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Thick gray lines are contours of constant enclosed masses in steps of 0. The thin gray lines are mass shells in steps of 0. In all cases, the trajectories from the two codes have a common starting radius and time and the same minimum radius and time. By design, the two explosion models thus agreed almost exactly in explosion energy and piston mass. Then using the innermost zone abundances, most models were scaled up slightly until the fallback trajectory value, so that the final disagreement of iron-group synthesis was a few percent at most.

The full tabulated list of all piston parameters for all explosion calculations is available at the MPA-Garching archive see footnote 4. The shaded gray region is bounded on the bottom by the total iron produced by P-HOTB outside the "special" trajectory orange , and on the top by the total iron ejected green.

In the remainder of the paper, the baryonic remnant masses, the kinetic energies at infinity of the ejecta, and the total iron-group synthesis are based on the 1D neutrino-powered explosions using P-HOTB. Inserting the standard "central engines" described in Section 3 in the various pre-SN stars resulted in a wide variety of outcomes depending on the properties of each progenitor, especially its mass and compactness, and the choice of 87A model used for the engine's calibration Figure Generally speaking, weaker central engines like W20 gave fewer SNe than stronger engines like N Successful explosions that make neutron stars are green, the explosions that make BHs through fallback are light blue, and the failures, which make BHs, are black lines.

The calibrators are listed by the engine strength, weakest at the bottom. Models heavier than This is an interesting point that warrants elaboration. Not every model for 87A will give equivalent, or even necessarily valid, results when its central engine is inserted in other stars. SN A was a blue supergiant in a galaxy with lower metallicity than the Sun. All pre-SN models considered here, except those that lost their envelopes before exploding, are red supergiants with an initially solar composition.

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The SN A models, at least those that made blue supergiant progenitors Table 1 , also used a different value for semiconvective mixing that affected the size of the carbon—oxygen core for that mass made the core smaller. One of the models, W18, included rotation, while the present survey does not. Our calculations are 1D, not 3D. The very similar results for "explodability" for models N20, W18, W15, and S They also justify the neglect of model set W20 in the surveys of nucleosynthesis carried out in Section 6.

A much larger SN rate would be required to make even abundant elements like silicon and oxygen. Qualitatively, the outcome seems more influenced by pre-SN structure than details of the central engine, provided that that engine is sufficiently powerful to explode many stars. Models are normalized to SN A here because it was a well-studied event with precise determinations for its explosion energy and 56 Ni mass, as well as its progenitor properties.

One could take a different tack and use an even more powerful central engine than N20 in order to achieve optimal agreement with the solar abundances. That was not done here. Also given in Figure 14 is the fraction of successful SNe above a certain main-sequence mass, but below Since heavier stars either fail to explode or explode after losing their hydrogen envelopes, this would be the fraction of SNe IIp. A Salpeter IMF has been assumed. Successful explosions above 30 do not make SNe II. The distribution of successful explosions in Figure 13 is not a simply connected set. Any parameter that samples the density gradient outside the iron core will correlate with explodability.

Recently, Ertl et al.

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No known single-parameter criterion works as well. The successful explosions are noted with gray vertical lines. Though this does not work as well as the two-parameter method of Ertl et al. As expected, the successful explosions are, by whatever measure, the outcome of core collapse in stars with steep density gradients around their iron cores. As shown in Figure 1 , has a very small value for stars below All of these small models are particularly easy to explode because they are essentially degenerate cores inside of loosely bound hydrogenic envelopes, especially at the lower-mass end.

All versions of the central engine explode stars lighter than this limit. From 22 to 25 , very few or no successful explosions were found for all central engines. From 30 on up to about 60 , nothing explodes, except for the strongest engines. Eventually, however, the large mass loss appropriate for such large solar-metallicity stars removes the hydrogen envelope and whittles away at the helium core, making it once again compact and easier to explode, at least for the stronger engines.

The results are therefore sensitive to the mass-loss prescription employed. On general principles, one expects correlations to exist among the explosion energy, 56 Ni production, and compactness parameter in successful explosions. The compactness parameter is a surrogate for the density gradient outside of the iron core.

After a short time, this internal energy converts into kinetic energy and becomes nearly equal to the final kinetic energy of the SN. A shock temperature in excess of about K is required for the production of 56 Ni, so for an explosion energy of 10 51 erg, most of the matter between the final mass cut and a point located at km in the pre-SN star will end up as 56 Ni. This is provided, of course, that the km point does not move a lot closer to the origin as the explosion develops, and the final mass cut lies inside of the initial km mass coordinate.

Both assumptions are generally valid, although fallback can occasionally reduce 56 Ni synthesis to zero. One expects then, for stars of similar initial compactness and final remnant mass, a weak positive correlation between explosion energy and 56 Ni production. A greater explosion energy increases , and this larger radius encompasses a greater mass.

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This correlation can be obscured, or at least rendered "noisy" by variations in the compactness, remnant masses, and explosion energies. In particular, the compactness of pre-SN stars below 12 is very small, i. These stars are also easy to explode and have substantially lower final energies than the heavier stars. The radius that reaches K is small, and the density gradient is also steep there. Thus, as has been known for some time, stars below 12 are not prolific sources of iron. These low-mass SNe, in fact, separate rather cleanly, in theory at least, into a separate class with low energy and low 56 Ni yield see Figures 8 and 9 —and as we shall see in Section 7 , shorter, fainter light curves.

A correlation is also expected between explosion energy and compactness, but which way does it go? Stars with more compact cores low values of are easier to explode and thus explode with lower energy, but the stars with extended cores larger values of might also have lower final energy simply because they are harder to explode. The neutrinos have to do more work against infall, and the explosion may be delayed. The mantle has greater binding energy that must be subtracted, but the additional accretion might increase the neutrino luminosity and may give a larger explosion energy.

Figure 16 shows the 56 Ni mass versus compactness parameter for the model series Z9. The clustering of points around with low 56 Ni and energy is expected, as is a transition region to higher values of both. Over most of the compactness parameter range, however, the explosion energy is roughly constant, with some small variation due to the effects just mentioned. With a constant explosion energy, the 56 Ni synthesis is slightly greater for the stars with shallower density gradients, i. Amount of nickel and explosion energies resulting from use of the Z9. The top panel shows a positive correlation of 56 Ni production with compactness.

More matter is heated by the SN shock for models with high. The explosion energy is low for stars with very small compactness parameter because their thin shells are inefficient at trapping neutrino energy and there is very little luminosity from accreting matter. The results for the W18 engine are not plotted, but closely follow the N20 points plotted here.

It is also interesting to compare the correlation between 56 Ni production and the explosion energy, especially since both can potentially be measured. Two classes of events are expected—the stars above 12 and stars below A slight positive correlation of 56 Ni with explosion energy is also expected in the more massive stars. Figure 17 supports these expectations, but shows that the variation of 56 Ni production in stars above 13 is really quite small.

While it may be tempting to draw a straight line through the full data set, this obscures what is really two different sorts of behavior. It is important to note that about half of all observable SNe in the current survey have masses below 12 Table 4. Amount of 56 Ni nucleosynthesis vs.

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  • Introduction.
  • Models below an initial mass of 12 half of all SNe explode easily and produce small amounts of nickel that correlate with their explosion energy. Higher-mass models, on the other hand, produce a nearly constant 0.

    The results for the W18 engine, though not plotted here for clarity, look very similar to those of N The observations Hamuy ; Spiro et al. Our lowest 56 Ni synthesis, in a successful SN that left a neutron star remnant, was 0. Table 7 gives a substantially different value for 56 Ni synthesis in the 9.

    Because the shock-produced 56 Ni is very small in these low-mass stars, the neutrino wind contribution is non-negligible. In that calculation all iron-group and trans-iron species in the neutrino-powered wind are represented by 56 Ni if neutrino interactions lead to. Otherwise, 56 Ni is replaced by a "tracer nucleus" for neutron-rich species.

    The network employed does not therefore track the composition in detail.

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    For heavier stars that make most of the elements, this small difference is negligible, but for these very light stars it is not. It is also possible to get still smaller values for 56 Ni in heavier stars that experience appreciable fallback. We note in passing that the 56 Ni produced by electron-capture SNe is mostly made in their neutrino-powered winds. On the upper end, one might be tempted to extend this correlation of 56 Ni production and kinetic energy to still more energetic events, including gamma-ray bursts and ultraluminous SNe Kushnir , but these other events likely have other explosion mechanisms, and the paper here is focused on nonrotating, neutrino-powered models.

    A distribution of gravitational masses can then be constructed by weighting the occurrence of each of our successful models according to a Salpeter IMF. Freire , in preparation in terms of overall spread and mean value. Our maximum neutron star mass for the W18 series is 1. Use of 10 km for the neutron star radius reduces these numbers to 1. Instead, the aim here is purely to provide a visual guide to the currently known measurements.

    Freire , in preparation. Several neutron stars with masses greater than the maximum mass plotted here have been observed, and the lightest observational masses have large error bars. In the top panel, the W18 calibration is used, and in the bottom, the N20 calibration. Both distributions show some weak evidence for bimodality around 1. Results have been color-coded to show the main-sequence masses contributing to each neutron star mass bin. Note that this is not a direct comparison to the observations, as the measured values may already include accretion i. Different mass neutron stars, for the most part, come from different ranges of main-sequence mass with lower-mass progenitors producing low-mass neutron stars.

    There is some overlap, however. Remnants from stars between 15 and 18 solar masses have slightly lower masses than some below 15 solar masses.