Explosion mechanism of core-collapse supernovae: role of the Si/Si-O interface

Several three-dimensional simulations have shown that progenitors with steep density gradients near the Si/Si-O interface often lead to explosions (Burrows et al., 2018; Vartanyan et al., 2021), confirming what had been argued in the past. In particular, as explained in Section 2, Ertl et al. (2016) formulated a criterion to predict the outcome of the explosion based on the pre-collapse structure of the progenitor star. The choice of $s=4$ $k_{\rm B}$ baryon^-1 in their criterion (later applied to a wider set of progenitors by Sukhbold et al. (2016)) was motivated by the fact that the Si/Si-O interface is generally located at entropies per baryon around that value.

However, Couch et al. (2020) and Boccioli et al. (2021) found the explodability as a function of progenitor mass to be very different from the one predicted by Ertl et al. (2016) and Sukhbold et al. (2016). Both of those studies employed STIR in 1D simulations with full $\nu$ transport. Hence, it appears that explosions achieved via $\nu$-driven turbulent convection do not obey the explosion criterion proposed by Ertl et al. (2016). Since 3D simulations from Burrows et al. (2020) seem to agree with the explodability predicted by STIR, we are confident that one can develop a simple explosion criterion based upon the shock dynamics during the accretion of the Si/Si-O interface.

In spherically symmetric simulations the accretion of the Si/Si-O interface creates a sudden decrease in the ram pressure. This induces a temporary expansion – or surge – of the shock. Several semi-analytical models have been used to investigate how fluctuations in pre-shock thermodynamic quantities affect the expansion of the shock (Blondin et al., 2003; Nagakura et al., 2013, 2019). In these studies, it was concluded that these fluctuations can both act as seeds for fluid instabilities that can potentially revive the shock, and also relieve the ram pressure on the shock enough for heating caused by $\nu$-driven turbulence to trigger an explosion. Therefore, in 1D simulations, where $\nu$-driven turbulence is not present, the shock is slowly pushed back down after the initial expansion, continuing its recession towards the PNS. This temporary expansion is almost always caused by the accretion of the Si/Si-O interface, although there are some exceptions.

If the surge is large enough and occurs between $\sim 70$ ms and $\sim 400$ ms after bounce one can expect a 3D simulation to produce an explosion for that progenitor. To quantify the surge we follow the methods outlined in Schneider et al. (2020). Since the surge is connected to a drop in the mass accretion rate, Schneider et al. (2020) defined a smooth shock radius $\widetilde{R}_{\rm shock}$ as a fit to the actual shock radius $R_{\rm shock}$ without the inclusion of the region where the accretion rate drops significantly, which is indeed the signature of a surge.

In our case, we instead define the start of the surge as the time when the density jump (as defined in Section 4) is accreted through the shock. The two definitions agree very well, which confirms that our definition of the density jump is physically sound since its accretion almost always causes a surge. Moreover, by using our definition we avoid all of the numerical complications derived from having to perform derivatives of the mass accretion rate.¹¹1Sometimes the accretion happens very close to the maximum of the shock radius, where the definition of the surge becomes ambiguous. At early times the mass accretion rate quickly decreases. Therefore, even if it drops as a consequence of the accretion of the jump, the overall trend won’t be affected significantly.

Our fitting procedure is as follows: if the accretion of the density jump happens earlier than 100 ms, we exclude the time window from the accretion until 50 ms after. We then fit a 2nd-degree polynomial to the shock radius versus time post-bounce. Otherwise, we do a linear fit to the inverse of the shock radius versus time post-bounce, for a time window of 100 ms.²²2Early accretions before 100 ms happen around the maximum of the shock radius. Therefore, one expects the shock radius to be roughly quadratically dependent on time. For late-time accretion, one expects $R_{\rm shock}\propto 1/t$ instead. An example of the fitting procedure, as well as the difference between the simulations with and without STIR, is shown in Figure 4. The surge is then defined as $R_{\rm surge}=R_{\rm shock}-\widetilde{R}_{\rm shock}$. The maximum of the surge is $\delta R_{\rm surge}$. The larger $\delta R_{\rm surge}$, the more likely is the explosion of that progenitor in realistic 3D simulations.

The above definition of $R_{\rm surge}$ can only be applied to non-exploding simulations. During an explosion, the shock expands further after the surge, and therefore it is not possible to quantify how much of the expansion is solely due to the accretion of the jump. Hence, in order to quantify $R_{\rm surge}$, we performed a standard 1D simulation for every progenitor, without the inclusion of $\nu$-driven turbulence, in addition to the simulations with STIR. To distinguish between these two sets of simulations, we refer to the simple spherically symmetric simulations as “1D”, and to the spherically symmetric simulations with STIR as “1D+” since they incorporate $\nu$-driven convection, a multi-dimensional effect.

It is worth pointing out that if one uses a loose definition of the density jump (e.g. the location of the $s=4$ $k_{\rm B}$ baryon^-1 layer), there is a mismatch between when the surge occurs and when that layer is accreted. Therefore, we give a more accurate description of how the jump is defined in the next Section.

As density decreases as a function of enclosed mass, entropy increases. In particular, in layers where the composition changes abruptly, the decrease in density and increase in entropy are very steep, as can be seen in Figure 3. Due to the small range of entropies involved (3-5 $k_{\rm B}$ baryon^-1) it is more practical to use the entropy to identify the zones that make up the jump. We define the starting zone of a jump as the $i$-th zone such that:

where s is the entropy per baryon. The end of a jump happens at the $i$-th zone such that:

The reason for the double condition in Equation (7) is that sometimes, along the jump, the entropy might not significantly increase in one zone, but keeps increasing after that. Using only the first condition would then underestimate the size of the jump since a fluctuation along the jump would wrongly be identified as the end.

After finding all the jumps in the progenitor, one has to determine which one will cause the surge. If the jump is located at low densities, it will be accreted very late. Therefore, it will not cause any surge since the shock is already overwhelmed by the ram pressure of the infalling material, and is inevitably receding. On the other hand, if the jump is located at high densities, it will be accreted right after bounce, when the shock is still expanding and neutrino heating is not yet fully developed. Therefore, it will not cause any surge. We then conservatively chose a range of densities between $9\times 10^{5}$ g cm^-3 and $2\times 10^{7}$ g cm^-3 where the entropy jump can be located. This allows us to include discontinuities that are accreted very early $\sim 60$ ms after bounce, and very late $\sim$ 600 ms after bounce. If no jump is present in this density range, the progenitor will not explode (see Section 6.1 for more details).

If more than one jump is present within this density range (which is often the case), we select the one where the maximum of $\delta\rho^{2}/\rho^{2}$ occurs. Here, $\delta\rho$ is the difference between the density at the start and the end of the jump. The reason for this choice will become clear when we discuss our results in Section 6. An example of a typical jump in entropy and density is shown in Figure 3. For the remainder of this paper, any quantity “$q$” calculated at the start (i.e. closer to the center of the star) of the jump will be labeled as $q_{*}$. If it refers to the end (i.e. closer to the atmosphere of the star) of the jump it will be labeled as $q_{*}^{\rm end}$. The size of the density jump will be labeled as $\delta\rho_{*}$.

To determine whether a progenitor explodes or not, we use the 1D+ simulations with $\alpha_{\rm MLT}=1.51$, as explained in Section 4. We can now formulate two explosion criteria: criterion (a) based upon dynamical properties ($\delta R_{\rm surge}$/$R_{\rm accr}$ and $t_{\rm accr}$); and criterion (b) based upon pre-collapse properties ($\delta\rho_{*}^{2}/\rho_{*}^{2}$ and $\widetilde{t}_{\rm accr}$).

The first part of our criteria reads:

1. a)

   if $t_{\rm accr}>0.4$ s the star will not explode. Otherwise, the star can explode, based on the discussion that follows.

2. b)

   if $\widetilde{t}_{\rm accr}>0.4$ s the star will not explode. Otherwise, the star can explode, based on the discussion that follows.

This leads to the prediction that that 17 progenitors do not explode. Of these, only the 22.8 M_⊙ KEPLER progenitor explodes despite having $\widetilde{t}_{\rm accr}=0.430$ s and $t_{\rm accr}=0.401$ s, and is therefore misclassified by both criteria. After determining that progenitors with $t_{\rm accr}>0.4$ s ($\widetilde{t}_{\rm accr}>0.4$ s) do not explode, we exclude them from the subsequent discussion. Moreover, we also exclude the progenitors that do not have any density jump satisfying the definition given in section 4. None of these progenitors explode, as expected.

Based on the discussion above, the remaining progenitors should follow Equation 10, and as can be seen from Figure 7 that is indeed the case. For completeness, progenitors with $\widetilde{t}_{\rm accr}>0.4$ s are shown as shaded symbols, but they are not included in the analysis. It is interesting to notice that some of them have very large $\delta\rho_{*}^{2}/\rho_{*}^{2}$ but small $\delta R_{\rm surge}$/$R_{\rm accr}$. However, for accretions at very late times, it is very hard to estimate the surge correctly, and therefore these variations are likely to be a consequence of numerical noise.

Interestingly, on the right side of Figure 7, there are some progenitors with $\delta R_{\rm surge}$/$R_{\rm accr}>0.25$ and others with $\delta\rho_{*}^{2}/\rho_{*}^{2}>0.4$ that seem to significantly deviate from the best-fit line, shown as a dashed black line. We verified that in the case of the progenitors with $\delta R_{\rm surge}$/$R_{\rm accr}>0.25$, $\delta R_{\rm surge}$ has been overestimated by our fitting algorithm. This also partially explains the vertical spread around the best-fit line of all progenitors, since as discussed in previous sections the estimation of $\delta R_{\rm surge}$ suffers from numerical noise. Analogously, for the progenitors with $\delta\rho_{*}^{2}/\rho_{*}^{2}>0.4$, $\delta R_{\rm surge}$ has been underestimated.

We find that our fitting procedure is relatively simple, physically justified, and has an overall good performance. Therefore, we believe that fine-tuning it to properly account for these outliers defeats the purpose of finding a simple criterion that can connect dynamical and pre-collapse properties. Moreover, none of these outliers are misclassified, and therefore changing the fitting procedure would not change the results. The only change would be a better correlation between $\delta R_{\rm surge}$/$R_{\rm accr}$ and $\delta\rho_{*}^{2}/\rho_{*}^{2}$, which we only use as a consistency check for Equation 10, but it does not enter in our criterion.

We can now formulate the second part of our explosion criteria:

1. a)

   If $\delta R_{\rm surge}$/$R_{\rm accr}>0.04$ the progenitor explodes. Otherwise, it results in a failed SN.

2. b)

   If $\delta\rho_{*}^{2}/\rho_{*}^{2}>0.08$ the progenitor explodes. Otherwise, it results in a failed SN.

Criterion (a), formulated using dynamical properties, produced $\sim 9\%$ false positives and $\sim 1\%$ false negatives. Some of these are located near the horizontal line in Figure 7, which represents $\delta R_{\rm surge}$/$R_{\rm accr}=0.04$, and can be considered statistical fluctuations. The imbalance between false positives and negatives is partially due to the definition of $\delta R_{\rm surge}$, which can be affected by numerical noise and lead to an overestimation of $\delta R_{\rm surge}$, especially when the accretion happens close to the maximum of the shock radius. This generates a significant number of false positives on the top left quadrant of Figure 7, that are however correctly classified by criterion (b). A few more false positives can be found on the top right quadrant of Figure 7, and are quite far from the horizontal line. All of them are KEPLER progenitors and, as explained later in the section, some of them explode at slightly larger values of $\alpha_{\rm MLT}$. They are also misclassified by criterion (b).

Criterion (b), formulated using pre-collapse properties, produced $\sim 5\%$ false positives and $\sim 4\%$ false negatives. Some of these are located near the vertical line in Figure 7, which represents $\delta\rho_{*}^{2}/\rho_{*}^{2}=0.08$, and can be considered statistical fluctuations. However, some misclassifications are quite far from the dividing line and deserve an explanation. Like in criterion (a), some misclassifications happen on the top-left quadrant, but in this case, these are false negatives, i.e. progenitors with $\delta\rho_{*}^{2}/\rho_{*}^{2}<0.08$ that nevertheless explode. These are all FRANEC progenitors, and the ones with the smallest values of $\delta\rho_{*}^{2}/\rho_{*}^{2}$ have all very low compactnesses $\xi_{\rm 2.5}<0.05$.

Because of their low compactness, these progenitors are characterized by very steep density profiles and low mass accretion rates. Therefore, the ram pressure exerted on the shock is small. This means that even density discontinuities accreted very late can be sufficient to trigger an explosion. Moreover, they generally tend to be easier to explode, and some of them can indeed explode very early without the need for the accretion of a density jump. A more nuanced criterion is likely needed to describe these progenitors, but this goes beyond the purposes of this paper and will be addressed in future work. Finally, several misclassifications of KEPLER progenitors happen in the top right quadrant, as already described for criterion (a). A more detailed analysis of why such misclassifications arise is given in section 6.3.

The analysis presented so far was done on the combined set of progenitors. However, it is interesting to apply the criteria to the KEPLER and FRANEC sets separately. The results are reported in Table 1. Criterion (a) performs equally well on both sets, with no remarkable differences. This is a consequence of the overestimation of $\delta R_{\rm surge}$ that occurs in some cases due to uncertainties in the fitting algorithm.

Criterion (b), instead, has significantly different performances on the two sets. The false positives are much higher for the KEPLER progenitors. This can be attributed to the outliers on the top-right quadrant in Figure 7 which, as discussed in Section 6.3, can be accounted for by tweaking $\alpha_{\rm MLT}$. By doing that, the false positives drop from 7.1 % to 2.6 %, which is much closer to the value of 3.1 % found for FRANEC progenitors.

The false negatives are instead much higher for FRANEC progenitors. This can be attributed to the finer resolution of FRANEC progenitors with masses $M<13$ M_⊙, compared to KEPLER. These progenitors have very low compactness and, as described above, are likely to explode even in absence of the accretion of a strong density jump. If one excludes from the analysis progenitors with $\xi_{2.5}<0.05$, the false negatives for FRANEC progenitors drop from 6.9 % to 3.3 %. This indicates that low compactness progenitors can explode even without the presence of a strong density jump.

Interestingly, KEPLER progenitors with $\xi_{2.5}<0.05$ obey the criterion since the 12.5 M_⊙ and 12.75 M_⊙ do not explode, contrary to the expectation that low compactness progenitors should be more susceptible to explosions. Therefore, one can only surmise that a more nuanced criterion is needed to account for the behavior of low compactness progenitors, but this goes beyond the scope of this paper.

A similar study was very recently carried out by Wang et al. (2022), who used 100 2D simulations to calibrate their criterion instead of our 1D+ approach. Moreover, they only applied this criterion to KEPLER progenitors, whereas we also included FRANEC models.

The first difference is in the fact that Wang et al. (2022) use the ram pressure $P_{\rm ram}$ instead of density as the main quantity in their criterion. However, the two are interchangeable, since $P_{\rm ram}=\rho v_{\rm infall}^{2}$, and since the infall velocity $v_{\rm infall}$ does not change when the jump is accreted one finds that $\delta P_{\rm ram}/P_{\rm ram}\approx\delta\rho/\rho$.

The selection of the density jump of Wang et al. (2022) is very similar to the one we adopted, described in section 4. In our case, we look for the maximum of $\delta\rho^{2}/\rho^{2}$ in the density range between $9\times 10^{5}$ g cm^-3 and $2\times 10^{7}$ g cm^-3, whereas Wang et al. (2022) look for the maximum of $\delta P_{\rm ram}/P_{\rm ram}$ in the region between the outer envelops and the Si-O interface.

In their paper, Wang et al. (2022) define the Si-O interface as “the density discontinuity closest to the inner boundary of the oxygen shell in which the oxygen abundance is above 15 percent”. Therefore, with this definition, the Si-O interface is sometimes well inside the silicon shell. For example, for the profile shown in Figure 3, the definition of the jump by Wang et al. (2022) agrees with ours, and the density jump is entirely inside the silicon shell. For that reason, in this paper, we have used the expression “Si/Si-O interface” rather than “Si-O interface” since the drop in density is not necessarily located at the bottom of the oxygen layer.

Quantitatively, our criterion agrees extremely well with the one by Wang et al. (2022). In their case, they predict explosions for progenitors where ${\rm max}(\delta P_{\rm ram}/P_{\rm ram})>0.28$. Since $\delta P_{\rm ram}/P_{\rm ram}\approx\delta\rho/\rho$, this condition becomes ${\rm max}\left(\delta\rho^{2}/\rho^{2}\right)>0.078$, which is almost exactly the same as our $\delta\rho_{*}^{2}/\rho_{*}^{2}>0.08$. The small discrepancy is due to slight differences in the definition of the density jump. Wang et al. (2022) define $\delta P_{\rm ram}=P_{\rm ram}(t+\delta t)-P_{\rm ram}(t)$, where $\delta t=10$ ms and $t$ is calculated using Equation (11). Basically, instead of calculating the density jump directly from the pre-collapse progenitor, as we do, they use the density profile to calculate the accretion time and look at density variations in 10 ms windows. Therefore, the actual values they obtain might be slightly different from ours, but the agreement is still excellent.

We can therefore conclude that this is a robust criterion to predict the explodability of massive stars. In their criterion, Wang et al. (2022) use 2D and 3D simulations computed with Fornax to determine the explodability of stars. In our criterion, we use 1D+ simulations computed with GR1D, a completely different code where $\nu$-driven turbulence is added through a time-dependent MLT model. This serves as further confirmation that the main ingredient missing from 1D simulations is $\nu$-driven turbulence, and that STIR does a good job simulating it in spherical symmetry.

To understand the origin of the top-right quadrant misclassification, i.e. progenitors resulting in failed SNe despite being expected to explode according to both criterion (a) and (b), it’s useful to analyze how the criteria change as a function of $\alpha_{\rm MLT}$. In this analysis, we do not require that FRANEC and KEPLER progenitors are simulated using the same value of $\alpha_{\rm MLT}$. The percentage of misclassifications is shown in Figure 8. Even without forcing the two progenitor sets to have the same $\alpha_{\rm MLT}$, both criteria give the best predictions in the range $1.5\leq\alpha_{\rm MLT}\leq 1.52$ for both KEPLER and FRANEC models.

The same best-fit range was derived in section 4 based on the calibrations of Boccioli et al. (2022), as well as on the total explosion fraction shown in Figure 2. This serves as further confirmation of the robustness and consistency of STIR. A more detailed analysis of Figure 8 shows that the best performance of both criteria is obtained with $\alpha_{\rm MLT}=1.51$ for FRANEC progenitors and $\alpha_{\rm MLT}=1.52$ for KEPLER progenitors. This partially explains the misclassifications on the top right quadrant of Figure 7. Two of those KEPLER progenitors explode at $\alpha_{\rm MLT}=1.52$, and three more have $\tau_{\rm adv}/\tau_{\rm heat}>1.1$, which means that they are very close to an explosion. Only two progenitors are then left unaccounted for in the top-right quadrant.

By fine-tuning $\alpha_{\rm MLT}$ and allowing $\alpha_{\rm MLT}=1.51$ for FRANEC progenitors and $\alpha_{\rm MLT}=1.52$ for KEPLER progenitors, one can improve both criteria. Additionally, one can go a step further and consider successful explosions even the 1D+ KEPLER simulations where the shock is not revived, but where $\tau_{\rm adv}/\tau_{\rm heat}>1.1$. The latter condition is based on the concept that even if the 1D+ simulation does not explode, a slightly more efficient $\nu$-driven convection would yield a successful explosion, because of the large $\tau_{\rm adv}/\tau_{\rm heat}$. That would improve both criteria, yielding a 7.5 % misclassification rate for criterion (a) and a 7 % misclassification rate for criterion (b).

This criterion was however developed based on very general principles. Therefore, too much fine-tuning does not add anything to the actual physical interpretation of the criteria, even though it explains the origin of a few KEPLER outliers. The more significant result is that both criteria yield very low misclassification rates even when using the same value of $\alpha_{\rm MLT}=1.51$. Therefore, it can be inferred that STIR does not depend on the pre-collapse history of the progenitor, but simply on the post-bounce thermodynamic conditions in the gain region, as expected.

The present study was partially motivated by a mismatch between the explodability found by Ertl et al. (2016) and the one found by Couch et al. (2020) and Boccioli et al. (2021). It is therefore useful to understand where the discrepancy comes from. In this section, we address this difference by focusing only on the KEPLER progenitors from Sukhbold et al. (2016) since they were used in Ertl et al. (2016).

In the simulations of Ertl et al. (2016) the explosion was triggered using the engine from Ugliano et al. (2012), which was calibrated from observations (i.e. explosion energy and ejected nickel mass) of core-collapse supernovae. In their model, they replace the PNS with an inner boundary from which neutrinos are emitted. The luminosity of the emitted neutrinos is calculated based on the mass accretion rate as well as on the thermodynamic properties of the infalling material. The inner boundary’s contraction follows an analytical prescription fitted to reproduce the explosion energy and ejected nickel mass of SN 1987a (Sonneborn et al., 1987), assuming a 20 M_⊙ progenitor. The luminosities in these models tend to be overestimated by a factor of $\sim 2-3$ compared to realistic 3D simulations.

Firstly, by looking at Figure 9, it’s very clear that our predicted explosions do not follow the criterion from Ertl et al. (2016). They find that there is a line in the $\mu_{4}-M_{4}\mu_{4}$ plane that separates explosions (below the line) from failed SN (above the line). We find almost the exact opposite, although no line can separate our predicted explosions from failed SN. The reason behind the mismatch is that large $\mu_{4}$ typically occur for progenitors with a large $\delta\rho_{*}/\rho_{*}$, which explode according to our criterion. A similar mismatch was found in Couch et al. (2020), whose Figure 13 is very similar to our Figure 9. Since they also use STIR to trigger explosions in 1D, this does not come as a surprise. Instead, it serves as further evidence that what is causing this mismatch is the inclusion of $\nu$-driven turbulence via STIR.

The bottom two panels of Figure 7 from Ertl et al. (2016) show that progenitors that have large mass accretion rates right after the infall of the layer with $s=4$ result in failed supernovae. This suggests that in their explosion models this is the most important feature. Indeed, to trigger the explosion one needs a small ram pressure ahead of the shock, a condition that is satisfied by small mass accretion rates. The results that they find are in line with the critical luminosity condition criterion. However, their luminosities are much larger than what is seen in 3D simulations, and most importantly their results suggest that all progenitors with mass accretion rates above a certain value lead to failed supernovae, regardless of the luminosity.

In our supernova simulations, it is $\nu$-driven turbulence that provides extra pressure behind the shock, and therefore even large ram pressures can be overcome if the neutrino heating (aided by turbulence) is large enough. An example of this can be seen in the progenitors with masses $22<M<25$ M_⊙. These have the largest mass accretion rates amongst the progenitors simulated, as a consequence of their large compactness $\xi_{2.5}>0.3$. According to the criterion by Ertl et al. (2016) these progenitors should not explode, whereas we find the opposite. For these progenitors, the mass accretion rates are counterbalanced by large neutrino heating. This is only possible with the inclusion of $\nu$-driven convection.

This is also evident in Figure 10, to be compared with Figure 6 from Müller et al. (2016a), who find qualitatively similar results to Ertl et al. (2016). For completeness, we show both the KEPLER progenitors on the left panel and the FRANEC progenitors on the right panel. It is clear that even progenitors with large $\xi_{2.5}$, predicted to fail by a simple criterion based on compactness, can successfully explode. Furthermore, no clear explodability trend can be seen in either the ZAMS mass or the compactness.

The mass range $22<M<25$ M_⊙ coincides with the largest compactness progenitors in the KEPLER set. Again, despite being predicted to fail by the criterion of Ertl et al. (2016), these models explode. We emphasize that it is $\nu$-driven convection that generates the heating necessary for these progenitors to sustain shock expansion after the accretion of the density jump. This can be inferred by the $\dot{M}-\dot{Q}_{\nu}$ plane in Figure 11. Panel (a) shows the tracks of a few 1D models (i.e. without STIR), whereas panel (b) shows the tracks for the same 1D+ models.

The reason we chose the $\dot{M}-\dot{Q}_{\nu}$ plane rather than $\dot{M}-L_{\nu}$ is that the latter does not show the influence of $\nu$-driven convection. The luminosity is determined by the thermodynamic properties of the PNS, whereas the contribution from $\nu$-driven convection is only present in the gain region, and therefore cannot be captured by the $\dot{M}-L_{\nu}$ plane.

The tracks in Figure 11 show that the largest difference between 1D and 1D+ is for progenitors with $\xi_{2.5}>0.3$. This indicates how, for these progenitors, the contribution of $\nu$-driven convection is particularly important. Instead, in the $\dot{M}-L_{\nu}$ plane there would be no significant difference between 1D and 1D+. The same holds for the FRANEC progenitors, shown for completeness in Figure 12. The difference, as can be seen in the right panel of Figure 10, is that for FRANEC progenitors $\xi_{2.5}>0.3$ corresponds to progenitor masses with $M>25$ M_⊙.

To summarize, it appears that in progenitors with large mass accretion rates, which can be achieved for large compactnesses $\xi_{2.5}>0.3$, $\nu$-driven convection is very efficient. This is the reason why in previous studies, like the one by Ertl et al. (2016), these progenitors did not explode. Their models did not account for this very important mechanism.

This analysis shows that different methods of triggering the explosion are more dependent on certain properties of the progenitor than others. In this case, the method of Ugliano et al. (2012) appears to strongly depend on $\mu_{4}$ and $M_{4}$, whereas our method strongly depends on $\delta\rho_{*}/\rho_{*}$. This might seem like a moot point, but it needs to be stressed that just because one method can produce an explosion using a physically motivated model, it does not mean that Nature operates in the same way.

There are still many uncertainties in stellar evolution prescriptions. These arise in part from using different codes (Chieffi & Limongi, 2020) as well as different reaction rates (Chieffi et al., 2021; Sukhbold & Adams, 2020). In addition to that, there are many uncertainties in the collapse and explosion phase affecting the nuclear matter equation of state, neutrino physics, and neutrino transport algorithms.

In this work we only considered radial perturbations in density, that remain roughly constant during the accretion onto the shock. However, previous studies (Kovalenko & Eremin, 1998; Lai & Goldreich, 2000; Takahashi & Yamada, 2014; Nagakura et al., 2019) have shown that non-radial perturbations (i.e. $l>0$) can significantly grow during accretion, with larger $l$ yielding larger amplifications. From this one can conclude that progenitor asphericities likely play a very important role in determining the outcome of the explosion, as shown by several 3D simulations (Müller et al., 2016b; O’Connor & Couch, 2018b; Fields & Couch, 2021).

Finally, in this study, we do not definitively claim that we can provide the true explodability of massive stars. By looking at the differences between the upper and bottom panels of Figure 1, it is clear that large uncertainties in the stellar models can significantly affect the explodability of supernovae. Our goal, however, has been to show that $\nu$-driven convection plays an important role in reviving the shock together with the accretion of large density discontinuities.