	0.27	0.023	0.013	0.084	0.047	0.0034	0.0015	0.29
M15cE1.81	0.10	0.047	0.047	0.075	0.041	0.0035	0.00021	0.19
M15cE1.86	0.25	0.10	0.048	0.075	0.041	0.0035	0.00021	0.19
M15cE1.90	0.25	0.10	0.051	0.075	0.041	0.0035	0.00024	0.19
M15cE1.94	0.25	0.10	0.051	0.075	0.041	0.0035	0.00024	0.20
M15cE2.06	0.25	0.099	0.054	0.075	0.042	0.0035	0.00027	0.22
M15cE2.24	0.25	0.094	0.059	0.075	0.042	0.0035	0.00031	0.24
M15cE2.60	0.25	0.090	0.066	0.075	0.042	0.0035	0.00034	0.27
M15cE3.43	0.25	0.089	0.079	0.078	0.042	0.0035	0.00033	0.29
M20aE0.53	0.29	0.0031	0.0011	0.00032	0.00015	3.1 × 10−5	5.7 × 10−5	0.
M20aE0.65	0.34	0.0085	0.0012	0.00037	0.00017	3.6 × 10−5	6.4 × 10−5	0.
M20aE0.81	0.40	0.041	0.0016	0.00041	0.00019	4.1 × 10−5	7.1 × 10−5	0.0
M20aE0.85	0.40	0.056	0.0016	0.00045	0.00020	4.1 × 10−5	7.3 × 10−5	0.
M20aE1.39	0.40	0.073	0.015	0.025	0.0048	8.5 × 10−5	8.4 × 10−5	3.8 × 10−7
M20aE1.47	0.33	0.0052	0.0016	0.00042	0.00020	4.6 × 10−5	7.4 × 10−5	0.
M20aE2.43	0.37	0.044	0.018	0.10	0.035	0.0042	0.00016	0.0066
M20aE2.50	0.50	0.070	0.037	0.092	0.018	0.00095	7.8 × 10−5	2.5 × 10−5
M20aE4.15	0.36	0.044	0.022	0.13	0.068	0.0072	0.0011	0.062
M20bE0.78	0.37	0.024	0.0014	0.00039	0.0018	4.3 × 10−5	7.4 × 10−5	0.
M20bE1.04	0.41	0.11	0.00030	0.00043	0.00020	4.3 × 10−5	7.4 × 10−5	0.
M20bE1.19	0.40	0.16	0.0073	0.00045	0.00021	4.4 × 10−5	7.7 × 10−5	0.
M20bE1.52	0.41	0.094	0.021	0.087	0.0017	5.5 × 10−5	8.1 × 10−5	1.9 × 10−7
M20bE2.60	0.40	0.059	0.022	0.088	0.030	0.0032	0.00013	0.000089
M20bE4.33	0.37	0.048	0.023	0.14	0.073	0.0075	0.00071	0.039
M20cE0.75	0.38	0.024	0.0014	0.00040	0.00018	4.0 × 10−5	7.0 × 10−5	0.
M20cE0.84	0.40	0.043	0.0016	0.00042	0.00019	4.1 × 10−5	7.3 × 10−5	0.
M20cE1.00	0.42	0.11	0.0029	0.00043	0.00020	4.3 × 10−5	7.4 × 10−5	0.
M20cE1.65	0.33	0.022	0.016	0.099	0.024	0.0032	0.00014	0.0070
M20cE2.76	0.31	0.018	0.017	0.13	0.050	0.0061	0.0012	0.064
M20cE2.85	0.29	0.014	0.016	0.13	0.047	0.0058	0.0012	0.085
M20cE5.03	0.30	0.014	0.019	0.13	0.071	0.0074	0.0013	0.15
M20cE8.86	0.30	0.015	0.023	0.15	0.088	0.0090	0.00056	0.21
M25bE1.86	0.68	0.28	0.018	0.00039	0.00011	5.2 × 10−5	5.7 × 10−5	0.0
M25bE1.92	0.88	0.38	0.083	0.00076	0.00020	7.1 × 10−5	7.5 × 10−5	4.0 × 10−9
M25bE8.40	1.05	0.13	0.19	0.41	0.037	0.00096	9.9 × 10−5	3.2 × 10−6
M25bE9.73	1.03	0.12	0.19	0.44	0.042	0.0011	0.00010	1.0 × 10−5
M25bE18.4	1.04	0.12	0.19	0.43	0.042	0.0011	0.00010	9.8 × 10−6
M25d1E3.30	0.99	0.11	0.17	0.37	0.061	0.0040	0.00012	0.0036
M25d1E4.72	0.99	0.11	0.17	0.37	0.061	0.0040	0.00012	0.0036
M25d1E7.08	0.99	0.11	0.17	0.37	0.061	0.0040	0.00012	0.0036
M25d2E2.53	0.99	0.11	0.17	0.37	0.061	0.0040	0.00012	0.0036
M25d2E2.64	0.99	0.11	0.17	0.37	0.061	0.0040	0.00012	0.0036
M25d2E2.78	0.99	0.11	0.17	0.37	0.061	0.0040	0.00012	0.0036
M25d2E3.07	0.84	0.090	0.17	0.41	0.072	0.0054	0.00063	0.43
M25d3E0.89	0.77	0.29	0.021	0.00049	0.00016	5.5 × 10−5	6.2 × 10−5	0.0
M25d3E0.92	0.73	0.078	0.15	0.37	0.048	0.0035	0.00055	0.35
M25d3E1.04	0.74	0.081	0.15	0.37	0.048	0.0034	0.00053	0.35
M25d3E1.20	0.73	0.078	0.15	0.37	0.048	0.0035	0.00055	0.35
M25d3E1.52	0.74	0.079	0.15	0.37	0.048	0.0034	0.00057	0.35

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Stellar yields, shock-driven nuclear burning in the supernova, and the amount of ejected material all play an important role in the nucleosynthetic yields of supernovae. To compare the different roles these three factors have on the nucleosynthetic yields, we focus on two explosions of our 25 M_⊙ progenitor. Figure 5 shows the abundance fraction of carbon, nitrogen, oxygen, silicon, and iron peak elements for model M15bE5.08 (a 15 M_⊙, 5 × 10⁵¹ erg explosion, see Table 1 for characteristics) at collapse and after the supernova shock has passed through it. Although the pre-collapse star sets the initial seeds for nuclear burning, this strong explosion alters the abundance fraction for the entire inner material in the star, producing roughly 0.5 M_⊙ of iron peak elements. In the weak explosion (model M15aE0.54: a 15 M_⊙, 0.5 × 10⁵¹ erg explosion), much of the material synthesized in the shock falls back onto the newly formed neutron star (Figure 6). The innermost ejecta are also determined by the supernova shock. In both cases, the outermost material is set by the pre-collapse progenitor.

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To further illustrate these effects, we compare the yields of a few key elements as a function of the explosion energy and progenitor mass. Figure 7 shows the abundance fraction of a few key yields: the neon, magnesium, sulfur, calcium, silicon, and iron as a function of enclosed mass for the 15 M_⊙ progenitor with 6 explosion energies ranging from 0.34 to 4.79 × 10⁵¹ erg. Again, the elements produced in the innermost part of the ejecta are most affected by explosion energy and fallback. The iron abundance profile (abundance fraction as a function of enclosed mass) for the different-energy explosions is fairly similar, but the weak explosions simply do not eject this mass. The silicon, sulfur, and calcium profiles are affected by fallback, but also by the shock strength. As the shock energy increases, it induces burning further out, producing more of these elements. Note, however, that as we increase the energy further, the effect is less and less. The 2.47, 3.63, and $4.79\times {10}^{51}$ erg explosions produce nearly the same silicon, sulfur, calcium, and iron yields. For these high energies, the shock strength is still growing in the innermost regions where these materials are produced and the temperatures are not significantly different, producing similar yields.

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Magnesium and neon demonstrate the variation of yields due to the destruction of elements as they are fused into heavier elements. The abundance of neon initially increases with explosion energy, but then decreases as it is destroyed with the highest explosion energies. The combination of destruction and production make it difficult to predict trends in the final yields. The ratio of elements can be even more complex, producing the range of abundance ratios.

Additional complexities occur in the nature of the explosion. Figure 8 shows magnesium, neon, silicon, sulfur, calcium, and iron abundances as a function of enclosed mass for two 20 M_⊙ explosions varying the injection time in the explosion and the core mass. Comparing two models with very similar injection and final energies, but different injection timescales, we see that the yields can vary dramatically. The longer injection timescale can change both the amount of fallback and the strength of the shock.

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Figure 9 shows the yields from explosions versus compact remnant mass, color coded by the progenitor zero-age main-sequence mass. Because the matter is not ejected, the yields decrease with increasing remnant mass, especially for the heavier elements. But there are noticeable deviations from this trend. First and foremost, because the shell burning layers have more mass in the larger progenitors, they can eject more mass, even when the remnant mass is larger. This is most evident in the 25 M_⊙ progenitor. For remnants less than 2.5 M_⊙, explosions of this progenitor can produce more mass of all the elements in our study. The effect of the supernova-induced nuclear burning is also evident in the range of yields for a given progenitor with the same final compact remnant mass. A strong impulse shock will produce more heavy elements than a slower, continuously driven shock. But the fallback in the continuously driven shock is much less and these heavy elements are more likely to be ejected.

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To better see the dependence of the yields on the explosion energy, Figure 10 shows the yields versus explosion energy. At low energies, the primary effect on the yield is the amount of fallback. Stronger shocks (higher energy) have less fallback as most of the material is moving well over the escape velocity. But a continously driven explosion can have very little fallback even with an explosion of modest energy. But the shock also dictates the amount of shock-induced burning. Especially in the 15 M_⊙ progenitor where remnant masses do not change much for explosion energies above 10⁵¹ erg, the trends in explosion energy are more evident: neon and oxygen are typically destroyed in stronger explosions, but many of the other elements increase slightly.

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Within the paradigm of the convective engine, we can produce a wide range of remnant masses and nucleosynthetic yields. Even so, some generic trends exist. First, even weak explosions are able to eject most of the star for the 15 M_⊙ star in this study because of its low binding energy (which is in agreement with most 15 M_⊙ stars). If an explosion occurs for such progenitors, the compact remnant is most likely going to be a neutron star. For more massive progenitors, late-time energy injection (either through fallback or magnetar-like activity) is required to make neutron star remnants with 10⁵¹ erg explosions. Without this late energy injection preventing fallback, more massive progenitors are likely to form black holes, even if a 10⁵¹ erg explosion occurs. For the 25 M_⊙ progenitor, an explosion with energy less than 2 × 10⁵¹ erg without late-time energy injection will produce a black hole. If we assume the kick mechanism is produced by asymmetries in the ejecta, these systems are likely to be the ones that form kicks in black hole systems (Willems et al. 2005; Fragos et al. 2009).

These abundances are inferred in the observations of a number of supernova remnants (e.g., Kumar et al. 2012) and abundance ratios have been used to place constraints on the supernova progenitor. Figure 11 shows the abundance ratio of a few key elements with respect to silicon: oxygen, neon, magnesium, sulfur, argon, calcium, and the iron peak. Each of our progenitors predict a range of yields based on the supernova explosion and in many cases, it will be difficult to discriminate the exact progenitor mass based on the yields. There are a few trends: more massive progenitors can produce more intermediate elements per silicon atom (oxygen, neon, magnesium) and less massive progenitors produce more iron elements versus silicon.

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With our parameterized study, we have found that the yields depend not only on the progenitor and final explosion energy, but on the nature of the explosion itself. Mass loss through winds or binary mass transfer can alter these trends, especially at high masses. Similarly, as we begin to better understand mixing and shell merger during convection (e.g., Arnett et al. 2010), the structure of the star can change considerably  (e.g., Young et al. 2001), altering its fate.

This project was funded in part under the auspices of the U.S. Department of Energy, and supported by its contract W-7405-ENG-36 to Los Alamos National Laboratory. Simulations at LANL were performed on HPC resources provided under the Institutional Computing program. This research was supported in part by the National Science Foundation under grant No. NSF PHY11-25915 and under LANL LDRD grant 20160173ER. The Joint Institute for Nuclear Astrophysics funded workshops that led to this research. The work of A.H. was supported by the Australian Research Council ARC FT120100363. The work of S.S.-H. was supported by NSERC (Natural Sciences and Engineering Research Council of Canada) and CSA (the Canadian Space Agency).