Monotonicity of the cores of massive stars

The evolution of single He stars is calculated using the HOSHI code (Takahashi et al., 2018; Takahashi & Langer, 2021). The initial chemical composition is set to pure helium, and the initial metallicity is zero. The input physics used in the code is almost the same as that used in Takahashi & Langer (2021), so we omit writing the details here and describe only the differences. First, while the code is capable of treating the time evolution of the stellar rotation and the magnetic fields, these are neglected in the present models. Also, wind mass loss is not taken into account in the current modeling. The nuclear reaction network includes 300 isotopes ranging from n and p to ⁸⁰Br. The complete list is given in Takahashi et al. (2018). The ¹²C($\alpha$,$\gamma$)¹⁶O rate of deBoer et al. (2017) is applied for our fiducial models. The effect of convective boundary mixing is treated as a diffusive process as described in Takahashi & Langer (2021), however, the effect is set to be minimal as a very small control parameter of $f_{\mathrm{ov}}=0.001$ is applied for the current models.

We calculate stellar evolution for a total of 128 models with different initial masses. The initial mass interval ($dM_{\rm ini}$) is changed for light, intermediate-mass, and heavy stars. For the lightest stars in the range $M_{\rm ini}/M_{\odot}\in[2.0,9.1]$, we calculate 72 models using $dM_{\rm ini}/M_{\odot}=0.1$, and for the intermediate mass range $M_{\rm ini}/M_{\odot}\in[9.2,14.0]$, 25 models are calculated with $dM_{\rm ini}/M_{\odot}=0.2$. For the heavier side $M_{\rm ini}/M_{\odot}\in[14.5,29.5]$, an increment of $dM_{\rm ini}/M_{\odot}=0.5$ is used (31 models).

For each model, the evolution from the He-zero-age-main-sequence (HeZAMS) phase until the central density, $\rho_{c}$, reaches $10^{10}$ g cm^-1 is calculated unless the simulation stops due to convergence problems. We have confirmed that the star has already lost hydrostatic stability with this final central density. Convergence problems happen for the lowest mass models with $M_{\rm ini}\leq 2.7M_{\odot}$, where two models stop during the shell carbon burning phase ($M_{\rm ini}/M_{\odot}=2.0,2.1$), five models stop after the formation of the ONe core ($M_{\rm ini}/M_{\odot}\in[2.2,2.6]$), and one model of $M_{\rm ini}/M_{\odot}=2.7$ stops during the shell O+Ne burning phase. After removing the 8 non-convergent models, we obtained in total of 120 progenitor models of CCSNe.

As a result of neglecting the wind mass loss, the CO core mass ($M_{\rm CO}$) distribution obeys a highly monotonic relation with the He star mass (Fig. 1). In this work, $M_{\rm CO}$ is defined as the lower mass limit of the region where the helium mass fraction exceeds 0.01. The usage of other definitions (such as based on the heating rate) is possible but the qualitative results would be unchanged. Hereafter, we will use the CO core mass, instead of the He star mass, as the model indicator.

In order to estimate the fate after core collapse, a semi-analytic code has been developed following the description in Müller et al. (2016). By integrating a few ordinary differential equations, Müller’s semi-analytic model provides the result of the post-collapse evolution including the fate and the explosion properties such as the explosion energy and the remnant (NS) mass if the progenitor is estimated to successfully explode.

This model relies on the delayed neutrino-heating mechanism for CCSNe, in which a fraction of the gravitational energy released by the accretion is converted into thermal energy as a result of the neutrino energy transport. Hence, the shock revival is assumed to happen if the neutrino heating timescale becomes shorter than the advection timescale, $\tau_{\rm heat}<\tau_{\rm adv}.$ Each of the timescales is estimated based on scaling relations and fitting formulae obtained from realistic simulations. Shock propagation after the revival is treated differently depending on whether the shock is strong enough to blow off the post-shock material. In the earlier phase, the post-shock material is still bound and a fraction of the shocked material is assumed to accrete onto the central remnant, leading to the growth of the remnant mass as well as the additional energy injection. On the other hand, all the shocked material is ejected in the later phase, and accordingly, the remnant mass becomes constant and the explosion energy is changed only by the explosive nucleosynthesis. The transition is assumed to take place when the post-shock velocity exceeds the local escape velocity, $v_{\rm post}>v_{\rm esc}.$

An outline of the flow that determines fate is as follows. First, it is assumed that stars that do not experience shock revival fail to explode and eventually form BHs. Of those that experience shock revival, stars that are affected by significant matter accretion after shock revival are also assumed not to explode and form BHs. The judgment on this is based on the evolution of either the proto-neutron star (PNS) mass, diagnostic explosion energy, or redshift correction at the surface of the PNS. Eventually, stars that experience shock revival and are affected by minimal matter accretion are assumed to successfully explode and form NSs. BH formation can be accompanied by matter ejection if an accretion disk surrounding the central BH is formed (e.g., Just et al., 2022; Fujibayashi et al., 2022) or if a large energy loss due to neutrinos occurs (Lovegrove & Woosley, 2013). However, the possibility of a successful explosion from a BH forming progenitor is not considered in our model as in Müller’s original work, given the huge uncertainties in the current theory.

We have carefully constructed the code and have confirmed that it yields largely consistent results with the original work. Hence, we believe that the conclusions regarding the fate presented in this work will be unaffected by the different implementations of this model. Nevertheless, some disagreements have remained. For the traceability, we provide our implementation of the model in the Appendix.

The advantage of utilizing a semi-analytic model is the low cost of the computation, which enables us to compare the explosion properties for hundreds of progenitors (e.g., Schneider et al., 2021; Aguilera-Dena et al., 2022). On the other hand, care must be taken since the model relies on many approximated relations. Comparison with the most realistic, and thus the most computationally expensive, simulations (e.g., Takiwaki et al., 2016, 2021; Müller et al., 2017; Nagakura et al., 2018; Vartanyan et al., 2019; Bollig et al., 2021) and with statistical properties derived from a number of multi-dimensional simulations (Nakamura et al., 2015; Summa et al., 2016; Suwa et al., 2016; Pan et al., 2016; Vartanyan et al., 2018; Ott et al., 2018; O’Connor & Couch, 2018; Burrows et al., 2020) will be needed for verification. For this purpose, a discussion of whether the trends obtained in this model are consistent with previous studies is given in the Appendix.

As noted by O’Connor & Ott (2011), care must be taken for the timing of evaluation of the compactness parameter because the value can change as the star collapses. To avoid this uncertainty, O’Connor & Ott (2011) uniformly evaluate the compactness parameter at the core bounce time. However, calculations until core bounce may be expensive for a stellar evolution simulation. Instead, it is beneficial if one has a criterion about the timing, after which the constancy of the compactness parameter is guaranteed.

Throughout this work, we set 2.5 $M_{\odot}$ as the reference mass coordinate to assign the compactness parameter for a given progenitor structure ($\xi_{2.5}$). The evolution of $\xi_{2.5}$ as a function of central density is shown in Fig. 2. For this purpose, a sequence of zero-metallicity stellar models is additionally calculated, and results of models with initial masses 11, 20, 30, 60, and 90 $M_{\odot}$ are plotted. Their ZAMS and the CO core masses are indicated in the figure as well. It shows that more compact models require a larger central density for the time evolution of $\xi_{2.5}$ to converge. Less compact models that have $\xi_{2.5}$ $\lesssim$ 1.0 when the central density reaches $10^{10}$ g cm^-3 do not change their compactness values thereafter. Therefore, the $\xi_{2.5}$ measured when the central density reaches $10^{10}$ g cm^-3 for models with $\xi_{2.5}$ $\lesssim$ 1.0 is expected to be maintained until core bounce. On the other hand, to measure converged values of compactness for more compact models characterized by $\xi_{2.5}$$\gtrsim$ 1.0, a central density greater than $10^{11}$ g cm^-3 is needed¹¹1Here, we have implicitly assumed that the convergence of compactness with respect to time evolution depends monotonically on compactness. This assumption can be confirmed from Fig. 2 and can be inferred from the monotonicity of the cores found in this study..

For our fiducial He star models, Fig. 3 shows the distribution of $\xi_{2.5}$ as a function of the CO core mass, which is evaluated at $\rho_{c}=10^{10}$ g cm^-3. Note that the results for models with $M_{\rm CO}\leq 1.4M_{\odot}$ ($M_{\rm He}\leq 2.7M_{\odot}$) are not included here since evolution simulations for these models are halted before their central densities reach $10^{10}$ g cm^-3. Given the results of the examination of the compactness convergence presented above, the central density $\rho_{c}=10^{10}$ g cm^-3 is large enough to converge $\xi_{2.5}$ for these He star models since all of these models have $\xi_{2.5}$ $<$ 1.0, especially $\xi_{2.5}$ $\lesssim$ 0.7 for $M_{\rm CO}\leq 15M_{\odot}$.

The feature of the $\xi_{2.5}$ distribution will be summarized as follows: $\xi_{2.5}$ monotonically increases with the mass in the less massive end with $M_{\rm CO}<2.6M_{\odot}$. Except for some offsets, the monotonic behavior is kept until $M_{\rm CO}<4.0M_{\odot}$. The compactness follows an interesting decreasing trend with significant scatter and reaches a local minimum at $M_{\rm CO}=6.2M_{\odot}$. After showing a steep increase, it shows a prominent peak at $M_{\rm CO}=6.9M_{\odot}$. Then a second decreasing trend follows, after which the second local minimum exists at $M_{\rm CO}=8.8M_{\odot}$. At last, the compactness follows a monotonically increasing trend. These are, in particular, very consistent with the KU series in Sukhbold & Woosley (2014), which is a CO core model series with a very metal-poor $10^{-4}$ $Z_{\odot}$ initial composition²²2More smooth $\xi_{2.5}$ distributions as a function of the CO core mass can be found in Fig. 21 and 22 of Limongi & Chieffi (2018). This may be because they have applied wide initial mass spacing between models and have mixed models with different metallicities in the figures. Also, the higher carbon mass fraction in their models (Chieffi & Limongi, 2020) could account for the difference..

The non-monotonic behavior has been described in detail by Sukhbold & Woosley (2014). In accordance with their analysis, we have confirmed that the first decreasing trend in $M_{\rm CO}\in[4.0,6.2]\ M_{\odot}$ is due to the decreasing widths of C burning convective regions (both the core and the shell convective regions), which is related to the transition of the convective to radiative nature of central C burning. We have confirmed that the sudden increase in the range of $M_{\rm CO}\in[6.2,6.9]\ M_{\odot}$ as well as the more gentle increase in $M_{\rm CO}\geq 8.8M_{\odot}$ result from outward migration of the third or second C burning shells in these mass ranges, and the transitional inward migration of the second C burning explains the decreasing trend between them.

There are changes of inclination at $M_{\rm CO}=2.6M_{\odot}$ and $15.6M_{\odot}$. We note that these changes are artificially introduced through the definition of the compactness parameter. Namely, the inclination in the compactness distribution is affected by the chemical composition of the location where it is estimated because layers of two different chemical compositions can have different density gradients. $\xi_{2.5}$ is evaluated at the enclosed mass of 2.5 $M_{\odot}$. While this location is inside the oxygen-carbon layer in the majority of cases, it is included in the helium layer for the less massive models with $M_{\rm CO}<2.6M_{\odot}$, and is included in the inner carbon-free layer for more massive models with $M_{\rm CO}>15.6M_{\odot}$. Therefore, the two changes of inclination should not mean qualitative changes in the density structure.

The compactness parameter may not be the unique indicator for characterizing the density structure of a progenitor model. To find another indicator, we utilize the free-fall time, which can be defined as a function of the mass coordinate as

The free-fall time is in proportion to the inverse of the average density, hence, it becomes a monotonically increasing function as long as the density distribution is monotonically decreasing. We define the mass coordinate at which the free-fall timescale exceeds a provided time reference as $M_{\rm ff}$. In this work, a reference time of 1 s is used³³3We do not have a strong motivation to decide the reference time of 1 s, though it is perhaps closer to the timescale of CCSN explosion than 0.1 or 10 s. In fact, the results shown later are insensitive to the choice, and the fact that there is no specific way to determine the reference time at least up to the zeroth order is an important property that supports the monotonicity of the core discussed later.. Owing to the monotonicity of the free-fall time, a unique solution is found for $M_{\rm ff}$.

Figure 4 shows the distribution of $M_{\rm ff}$ evaluated when $\rho_{c}=10^{10}$ g cm^-3. The similarity to the compactness distribution is apparent. For instance, both $\xi_{2.5}$ and $M_{\rm ff}$ follow decreasing trends for $M_{\mathrm{CO}}\in[4.0,6.2]\ M_{\odot}$ and $M_{\mathrm{CO}}\in[6.9,8.8]\ M_{\odot}$, which are overlaid with the cyan bands as well as the sudden increasing trends between them overlaid with the red band. The accurate correspondence may not be so surprising because both quantities are merely determined by ratios between (powers of) the mass coordinate and the radius.

Nevertheless, there are several benefits of utilizing $M_{\rm ff}$ instead of $\xi_{2.5}$. Firstly, the value of $M_{\rm ff}$ is more intuitive since it provides a rough estimate of the averaged mass accretion rate during the formation of the PNS. At the same time, this value can be regarded as an estimate of the remnant mass, assuming that shock revival occurs in about 1 s after core collapse and that subsequent mass accretion is negligible. Secondly, $M_{\rm ff}$ is applicable to a less massive progenitor model in which the reference mass coordinate ($M$ of $\xi_{M}$) is outside the CO core. In such a case, the compactness parameter is evaluated to be extremely small, while $M_{\rm ff}$ is basically set to be the CO core mass. Similarly, the monotonically increasing trend in more massive models of $M_{\rm CO}>15.6M_{\odot}$ is now linearly followed. Hence, fewer artifacts are included in the $M_{\rm ff}$ distribution. Thirdly, $M_{\rm ff}$ has a better convergence on the evolutionary phases than $\xi_{2.5}$. The evolution of $M_{\rm ff}$ as a function of the central density is shown in Fig. 5 for the same zero-metallicity models as in Fig. 2, which shows better convergence of $M_{\rm ff}$ for the most massive $M_{\rm CO}=34.5M_{\odot}$ model than $\xi_{2.5}$.

A massive star is considered to form an onion-like chemical structure, in which layers composed of heavier elements are located closer to the stellar center. In addition to the density distribution, the chemical distribution is also fundamental to the progenitor structure. In order to characterize the chemical distribution, we define base masses, that is, the mass coordinates of bases of chemically defined layers. For example, the silicon base mass, $M_{\rm Si,base}$, is defined as the innermost mass coordinate where the mass fraction of ²⁸Si firstly exceeds $X(^{28}\rm{Si})$ = $10^{-2}$. Similarly, the oxygen base mass ($M_{\rm O,base}$) and the carbon base mass ($M_{\rm C,base}$) are defined by the conditions of $X(^{16}\rm{O})$ = $10^{-3}$ and $X(^{12}\rm{C})$ = $10^{-3}$. Although the reference mass fractions are set arbitrarily, the base masses defined here will correspond to the traditional core masses such as the Fe and Si core masses.

The relation between the base masses and $M_{\mathrm{CO}}$ is shown in Fig. 6. Similar to the $M_{\rm ff}$ distribution, the distributions of the three base masses clearly share the basic features of non-monotonicity obtained for the $\xi_{2.5}$ distribution. The correlation of $M_{\rm Si,base}$ is comparable to the correlation between the compactness parameter and the iron core mass shown in O’Connor & Ott (2011). In addition, we find that $M_{\rm O,base}$ and $M_{\rm C,base}$, which are defined at outer layers than $M_{\rm Si,base}$, also show strong correlations with $\xi_{2.5}$ and $M_{\rm ff}$.

The time evolution of $M_{\rm O,base}$ and $M_{\rm Si,base}$ as a function of central density is shown in Fig. 7. $M_{\rm O,base}$ is basically kept constant if $\rho_{\mathrm{c}}>10^{9}$ g cm^-3. An exception is the model with $M_{\rm CO}$ $=1.54\ M_{\odot}$, but the change is $\sim 15$% and it approaches convergence if $\rho_{\mathrm{c}}\gtrsim 10^{10}$ g cm^-3. On the other hand, $M_{\rm Si,base}$ is farther from convergence since it is defined more inside than $M_{\rm O,base}$, where the temperature is higher and the nuclear reaction timescale is shorter. Conversely, we have confirmed that $M_{\rm C,base}$ becomes nearly constant at least for $\rho_{\mathrm{c}}>10^{8}$ g cm^-3.

Ertl et al. (2016) have analyzed the interaction between matter accretion and neutrino heating and have proposed a criterion to judge the explodability of CCSN progenitors, which can discriminate between explosion and non-explosion with high accuracy of only a few percent exceptions. They have defined two parameters, $M_{4}$ and $\mu_{4}$, which are related to both the density and entropy distributions, and have used $M_{4}$ and the product of the two, $M_{4}\mu_{4}$, for the criterion. $M_{4}$ is defined as the mass coordinate at which the entropy per baryon firstly exceeds the reference value of 4 $\mathrm{k_{B}}$. For clarity, we express the parameter by $M(s_{k}\!=\!4)$ hereafter in this work. The other parameter, $\mu_{4}$, is the normalized mass derivative, (dM/dr)/($M_{\odot}$/1000 km), evaluated at $M(s_{k}\!=\!4)$. Again, we express the parameter by $\mu(s_{k}\!=\!4)$. In practice, $\mu(s_{k}\!=\!4)$ has been evaluated by numerical differentiation as $\mu(s_{k}\!=\!4)$ $=\Delta M/[r(M(s_{k}\!=\!4)+\Delta M)-r(M(s_{k}\!=\!4))]$ with the mass interval of $\Delta M=0.3M_{\odot}$ in the original work, and has not been directly related to the density at $M(s_{k}=4)$ that is implied by the formal definition.

The distributions of $M(s_{k}\!=\!4)$ and $\mu(s_{k}\!=\!4)$ as a function of $M_{\rm CO}$ are shown in Fig. 8. Both parameters exhibit non-monotonic CO core mass dependencies, which are very similar to the $\xi_{2.5}$ and $M_{\rm ff}$ distributions in the less massive models characterized by $M_{\rm CO}$ $\lesssim 17\ M_{\odot}$. This property originates from the fact that $M(s_{k}\!=\!4)$ is almost identical to $M_{\rm O,base}$ on the less compact side $M_{\rm ff}$ $\lesssim 3.5\ M_{\odot}$ corresponding to $M_{\rm CO}$ $\lesssim 17\ M_{\odot}$, as Fig. 9 plotting the distribution of $M(s_{k}\!=\!4)$ and $M_{\rm O,base}$ as a function of $M_{\rm ff}$ shows. This is because oxygen burning forms a strong entropy jump; in fact, $M(s_{k}\!=\!4)$ has been used as an indicator to specify the O-burning shell (cf. Heger & Woosley, 2010). Hence, the correlation between $M(s_{k}\!=\!4)$ and $M_{\rm ff}$ shown here is essentially identical to the one between $M_{\rm O,base}$ and $M_{\rm ff}$, which has been discussed earlier. Meanwhile, the $M(s_{k}\!=\!4)$ and $\mu(s_{k}\!=\!4)$ distributions in models characterized by $M_{\rm CO}$ $\gtrsim 17\ M_{\odot}$ show completely different trends. The distribution of $M(s_{k}\!=\!5)$ as a function of $M_{\rm ff}$ in Fig. 9 shows that, in models with $M_{\rm ff}$ $\gtrsim 3.5\ M_{\odot}$ corresponding to the heavier models, the indicator tracing $M_{\rm O,base}$ shifts from $M(s_{k}\!=\!4)$ to $M(s_{k}\!=\!5)$ explaining the changes in the trends. The shift is due to the effect that the entropy of the entire core is larger for more compact models (see Section 3.5), and the entropy after the jump exceeds $s_{k}=5$.

Similar to Fig. 5, the evolution of $\mu(s_{k}\!=\!4)$ in the later evolutionary phase is shown in Fig. 10. It shows that the less massive models with $M_{\rm CO}$ $\lesssim 20M_{\odot}$ and $\mu(s_{k}\!=\!4)$ $\lesssim 0.2$ keep $\mu(s_{k}\!=\!4)$ nearly constant in the later phase of $\rho_{c}>10^{9}$ g cm^-3. Besides, we have confirmed that $M(s_{k}=4)$ stays constant independent of the CO core mass, which is consistent with the identity between $M(s_{k}\!=\!4)$ and $M_{\rm O,base}$. Hence, as far as the identity between $M(s_{k}\!=\!4)$ and $M_{\rm O,base}$ is established, both $\mu(s_{k}\!=\!4)$ and $M(s_{k}\!=\!4)$ stay constant during the collapsing phase. The figure also shows that $\mu(s_{k}\!=\!4)$ changes by $\sim$ 30% for the more massive models. However, this result may not be relevant to us, because $\mu(s_{k}\!=\!4)$ of such massive models will have incompatible characteristics with that of the less massive models because it lacks the identity to $M_{\rm O,base}$.

In Fig. 11, the entropy of the stellar center is shown as a function of CO core mass. Given the similar shapes of the distributions, this figure shows that the central entropy is strongly correlated with the base masses of the Si, O, and C layers as well as $M_{\rm ff}$. It is noteworthy that Schneider et al. (2021) attributes the correlation between the iron core mass and the central entropy to the property of quasi-isentropic iron cores. Our results are consistent with this understanding, but more so, imply that this correlation can be traced back to earlier evolutionary phases as the correlation extends not only to the base masses of the inner Si and O layers but also to the base mass of the C layer.

However, the correlation is not so trivial from the point of view of stellar evolution because the core entropy should initially correlate with the total mass of the star, and the total mass is not correlated well with $M_{\rm ff}$. In order to develop the monotonic dependence on $M_{\rm ff}$, the entropy order must reverse in some models during the stellar evolution. The example of the reversal is illustrated by the evolution in the central density and temperature plane shown in Fig. 12, in which results of models with $M_{\rm CO}=$ 3.12, 7.82, and 8.70 M_⊙ are compared. They have $M_{\rm ff}$ = 1.94, 2.51, and 1.94 $M_{\odot}$ and the mass coordinates of the last shell C burning, $M_{\rm C,base}$, of 1.85, 2.96, and 1.93 $M_{\odot}$ for models with $M_{\rm CO}$= 3.12, 7.82, and 8.70 $M_{\odot}$, respectively. In the beginning, the entropy order coincides with the mass order as expected. The lowest mass model follows the track of the lowest entropy, in which several bumpy features (e.g., a bump at $T_{c}\sim 10^{8.8}$ K, a loop at $T_{c}\sim 10^{9.2}$ K, bumps at $T_{c}\sim 10^{9.3}$ and $\sim 10^{9.6}$ K, respectively due to the C, Ne, O, and Si burnings) appear due to the relatively high electron degeneracy. The higher mass models initially follow higher entropy tracks, which have fewer bumpy features. However, the entropy order reverses after the central Ne burning ($T_{c}\sim 10^{9.2}$ K). After this phase, the most massive model with $M_{\rm CO}$= 8.70 M_⊙ eventually follows a converging evolution on the track of the lowest mass model with $M_{\rm CO}$= 3.12 M_⊙.

Based on the rough concurrence of the start of the last C burning with the start of the reversal of the entropy order, we speculate that the late evolution after the central Ne burning can be described as an evolution of a core that has an effective core mass given by the base mass of the last C burning shell. The reversal taking place in the model with $M_{\rm CO}$= 8.70 M_⊙ would be understood as a relaxation process, in which a core having a high initial entropy eventually cools to adopt a lower entropy that is required to contract with a given small effective mass.

The late-time evolution of the central entropy is plotted in Fig. 13. It shows that the central entropy becomes nearly constant for $\rho_{\mathrm{c}}\gtrsim 10^{9}$ g cm^-3 irrespective of $M_{\rm CO}$. This result indicates that the rate of change of $Y_{e}$ and the accompanied neutrino emission do not significantly affect the entropy in the central NSE region during the early collapsing phase. We note that the entropy change after $\rho_{\mathrm{c}}\gtrsim 10^{11}$ g cm^-3 shown in the figure is unreliable. This is because our stellar evolution code does not treat neutrino trapping, which will take place in such a high-density region. Both the $Y_{e}$ evolution and neutrino emission will be overestimated in the high-density regions. The entropy evolution will be less substantial in the later collapsing phase if the neutrino processes are properly treated.

So far, we have investigated the behavior of parameters that characterize the progenitor structure such as $\xi_{2.5}$ and $M_{\rm ff}$. Although interesting correlations among these parameters have been found, relevance to the global structure is not clear. In order to link these parameters to the global progenitor structure, distributions of the entropy, density, and chemical elements are plotted in Fig. 14, in which the two models with $M_{\rm CO}$= 3.12 and 8.70 M_⊙, the converging evolutions of which have been discussed in the previous section, are compared.

In spite of the difference in the CO core masses, these models show a striking resemblance of the entropy distributions, in particular for the inner regions of $\sim 1.9\ M_{\odot}$, which is shown in the top panel. The most important feature is the coincidence of the significant jumps at $\sim 1.6\ M_{\odot}$, which trace the strong heating of the shell O burning. Not only the locations but also the level differences are similar. The entropy structures inside the jump are also similar, though less significant saw-shaped bumps, which are remnants from previous shell Si burning phases, are involved. Mass coordinates of the last shell C burnings that are indicated by more or less significant jumps outside the O burning bases are close as well. Meanwhile, the outer distributions of shell C burning are totally different. The entropy of the convective C burning layers are $\sim 5.1$ and $\sim 6.2$ respectively for the less and more massive models. This order originates from the entropy order of the CO cores. Furthermore, while the plot includes the high entropy helium envelope surrounding the CO core of $3.12\ M_{\odot}$ for the less massive model, the corresponding structure is out of the plot for the more massive model as it has a more extended CO core of $8.70\ M_{\odot}$.

The density distributions shown in the middle panel also validate the close resemblance of the progenitor’s internal structures. As we have selected the progenitor models having the same central density of $10^{10}$ g cm^-3, they show nearly perfect agreement up to $\sim 1.9M_{\odot}$, where carbon layers begin. Inside the carbon layer, density jumps locate at $\sim 1.6M_{\odot}$ in both models, which, of course, coincide with the entropy jump due to the shell O burning. On the other hand, density distributions outside the carbon layers are totally different, as they must connect with the helium layers at different locations.

Crosses in the top and middle panels indicate the locations of $M_{\rm ff}$ for both models. By chance, the locations roughly correspond to the bases of the shell C layers at $\sim 1.9M_{\odot}$. The $M_{\rm ff}$ depends on the mass and radius distributions, hence only on the density distribution. Therefore, by having the nearly same density structures of inner $\sim 1.9M_{\odot}$ regions, these two models give the same $M_{\rm ff}$ for reference times shorter than 1 s. In turn, these models are also expected to have nearly the same mass accretion histories up to $\sim 1$ s after core collapse.

Consistent with the similarities confirmed for the entropy and density distributions, the chemical distributions shown in the bottom panel are also similar for these two models. Accordingly, the models have similar base masses of ($M_{\rm Si,base}$/$M_{\odot}$, $M_{\rm O,base}$/$M_{\odot}$, $M_{\rm C,base}$/$M_{\odot}$) = (1.49, 1.63, 1.85) and (1.50, 1.64, 1.93), respectively.

The similarities discussed in the above section imply that it is possible to represent the global core structure based on the parameter $M_{\rm ff}$, which is calculated from only partial information about the core structure. To further investigate this implication, density distributions of all models are projected by using a color coordinate in Fig. 15. Models are sorted according to $M_{\rm CO}$ in the top panel, and each horizontal line shows the density distribution of one model. The location of $M_{\rm ff}$ is shown by the black dashed line, and locations of base masses of $M_{\rm Si,base}$, $M_{\rm O,base}$, and $M_{\rm C,base}$ are overplotted by the black, red, and blue solid lines. Because the central densities of our models are adjusted to be $\rho_{\mathrm{c}}=10^{10}$ g cm^-3, inner density structures of $\rho\gtrsim 10^{9}$ g cm^-3 are nearly identical. On the other hand, density structures marked by the color boundaries of $\rho=$ $10^{8}$, $10^{7}$, and $10^{6}$ g cm^-3 show similar $M_{\rm CO}$ dependencies to $\xi_{2.5}$ and $M_{\rm ff}$. The models are sorted according to the $M_{\rm ff}$ order in the bottom panel. This plot demonstrates that, by sorting the models with $M_{\rm ff}$, the density structure approximately inside $M_{\rm C,base}$ can be sorted into a highly monotonic sequence for the wide range of $M_{\rm CO}$.

In addition to the density structure, the thermal structure follows a monotonic sequence once the models are sorted with $M_{\rm ff}$. This is shown by Fig. 16, in which the temperature distributions ordered by $M_{\rm ff}$ are shown in the top panel, and the entropy distributions are in the bottom. The high monotonicity of the temperature distributions will explain the strong correlations between $M_{\rm ff}$ and the chemically defined base masses. This is because the temperature is the chief determinant of nuclear reaction rates so the locations of the elemental bases are well traced by the contour of constant temperature, such as $M_{\rm Si,base}$ by $T\sim 10^{9.6}$ K, $M_{\rm O,base}$ by $T\sim 10^{9.5}$ K, and $M_{\rm C,base}$ by $T\sim 10^{9.3}$ K. Although the monotonicity in the entropy distributions is much more complicated than in the density and temperature distributions, the figure shows that the central entropy follows the $M_{\rm ff}$ order as discussed in the previous section. Also, note that the coincidence of $M_{\rm O,base}$ and the color boundary of the entropy of $s_{k}=4.0$ for models with $M_{\rm ff}$ $\lesssim 3.5M_{\odot}$ or $s_{k}=5.0$ for models with $M_{\rm ff}$ $\gtrsim 2.5M_{\odot}$ indicates that the location of the most significant entropy jump in a collapsing star also correlates well with $M_{\rm ff}$ as discussed above.

From this result, we further deduce that the inner structure of a collapsing star can be identified as first order by specifying only a single parameter, even though the late-time stellar evolution, especially the convective evolution, of CCSN progenitors is quite complicated. The density-dependent parameter $M_{\rm ff}$, or equivalently $\xi_{2.5}$, is applicable for the sorting. Besides, provided the strong correlations, other parameters such as base masses of $M_{\rm Si,base}$, $M_{\rm O,base}$, and $M_{\rm C,base}$ or the central entropy are also plausible.

Once an evolutionary phase is properly defined, the remaining time from that phase to core collapse can be evaluated based on a stellar evolution calculation. In this work, we define evolutionary phases mainly based on chemical composition. The location of the maximum temperature is first taken as a reference position for one time-snapshot (most of the time, it is at the stellar center). The evolutionary phase, $iphase$, is set according to the chemical composition at the reference position as

and the initial value of $iphase=1$ is set for the start of the simulation, the He ZAMS phase. For each $iphase$, the reference element is defined as

and furthermore, the mass fraction of the reference element, $X_{\mathrm{ref}}=$(mass fraction of $elem(iphase)$ at the reference position), is defined.

The remaining time till core-collapse estimated for our model set is shown in Fig 17. Similar to Fig. 15, the remaining times are shown according to the $M_{\rm CO}$ order in the top panel and shown according to the $M_{\rm ff}$ order in the bottom panel. The rightmost black line indicates the start of the simulation of the He ZAMS phase. The leftmost lines of each color, corresponding to the boundaries between different $iphase$, are set to indicate the depletion times of the corresponding elements. Other thinner lines indicate the depleting processes; in the beginning of each $iphase$, the initial reference mass fraction $X_{\mathrm{ref,0}}$ is recorded, and lines are drawn when $X_{\mathrm{ref}}/X_{\mathrm{ref,0}}=$ 0.9, 0.8, …, and 0.1. Hence, the thinner lines of $X_{\mathrm{ref}}/X_{\mathrm{ref,0}}=0.9$ roughly indicate the beginning of each nuclear-burning phase but carbon. Since the mass fraction of ¹²C has already begun to decrease due to the ¹²C($\alpha$,$\gamma$)¹⁶O reaction in the late core He burning phase, $X_{\mathrm{ref}}/X_{\mathrm{ref,0}}\sim 0.7$ may provide a better proxy for the initiation of the core C burning. Also note that a depletion line indicates the time when the reference element is depleted at the reference location for the first time, but it does not necessarily mean the complete depletion of the reference element from the whole core. On the contrary, the depletion is usually followed by successive shell burnings. In particular, the shell C burning phase starts after central C depletion.

The remaining times for the He burning phase show clear correlations to the CO core mass, thus, to the He core mass and presumably to the ZAMS mass. Similarly, the remaining times of the C burning phase also show strong correlations to $M_{\mathrm{CO}}$, though the mass dependency is stronger than the He burning phase. The least massive models take $\sim 10^{4}$ yr from the initiation of core C burning till collapse, while it takes only $\sim 1$ yr for the most massive models in our sample. This huge difference is due to the significant temperature dependency of the neutrino cooling rate. Besides, a jump at $M_{\mathrm{CO}}\sim 5M_{\odot}$ indicates a transition from the convective to the radiative nature of the central C burning. Above this transition, the duration of the central C burning is reduced because convective transport no longer supplies nuclear fuel to the center.

The later Ne, O, and Si burning phases show peak structures; the durations are longest locally around $M_{\mathrm{CO}}\sim 6M_{\odot}$, decrease towards the local shortest peak at $M_{\mathrm{CO}}\sim 7M_{\odot}$, increase until $M_{\mathrm{CO}}\sim 9M_{\odot}$, then decrease constantly. These features are quite consistent with the trends obtained for the $\xi_{2.5}$ and $M_{\rm ff}$ distributions, which are indicated in the top panel as cyan and red bands. Therefore, clear monotonic correlations can be manifested when the remaining times are expressed as a function of $M_{\rm ff}$, as shown in the bottom panel.

The plot still involves a huge scatter, especially in the less massive range with $M_{\rm ff}$ $\in[1.6,2.3]\ M_{\odot}$. This scatter is not due to the shuffling of models having smaller and larger $M_{\rm CO}$, but rather originates from the scatter seen in the less massive models with $M_{\rm CO}$ $\in[3,6]\ M_{\odot}$. At present, it is unclear whether this kind of scatter is realistic or not. The large scatter is likely induced by the highly non-linear interplay of nuclear burning and convective mixing, which can significantly affect the lifetimes of nuclear burning phases. Hence, real stars would show the same scatter. However, from a numerical point of view, such behavior might be enhanced due to coarse resolutions both in space and time. To answer the question, further investigation is needed.

It is noteworthy that the least massive models with $M_{\rm ff}$ $\leq 1.56\ M_{\odot}$ ($M_{\rm CO}$$\leq 1.72\ M_{\odot}$) start the ‘core Si burning’ $\sim 1$ year before core-collapse. This Si burning is induced by the off-center O+Ne flash, which takes place in a low mass oxygen core having a high electron degeneracy (Umeda et al., 2012; Woosley & Heger, 2015). The relevance of observations is discussed later.

The radius evolution of all models is plotted in Fig. 18, in which the evolutionary phases are also shown. The mass dependence of the radius evolution up to core C depletion is rather simple. As a common feature, a He star model first expands and then contracts during the core He burning phase. In this phase, the smaller the initial mass is, the smaller the stellar radius is. Later, this relation is reversed by the shell He burning; the helium envelope expands more for less massive models, while it keeps contracting for more massive models after core He depletion. The expansion/contraction bifurcation takes place at $M_{\mathrm{CO}}\sim 15$ $M_{\odot}$.

On the other hand, the evolution after core C depletion has a more complicated mass dependency. The less massive models with $M_{\mathrm{CO}}\lesssim 4.3$ $M_{\odot}$ except for the least massive model hardly change the radii for the later phases, thus $|R_{\mathrm{collapse}}/R_{\mathrm{Cdep}}|<$ 0.05 dex, where $R_{\mathrm{collapse}}$ and $R_{\mathrm{Cdep}}$ are the stellar radii at the core-collapse and core C depletion phases. Models with $M_{\mathrm{CO}}\sim$ 4.5–6.8 $M_{\odot}$ expand their radii during Ne and O burning phases. This is particularly true for models with $M_{\mathrm{CO}}\sim$ 5.6–6.4 $M_{\odot}$, in which $R_{\mathrm{collapse}}/R_{\mathrm{Cdep}}$ can be as large as +0.1 dex. Conversely, models with $M_{\mathrm{CO}}\sim$ 7.1–8.4 $M_{\odot}$ contract during the later phases, resulting in $R_{\mathrm{collapse}}/R_{\mathrm{Cdep}}\sim-0.1$ dex. More massive models with $M_{\mathrm{CO}}\gtrsim$ 8.7 $M_{\odot}$ expand after the core Ne burning phase. Among them, less massive models with $M_{\mathrm{CO}}\sim$ 8.7–15.0 $M_{\odot}$ expand also after the O burning phase similar to the models with $M_{\mathrm{CO}}\sim$ 4.5–6.8 $M_{\odot}$, while the radii decrease during the later phases for more massive models with $M_{\mathrm{CO}}\gtrsim$ 15.0 $M_{\odot}$.

As a result, the stellar radius shows a smooth relation with the CO core mass up to C depletion, but eventually forms distinctive peaks ($M_{\mathrm{CO}}\sim$5–7 $M_{\odot}$ and $\sim$ 9–15 $M_{\odot}$) and a valley between them by core-collapse. In the top panel of Fig. 19, the distribution of the stellar radius at the core-collapse phase and the C depletion is shown. The coincidence between the first peak (valley) and small (large) $M_{\rm ff}$ at $M_{\mathrm{CO}}\sim$ 5–7 (7–9) $M_{\odot}$ strongly indicates that this structure originates from different inner core evolutions. Meanwhile, such a structure does not develop significantly for the luminosity distribution shown in the bottom panel. Consequently, the peak-valley structure in the radius distribution appears as a valley-peak structure in the effective temperature distribution.

Figure 20 shows the relation between the explodability and density indicators ($\xi_{2.5}$ and $M_{\rm ff}$) and the Ertl’s parameters. The explodability is estimated based on the semi-analytic model developed by Müller et al. (2016), with which we assign ‘explosion’ for models experiencing the shock revival and forming NSs and ‘implosion’ for models never experiencing the shock revival or forming BHs due to the late time accretion. The critical value of each indicator is determined as the value that would minimize the false identification number of

where $x\in\{\xi_{2.5},M_{\mathrm{ff}},\mu(s_{k}=4),M\mu(s_{k}=4)\}$. The false identification rate is the ratio of the false identification number to the total model number.

As for $\xi_{2.5}$, the figure clearly shows that the absolute value can be used to determine the explodability, which is consistent with previous works. Besides, we also find that $M_{\rm ff}$ is as useful as the compactness for the identification, showing the same false identification number. Originally, the Ertl’s parameters of $\mu(s_{k}\!=\!4)$ and $M\mu(s_{k}\!=\!4)$ (product of $M(s_{k}\!=\!4)$ and $\mu(s_{k}\!=\!4)$) were used in combination for the fate identification in Ertl et al. (2016). However, Müller et al. (2016) has reported that a single-parameter classification only depending on $\mu(s_{k}\!=\!4)$ can yield a better false identification with their semi-analytic model because the possibility of the late time BH formation after the shock revival has been neglected in Ertl et al. (2016). This is why we compare the results of the fate classification utilizing one of $\mu(s_{k}\!=\!4)$ and $M\mu(s_{k}\!=\!4)$ in this work. These parameters are also capable of identifying the fate, resulting in similar false identification rates. In summary, we have found that, based on any of these indicators, explodability can be judged with roughly equal accuracy.

Although we compute the false identification number and rate, these are only for determining the optimal value to identify different fates, and it is not our purpose to determine the precise values. This is because we do not expect that a complete identification is possible from approximate methods such as those performed in this work. False identification rates of about 10% are obtained for all the indicators in this work, and it should be interpreted as a typical accuracy when using such an approximate method. Furthermore, the critical values derived in this work are not accurate enough for quantitative comparisons. This is because we have found that the critical values are sensitive to the method applied for the fate estimate. For example, implosion more likely takes place if $\xi_{2.5}$ $>$ 0.36 for our model set, which is larger than 0.278 obtained in Müller et al. (2016). However, we speculate this is not due to the different model set but chiefly due to the different implementations of Müller’s semi-analytic prescription since a critical value of 0.33, which is closer to ours than that of Müller et al. (2016), is obtained even if we apply our implementation to the same progenitor models used in Müller et al. (2016). If another method based on, for example, 1D hydrodynamical simulations were used, even different values could be obtained. Therefore, we conclude that the qualitative feature of being able to identify fate is more robust and reliable than the quantitative features including the false identification rate and the critical values.

Explosion properties of (baryonic) PNS mass, explosion energy, nickel ejecta mass, and shock revival time are presented as a function of $M_{\rm ff}$ and $M_{\mathrm{CO}}$ in Fig. 21⁴⁴4The PNS mass, explosion energy, and nickel ejecta mass are $M_{\rm PNS}$, $E_{\rm diag}$, and $M_{\rm Ni}$ calculated at the simulation end, respectively, and the shock revival time is the time when the condition $t_{\rm heat}<t_{\rm adv}$ is met. For detailed definitions, see the Appendix.. It shows that these explosion properties, especially the PNS mass, have strong positive correlations with $M_{\rm ff}$. These correlations suggest that the progenitor density structure would determine not only the explodability but also the detailed properties of the supernova explosions. Taking the fact that the supernova explosion is a genuine non-linear phenomenon into consideration, the existence of this kind of correlation is non-trivial and thus interesting. Since the present analysis is based on the approximate model, further investigations with more realistic simulations are required. Nevertheless, it is noteworthy that the correlations shown here are consistent with an interesting correlation between the mass and the entropy of PNSs that is found from more realistic and systematic 1D explosion simulations (da Silva Schneider et al., 2020). This is because, provided a likely correlation between the entropies of the nascent NS and the progenitor’s iron core, the aforementioned correlation results from the correlation between the PNS mass and the entropy of the progenitor core.

$M_{\mathrm{CO}}$ is probably a more accessible parameter by observations than $M_{\rm ff}$ as it could correlate with the total ejecta mass both in cases of type II SNe and SE-SNe. The right column of Fig. 21 indicates that explosion properties show different tendencies depending on $M_{\mathrm{CO}}$. In the lower end of $M_{\mathrm{CO}}\lesssim 4M_{\odot}$, explosion properties, in particular the PNS mass and the explosion energy, obey linear correlations with $M_{\mathrm{CO}}$. This is due to the linear correlation between $M_{\mathrm{CO}}$ and $M_{\rm ff}$ for these less massive progenitors. Because the CO core mass range roughly corresponds to the ZAMS mass of $M_{\mathrm{ZAMS}}\lesssim 20M_{\odot}$, and because most SNe may emerge from the less massive range considering the nature of the initial mass function, this coincides with the correlations observed for type II SNe (e.g., Müller et al., 2017). An island of explosion exists for $M_{\mathrm{CO}}\in[8.1,11.6]\ M_{\odot}$, which is consistent with earlier theoretical studies (e.g., Ugliano et al., 2012). These massive exploding models are estimated to yield explosions with relatively larger NS masses, explosion energies, and ⁵⁶Ni ejecta masses, and this could be consistent with observations suggesting the positive correlation between the total ejecta mass, the ⁵⁶Ni ejecta mass, and the kinetic energy of SE-SNe (e.g., Taddia, F. et al., 2018).

In this subsection, we aim to check the degree to which the monotonic relation between the indicator, $M_{\rm ff}$, and the global density and temperature distributions is robust. For this purpose, a similar analysis has been performed for four additional sets of models, in addition to the set we have described so far (hereafter referred to as H1). Three of them, H2, H3, and H4 are calculated using the same stellar evolution code but with different initial compositions; they have pure helium (H2), solar- (H3), or zero-metallicity (H4) compositions initially (full stellar evolution with hydrogen envelopes are treated in H3 and H4). Another difference is that a reaction rate of ${}^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ of Caughlan & Fowler (1988) multiplied by a factor of 1.2 is applied for models in these sets. The fourth set is the one provided by Müller et al. (2016), which consists of models with solar-metallicity calculated by the stellar evolution code KEPLER applying ${}^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ of Buchmann (1996) multiplied by a factor of 1.2. This set is referred to as M16 hereafter. In addition, these model sets use different termination conditions for stellar evolution simulations; H2 uses the same condition as H1, stopping the simulation at $\rho_{c}=10^{10}$ g cm^-3, and H3 and H4 use the condition $T_{c}=10^{9.9}$ K. On the other hand, simulations in M16 terminate when the collapse velocity anywhere in the core exceeds 1,000 km s^-1 (Heger, private communication).

The $M_{\mathrm{CO}}$–$M_{\rm ff}$ relations compared in Fig. 22 show different properties among the model sets. In particular, the locations, widths, and heights of the peaks seen at $M_{\rm CO}$$\sim$ 5–9 $M_{\odot}$ are different for all sets (The major peaks are around 6–9 $M_{\odot}$ for H1, 5–7 $M_{\odot}$ for H2, 5–8 $M_{\odot}$ for H3, 6–9 $M_{\odot}$ for H4, and 5–6 $M_{\odot}$ for M16). The diversity seen in models H1 to H4 indicates that the $M_{\mathrm{CO}}$–$M_{\rm ff}$ relation is sensitive to the different computational settings of hydrogen envelopes, metallicity, and ${}^{12}\mathrm{C}(\alpha,\gamma)^{16}\mathrm{O}$ rate. This result is understandable since any of those differences result in different C/O ratios that the CO cores have at their birth (e.g., Sukhbold & Woosley, 2014; Patton & Sukhbold, 2020; Sukhbold & Adams, 2020). Moreover, more significant offsets between H models and model M16 may indicate that the difference in the stellar evolution code including the reaction network, EOS, opacity, convective boundary mixing, etc, is as influential as the other settings. We do not perform a comprehensive analysis in this work as it is beyond its scope, however, performing such an analysis is clearly important for future realistic predictions.

Figure 23 shows the projections of density and temperature distributions similar to Fig. 15 and 16 but for other model sets. Differences exist in the details. For example, in the M16 model set, the $M_{\rm O,base}$ (shown by the red line) traces a constant temperature $T\sim 10^{9.6}$ K, whereas, in other model sets calculated with the HOSHI code, it traces a lower constant temperature $T\sim 10^{9.5}$ K. This would indicate the strong impact of applying different reaction rates, reaction network, or QSE/NSE treatments on determining the innermost stellar structures. Also, due to different termination conditions, the density and temperature structures below $\sim$1 $M_{\odot}$ show different trends for different model sets, i.e., when compared at the same $M_{\rm ff}$, the H1 and H2 models show higher densities and temperatures in the inner regions than the H3, H4, and M16 models. Nevertheless, the $M_{\rm ff}$-based sorting clearly reveals a significant correlation in density, temperature, and compositional structure for all model sets. Hence, this correlation is likely to be universal and independent of the prescriptions for stellar evolution simulations.

Recent high-cadence SN surveys have revealed that many CCSN progenitors experience enhanced mass loss in the final years before core-collapse, which leads to the formation of dense circumstellar medium (CSM) (e.g., Bruch et al., 2021). The CSM-ejecta interaction is believed to be the origin of the narrow lines of Type IIn SNe (e.g., Chevalier & Fransson, 1994) as well as Type Ibn SNe, and the existence of a dense CSM can also be inferred from the so-called ‘flash-spectroscopy’ (Khazov et al., 2016; Yaron et al., 2017). More direct evidence can be obtained from pre-explosion images. Ofek et al. (2014) has conducted a systematic search for the precursor eruptions for progenitors of Type IIn SNe and concluded that most Type IIn progenitors undergo the precursor eruptions prior to the SN explosion (see also Strotjohann et al., 2021). Furthermore, it will be possible to estimate the onset time of the enhanced mass loss by monitoring the evolution of the spectroscopic features changing from optically thick to optically thin.

Several theoretical explanations have been proposed for the mechanism of enhanced mass loss. The high mass loss rate may be achievable by line-driven winds (Vink, Jorick S. & de Koter, A., 2002) or super-Eddington continuum-driven winds (Shaviv, 2001; Van Marle et al., 2008). However, considering the peculiar proximity to the core collapse, other mechanisms such as wave-driven mass loss (Quataert & Shiode, 2012; Shiode & Quataert, 2013) or mass ejection powered by off-center nuclear flashes (Dessart et al., 2010) might be more plausible because these mechanisms will operate for only later evolutionary phases.

The convective motion inside the star will excite waves when it hits the convective boundary layers. After the waves are transported to the surface evanescent region, some of the energy will be dissipated leading to heating in the stellar envelope. The convective motion is more energetic for the later evolutionary phases, so at some point, this energy transfer may result in mass ejection from the surface. Theoretical studies have estimated that this wave-driven mass loss can operate during and after the Ne burning phase (Quataert & Shiode, 2012; Shiode & Quataert, 2013; Fuller, 2017; Fuller & Ro, 2018). As we have shown, the remaining lifetime till collapse for the later burning phases of Ne, O, and Si burnings have rough anti-correlations to $M_{\rm ff}$ (Fig. 17). Hence, we expect that the onset time of the wave-driven mass loss will also show anti-correlations to $M_{\rm ff}$⁵⁵5This expectation should be consistent with the anti-correlation between the onset time and the He core mass indicated by Shiode & Quataert (2013)..

In addition, Figure 17 illustrates that the least massive models of $M_{\mathrm{CO}}\in[1.42,1.72]\ M_{\odot}$ ($M(\tau_{\mathrm{ff}}=1s)\in[1.40,1.56]\ M_{\odot}$) experience off-center O+Ne flashes due to the high electron degeneracy, which takes place $\sim$ 1–10 yr before core-collapse. Depending on the injected energy, such flashes may result in mass ejection (Woosley & Heger, 2015), which itself could be observed as SN-like transients or SN impostors (Dessart et al., 2010). From the small $M_{\rm ff}$s, it is expected that the additional transients triggered by the off-center flashes will be associated only with the least energetic CCSNe that finally form the least massive NSs (Suwa et al., 2018). In the coming decades, the number of SNe, in which both the explosion properties and the onset time of the final enhanced mass loss are estimated, will significantly increase thanks to the large surveys such as the Rubin Observatory LSST (Ivezić et al., 2019). We expect that the further correlations linked via the fundamental correlations with $M_{\rm ff}$ will be verified with future statistics.