ADVANCED BURNING STAGES AND FATE OF 8–10 M_☉ STARS

The evolution of all the models in the Hertzsprung–Russell diagram and the ρ_c–T_c plane are shown in Figures 2 and 3, respectively. Carbon is ignited centrally in all but the 8.2 M_☉ model, in which it is ignited at a mass coordinate of 0.15 M_☉ away from the center and the C-burning front propagates to the center (see Figure 4(a)) in the manner of a canonical SAGB flame (Nomoto 1984).

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Following the core He-burning stage, core contraction is accompanied by an expansion of the envelope seen in Figure 4 as a deepening of the base of the convective envelope in mass. Core contraction and the related envelope expansion continue until they are halted by the ignition of carbon. After the exhaustion of carbon in the center, carbon burning proceeds in shells and from this point onward the behavior of the envelope begins to diverge across the 8–12 M_☉ mass range.

In our models with M ⩽ 8.8 M_☉, the timescale for expansion of the H-envelope is comparable to the evolutionary timescale. Owing to the higher degree of degeneracy in the core, the envelope in the 8.2 M_☉ model has time to engulf almost the entire helium shell, whereas in the 8.7, 8.75, and 8.8 M_☉ models gravo-thermal energy release induces convection in the helium shell that merges with the envelope, referred to as dredge-out (Iben et al. 1997; Siess 2007). In the 8.8 M_☉ model, as much as 0.8 M_☉ of He-rich material is mixed into the envelope. Aside from the huge increase in the amount of helium that now resides at the surface following this deep mixing event, there are many other observable quantities resulting from dredge-out. In particular, the dredge-out is accompanied by a large increase in luminosity, inducing luminosities at the pre-SN stage larger than for the 12 M_☉ model as shown in Figure 2 (see also Eldridge & Tout 2004).

In the 12 M_☉ model, the evolution of the core is accelerated by plasma neutrino-energy losses whereas the envelope expands on a thermal timescale. As a result, the convective envelope remains unaltered after carbon burning. With decreasing initial mass, the core is more degenerate and compact following carbon burning and thus contraction is slower. This provides further energy and time for the expansion of the envelope, as can be seen at log₁₀(t*/yr) ≈ 4–3 in Figures 4(a)–(e).

The 8.2 M_☉, 8.7 M_☉, and 8.75 M_☉ models develop cores with masses that fall short of the critical mass for neon ignition (see Section 3.3) following 2DUP (M_CO = 1.2670, 1.3509, and 1.3621 M_☉, respectively), developing thin (of the order of 10⁻⁵–10⁻⁴ M_☉) He shells that soon develop a recurrent thermal instability producing transient He-fueled convection zones (TPs). The 8.2 M_☉ star expels its envelope to become an ONe white dwarf (WD). It is uncertain whether the 8.7 M_☉ star would produce an ONe WD like the 8.2 M_☉ star, or whether its core would reach the critical central density for electron captures on ²⁴Mg, ρ ≈ 10^9.6 g cm⁻³, before the envelope is lost. We have modeled the 8.75 M_☉ star through the entire TP-SAGB phase (about 2.6 × 10⁶ time steps) including the URCA process and electron captures by ²⁴Mg and ²⁰Ne (see Figure 3). It becomes an EC-SN.

The outcome of these models is highly sensitive to the mass–loss prescription on the SAGB and the rate at which the core grows (Poelarends et al. 2008). We have modeled the TP-SAGB phase of the 8.7 M_☉ star for about 240 pulses, at which point ρ_c = 10^9.34 g cm⁻³. Though still far from ρ_crit(²⁴Mg + e⁻), the central density has exceeded the thresholds for both major URCA process reactions, accelerating the contraction of the core toward ρ_crit(²⁴Mg + e⁻). Due to this acceleration in contraction and comparison with literature (Nomoto 1984, 1987; Ritossa et al. 1999; Poelarends et al. 2008), the most probable outcome for the 8.7 M_☉ model is an EC-SN.

In order to maintain numerical stability in the 8.75 M_☉ model, after the depletion of ²⁴Mg at the center by electron captures, the input physics assumptions were modified. First, the effects of mass loss were excluded from the calculation and second the surface was relocated to a region where the optical depth is an order of magnitude greater than that at the photosphere (which is where the surface had previously been defined). Choosing to set the boundary at a larger optical depth is one way to deal with the inappropriate way we are simulating the final stages of these massive SAGB envelopes. In a one-dimensional code (and probably in the real star), large pulsations occur signaling an increasing instability of the envelope which may lead to enhanced mass loss or even ejection phases, such as the super-wind. These issues have been alluded to recently by Lau et al. (2012). Choosing the photosphere to be at a larger optical depth indeed lets the star be hotter and smaller, and the mass loss calculated from the stellar parameters, if it were still included, will not be the same as for the default photosphere parameters. Through this treatment, the details of the envelope evolution are increasingly inaccurate from this point. When these changes were made, the remaining envelope mass was 4.48 M_☉ and the central density ρ_c = 4.67 × 10⁹ g cm⁻³. For further discussion of numerical instabilities and their physical interpretation, we refer the reader to Wagenhuber & Weiss (1994) and Lau et al. (2012). A simple calculation involving the mass of the envelope at the first TP of the 8.75 M_☉ model (see Table 1) and the time spent on the TP-SAGB yields a critical mass-loss rate of $\dot{M}_\mathrm{crit}=6.75\times 10^{-4}\,M_\odot \,\mathrm{yr}^{-1}$. That is to say, a mass-loss rate higher than $\dot{M}_\mathrm{crit}$ would have reduced the star to an ONe WD before it could produce an EC-SN. This critical mass-loss rate is within the wide realms applied to SAGB stars (see Poelarends et al. 2008, and references therein).

Table 1. Summary of Model Properties

	8.2 M☉	8.7 M☉	8.75 M☉	8.8 M☉	9.5 M☉	12.0 M☉
$M^\mathrm{C}_\mathrm{ign}/M_\odot$a	0.15	0.00	0.00	0.00	0.00	0.00
$M^\mathrm{Ne}_\mathrm{ign}/M_\odot$b	...	...	...	0.93	0.42	0.00
$T^\mathrm{Ne}_\mathrm{ign}/{\rm GK}$c	...	...	...	1.318	1.311	1.324
$\psi ^\mathrm{Ne}_\mathrm{c}$d	...	...	...	46.0	15.2	5.6
$\rho ^\mathrm{Ne}_\mathrm{c}/\mathrm{g\, cm}^{-3}$e	...	...	...	3.343 × 108	7.396 × 107	1.730 × 107
Mtot/M☉f	7.299	7.910	8.572	8.544	9.189	11.338
Menv/M☉g	6.031	6.559	7.210	7.174	6.702	8.023
MHe/M☉h	1.26721	1.35092	1.36230	1.36967	2.48733	3.31580
MCO/M☉i	1.26695	1.35086	1.36227	1.36964	1.49246	1.88602
Remnant	ONe WD	ONe WD/NS	NS	NS	NS	NS
SN type	...	.../EC-SN (IIP)	EC-SN (IIP)	EC-SN (IIP)	CC-SN (IIP)	CC-SN (IIP)

Notes. ^aMass coordinate of carbon ignition. ^bMass coordinate of neon ignition. ^cTemperature at locus of neon ignition. ^dCentral degeneracy at time of neon ignition. ^eCentral density at time of neon ignition. ^fTotal mass at time of first thermal pulse or neon ignition. ^gEnvelope mass at time of first thermal pulse or neon ignition. ^hHelium core mass (H-free core mass) at time of first thermal pulse or neon ignition. ⁱCarbon–oxygen core mass (He-free core mass) at time of first thermal pulse or neon ignition.

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In contrast to the 8.8 M_☉ model, which is discussed in Section 3.3, there is no significant Y_e reduction in the outer core, since there was no Ne–O flash. Instead, the contraction is driven by the steady growth of the core during each TP and the contraction is slower and heating competes with neutrino losses so that the core resumes cooling until electron captures by ²⁴Mg are activated (see Figure 3). The difference can again be seen following the depletion of ²⁴Mg at the center of both models, where the 8.8 M_☉ model continues to heat while the 8.75 M_☉ model again cools down. This difference in temperature between the center of the 8.8 M_☉ and 8.75 M_☉ models is important when considering the next phase of their evolution—electron captures by ²⁰Ne.

The mass of the CO core, M_CO, continues to grow for the entire lifetime of the secondary C-burning shells in all models due to helium shell burning. Previous studies (see Nomoto 1984, and references therein) show that the core mass limit for neon ignition is very close to 1.37 M_☉, which our models confirm. Indeed, in all models with initial mass greater than 8.8 M_☉, a CO core develops, with a mass that exceeds the limit for neon ignition, M_CO(8.8 M_☉, 9.5 M_☉, 12.0 M_☉) = 1.3696, 1.4925, 1.8860 M_☉.

A temperature inversion develops in the core following the extinction of carbon burning in both the 8.8 M_☉ and 9.5 M_☉ models. The neutrino emission processes that remove energy from the core are (over)compensated by heating from gravitational contraction in more massive stars. However in these lower-mass stars the onset of partial degeneracy moderates the rate of contraction and hence neutrino losses dominate, cooling the central region. As a result, the ignition of neon in the 8.8 and 9.5 M_☉ models takes place off-center, at mass coordinates of 0.93 M_☉ and 0.40 M_☉, respectively. This result confirms the work of Nomoto (1984) (case 2.6), but diverges from that of Eldridge & Tout (2004), which we will discuss later. In both models, the temperature in the neon-burning shell becomes high enough to also ignite ¹⁶O + ¹⁶O. As we mention in Section 2, owing to the high densities in the cores of these stars, the products of neon and oxygen burning are more neutron-rich than in more massive stars. This results in an electron fraction in the shell of as low as Y_e ≈ 0.48 (see Section 3.4 and Figure 5). Such low Y_e causes the adiabatic contraction in the following way. If the temperature is high during the flash, the flashing outer layer expands and exerts lower pressure (less weight) on the central region (as can be seen in Figure 3 labeled "Ne-flash," ρ_c decreases due to the almost adiabatic expansion of the central region). However, when the flashed region has cooled down by neutrino emission following the extinction of nuclear burning, the outer layer shrinks and exerts more weight on the core, which is less able to provide support than before the flash because there are fewer electrons available to contribute to the degeneracy pressure. The center then reaches higher densities, and hence temperatures, than before. As mentioned above, for this reason the reduction in Y_e is important for cores so close to M_Ch.

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As illustrated in Figure 4(e), following the neon shell flashes the 9.5 M_☉ model recurrently ignites neon and oxygen burning in shells at successively lower mass coordinates that eventually reach the center, following which Si-burning is ignited off-center. Although neon (and oxygen) burning in the 8.8 M_☉ model begins as a flash and later propagates toward the center, the evolution of the 8.8 M_☉ model diverges from that of the 9.5 M_☉ star when its center reaches the conditions necessary for the first URCA process pair to become significant (whereas the 9.5 M_☉ model avoids such dense conditions). More details of the neon and oxygen shell burning episodes are discussed in Sections 3.3.1 and 3.3.2.

The CO core (or equivalently He-free core) in the 8.8 M_☉ model at the time of neon ignition is 1.36964 M_☉, very close to M_Ch, while that of the 9.5 M_☉ model is 1.49246 M_☉ (see Table 1). Under these conditions, the 8.8 M_☉ model experiences a much more marked contraction due to the reduction in Y_e. The central density at this time is as high as 3.43 × 10⁸ g cm⁻³, which is exceedingly close to the threshold density for ²⁷Al(e⁻, ν)²⁷Mg. Although there is no cooling effect from the A = 27 pair because the decay channels are blocked, the further removal of electrons from the core causes contraction toward the threshold densities of the second and third URCA pairs (A = 25 and A = 23, respectively). The cooling effect supplied by the A = 25 URCA pair (and later the A = 23 pair, shown in Figure 10) allows for a small amount of contraction but again it is the associated change in the electron fraction that enables the largest contraction when the core is so close to the Chandrasekhar limit ($M_\mathrm{Ch}\propto Y_\mathrm{e}^2$). The core of the 8.8 M_☉ model continuously contracts until the center reaches the critical density for electron captures by ²⁴Mg, quickly followed by further contraction to the critical density for those by ²⁰Ne (see Figure 3).

There is a significant discrepancy between the URCA process trajectories of our models and those of Ritossa et al. (1999). This is due to the undersampling of weak reaction rates for the URCA process that we employ in the MESA code (Oda et al. 1994). In Section 4.1, we discuss the implications of this undersampling and show that, by using new well-sampled weak rates (Toki et al. 2013), the URCA process central trajectory of Ritossa et al. (1999) is qualitatively reproduced in the 8.8 M_☉ case. The difference between the Oda et al. (1994) compilation and the newly calculated Toki et al. (2013) rates that are available to use in our calculations for the A = 25 pair is shown in Figure 6.

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This central evolution is significantly different from that for the 8.75 M_☉ model, which is described in Section 3.2. The energy release from both the rapid contraction and the γ-decays from electron capture products raises the temperature high enough to ignite neon and oxygen in quick succession. We have followed from the resulting oxygen deflagration onward with the AGILE-BOLTZTRAN hydrodynamics code and can confirm that the model results in core collapse (T. Fischer 2013, in preparation). Although Eldridge & Tout (2004) report the same fate for their 10 M_☉ model in which a limited network was used, there is no neon shell flash following the completion of the 2DUP. In these models, neon burning was found to take place at the edge of the core during the last carbon shell flash, reducing the core mass to M_Ch (Eldridge 2005; J. J. Eldridge, 2013, private communication). Subsequently, the core contracted directly to central densities of about log₁₀(ρ_c/g cm⁻³) = 9.8 (roughly the critical density for electron captures by ²⁰Ne to start) with no further neon shell flashes, though electron captures were not included in the nuclear reaction network. Neon-burning reaction rates were artificially limited to prevent numerical problems and a low spatial resolution was used. We believe these two caveats to be the reason that the neon–oxygen shell flashes we find to occur in such stars were not present in these earlier models. In this work, we were able to follow the evolution all the way to oxygen deflagration by using a very large network of 114 nuclei including all the relevant fusion and weak reactions. Our models thus highlight the importance of neon-shell burning in determining the path to collapse.

As mentioned above, the 9.5 M_☉ model starts silicon-burning off-center in a shell that later propagates toward the center. This is another example of the continuous transition toward massive stars, in which all the burning stages begin centrally. Although we have not evolved this model to its conclusion, we expect that silicon burning will migrate to the center, producing an iron core, and that it will finally collapse as an FeCCSN. Such a low-mass progenitor will make for interesting explosion simulations (Müller et al. 2012). The 12 M_☉ is the canonical massive star in our grid, igniting C-, Ne-, O-, and Si-burning centrally (see Figure 4(f)). It eventually collapses, and would produce a type-II FeCCSN.

3.3.1. Neon–Oxygen Flashes

As briefly mentioned above, following the extinction of the final carbon-burning shell, a degeneracy/neutrino-induced temperature inversion arises in the core in a similar way to the temperature inversion in SAGB stars. This causes the ignition of carbon to take place away from the center. Neon is thus ignited off-center at mass coordinates of 0.93 M_☉ and 0.40 M_☉ for the 8.8 M_☉ and 9.5 M_☉ models, respectively. Some of the important model properties are given in Table 1 at this time.

At the point of Ne-shell ignition, the density profile of the two stars is very different (see Figure 7). While the 8.8 M_☉ model is structured more like an SAGB star due to the previous dredge-out episode, the 9.5 M_☉ model resembles more a massive star, with a distinct He-shell and C-shell still present.

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When neon and oxygen is first ignited, fuel is abundant and a convective shell quickly develops. The sharp increase in energy production briefly halts the contraction of the core and causes the center to expand and cool (see Figure 3), while convection brings in fresh fuel to be burnt at the base of the shell. Conduction at the base of the shell is slow and so when the fuel is depleted the shell is extinguished and the core contracts. This contraction continues until the temperature becomes high enough where neon and oxygen is abundant, re-igniting the nuclear burning and producing a new convective shell. After a few flashes, the region previously engulfed by the shell convective shell as it extended radially outward has become heavily depleted in Ne and O and so the closest fuel is in the direction of the center. At this point a new regime, the Ne–O flame, is begun.

3.3.2. Neon–Oxygen Flame

In this section, and throughout the remainder of the manuscript, it should be noted that we use the term flame to describe the inward propagation of a nuclear burning shell, driven by either compressional heating or other form of local heat transport toward the center. After the last flash has extinguished and contraction begins, the two models begin to diverge, as best illustrated in Figure 3 and Figures 4(d) and (e). The 9.5 M_☉ star once again contracts and a thin shell of neon and oxygen is ignited below the base of the previously convective shell. Any convection developing at this time does not bring any fresh fuel (only the ashes of the previous shells) into the burning region. The core is so dense that the photon mean free path is too short for radiative transfer to play an important role in the inward propagation of the flame and instead compressional heating due to core contraction and local heating due to electron conduction are largely responsible for intermittent periods of nuclear energy production that move toward the center.

It is a different story for the 8.8 M_☉ star. Contraction, following the final ONe shell flash, at first acts to heat the material locally and to burn neon and oxygen moderately as in the 9.5 M_☉ model, except that the core is more degenerate in the 8.8 M_☉ star. Electron conduction is therefore much more efficient and the localized effect of heat generation due to contraction and any subsequent nuclear burning is instead diluted across the core. This smoothing of the temperature profile across the core prevents the region directly below the previously ONe-burning shells from reaching temperatures in excess of the Ne-burning threshold. Instead of a flame developing as in the 9.5 M_☉ star, the core contraction, driven by the neutron-rich composition in the NeO shell, causes local heating much further from the center where the degeneracy is lower, and a new neon- and oxygen-burning shell ignites (where the fuel is still abundant) above the outermost extent of the previous ONe shells.

Figure 8 shows the opacity profiles following the extinction of the last neon–oxygen flash and at a later time in each model. Although electron conduction dominates the heat transfer in both cases, it is more efficient (lower κ) by a factor of about three in the 8.8 M_☉ model's early flame and by a factor of more than 10 later, meaning that any energy production from subsequent radiative neon–oxygen burning or contraction is diluted across the majority of the core. In contrast, the higher conductive opacities in the 9.5 M_☉ model allow for the nuclear and compressional energy to take effect much more locally, heating the underlying shell of material to ignition temperatures and causing the development of a nuclear flame. These two contrasting paths are further illustrated in Figure 9, which shows clearly a flame front developing in the 9.5 M_☉ model and the dilution of heat across the core of the 8.8 M_☉ model. The effects of spatial resolution on flame development and energy transport are discussed in Section 4.3.

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In summary, the propagation of the flame in the 8.8 M_☉ model is more difficult because of the lower opacity. Furthermore, the combined effect of electron captures in the center and the low Y_e in the shell due to Ne- and O-burning leads to the core contraction on a shorter timescale than the evolution of the flame.

The structure of the progenitor star, in terms of density and electron-fraction profiles of the stellar core, has a strong impact on the timescale at which the later SN explosion may develop as well as on the explosion energetics. Core-collapse SN explosions are related to the revival of the stalled shock wave, which forms when the contracting core reaches normal nuclear matter density and bounces back. For massive iron-core stars, the structure of the core at the onset of contraction is determined by the mass enclosed inside the carbon shell. In general, a sharp density gradient separating iron-core and silicon layer results in a strong acceleration of the bounce shock at the onset of shock revival early after core bounce on a timescale of only few 100 ms. Progenitors with a shallower density gradient suffer from a more extended mass accretion period after core bounce, during which the standing bounce shock oscillates, driven by neutrino-energy deposition behind and mass accretion from above. This results in a delayed onset of shock revival by several 100 ms and more energetic explosions due to the larger heat deposition behind the shock via neutrinos before shock expansion. For a recent review of the connection between progenitor structure and recent axially symmetric SN explosion models, see Janka et al. (2012).

In addition to the standard iron-core progenitors commonly explored in core-collapse SN studies, we provide a selection of new models of lower zero-age main-sequence mass that belong to the SAGB class as well as to low-mass massive stars. Therefore, in Figure 7, we compare the structures of our SAGB model (8.75 M_☉) after central ²⁴Mg depletion, EC-SN progenitor (8.8 M_☉, failed massive star) at ignition of oxygen deflagration, low-mass massive star (9.5 M_☉) at the point of neon-shell ignition, and standard iron-core progenitor (12 M_☉) at the onset of core contraction. Note that the 9.5 M_☉ progenitor is not then as evolved as the other models and hence its central density is still lower than those of the other models. It is therefore only used as a reference case. The major difference between the low-mass (8.75 and 8.8 M_☉) and the more massive iron-core progenitors is the very steep density gradient separating the core and the envelope. There the density drops about 16 orders of magnitude, from about 10⁸ to 10⁻⁸ g cm⁻³.

Distinguishing the 8.75 M_☉ and 8.8 M_☉ progenitor structures becomes clearer when inspecting the density profiles with respect to radius (Figure 7(b)). The bulge from log₁₀(R/km) ≈ 3.2 to 3.8 that features in the 8.8 M_☉ structure but is absent in the 8.75 M_☉ structure, is a carbon-burning shell. One would expect that, since the 8.8 M_☉ model experienced several neon–oxygen flashes, the structure within the core should be significantly different from that of the SAGB model. Aside from the abundance profiles showing a large region in which the composition is dominated by Si-group isotopes, the most striking difference is in the electron fraction, Y_e, which is shown in Figure 5.

In Figure 7, we have included the progenitor structures of the Nomoto (1987; SAGB-like) 8.8 M_☉ and the Woosley et al. (2002) 12 M_☉ models for comparison. The Nomoto (1987) structure is at a later evolutionary stage compared to our models. A fraction of the core has already been burnt to nuclear statistical equilibrium (NSE) composition, but the core structure is qualitatively similar to our 8.75 M_☉ SAGB model. It is also clear from Figure 7, bottom panel, that there are differences in the structure of the Nomoto (1987; SAGB-like) model and our 8.8 M_☉ (failed massive star) model, where there is a CO-rich layer at the edge of the core. As discussed previously, there is a neutron-rich layer in our 8.8 M_☉ model where the Ne–O shell flash consumed previously that is not a feature of the Nomoto (1987) model. There is a clear clustering of the SAGB EC-SN progenitor structures and the CCSN progenitor structures in the density profiles as a function of radius (Figure 7, bottom panel), while the 8.8 M_☉ model lies in-between.

The iron-core progenitors have extended high-density silicon as well oxygen and carbon layers above the core. These result in a shallower transition from iron core to helium envelope. The density decreases steadily stepwise according to the different composition interfaces (see Figure 7, top panel). Moreover, different evolutionary tracks for the 8.75 and 8.8 M_☉ progenitor cores lead to low-mass cores of only about 1.376 M_☉, which is significantly lower than for the 12 M_☉ model of 1.89 M_☉ (see Table 1). Note that the 12 M_☉ iron-core results are in qualitative agreement with those of the KEPLER code (Woosley et al. 2002) and, as a function of radius, match very well. The reason for the discrepancy between the two as a function of mass is the difference in assumption for convective overshooting, which has led to the production of larger cores in the MESA model. We are currently working on a code comparison study of MESA, KEPLER, and the Geneva stellar evolution code (Hirschi et al. 2004) for the evolution, explosion, and nucleosynthesis of massive stars in order to quantify some of the related uncertainties. We expect that the resulting steep density gradient at the edge of the core of our EC-SN progenitor models will accelerate the SN shock on a short timescale after core bounce, producing a weak explosion with little ⁵⁶Ni ejecta. Such an explosion should produce qualitatively similar results as obtained for the 8.8 M_☉ progenitor from Nomoto (1987; for details about EC-SN explosions see Kitaura et al. 2006; Janka et al. 2008; Fischer et al. 2010). The split between weaker, more rapid EC-SN explosions and stronger, slower FeCCSN explosions is a possible explanation for the observed bimodality in the spin period and orbital eccentricity of X-ray binaries, although it is not clear how this is manifested (Knigge et al. 2011).

During the very late stages of the 8.75 and 8.8 M_☉ stars electron captures on sd-shell nuclei become crucial to their fate. In the degenerate core, there is a very sharp jump in the rates of these electron captures, which corresponds to a threshold density at which the electron Fermi energy, _F, exceeds the threshold energy for the reaction to proceed. Tabulated electron capture rates that we use as input for the models must properly resolve this steep transition if we want to know at what density the oxygen deflagration is ignited. We should want to know that density so that it can be determined whether nuclear energy release from burning the core to NSE composition is high enough to exceed the gravitational binding energy of the core and thus lead to its explosion (Gutierrez et al. 1996). Otherwise, the core would collapse to a neutron star following its deleptonization through electron captures on Fe-group isotopes.

There are still more shortcomings of calculations involving electron capture rates that are poorly resolved in the ρ–T plane. For example, the vast majority of widely used rate tables for sd-shell nuclei possess a grid spacing of 1 dex in ρY_e. As an example one of these crucial reactions, ²⁴Mg + e, jumps by about 20 orders of magnitude from log₁₀(ρY_e/g cm⁻³) = 9.0 to 10.0 at the temperature of interest (T ≈ 0.4 GK). This is not only a problem for resolving the rate at the threshold density, because at lower densities the rate, λ/s⁻¹, is significantly underestimated through linear interpolation of log₁₀(λ/s⁻¹) (see Figure 11).

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It is clear from Figure 3 that the central evolution of the 8.75 and 8.8 M_☉ models is dominated by weak reactions. The onset of the URCA process disrupts the propagation of the neon–oxygen flame and aids the central contraction. In fact, the same is true for all EC-SN progenitors and thus it is imperative to treat the URCA process as accurately as possible to best predict the fate of 8–12 M_☉ stars. Toki et al. (2013) have produced well resolved (Δlog₁₀ ρY_e/g cm⁻³ = 0.02 and Δlog₁₀ T/K = 0.05) reaction and neutrino-loss rates for the A = 23, 25, and 27 URCA pairs under the conditions 7.0 ⩽ log₁₀ (T/K) ⩽ 9.2 and 8.0 ⩽ log₁₀ (ρY_e/g cm⁻³) ⩽ 9.2. The differences between these new rates and those of Oda et al. (1994) is shown in Figure 6 for the A = 25 pair at T₉ = 0.4. The impacts of these new, well-resolved rates compared to those of Oda et al. (1994) are shown in Figure 12. Not only is the cooling effect more pronounced, the reaction thresholds are more clearly identifiable and occur at higher densities than with the rates of Oda et al. (1994).

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Because the rates are so sensitive to density, any form of interpolation cannot properly represent the physical situation without some input from knowledge of the nuclear physics. This is why several groups employ an interpolation of effective log ft values (Fuller et al. 1985). An effective log ft value for a reaction is related to its raw rate by the relationship in Equation (2),

$$\begin{equation} ft=\frac{\phi }{\lambda }\,\mathrm{ln}\,2, \end{equation} \tag{ 2 }$$

where ϕ is the ground-state to ground-state phase space integral. The aim is to produce a quantity that varies smoothly with T and ρ from which the raw rate may be obtained within a stellar evolution calculation by approximation of the phase space integral at the desired conditions. This method is relatively robust for those weak rates for which ground-state to ground-state transitions dominate. However, this is not the case for the reactions of interest in EC-SN progenitors. The change in Y_e is not the only important facet of the electron captures; they also possess a strong heating effect due to the γ-decay following transitions to excited states of the daughter nuclei. Hence, this demonstrates the importance of excited states when we attempt to normalize the reaction rate using simplifications or approximations.

Therefore, we conclude that there are two possible sets of desired quantities, either grids of weak reaction rates for sd-shell nuclei that are appropriately resolved through the threshold density or logft values that incorporate all important transitions in the normalization of the rate. There are contributions from many states of the parent and daughter nuclei for these reactions and to perform phase space integral routines within a stellar evolution code to account for this could be exceptionally inefficient. It is also important to use β^±-decay and neutrino-loss rates calculated with the same physics and grid resolution to ensure consistency when we look at the impact of the URCA process on the evolution. The most up-to-date rates should also include the effects of Coulomb screening, which has been shown to increase the threshold density for electron captures (Gutierrez et al. 1996). An increase in the threshold density of ²⁰Ne(e⁻, ν)²⁰F would cause the oxygen deflagration to ignite under denser conditions in the SAGB progenitors. However in the failed massive star case the center is approaching the ignition temperatures of Ne and O adiabatically, and so the oxygen deflagration could be ignite before ²⁰Ne + e⁻ becomes significant if there were an increase in the threshold density.

Still one of the largest uncertainties in any one-dimensional stellar evolution calculation is the treatment of convection. Extra mixing at convective boundaries may explain many observed phenomena, for example, the s-process abundance patterns in AGB stars, and hence we include such mixing in our models. Due to the turbulent and advective nature of convection, it is physically plausible to infer some extra mixing across the boundary between convective and radiative layers but without the benefit of three-dimensional hydrodynamical simulations of the physical conditions it is difficult to quantify its extent. We use the term convective boundary mixing rather than overshooting for the advanced evolution phases of the deep stellar interior, such as convective shells. This is because the term overshooting suggests a physical picture in which coherent convective structures or blobs cross the Schwarzschild boundary before they notice the reversal of buoyancy acceleration. However, in the deep interior other hydrodynamic instabilities, such as Kelvin–Helmholtz or internal gravity wave induced turbulence dominate mixing at the convective boundary. Largely, the effect of including convective boundary mixing is to shift the transition masses due to increased core sizes. However it is intuitive to hypothesize that increased amounts of extra mixing below the ONe-burning shells would have a crucial effect on their inward propagation. To test this, we assumed extra mixing below the convective ONe-burning shells to behave as an exponentially decaying diffusion process as outlined in Equation (1) with f_flame = 0.005 (our original assumption), 0.014, 0.028, and 0.100. The central density–temperature evolution from the flame's ignition for all of these assumptions is shown in Figure 13. It should be noted that setting f_flame = 0.100 is an extremely unphysical assumption that we adopt simply to test the uncertainty of our conclusions.

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Although the central evolution behaves slightly differently for each mixing assumption, all the models reach central densities of ρ_c = 10^9.6 g cm⁻³ at temperatures well below the neon-ignition threshold. The model with the largest amount of mixing (f_flame = 0.100) is unique because although all models undergo a few flashes after the extent of the URCA process has been exhausted, it is the only one to re-ignite an ONe shell at a mass coordinate in-keeping with the original location of the flashes.¹³ At this point, the center is already extremely close to the threshold density for ²⁴Mg(e⁻, ν)²⁴Na at ρ_c ≈ 10^9.6 g cm⁻³. The change in extent of the convective boundary mixing between the f_flame = 0.005 and f_flame = 0.100 models is shown in Figure 14. The deep mixing in the extreme (f_flame = 0.100) model replenishes fuel at the flame front, allowing it to re-ignite.

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Convective boundary mixing is at present still a very uncertain phenomenon. While the timescales for stellar evolution restrict theoretical models to only one dimension, there is an emergence of effort to explore specific phases of the evolution in two (Herwig et al. 2006, 2007) and three (Mocák et al. 2011; Herwig et al. 2011) dimensions in order to properly quantify the extent of convective mixing and its behavior at the boundary with a radiative zone. We plan to perform an in-depth parameter study of convective boundary mixing in 8–10 M_☉ stars. The long term goal is to constrain the parameters of our diffusive treatment by analyzing the data from three-dimensional simulations.

The development and propagation of a nuclear flame front is highly sensitive to the spatial resolution due to the thin flame width. Figure 9 showed the evolution of the T-profile at the time of flame development/propagation in both models. Each red dot represents a mesh point in the calculation. It can be seen that the model possesses a spatial resolution much finer than the width of the flame front, however in the transition at the base of the flame it is evident that there is a less than desirable resolution very early on in its development.

To examine the effect of spatial resolution on the outcome of the 8.8 M_☉ model, we increased the resolution of the model 10-fold and then 20-fold at the base of the Ne + O-burning convection zone, from before the ignition of the Ne + O flame. In a second test, we increased the resolution 10-fold in the regions where energy production from ¹⁶O(α, γ)²⁰Ne, ²⁰Ne(α, γ)²⁴Mg, ¹⁶O(¹⁶O, γ)³²S, ¹⁶O(¹⁶O, p)³¹P, or ¹⁶O(¹⁶O, α)²⁸Si became significant (greater than 10⁴erg g⁻¹s⁻¹). Neither of the enhanced resolutions at the base of the convective shell alter the outcome of the 8.8 M_☉ model (EC-SN), and nor does the re-meshing based on energy production. This demonstrates that our results concerning Ne–O flame propagation are robust and not due to an underresolved flame front.