Evolution and Final Fate of Solar Metallicity Stars in the Mass Range 7–15 M ⊙. I. The Transition from Asymptotic Giant Branch to Super-AGB Stars, Electron Capture, and Core-collapse Supernova Progenitors

According to a standard initial mass function, stars in the range 7–12 M ⊙ constitute ∼50% (by number) of the stars more massive than ∼7 M ⊙, but in spite of this, their evolutionary properties, and in particular their final fate, are still scarcely studied. In this paper, we present a detailed study of the evolutionary properties of solar metallicity nonrotating stars in the range 7–15 M ⊙, from the pre-main-sequence phase up to the presupernova stage or an advanced stage of the thermally pulsing phase, depending on the initial mass. We find that (1) the 7.00 M ⊙ star develops a degenerate CO core and evolves as a classical asymptotic giant branch (AGB) star in the sense that it does not ignite the C-burning reactions, (2) stars with initial mass M ≥ 9.22 M ⊙ end their lives as core-collapse supernovae, (3) stars in the range 7.50 ≤ M/M ⊙ ≤ 9.20 develop a degenerate ONeMg core and evolve through the thermally pulsing super-AGB phase, (4) stars in the mass range 7.50 ≤ M/M ⊙ ≤ 8.00 end their lives as hybrid CO/ONeMg or ONeMg WDs, and (5) stars with initial mass in the range 8.50 ≤ M/M ⊙ ≤ 9.20 may potentially explode as electron-capture supernovae.


Introduction
In the general picture of stellar evolution, stars with M  7 M e evolve along the thermally pulsing asymptotic giant branch (TP-AGB) phase and end their lives as CO white dwarfs (CO WDs).On the contrary, stars with 12 M e , the so-called massive stars (MSs), evolve through all of the major stable nuclear burning stages and eventually explode as core-collapse supernovae (CCSNe) leaving a compact remnant, i.e., either a neutron star or a black hole.Stars between ∼7 and ∼12 M e have a much more complex evolution.The lower masses ignite C off-center in an electron-degenerate environment, and their ability to remove the degeneracy totally relies on the capability of the thermally unstable zones (the convective zones) to heat the layers below the main burning front.The higher masses, on the contrary, ignite C burning centrally.After the C-burning phase, the lower-mass stars (M  10 M e ) develop an inert ONeMg electron-degenerate core and enter the TP phase along the AGB.These stars are generally referred to as super-AGB (SAGB) stars.The final fate of the SAGBs depends on the competition between the core growth and the mass loss (Nomoto 1984).If mass loss dominates, the envelope is completely lost, and the result is an ONeMg WD.On the contrary, if the core grows in mass enough to achieve central densities close to the threshold density for electron capture (EC) , deleptonization and thermonuclear instability develop to induce the electron-capture supernova (ECSN).The final fate of such stars depends on the competition between the energy released by the nuclear burning front and the loss of pressure due to the deleptonization occurring in the central zones.If the energy released by nuclear burning prevails, degeneracy is removed, and the star explodes as a thermonuclear ECSN (Miyaji et al. 1980;Isern et al. 1991;Nomoto & Kondo 1991;Jones et al. 2016;Nomoto & Leung 2017).If, on the contrary, the deleptonization dominates, the collapse cannot be halted, and the star collapses into a neutron star (core-collapse ECSN; Miyaji et al. 1980;Miyaji & Nomoto 1987;Nomoto 1987;Kitaura et al. 2006;Fischer et al. 2010;Jones et al. 2016;Radice et al. 2017;Zha et al. 2019Zha et al. , 2022)).Which outcome is realized from ECSNe depends on the details of the modeling of both the presupernova evolution and the explosion.
Stars in the mass range ∼10-12 M e ignite Ne burning offcenter that develops in a similar fashion as the off-center C burning (Nomoto 1984).The final fate of these stars depends, once again, on the behavior of the off-center Ne burning (Nomoto & Hashimoto 1988).If the Ne burning is ignited far enough off-center, the contracting core may achieve densities sufficiently high for the URCA processes to be activated until the conditions for an ECSN are reached before the Ne burning front is able to reach the center.If, on the contrary, the Ne burning front reaches the center before the activation of the URCA processes, O burning first and Si burning later are ignited off-center, ultimately leading to a CCSN (Jones et al. 2013(Jones et al. , 2014;;Nomoto 2014;Woosley & Heger 2015).
Stars in the range 7-12 M e constitute roughly 50% of the stars (by number) more massive than ∼7 M e according to a standard initial mass function (IMF).This means that proper knowledge of how they evolve and die is crucial for many astrophysical subjects.Determining the mass boundaries between stars that form CO WDs, ONeMg WDs, ECSNe, and CCSNe is mandatory to understand the relative frequencies among these objects.If a substantial fraction of stars does indeed explode as ECSNe, then they may contribute significantly to the overall SN rate and also to the population of neutron stars.ECSNe have also been proposed as potential sites for the r-process (Ning et al. 2007), which is responsible for the synthesis of roughly half of the nuclei above the Fe group.The ECSN has been proposed for SN 1054, which formed the Crab Nebula, in consideration of the similarity of low explosion energies and the small amount of heavy elements between the Crab Nebula and the ejecta of the ECSN (Nomoto et al. 1982).ECSNe may explain the observations of subluminous type II plateau supernovae (SNIIP) with a low amount of 56 Ni ejected (Smartt 2009).Recently, Hiramatsu et al. (2021) observed SN II 2018zd and found that its several observed features can be well explained by an ECSN but not by an Fe-CCSN.In the context of the chemical evolution of the galaxies, because of the shape of the IMF, these stars should contribute significantly to the production of some specific isotopes.Also, the chemical yields produced by stars in this mass interval are either completely ignored or obtained by means of an arbitrary interpolation between the yields produced by the AGB stars and those produced by the MS.In both cases, a significant error can be made.Moreover, a large fraction of MSs are part of binary or multiple systems (Duchêne & Kraus 2013).Binary interactions crucially affect the amount of mass lost by these stars, and hence even more massive progenitors may contribute to produce ECSNe (see, e.g., Brinkman et al. 2023, and references therein).Unfortunately, the binary scenario remains mostly unexplored so far.
Despite their astrophysical relevance, not many models in the range 7-12 M e that cover the full evolution are available at present.In particular, none of the papers found in the literature on this subject (e.g., Ritossa et al. 1999;Siess 2010;Ventura et al. 2011;Karakas et al. 2012;Lau et al. 2012;Gil-Pons et al. 2013;Jones et al. 2013Jones et al. , 2014;;Takahashi et al. 2013Takahashi et al. , 2019;;Ventura et al. 2013;Doherty et al. 2015;Woosley & Heger 2015;Zha et al. 2019) present a homogeneous, detailed, and comprehensive study of the full evolution of stars in the mass range 7-12 M e , with a rather fine mass grid.In some cases, as has been reported in a number of recent papers, the TP phase cannot be followed due to numerical problems, and the core growth is treated parametrically, assuming an arbitrary accretion rate, or even neglected (Woosley & Heger 2015;Takahashi et al. 2019).Moreover, the full evolution of these stars with rotation has never been computed, as far as we know.The main reason for the paucity of models in this mass range is that the computation of their evolution is extremely challenging from a numerical point of view, and in general, it also requires an enormous amount of computer time and memory.As already mentioned above, in these stars, depending on the initial mass, C, Ne, O, and Si burning ignite off-center, and the burning front propagates inward in mass accompanied by a number of convective shells.A proper treatment of the heat transfer from the burning front to the underlying layers requires an extremely fine spatial resolution of the order of a few kilometers.The full coupling of convective mixing and nuclear burning, possibly coupled to the structure equations, is also necessary to properly follow these stages and avoid numerical instabilities.The adoption of a quite extended nuclear network, including at least 100 or more nuclear species fully coupled to the stellar evolution, is also needed to properly compute the energy generation and trace the abundance of the various isotopes, in particular the electron fraction.A very large number of thermal pulses is expected for SAGB stars, ranging from several tens to thousands.Typically, the thickness of the intershell region, i.e., between the He and H shells, is of the order of 10 −4 -10 −5 M e .Resolving all of these phenomena, both in space and in time, is extremely challenging and generally requires several thousands of zones per model and hundreds of thousands, if not millions, of models to cover the entire evolution.Coupling an extended network to these kinds of models implies the need for an enormous amount of computing time to calculate a full single evolution.This is the first paper of a series in which we aim to study in detail the evolutionary properties of stars in the transition from AGB stars to CCSN progenitors and how they change with initial metallicity and rotation velocity.In this paper, we present the detailed evolution of solar metallicity nonrotating stars in the range 7-15 M e from the pre-main-sequence phase up to the presupernova stage or an advanced stage of the TP phase, depending on the initial mass.The main goals are to (1) study how the evolutionary properties of these stars change in the transition from AGBs to MSs, (2) determine the limiting masses between AGB and SAGB stars and between SAGB stars and MSs, (3) determine their final fate, and ultimately, (4) identify the limiting masses that mark the transitions from the various final outcomes, i.e., CO WDs, ONe WDs, ECSNe, and CCSNe.

Stellar Evolution Code and Nuclear Network
In the last 20 yr, we have developed and continuously improved upon our stellar evolution code FRANEC (Chieffi & Limongi 2013;Limongi & Chieffi 2018).One of the strengths of the code is that it can automatically manage nuclear networks, including an arbitrarily extended list of isotopes and associated reactions.Another feature that makes this code unique in the panorama of the stellar evolution codes is that all of the equations describing (a) the physical structure of the star, (b) the chemical evolution due to the nuclear reactions, and (c) the chemical mixing due to a variety of instabilities (thermal and mechanical) are coupled together in a single system of equations and solved simultaneously.Last, but not least, FRANEC includes the treatment of the stellar rotation, the transport of the angular momentum, and the rotation-induced mixing of the chemicals.The multithreads/parallel solver algorithms used to invert the large sparse matrices needed to solve the system of equations are the most sophisticated and fastest presently available.All of the features mentioned above make this code extremely robust and fast from a numerical point of view and suitable to study essentially all of the evolutionary phases of stars, even the most challenging ones, like the advanced burning stages of MSs; the off-center C, Ne, O, and Si burning; and the thermal pulses in AGB and SAGB stars.
All of the models presented in this paper have been computed by means of the latest version of the FRANEC code.This version is essentially the one adopted in Chieffi & Limongi (2013) and Limongi & Chieffi (2018), with the following updates/differences.
The induced overshooting occurring during the core Heburning phase, because of the transformation of He into C and O, is properly taken into account following Castellani et al. (1985).During the late stages of core He burning, the occurrence of the breathing pulses has been properly inhibited as described in Caputo et al. (1989) and Chieffi & Straniero (1989).
The equation of state adopted is the one provided by F. X. Timmes and is available on his web page. 8It takes into account all stages of Saha ionization, plus a simple density ionization model, for elements between H and Zn for all degrees of degeneracy and relativity.In addition to that, electron positron pairs and Coulomb corrections are also taken into account.
Since in this work, we are mainly interested in studying the physical properties of the stars in the transition between AGB and MSs, and since the calculation of these models requires an enormous amount of computer time and memory, we have chosen a nuclear network that includes the minimum number of isotopes but at the same time guarantees the calculation of the nuclear energy with great accuracy.The 112 isotopes included in the adopted network are reported in Table 1.All of these isotopes are coupled among each other by the most efficient reactions due to the weak and strong interactions, for a total of about 466 reactions.Among them, we have also taken into account the following 20 URCA processes, due to their crucial role played in some of the models presented in this work: The energy release/absorption associated with these reactions has been treated as described in Miyaji et al. (1980).
The nuclear cross sections have been updated by taking into account the most recent experimental data and theoretical calculations, as described in Roberti et al. (2024).In particular, since the URCA rates must be carefully calculated (Toki et al. 2013;Nomoto & Leung 2017), we have adopted for these processes the refined EC and beta decay rates provided by Suzuki et al. (2016).
The zones unstable for convection are determined according to the Ledoux criterion in the H-rich layers and according to the Schwarzschild criterion in all of the other cases.As is well known, convective core overshooting during core H burning determines the size of the He core at core H depletion that, in turn, drives all of the following evolution of the star (Bressan et al. 1981;Bertelli et al. 1986;Temaj et al. 2013), unless the He core is further reduced by the mass loss during core He burning (Limongi & Chieffi 2006;Chieffi & Limongi 2013).In the present version of the code, we use the same prescription used in the previous version adopted in Limongi & Chieffi (2018); i.e., during core H burning, we assume 0.2 H P of convective core overshooting by computing ( ) r = P g H P at the outer edge of the formal convective core, defined by the stability criterion mentioned above.Some amount of extra mixing is assumed at the lower edge of the convective envelope and the convective shells that form within the electrondegenerate CO core and that are associated with C, NeO, and Si off-center nuclear burning.The mixing efficiency in these zones is determined by assuming an exponentially decaying convective velocity given by where the subscript 0 refers to values corresponding to the lower border of the convective zone, v conv (r 0 ) is computed in the context of the mixing-length theory, and the free parameter f is assumed equal to 0.014.The diffusion coefficient is then computed as When a fuel is ignited off-center, because of the temperature inversion due to the degeneracy of the matter (see, e.g., Figures 1-3 in Nomoto 1984), a convective shell develops, and a sharp discontinuity in the temperature forms at the base of the convective zone, where burning is occurring.The capability of the burning front to propagate inward in mass, i.e., as a continuous flame or by recurrent flashes, as well as its speed, is in general controlled by the coupling of convective mixing and heat transfer from the hot zones at the base of the convective shell, where burning is occurring, to the radiative cooler and  Nomoto & Hashimoto 1988) through a sequence of convective shells that form where the fuel is abundant and disappears as the fuel is locally exhausted.The treatment of such a phenomenon is not trivial, and different approaches and assumptions can be followed (see, e.g., Nomoto & Hashimoto 1988;Jones et al. 2014;Woosley & Heger 2015 and references therein).In this work, we assume that every time a major fuel (C, NeO, Si) is ignited off-center, burning propagates as a convectively bounded flame (CBF).
More specifically, once the speed and width of the burning flame are assumed, a given amount of energy in the radiative layers below the convective bound is deposited following the prescription of Woosley & Heger (2015).
The initial composition adopted for the solar metallicity is the one provided by Asplund et al. (2009), which corresponds to a total metallicity of Z = 1.345 × 10 −2 .The adopted initial He mass fraction is 0.265.

Results
We computed the evolution of solar metallicity nonrotating stars with initial mass between 7 and 15 M e from the premain-sequence phase up to the presupernova stage or an advanced stage during the TP phase, depending on the initial mass.The main evolutionary properties of all of the computed models are reported in Table 2 for stars with initial mass between 7 and 9.20 M e and Table 3 for stars with initial mass between 9.22 and 15 M e .The meanings of the various entries (in most cases) are as follows: M CC is the maximum size of the convective core in units of M e ; t is the evolutionary time in units of years; 12 C is the central carbon mass fraction (this entry is reported only for the He-burning phase and refers to the value at core He depletion); M Fe , M Si-S , M ONe , M CO , M He , M CE , and M tot are the iron core mass, Si-S (O-depleted) core mass, ONe core mass, CO core mass, He core mass, convective envelope mass, and total mass, respectively, in units of M e ; ψ c , T c (kelvin), ρ c (g cm −3 ), and Y e,c are the central values of the degeneracy parameter, temperature, density, and electron fraction; T ign (kelvin), ρ ign (g cm −3 ), ψ ign , and M ign (M e ) are the temperature, density, degeneracy parameter, and mass coordinate corresponding to an off-center nuclear ignition; M C , reported among the quantities at the end of the second dredge-up in Table 2 and at neon ignition in Table 3, refers to the mass coordinate, in units of M e , marking the central zone where the carbon mass fraction exceeds 0.01 (this quantity is relevant for those stars that form hybrid CO/ONeMg cores, i.e., cores mainly composed of ONeMg but with a central region still rich in C).

Evolution during Core H and He Burning
The evolution of all of the computed models during core H burning is characterized, as usual, by the formation of a convective core that progressively recedes in mass.As a consequence at core H depletion, a He core is formed, surrounded by a zone with a gradient of chemical composition (Figure 1).As H burning shifts progressively in a shell, all of the stars move to the red side of the H-R diagram (Figure 2) and become red giants.During this phase of the evolution, the surface temperature decreases, inducing the formation of a convective envelope that penetrates progressively in mass.When the convective envelope reaches the region of variable composition left by the receding H convective core, a dredgeup to the surface of the products of the core H burning begins (first dredge-up; see Figure 1).
In all of the models, the core He burning starts after the first dredge-up, i.e., when the star is a red giant, and develops in a convective core that, at variance with core H burning, increases progressively in mass (Figure 1).Such an increase, due to the increase of the opacity as a result of the conversion of He into C and O, produces a sharp discontinuity in the radiative gradient at the edge of the convective core that drives the so-called induced overshooting; i.e., the zone homogenized by convection extends beyond the formal border of the convective core, until the neutrality of the radiative and adiabatic gradients is realized (Castellani et al. 1985).During the late stages of core He burning, i.e., when the central He mass fraction decreases below ∼0.1, the fresh He engulfed by the increasing convective core becomes comparable to the central He mass fraction, and this produces a burst of nuclear energy that, in turn, drives a so-called breathing pulse, i.e., a progressive increase of both the convective core and the central He mass fraction.This occurs until the He ingested no longer produces a substantial increase of the nuclear energy (Castellani et al. 1985).Since the occurrence of the breathing pulses is still highly uncertain, they have been suppressed, as already mentioned in Section 2.
During core He burning, models with mass lower than 9.20 M e perform an extended blue loop in the H-R diagram (Figure 2) that is accompanied by a temporary recession and disappearance of the convective envelope when the stars cross the blue side of the H-R diagram (Figure 1).The H shell during core He burning is active and advances in mass, progressively increasing the size of the He core.At core He depletion, all of the models are red giants, and their He core mass is increased by ∼50%, with respect to the one at core H depletion (Figure 3).The CO core at core He depletion increases with the initial mass (see Figure 4, black line, and Tables 2 and 3), ranging between ∼0.5 and ∼2.6 M e , while the 12 C mass fraction left by the core He burning decreases smoothly with the initial mass and ranges between ∼0.5 and ∼0.38 (see Figure 5, black line, and Tables 2 and 3).

Evolution after Core He Depletion
The evolution after core He depletion is characterized by the following processes: (1) the shift of the He-burning shell, outward in mass, that progressively increases the size of the CO core and switches off the H-burning shell; (2) the substantial energy loss from the central zones due to the neutrino emission; (3) the energy deposition in the more external zones of the CO core due to compressional heating induced by the advancing of the He-burning shell; and (4) the progressive penetration of the convective envelope that may eventually produce the second dredge-up.The interplay and timing of these processes and the final fate of the star depend on the CO core mass at core He depletion that, in turn,  Note.In the H-and He-burning sections, all of the quantities refer to the values at the end of each nuclear burning, the only exception being M CC , which is the maximum extension in mass of the convective core during the nuclear burning, and t, which is the evolutionary time of each burning stage.In stars with initial mass M 7.0 M e (that form CO cores at core He exhaustion with M CO 0.69 M e ), the maximum temperature within the CO core does not reach the threshold value for C ignition, the CO core becomes progressively more and more degenerate, and eventually, these stars enter the TP-AGB phase.We followed the evolution of the 7 M e model along 27 thermal pulses.The evolution of solar metallicity AGB stars with mass around 7 M e has already been discussed in detail in the literature (see Section 1); therefore, we will not address them here, but we report in Table 2 the main properties of the model at various key times during the evolution.Figures 2 and 6 show the evolutionary path of the model in the H-R diagram and the central temperature-central density plane.It is worth mentioning the occurrence of the second dredge-up starting after core He depletion and going to completion before the onset of the TPs.The effect of the second dredge-up is that of reducing the He core from 1.76 M e (value at core He depletion) to 1.01 M e (Figure 3).
The main properties of the thermal pulse phase are reported in Table 4 and the upper left panels of Figures 7-10.The values reported in the table, as well as the behavior of the various quantities shown in the figures, are consistent with what has been found in the literature (see Section 1).Let us note only that the maximum temperature at the base of the convective envelope increases progressively from ∼40 MK at the beginning of the TP phase to a plateau value corresponding to ∼80 MK after the first ∼16 TPs (upper left panel of Figure 9).By the way, Nomoto & Sugimoto (1972) investigated in their Figures 2 and 3 how the temperature at the base of the convective envelope and the depth of mixing depends on the luminosity and mass of the CO core.The third dredge-up, due to the penetration of the convective envelope into the He core during the quiescent shell He-burning phase, i.e., when the H-burning shell is switched off, occurs after a few TPs and induces from one side a progressive reduction of the rate at which the CO core increases (upper left panel of Figure 8) and, at the same time, a progressive enrichment of the surface carbon abundance (upper left panel of Figure 10).Note, however, that such an enhancement is very mild; in fact, the surface carbon abundance has increased at the end of this phase by a factor of ∼1.1 compared to the value at core He depletion.For this reason, we decided to not take into account carbonenhanced opacity tables.By the way, let us remember that, as already mentioned in Section 2, we assume some amount of extra mixing at the base of the convective envelope; therefore, this is also applied during the third dredge-up.Figure 11 shows the behavior of the convective zones during the last thermal pulses before the end of the calculation, where one can appreciate the size of the He convective shell that forms after the He shell ignition and the efficiency of the third dredge-up, in particular the quantity λ = ΔM dredge /ΔM H ∼ 0.78, where ΔM H is the increase of the core mass during the interpulse phase and ΔM dredge is the maximum penetration of the convective envelope following the pulse (see, e.g., Figure 5 in Doherty et al. 2017).

Evolution during C Burning: Stars with Initial Mass
M 7.5 M e (M CO 0.76 M e ) In stars with initial mass M 7.5 M e (CO core masses at core He depletion M CO 0.76 M e ), the maximum temperature within the CO core reaches the threshold value for the C-burning ignition.Depending on the initial mass, C ignition may occur in the center or off-center in a (partial) degenerate environment.
In stars with initial mass 7.5 M/M e 9.5 (0.76 M CO /M e 1.16), the interplay between processes 2 and 3 (see above) induces the formation of an off-center peak temperature that progressively increases locally and moves outward in mass while the CO core becomes progressively more degenerate.Therefore, for these stars, the C ignition occurs off-center, and, in general, as the initial mass increases, the mass coordinate corresponding to the ignition point becomes closer to the center, ranging from 0.59 to 0.02 M e for the 7.5 and 9.5 M e models, respectively (black line in Figure 12).The off-center ignition is generally accompanied by the formation of a convective zone driven by the high flux produced by the C-burning reactions in a (partial) degenerate environment (ψ ign ∼ 2.5; see Table 2).When the energy flux produced by the nuclear burning reduces, the convective zone vanishes.The disappearance of this convective shell is then followed by a number of other convective zones associated with the C-burning front that progressively moves toward the center and locally removes the degeneracy.In general, the number of convective zones that follow the first one decreases as the initial mass of the star increases (Figure 1).Typical internal structures corresponding to two different stages during this phase are shown in Figures 13 and 14 for the 8.5 M e model.Note that 12 C is not completely exhausted in the zones where the C-burning front has passed.The C-burning front, marked by the maximum temperature, reaches the center in stars with initial mass M 9.2 M e and then begins moving outward in mass where 12 C is still abundant, again inducing the formation of a number of successive convective shells (Figure 1).In the lower-mass models, on the contrary, the maximum temperature never reaches the center.However, the following shell C-burning phase develops in these stars similarly to the more massive ones.In all of these models, after the last C convective episode, C burning proceeds in a radiative shell, progressively reducing the CO-rich zone confined between the ONeMg core resulting from the shell C burning and the He-rich zone (the red zone in Figure 15).It is worth noting that in all of these models, some 12 C remains unburned in the central zone.The mass fraction of the unburned 12 C, as well as the mass of the core where this quantity is larger than 0.01, decreases with increasing initial mass.In particular, Figure 5 (blue line) shows that the mass of the central zone where the 12 C mass fraction is larger than 0.01 decreases from ∼0.33 M e in the 7.5 M e model to ∼0.01 M e in the 8.8 M e model and disappears for larger initial masses.
Figure 1 shows how the configuration of the ONeMg (green shaded areas)/CO (red shaded areas) cores changes as a function of the initial mass.
In stars with initial mass M 9.8 M e (CO core masses at core He depletion M CO 1.2 M e ), C burning is ignited in the center and develops in a convective core.Once the 12 C is depleted in the center, C burning shifts in shell and, as in the case of the lower-mass models, induces the formation of a number of convective shells (Figure 1).These stars behave like the classical MSs (Limongi & Chieffi 2018;Chieffi & Limongi 2020).

Evolution of the Convective Envelope: Stars with Initial
Mass M 10.0 M e (M CO 1.20 M e ) As already mentioned in Section 3.2, after core He depletion, the H-burning shell is switched off by the advancing Heburning shell, and this may induce the convective envelope to penetrate into the He-rich layer (second dredge-up).This phenomenon occurs only in stars with initial mass M 10 M e , although in different evolutionary stages (see also Section 4.2 and Figure 4.3 of Sugimoto & Nomoto 1980, for a discussion on the mechanism of the convective penetration).More specifically, the second dredge-up begins (1) before carbon ignition and goes to completion after carbon burning in stars Figure 1.Convective (green shaded areas) and chemical (color codes reported in the color bar) internal history of selected models from the main-sequence phase up to an advanced stage of the TP-SAGB phase (top and middle left panels) or the onset of the iron core collapse prior to the CCSN explosion (middle right and bottom panels).The dashed line in the top and middle left panels marks the onset of the TP phase.The x-axis reports the logarithm of the time until the end of the evolution (t fin − t) in units of years.
with initial mass 7.5 M/M e 8.0 (compare the black line with the cyan and blue lines in Figure 3) and (2) after carbon ignition and goes to completion after carbon burning in stars with initial mass 8.5 M/M e 10.0 (compare the black line with cyan, blue, and magenta lines in Figure 3).The second dredge-up for stars with initial mass M 7.0 M e has already been discussed in Section 3.3 and will not be repeated here.The blue line refers to the AGB star, i.e., the one that develops a degenerate CO core, does not ignite C burning, and enters the TP phase.The red lines refer to SAGB stars, i.e., those that ignite C burning and then do not ignite Ne burning; these stars develop a degenerate ONeMg core and enter the TP phase.The black lines refer to those stars that ignite Ne burning and eventually explode as CCSNe.The horizontal green dashed line marks the final luminosity of the lowest mass that explodes as CCSNe, i.e., the expected minimum luminosity of CCSNe (Smartt 2015).In general, during the second dredge-up, the convective envelope penetrates into the He core and therefore brings to the surface material that has been processed by H burning through the CNO cycle.This means that during this phase, the surface 12 C mass fraction decreases, while the surface 14 N mass fraction increases (Figure 10).
In stars with initial mass 8.50 M/M e 9.20, the C-burning shell is efficient enough to progressively switch off the He-burning shell.As a consequence, in these stars, the convective envelope penetrates deep enough to dredge up the products of the He burning.Once this deep penetration occurs, the surface abundance of 12 C suddenly increases, while that of 14 N progressively decreases (see Figure 10).In general, the larger the mass, the larger the increase of the surface 12 C abundance.
It is worth noting that in stars with 8.8 M/M e 9.20 (0.98 M CO /M e 1.08), a He convective shell forms during the second dredge-up.Such an occurrence slightly slows down the penetration of the convective envelope into the He core.Once the He convective shell disappears, the penetration resumes (Figure 15), and the products of the He burning, mainly (primary) 12 C (Figure 10), are mixed up to the surface.Note that, in this case, the increase of the surface 12 C abundance at the end of the second dredge-up is mainly due to the penetration of the convective envelope into the extinguished He convective shell and, to a lesser extent, the penetration into the CO core.The dredge-up of the He-burning products has been named "corrosive second dredge-up" by Gil-Pons et al. (2013) and Doherty et al. (2014), but we instead refer to it as an early third dredge-up (E3DU) because this phenomenon is identical to the one occurring during the interpulse phase of TP-AGB stars (see above), although in this case it occurs well before the beginning of the onset of the thermal pulses (see below).It is worth noting that in none of these models do the He convective shell and the convective envelope interact with each other; i.e., we do not find the socalled "dredge-out" episode, a phenomenon found for the first time by Ritossa et al. (1999) and later in other studies (see, e.g., Siess 2007;Gil-Pons et al. 2013;Doherty et al. 2015;Jones et al. 2016).This is probably due to the fact that while in the previously mentioned studies, a convective overshooting is considered for any convective zone (cores and shells), in these calculations, we assume some amount of extra mixing only at the edge of the H convective core, at the base of the convective envelope, and at the base of the convective shells during any off-center ignition (see Section 2).As a final comment, we point out that in all of the models mentioned above, the increase of the surface carbon abundance due to the E3DU is in any case very mild.In the 9.2 M e model, where the E3DU produces the largest effects, the enhancement of the surface Note.The columns are as follows.TP is the pulse number; Δt pulse is the duration of the pulse in years; is the maximum luminosity provided by the He-burning reactions during the pulse in solar luminosity; DM pulse max is the maximum extension of the He convective shell during the pulse in solar masses; T He max is the maximum temperature of the He-burning shell during the pulse in kelvin; λ is the measure of the efficiency of the third dredge-up and is defined in Section 3.3; Δt inter is the time elapsed between two consecutive thermal pulses in years; and ΔM He and ΔM CO are the increase of the He and CO core, respectively, in solar masses between two consecutive thermal pulses.The full machine-readable table containing the properties of the 7.00, 7.50, 8.00, 8.50, 8.80, 9.00, 9.05, 9.10, 9.15, and 9.20 M e models is available online.
(This table is available in its entirety in machine-readable form.) carbon abundance is a factor of ∼2 compared to the value at core He depletion.Since during the first dredge-up, the surface carbon abundance decreases by a factor of ∼2 compared to the initial one, the result is that the surface carbon mass fraction after the E3DU is roughly equal to the initial one.For this reason, also in this case, we decided to not take into account carbon-enhanced opacity tables.
In all of the stars with initial mass 7.5 M/M e 9.2, during the second dredge-up, the envelope expands and cools down in order to reabsorb the energy produced by the more internal nuclear burning shells.However, the rate at which the convective envelope penetrates in mass is higher than the rate at which it cools down; therefore, the temperature at the base of the convective envelope increases progressively until the  H-burning shell is reignited.After the convective envelope reaches its maximum depth, the H-burning shell begins to advance in mass.In this phase, the He-burning shell is progressively reignited in those stars where it was switched off.From this stage onward, for all of the stars, the evolution is characterized by a double shell burning, where the two H-and He-burning shells advance in mass with a similar rate.In fact, in this phase, the H-burning luminosity is a factor of ∼7 higher than that of the He burning (see the leftmost values in Figure 7), which corresponds roughly to the ratio between the energy provided by the CNO cycle and the energy provided by the 3α reactions.This means that the two shells burn the same  amount of matter per second.This stage ends with the onset of the thermal pulses (see next section and also Sugimoto & Nomoto 1975, for discussion on the evolution of this phase).
In stars with initial mass 9.22 M/M e 10.00, neon burning is ignited off-center during the second dredge-up and before the bottom of the convective envelope enters into the region enriched by the He-burning products (see also Nomoto 1984).Once neon is ignited, the evolution of the star becomes fast enough that the zones above the He cores remain essentially frozen.For this reason, in these stars, the products of He burning are never brought to the surface; therefore, no increase of the surface 12 C is found.After the H-burning shell has been reignited, stars in the mass range 7.5 M/M e 9.20 enter a classical TP phase, where the two H and He shells alternatively activate above a degenerate ONeMg core that is surrounded by a thin zone enriched in CO, the latter left by the He-burning shell.The general properties of these stars during this phase, named TP-SAGB, have been reviewed and described in detail in the literature (see Section 1); therefore, we will focus here mainly on how these properties change as a function of the initial mass.The main evolutionary properties during the TP-SAGB phase of stars in this mass interval (7.5 M/M e 9.20) are reported in Table 4.
In general, each thermal pulse is characterized by the following phenomena: (1) a strong activation of the He-burning reactions followed by the formation of a He convective zone and a peak in the He luminosity (L He ); (2) the disappearance of the He convective zone and the steady He-shell burning phase that accretes the CO core; (3) the switching off of the H-burning shell; (4) the penetration of the convective envelope that may in some cases erode the He core (third dredge-up); (5) the reactivation of the H shell and the switching off of the Heburning shell; and (6) the steady H-shell burning phase, where the He core increases and the convective envelope recedes in mass (interpulse phase) until the next pulse is ignited.A schematic view of this phase can be found, e.g., in Doherty et al. (2017; Figure 5).
Figure 7 shows the luminosity of the H-and He-burning shells as a function of time for the AGB and selected SAGB models.Moving from AGB stars (M = 7.0 M e ) to SAGB stars (7.5 M/M e 9.20), the maximum luminosity of the Heburning shell reached during each thermal pulse decreases, while the frequency of the TPs increases.This is due to the fact that the core mass becomes more massive and hotter as the initial mass of the star increases (see Doherty et al. 2017 and references therein).The increase of both the 4 He and 12 C abundance after the second dredge-up contributes to increasing the frequency of the thermal pulses in stars with initial mass M 8.5 M e because it makes the H shell more efficient.
Figure 11 shows a zoom of selected models during the last few computed thermal pulses.Moving from the 7 to the 9.2 M e model, the following are worth noting: (1) the reduction of the size of the He convective shell from ∼10 −4 to ∼10 −5 M e , (2) the strong reduction of the interpulse time from ∼10 3 to ∼10 yr, and (3) the progressive reduction of the third dredge-up that disappears in stars with mass M 9.0 M e .It is also worth mentioning that, in general, the higher the mass, the higher the number of thermal pulses occurring before the beginning of the formation of a He convective shell associated with each thermal pulse (see Table 4).
Figure 9 shows that the maximum temperature reached at the base of the convective envelope (T BCE ) is in the range 80-110 MK and scales roughly with the initial mass; i.e., the larger the mass, the larger the T BCE .In general, this quantity increases slightly during the TP phase, but it may also show a nonmonotonic behavior as a function of time if some other energy sources are activated inside the CO core, as in the case of the more massive models (M 9.05 M e ), where the URCA processes become efficient (see below).
The mass loss during this phase plays a key role because it competes with the increase of the CO core in reducing the H-rich envelope and therefore in determining the duration of the TP phase.The typical mass-loss rate averaged over the last few thermal pulses is in the range 1-3 ∼ 10 −5 M e yr −1 , the higher values reached by the more massive models.
The computations are stopped after a sufficient number of thermal pulses have been computed to safely extrapolate the evolution of these stars during the TP phase (see Section 3.8).

Stars with Mass 9.05 M/M e 9.20: URCA Processes
In stars with more massive ONeMg degenerate cores (9.05 M/M e 9.20), the density increases enough (Figure 16) that the Fermi energy becomes close to the threshold value for the ECs on a number of nuclear species that are quickly followed by beta decays.In this situation, given two generic nuclear species, N(A, Z) and M(A, Z − 1), the two reactions N(A, Z) , written in a compact form as A (N, M), are in equilibrium.The reaction pair A (N, M) is called the URCA process.
The effect of the activation of an URCA process can be explained with the aid of Figure 17, which shows the properties of a model in which a generic URCA pair A (N, M) is efficient.If we define λ as the number of captures/decays per unit time, the lower left panel shows that as the density decreases (i.e., the interior mass increases), the EC rate (λ ec ; black solid line) decreases, while the beta decay rate (λ β ; black dashed line) increases.The density at which λ ec = λ β is called the URCA shell (ρ crit ) and is marked in all of the panels of Figure 17 by a vertical dashed line.Inside the mass coordinate corresponding to the URCA shell, ρ > ρ crit and λ ec ?λ β .Outside the URCA shell, on the contrary, ρ < ρ crit and λ ec = λ β .As mentioned above, in this situation, the two reactions are in equilibrium, which means that the number of reactions occurring per unit mass and unit time r of the two processes coincide (r ec = r β ) and show a maximum corresponding to the URCA shell.Since  r = λY (where Y = X/A is the abundance by number, X is the abundance in mass fraction, and A is the atomic weight), this also implies that the equilibrium abundances of the two nuclei satisfy the relation Y(N)/Y(M) = λ β /λ ec .As a consequence, X (M) ?X(N) inside the URCA shell, while X(N) ?X(M) outside the URCA shell (upper left panel in Figure 17).
The energy released by the EC, E ec , and the beta decay, E β , are given by (see Miyaji et al. 1980;Suzuki et al. 2016) where Q nuc is the mass defect between reactants and products, E ν is the neutrino energy loss (in absolute value), and μ e is the chemical potential of the electrons.When an URCA pair is in equilibrium r ec = r β , therefore, the total net energy released per unit mass and unit time in this case will be simply , ; i.e., it will be always negative and will show a deep minimum corresponding roughly to the URCA shell (lower right panel of Figure 17).Thus, in general, we can identify inside a model various cooling zones associated with the URCA shells of the various URCA pairs.It goes without saying, however, that only URCA pairs involving nuclear species with sizable abundances will have some effect on the evolution of the model.In addition to that, as the core of the star contracts, the density increases, and the URCA shell of any given URCA pair shifts outward in mass and constitutes an outward-moving "cooling wave." Having said this, in the following, we will first describe the evolution of the 9.2 M e model.As the central density increases above .The effect of each URCA pair episode at the center is that of a cooling phase, followed by a roughly isothermal evolution (black line in Figure 16).The cooling phase corresponds to the stage when the central density is close to the URCA shell (ρ c ∼ ρ crit ), while the isothermal evolution corresponds to the stage when the URCA shell leaves the center and shifts outward in mass.In order to describe these phases in more detail, we show in Figure 18 some properties of the center of the model during the activation of the 25 (Mg, Na) pair, i.e., the first important URCA pair.
As the central density approaches the URCA shell, the nuclear energy (ε n ) that, as already mentioned above, is dominated by the neutrino emission due to the URCA pair (−E ν ) decreases dramatically.During this phase, the gravitational energy (ε g ) increases, while the (thermo)neutrino losses (ε ν ) progressively decrease due to the lowering of the central temperature, and the net result is that the total energy is negative.This implies a substantial reduction of the central temperature (T c ).As the central density continues to increase (ρ c > ρ crit ), the URCA shell leaves the center and moves outward in mass (driving an outward-moving cooling wave); therefore, in the center, the nuclear energy begins to increase toward the values it had before the activation of the URCA pair (i.e., it tends toward ∼0).During this phase, the gravitational energy decreases progressively, partially reabsorbing the increase of the nuclear energy, while the neutrino energy losses become negligible compared to the nuclear and gravitational energies because of the low temperature.The net effect is that in this phase, the total energy progressively increases toward less negative values.The energy imbalance between the center and the location of the URCA shell also produces an increase in the radiative gradient in the core (see dashed, dotted, and long-dashed lines in Figure 19).When the central density becomes higher than , the total net energy becomes positive, the radiative gradient overcomes the adiabatic one, and the center of the star becomes convective.Note that the gradient of chemical composition around the center is not high enough to stabilize the zone against the onset of convection (Figure 19).This is also confirmed by a test evolution in which we adopted the Ledoux criterion during this phase.Jones et al. (2013), Takahashi et al. (2013), andZha et al. (2019) did not find the formation of the convective core in their models.Such a difference might be due to the difference in the treatment of convection and the zoning, which could affect the gradient of the chemical composition, in the evolutionary codes.
When the convective core sets in, it has a strong effect on the equilibrium of the URCA pair reaction.The reason is the following.In a radiative environment, the 25 Mg abundance in the central zones is the result of the equilibrium between EC and beta decay and has an increasing profile from the center toward the URCA shell (like the one shown in the upper left panel of Figure 17).Once convection sets in, it forces the 25 Mg abundance to increase in the inner zones and decrease in the outer ones compared to the radiative case (compare the upper left panels of Figures 17 and 20).As a consequence, the two reactions of the 25 (Mg, Na) pair are no longer in equilibrium, but, on the contrary, r ec > r β in roughly the inner half of the convective core, while r ec < r β in the outer half (solid and dashed green lines in the figure).In this case, the total net energy released per unit mass and unit time is given by r ec E ec + r β E β .Since E ec is positive in roughly the inner half of the convective core and negative outward in mass, while E β is always negative (see the solid and dashed red lines in the lower left panel of Figure 20), the total energy released by the 25 (Mg, Na) pair is positive in roughly the inner half of the convective core and negative in the remaining half (see the orange line in the lower right panel in the figure), with the zero value roughly corresponding to the mass coordinate where r ec = r β .The continuous ingestion of a higher 25 Mg abundance from the outer radiative layers produces an increase of the nuclear energy close to the center that induces the convective zone to extend even more, driving in this way a progressive increase of the convective core (Figure 19).During the phase characterized by the increase of the convective core, the central density increases progressively at almost constant temperature (Figure 20).
When the central density approaches 9.248, the URCA pair 23 (Na, Ne) starts activating and producing some effects on the structure of the star.The evolution of the center during this phase is similar to that already discussed for the 25 (Mg, Na) pair.The initial phase is characterized by a cooling, due to the EC on 23 Na, that makes the center of the star radiative and forces the convective core driven by the 25 (Mg, Na) pair to become a convective shell that shifts progressively outward in mass.Then, after the URCA shell of the pair 23 (Na, Ne) leaves the center and moves outward in mass, a convective core forms that increases progressively in mass while the center contracts at almost constant temperature (Figure 16).A typical model during this phase is shown in Figures 21 and 22.The inner 0.02 M e zones are convective and show the typical behavior already discussed in the case of the 25 (Mg, Na) pair.In particular, the rate of the EC on 23 Na dominates over the decay of the 23 Ne in approximately the inner half of the convective core, while the 23 Ne decay prevails on the EC on 23 Na in the remaining half.As a consequence, the nuclear energy is positive in the zones where the EC dominates and negative where the beta decay prevails.Inside the convective core, 23 Ne is much more abundant than 23 Na, which, on the contrary, dominates in the outer radiative layers.During this phase, the convective shell driven by the 25 (Mg, Na) URCA pair is roughly confined between the 23 (Na, Ne) URCA shell at the bottom and the 25 (Mg, Na) URCA shell at the top.Also note the radiative zone that separates the convective core and the convective shell; in this zone, r ec = r β .It is worth mentioning at this point that a similar evolution has already been found by Ritossa et al. (1999).In particular, in their Figure 25, they show the properties of their last computed model characterized by a convective core driven by the 23 (Na, Ne) pair and a convective shell driven by the 25 (Mg, Na) pair.The chemical composition and the various contributions to the total energy generation are extremely similar to what we find.In particular, they also find that in each convective region, the URCA pair releases positive energy in the inner zone and negative energy in the outer layers (see panel (c) in their Figure 25).At variance with what we and Ritossa et al. (1999) find, the formation of a convective core and shell during this phase is not addressed by Jones et al. (2013), Takahashi et al. (2013), andZha et al. (2019).We do not have a clear explanation for that; hence, what we can say is that the origin of such a difference could be due to the difference in the numerical treatment of convection in the stellar evolution code.
The evolution of the center, following the formation of the convective core driven by the 23 (Na, Ne) pair, is characterized by an increase of the central temperature at almost constant density interspersed with phases where the density increases at almost constant temperature (black line in Figure 16).The reason for such a behavior is due to the fact that the convective core, after it is formed, begins to progressively increase in mass because of the increase of the nuclear energy produced close to the center due to the ingestion of fresh 23 Na present in the outer radiative zones.When the 23 Na abundance mixed into the convective core is comparable to or even larger than the one initially present, the increase of the nuclear energy is not reabsorbed; on the contrary, it drives a greater increase of the convective core on a very short timescales, compared to the previous evolution.This is a runaway that looks like the breathing pulse phenomenon occurring during the core He burning (see above).During this phase, the central 23 Na mass fraction increases to values as high as ∼3 × 10 −3 , i.e., more than 2 orders of magnitude, compared to the abundance present before the beginning of this process (see the first increase of the central 23 Na abundance; green line in Figure 23).Since the matter is highly degenerate, the increase of the nuclear energy due to this process induces an increase of the central temperature at constant density (Figure 23).We call this phenomenon "temperature increase due to a runaway" (TIR).The increase of the convective core eventually ceases when the 23 Na ingested from the radiative zones is such that it does not significantly alter the nuclear energy.This happens when the mass of the convective core is ∼0.06 M e .During the following evolution, the excess of the nuclear energy is progressively reabsorbed, and the mass of the convective core and the central temperature remain essentially constant, while the central 23 Na abundance progressively decreases toward values similar to those corresponding to the beginning of this process.This stage also coincides with the onset of the thermal pulses.The following evolution of the star is characterized by two other similar processes (see the last two sharp increase of the central temperature in Figure 23) that raise the central temperature to values as high as . During the phase characterized by the thermal pulses, the convective shell driven by the 25 (Mg, Na) increases in mass up to ∼0.9 M e , but this has little effect on the interior of the star (lower right panel in Figure 24).
The calculation of the evolution of this star is then stopped after 102 thermal pulses.The final fate of this star is discussed in the following; however, we anticipate here that, on the basis of the results obtained, it is difficult to envisage whether the center of the star will reach the threshold temperature for the activation of the 20 Ne photodisintegration, or the increase of the central temperature will stop and the core will restart contracting until the density thresholds for the activation of the ECs on 24 Mg first and on 20 Ne later are reached.Moreover, an interaction between the convective core and the convective shell cannot be excluded, with consequences for the evolution of the star that are difficult to predict.
It is interesting to note that the evolution of the center prior to the onset of the TIR discussed above, i.e., until the central density approaches the value , is not affected by the efficiency of mixing in the convective zones.Figures 25 and 26, in fact, show that, as long as , the evolution of both the central density and the central temperature of the standard model (red lines) is almost identical to the one obtained in a test model in which the mixing is artificially suppressed (blue lines).The differences between the two models appear only when the TIR begins in the standard model, i.e., when the red and blue lines begin to differ from each other.The occurrence of this phenomenon has two main effects: it slows down the contraction of the core (Figure 25) and induces an increase of the central temperature (Figure 26) compared to the case in which the chemical mixing is suppressed.For sake of completeness, we also report the results obtained for a test Figure 19.Temperature gradients (see the legend) as a function of the interior mass of the 9.2 M e star at selected times during the formation of the convective core associated with the 25 (Mg, Na) URCA pair, marked by the values of the central densities in units of g cm −3 .According to the adopted stability criterion (see text), convection sets in when the radiative gradient becomes larger than the adiabatic one.Let us remember that the actual temperature gradient used is the adiabatic one in the convective zones and the radiative one in the radiative layers.
model in which the URCA processes are not included (green lines in Figures 25 and 26).In this case, the contraction of the core is slower than in the case of the model where the URCA processes are taken into account and mixing is suppressed and similar to the reference case.Moreover, as expected, the activation of the two URCA pairs 25 (Mg, Na) and 23 (Na, Ne) reduces the central temperature by a factor of ∼3 (at ) compared to the model in which the URCA processes are not included.Let us eventually note that, as mentioned above and shown in Figure 26, the TIR is associated with the presence of a convective core and occurs in an advanced phase after its formation.For this reason, this phenomenon is not found either by Ritossa et al. (1999), because they stop the calculation too early, or by Jones et al. (2013), Takahashi et al. (2013), andZha et al. (2019), because they do not find the formation of the convective core in their models.
The evolution of the center of the 9.15 M e model is shown in Figure 16 (red line).As in the case of the 9.20 M e model, the first cooling phase is due to the activation of the 25 (Mg, Na) URCA pair.The cooling phase ends when the URCA shell shifts from the center outward in mass.Such an occurrence drives the formation of a convective core that progressively increases in mass.At variance with the 9.20 M e model, in this case, the 25 Mg ingested from the radiative zones above the convective core is high enough to induce a TIR before the threshold density for the activation of  27).As for the 9.20 M e model, in this case the TIR is followed by a phase in which the central abundance of the leading isotope (in particular 23 Na for the 9.20 M e and 25 Mg for the 9.15 M e cases, respectively) decreases progressively, and the extra energy provided by the TIR is progressively reabsorbed.This stage also coincides with the onset of the thermal pulses.We stopped the calculation during this phase after the completion of 193 thermal pulses (see Table 4).
The evolution of the 9.10 M e star is similar to the one of the 9.15 M e star (Figure 28), and it is followed for 159 thermal pulses.
3.8.Final Fate of Stars with Initial Mass 7.50 M/M e 9.20 During the thermal pulse phase, the CO core is continuously increased by the alternate advancing of the He-and H-burning shells.Such an occurrence induces an increase of the central density.
During the same stage, however, the star loses mass due to stellar wind, and this induces a progressive reduction of the H-rich envelope.If the CO core mass reaches the value (M CO-ec ) corresponding to a central density close to the before the H-rich envelope is completely lost, the core contracts rapidly until the density approaches the threshold value for the activation of the , and then the star can potentially explode as an ECSN (Miyaji et al. 1980;Nomoto 1987;Zha et al. 2019).If, on the contrary, the H-rich envelope is completely lost before the activation of the ECs on 24 Mg, then the final fate of the star will be as an ONeMg WD (Nomoto 1984).A self-consistent determination of the competition between the increase of the CO core mass and the reduction of the H-rich envelope due to the mass loss would require the calculation of several thousand thermal pulses, which, at present, is not feasible.Therefore, an estimate of the final fate of these stars must necessary rely on an "extrapolated" evolution.
Figure 29 shows the time evolution of the CO core mass (left panels) and the total mass (middle panels) for some selected models, i.e., 8.00, 8.50, 8.80, and 9.00 M e , starting from the beginning of the thermal pulse phase.These two quantities show an almost linear behavior in the last part of the evolution that can be very well approximated by a linear regression (red lines in the left and middle panels of the figure).In the abovementioned panels, we also show the average values of the CO core mass growth rate and mass-loss rate obtained by such a linear regression.Under the assumption that the evolution following the last computed model will remain self-similar, we can easily extrapolate these quantities at late times (dashed lines in the right panels in the same figure).We are aware that when the envelope becomes sufficiently small, the strength of the pulses may change, and therefore the behavior of the CO core mass and the total mass may change accordingly.However, the importance of these effects, if they really exist, is difficult to predict a priori; therefore, as a working hypothesis, we assume a self-similar behavior of the relevant quantities up to the end of the evolution.The intersection of the two (extrapolated) lines, corresponding to the total mass and the CO core mass, is the maximum CO core mass ( -M CO max , marked in the right panels of the figure with a black dot) that can be potentially formed before the envelope of the star is completely removed by the stellar mass loss.This quantity should be compared with M CO-ec , as defined above.An estimate of this last quantity can be obtained by solving the stellar structure equations for a completely degenerate star with a given mass M and a chemical composition typical of the zones interior to the CO core.In particular, we have taken the internal composition of the 9.00 M e star model as a representative one, being that slight variations of the chemical composition do not significantly affect the total mass-central density relation obtained in this way.By adopting the public code provided by F. X. Timmes,9 we find that the mass  30.It is worth noting that for any given mass, the density obtained assuming that the structure is fully degenerate is the minimum one, the reason being that a progressive departure from degeneration allows for a progressively higher contraction and therefore larger central densities (for the same mass).This implies that the value of M CO-ec marked by the blue dashed lines in the abovementioned figures constitutes an upper limit to this quantity.
Figure 30 shows the -M CO max as a function of the initial mass compared to the M CO-ec .Also shown in the figure is the final ONeMg core mass as a function of the initial mass under the assumption that the ONeMg core does not increase during the thermal pulse phase because the accretion rate of the CO core is not high enough to induce further C burning (Nomoto & Iben 1985).
Taking into account all possible uncertainties, we conclude that stars in the range 7.50 M/M e 8.00 will lose their H-rich envelope before the threshold density for the EC on 24 Mg is achieved; therefore, they will produce an ONeMg WD.Stars in the range 8.50 M/M e 9.20, on the contrary, will reach such a critical density before the H envelope reduces enough to definitely quench the H-burning shell.Once the ( ) n -Mg e , Na 24 24 is activated, the final fate of these stars (explosion or collapse to a neutron star) depends on the details both of the explosion modeling and of the initial conditions (see Section 1) and cannot be predicted with certainty in this work.As a final comment, we point out that in stars with initial mass 9.10 M/M e 9.20, the central temperature increases substantially due to the TIR, and therefore the ignition of the

20
The Astrophysical Journal Supplement Series, 270:29 (28pp), 2024 February  Limongi et al.     20 Ne photodisintegration before the activation of the EC on 24 Mg cannot be excluded.In that case, it is difficult to predict, a priori, the final fate of these stars.
3.9.Evolution toward Core Collapse: Stars with M 9.22 M e (M CO 1.08 M e ) Stars with initial mass M 9.22 M e form an ONeMg core in which the maximum temperature reaches the threshold value for the Ne ignition.The thermal behavior of the ONeMg core depends on both the behavior of the C-burning shell and the convective history of the CO core, which, in turn, depend in general on the CO core mass at core He depletion.In the present set of models, we find that the minimum CO core mass at core He depletion for the activation of Ne burning is M CO = 1.08 M e , which corresponds to CO and ONeMg core masses at Ne ignition of M CO = 1.363 and M ONeMg = 1.349M e , respectively (Figure 31 and Table 3).It is interesting to note that Ne ignition is activated before the ONeMg core (which coincides with the C-burning shell by definition) approaches the CO core (i.e., the He-burning shell), as happens in all of the models that do not ignite Ne and evolve through  the SAGB phase.This is the reason why the ONeMg core mass versus the initial mass relation shows a small bending in the transition between SAGB stars and stars that do ignite Ne burning (Figure 31).As in the case of C ignition, the mass coordinate corresponding to the Ne ignition decreases progressively as the initial mass increases, ranging from 0.966 M e for 9.22 M e to 0 for 13 M e , which is the lowest mass that ignites Ne at the center (Figure 12 and Table 3).Off-center Ne burning is ignited under conditions of sizable degeneracy (ψ ∼ 7-5, in the mass range 9.22-12.00M e ); therefore, the local nuclear energy release drives a progressive increase of both the temperature and the luminosity and, as a consequence, the formation of a convective zone.Such a convective zone reaches a maximum extension and then tends to recede in mass as the Ne is progressively depleted.In the lower-mass models (9.22-9.30M e ), during this phase, the temperature approaches values as high as ( ) T log K 9.3; therefore, O burning is also ignited before convection quenches.This drives the convective zone to increase again up to a maximum extension.After the O ignition, the Ne/O burning proceeds simultaneously in a convective shell that progressively moves toward the center as the fuel is locally exhausted, the temperature is increased, and the degeneracy is significantly removed (left panel of Figure 32).In the more massive models (9.50-12.0M e ), on the contrary, the local temperature does not reach the threshold values for O ignition; therefore, the first convective zone quenches and disappears as the Ne is depleted locally.After this first convective episode, contraction resumes, and another convective zone forms.From this time onward, the evolution of the Ne/O burning front in these more massive models is similar to the one described above for the lower-mass stars (see right panel of Figure 32).
Figures 33 shows the main properties of a typical model during the propagation of the Ne/O burning front toward the center.The burning is occurring at the base of the convective shell, marked by the gray area, which is at a high temperature compared to the inner, much cooler radiative zones.Because of the efficient ECs, the main products of the Ne/O burning within the convective shell are 34 S, 28 Si, 30 Si, and 32 S. The efficiency of the ECs, however, decreases as the initial mass of the star increases; therefore, the chemical composition left by the Ne/O burning tends to be dominated by less neutron-rich isotopes as the initial mass of the star increases.Figure 34 shows the chemical composition of selected models once the Ne/O burning front has reached the center.
In the 13.0 M e model, Ne burning is ignited at the center and develops in a convective core.Once Ne is depleted in the center, the burning shifts outward in mass, in the region with a variable composition left by the receding convective core, and drives the formation of a convective shell at a mass coordinate of ∼0.18 M e .During this phase, the temperature in the shell increases enough that O burning is ignited.Ne and O burning then proceed simultaneously in such a shell, which increases progressively in mass up to a maximum extension of 0.17-1.00M e .Once O is exhausted in the shell, the burning shifts inward and drives the formation of a convective core that reaches a maximum extension of ∼0.07 M e before disappearing at O depletion.Ne and O burning develop in the 15.0 M e model as in a typical MS.
It is interesting to note at this point that, at variance with offcenter C burning, no hybrid core is formed as a result of the offcenter Ne ignition.All of the stars that ignite off-center Ne burning form an O-depleted core; i.e., in all of these models, the ONe burning front reaches the center.This result is consistent with what has been found by Woosley & Heger (2015) and can be understood because we are using a similar approach to treat the CBF.On the contrary, Jones et al. (2013) found a case in which the ONe burning front does not propagate toward the center, leading the star to reach central densities high enough for the activation of the EC on 20 Ne and then to explode as an ECSN.This different behavior can be due to the fact that Jones et al. (2013) do not include in the code any specific treatment for the CBF; therefore, their models cannot be directly compared to ours.
In the 9.22 M e star, after the Ne/O burning front has reached the center, the most abundant nuclear species in the O-exhausted core are 34 S (∼0.48), 38 Ar (∼0.22), 28 Si (∼0.16), and 30 Si (∼0.13; left panel of Figure 34).O burning that shifts in a shell settles at a mass coordinate of ∼0.6 M e , where O is still quite abundant.Shell O burning moves outward in mass, inducing the formation of three consecutive convective shells.During this phase, in the inner core, 38 Ar and 28 Si are converted into 34 S and 30 Si, which increase to ∼0.70 and ∼0.28 in mass fraction, respectively.When the O-burning shell has reached ∼1.30M e , nuclear burning is ignited at ∼0.95 M e at a temperature of ∼3 × 10 9 K (Figure 35 shows the physical and chemical structure of the star at this stage).
During the initial phase of this burning, the rearrangement of the matter is such that 34 S and 30 Si are depleted while 28 Si, 52 Cr, 54 Fe, and 56 Fe are produced in a convective shell that increases progressively in size.Once 34 S and 30 Si are exhausted in the shell, convection quenches and the nuclear burning front shifts inward in mass, where 34 S and 30 Si are still abundant, and induces the formation of a convective shell that, once again, reaches a maximum extension and then quenches.The burning front, then, propagates in this way progressively toward the center.A typical model during this phase in shown in  Figure 36.As the 34 S-30 Si burning front moves inward in mass, the interplay between local burning and convective mixing is such that 52 Cr tends to be the dominant nuclear species, followed by 30 Si and 34 S, not completely depleted, and finally by 56 Fe.The physical and chemical structure of the star once the burning front has reached the center is shown in Figure 37.The residual 28 Si (∼0.02 in mass fraction) is then eventually burned in a convective core that increases in size up to ∼0.9 M e and leaves a chemical composition dominated by 52 Cr (∼0.60) and 56 Fe (∼0.28).
Figure 29.CO core mass (left panels) as a function of time, total mass (middle panels) as a function of time, and total mass as a function of CO core mass (right panels) for the 8, 8.50, 8.80, and 9.00 M e models (top to bottom).The time has been reset at the beginning of the thermal pulses.The solid red line in the left and middle panels refers to the linear regression of the black line over the last few thermal pulses, superimposed on the black line itself.The values reported in the plots (MCO dot and M dot ) refer to the rate of growth of the CO core and the rate of mass loss obtained with the linear regression.The dashed lines in the right panels refer to the extrapolation at late times of the various quantities shown in the figure, obtained with the linear regression mentioned before.The vertical blue dashed line marks the CO core mass corresponding to the central density threshold for the activation of the EC on 24 Mg derived as discussed in the text (Zha et al. 2019).
During the following evolution, the core contracts and heats up, and the matter is converted to iron peak (Fe) isotopes.The composition of the Fe core is dominated by the most abundant isotopes of matter at the nuclear statistical equilibrium corresponding to progressively higher values of the temperature and density and progressively lower values of the electron fraction due to the efficient ECs.The Fe core mass at the presupernova stage is M Fe = 1.257M e , and its composition is dominated by 50 Ti, 54 Cr, and 58 Fe (Figure 38).All of the other relevant physical quantities of the model at the presupernova stage are reported in Table 3.
The evolution of the stars in the range 9.25-12.0M e after the Ne/O burning front has reached the center and up to the presupernova stage is similar to that of 9.22 M e .The only difference is the mass coordinate corresponding to the Si-S ignition.In particular, in the models with mass in the range 9.25-9.50M e , the Si-S is ignited at a mass coordinate that progressively decreases as the mass increases; i.e., it is ∼0.52, ∼0.07, and ∼0.005 M e for 9.25, 9.30, and 9.50 M e , respectively (see Table 3).
In the models with mass in the range 9.80-12.0M e , the 28 Si is not completely exhausted in the inner core during the shell O burning, as it happens in the lower-mass models, and it shows a gradient (see Figure 39).The sizable abundance of 28 Si and its profile induces an off-center nuclear ignition at a mass coordinate that decreases as the initial mass increases, ranging from ∼0.387 M e for 9.80 M e to 0 for 13.0 M e , which is the lowest-mass model that ignites Si burning centrally and behaves during this phase as a typical MS.The 10.0 M e model is an outlier in this scheme because for some reason, difficult to understand, a sizable abundance of 28 Si ∼ 0.19 (in mass fraction) is left in the center (in the inner ∼0.020 M e ) at the end of the shell O-burning phase; therefore, in this model, the Si ignition point is more internal than in either the 9.80 or 11.0 M e model (Table 3).
Table 3 reports all of the main physical properties of all of these models at the presupernova stage.

Summary and Conclusions
In this paper, we computed the evolution of stars with initial mass in the range 7.00-15.00M e from the pre-main-sequence phase up to the presupernova stage or an advanced stage of the TP phase, depending on the initial mass.The main goal of these calculations is to study in detail the evolutionary behavior of stars across the transition from AGB and SABG stars to ECSN and CCSN progenitors.
A summary of our results is shown in Figure 40 and discussed below.
All the stars in the mass range studied here evolve through the core H-and He-burning stages.
Stars with initial mass M < 7.50 M e develop a degenerate CO core in which the temperature remains below the threshold value for the ignition of the C-burning reactions.These stars then evolve through the TP-AGB phase and eventually end their evolution by forming a CO WD surrounded by material ejected during the previous evolutionary phases, i.e., a planetary nebula.
In stars with initial mass M 7.50 M e , on the contrary, the temperature in the CO core becomes high enough to allow the ignition of the C-burning reactions.In particular, stars with initial mass in the range 7.50-9.50M e ignite C off-center, with the C ignition point decreasing from 0.588 M e for 7.50 M e to 0.022 M e for 9.50 M e .Stars with initial mass M > 9.50 M e ignite C centrally.This feature is mainly due to the fact that the degree of degeneracy in the CO core decreases progressively as the initial mass increases.In all of the stars, the result of the C burning is the production of an ONeMg core, with the exception of the 7.50 M e star, in which the C-burning front quenches before reaching the center, and therefore a sizable amount of 12 C is left unburned in the inner ∼0.3 M e .In this case, a hybrid CO core is formed, i.e., a CO core in which the central part is enriched by a mixture of O and Ne, resulting from the quenching of the off-center C burning.
After core He depletion, in stars with initial mass M < 11.00 M e , the convective envelope penetrates into the He layer, and the second dredge-up takes place.The evolutionary stage at which this phenomenon begins and goes to completion (i.e., when the convective envelope reaches the maximum depth) depends on the initial mass.In particular, it begins (1) after core He depletion for the 7.00 M e star, (2) before C ignition for the 7.50-8.00M e stars, and (3) after C ignition for the 8.50-10.00M e stars.The convective envelope  reaches its maximum depth during the second dredge-up (1) before the beginning of the TP phase for the 7.00 M e star and (2) after the C-burning phase for the 7.50-10.00M e stars.
In stars with initial mass in the range 7.50-9.20M e , the maximum temperature in the ONeMg core (the hybrid CO core for 7.50 M e ) does not reach the threshold value for the ignition of Ne burning.Therefore, these stars evolve through the TP-SAGB phase.As the initial mass increases, the maximum luminosity of the He-burning shell reached during each pulse decreases, while the frequency of the thermal pulses increases.This is due to the increasing size of the ONeMg core with the initial mass.This also implies that the third dredge-up, i.e., the penetration of the convective envelope into the He core, decreases progressively as the initial mass increases, disappearing for stars with M 9.00 M e .In stars with initial mass in the range 9.05-9.20 M e , the central density becomes high enough that the URCA pair 25 (Mg-Na) is activated.This induces a cooling of the center of the star while the core is still contracting followed by a phase of contraction at constant temperature.In the 9.20 M e star, the density increases enough to reach the threshold for the activation of the URCA pair 23 (Na-Ne).Also in this case, the center initially cools down and then evolves at constant temperature.During these phases, a convective core forms and progressively increases in mass, causing, in stars with mass 9.10-9.20 M e , a phenomenon similar to the breathing pulses in core He burning.This phenomenon happens after the activation of the 25 (Mg-Na) URCA pair, in stars with initial mass 9.10-9.15M e , and after the activation of the 23 (Na-Ne) URCA pair in the 9.20 M e model and induces a substantial increase of the central temperature (TIR).The final fate of all of these stars that do not ignite Ne burning depends on the competition between the increase of the CO core, which may lead to the potential explosion of the star once the central density reaches the threshold value for the ignition of the EC on 24 Mg, and the reduction of the envelope due to the mass loss.The detailed calculation of such a competition would require the calculation of several thousands of thermal pulses (together with the URCA pairs in the more massive ones), which is not feasible with the network adopted in this work and the computers presently available.For this reason, the final fate of these stars has been estimated by means of "extrapolated" evolutions.According to these extrapolations, and taking into account all of the possible uncertainties, we predict that in stars with initial mass in the range 7.50-8.00M e , the mass loss is efficient enough to reduce the total mass before the CO mass reaches the critical value for the activation of the EC on 24 Mg.These stars, therefore, will end their lives producing ONeMg WDs (a hybrid CO WD in the 7.50 M e star case).Stars with initial mass in the range 8.50-9.20 M e develop CO cores massive enough to reach the activation of the EC on 24 Mg before the envelope is completely removed by the mass loss and therefore can explode as ECSNe or collapse to a neutron star; the actual outcome depends on the details of the explosion modeling and the initial conditions and cannot be predicted with certainty in this work.Let us eventually remark that in stars with initial mass 9.10-9.20 M e , the increase of the central temperature due to the TIR up to the threshold value for the ignition of the 20 Ne photodisintegration, before the activation of the EC on 24 Mg, cannot be excluded.In such a case, the prediction of the final fate of these stars is difficult to predict a priori.
In stars in the mass range 9.22-15.00M e , the maximum temperature in the ONeMg core reaches the threshold value for the ignition of Ne burning.In stars with initial mass in the range 9.22-12.00M e , Ne burning is ignited off-center, with the mass coordinate of the ignition point decreasing progressively with increasing mass.The off-center Ne ignition induces the temperature to increase above the threshold value for the ignition of O burning; therefore, in these stars, Ne and O burning occurs simultaneously.The Ne/O burning front then shifts progressively toward the center until an O-exhausted core is formed.Note that, at variance with the off-center C ignition, no hybrid core is formed as a result of the off-center Ne/O burning.In stars with mass M 13.00 M e , the Ne burning is ignited centrally.While in the 13.00 M e star, Ne and O burning occur simultaneously, in the 15.00 M e star, they develop in two different stages, as happens in the classical MSs.Also in these stars, the final result of Ne and O burning is the formation of an O-exhausted  The evolution after either center and off-center Ne/O burning is characterized by the O-shell burning that shifts progressively outward in mass and leads to the Si-S ignition.This burning starts off-center in stars with initial mass in the range 9.22-12.00M e , and the main fuel is 34 S and 30 Si in the lower-mass models (9.22-9.50M e ) and 28 Si in the more massive ones.This is due to the fact the lower-mass models evolve at lower entropy and therefore in these stars, the ECs are more efficient in reducing the electron fraction.As in other previous off-center burning, in this case the Si-S burning front propagates toward the center, followed by a shell Si-S burning phase, until an Fe core is formed.Also in this case, the Si-S burning front does not quench before reaching the center; therefore, no hybrid Si-S core is formed.Si burning is ignited centrally in stars with initial mass M 13.00 M e and is followed by a shell Si-burning phase like in the classical MSs until an Fe core is formed.The final fate of all of the stars in the mass range 9.22-15.00M e is therefore explosion as CCSNe.As a final comment, let us note that the luminosity of the lowermass star that explodes as a CCSN (see Figure 2) is compatible with the estimate of the minimum luminosity for the progenitors of SNIIP derived from the analysis of the high-

Figure 2 .
Figure2.Evolutionary path in the H-R diagram of selected models.The blue line refers to the AGB star, i.e., the one that develops a degenerate CO core, does not ignite C burning, and enters the TP phase.The red lines refer to SAGB stars, i.e., those that ignite C burning and then do not ignite Ne burning; these stars develop a degenerate ONeMg core and enter the TP phase.The black lines refer to those stars that ignite Ne burning and eventually explode as CCSNe.The horizontal green dashed line marks the final luminosity of the lowest mass that explodes as CCSNe, i.e., the expected minimum luminosity of CCSNe(Smartt 2015).

Figure 3 .
Figure 3.He core mass at various evolutionary stages (see legend).

Figure 4 .
Figure 4. CO core mass at various evolutionary stages (see legend).

Figure 5 .
Figure 5. Central 12 C mass fraction at core He depletion (black line and dots) and at first thermal pulse (red line and dots).The blue line refers to the mass of the central zone where the 12 C mass fraction is larger than 0.01.The figure shows that the models with initial mass between 7.5 and 8.8 M e form a hybrid degenerate CO core.

Figure 6 .
Figure6.Evolution of the central temperature and density of all of the computed models.

Figure 7 .
Figure 7. Evolution of the H (blue line) and He (red line) luminosities as a function of time during the AGB phase (for the 7.0 M e model; upper left panel) and the SAGB phase (for the 8.0, 8.5, and 9.2 M e models; upper right, lower left, and lower right panels, respectively).The time has been reset at the core He exhaustion.

Figure 8 .
Figure 8. Evolution of the He (blue line) and C (red line) core masses as a function of time during the AGB phase (for the 7.0 M e model; upper left panel) and the SAGB phase (for the 8.0, 8.5, and 9.2 M e models; upper right, lower left, and lower right panels, respectively).The time has been reset at the core He exhaustion.

Figure 9 .
Figure 9. Evolution of the temperature at the base of the convective envelope as a function of time during the AGB phase (for the 7.0 M e model; upper left panel) and the SAGB phase (for the 8.0, 8.5, and 9.2 M e models; upper right, lower left, and lower right panels, respectively).The time has been reset at the core He exhaustion.

Figure 10 .
Figure10.Evolution of the surface 12 C (red line) and 14 N (blue line) mass fraction and the mass coordinate of the bottom of the convective envelope (green line) as a function of time during the AGB phase (for the 7.0 M e model; upper left panel) and the SAGB phase (for the 8.0, 8.5, and 9.2 M e models; upper right, lower left, and lower right panels, respectively).The time has been reset at the core He exhaustion.

Figure 11 .
Figure 11.Evolution of the convective envelope (green shaded area) and the He convective shell (red line) as a function of time during the last four thermal pulses for the AGB 7.0 M e model (upper left panel) and the 8.0, 8.5, and 9.2 M e SAGB models, (upper right, lower left, and lower right panels, respectively).The time has been reset at the core He exhaustion.

Figure 12 .
Figure 12.Ignition mass coordinate of C (black line and dots) and Ne burning (red line and dots) as a function of the initial mass (see legend).

Figure 13 .
Figure13.Selected interior properties of a 8.5 M e model during the off-center C burning (see the legend).The chemical composition and degeneracy parameter are reported on the left y-axis, while the temperature is reported on the right y-axis.The degeneracy parameter is divided by 10 in order to improve the readability.

Figure 14 .
Figure14.Selected interior properties of a 8.5 M e model during the off-center C burning (see the legend).The chemical composition and degeneracy parameter are reported on the left y-axis, while the temperature is reported on the right y-axis.The degeneracy parameter is divided by 10 in order to improve the readability.

Figure 15 .
Figure 15.Convective (green shaded areas) and chemical (color codes reported in the color bar) internal history of the 9 M e model during the late phase of the second dredge-up.The x-axis reports the logarithm of the time until the end of the evolution (t fin − t) in units of years.
pairs activate and produce some effect on the interior of the star.The first one is 25 (Mg, Na), at density [

Figure 16 .
Figure 16.Central temperature as a function of the central density during the late stages of models in which the URCA processes are activated.

Figure 17 .
Figure 17.Selected properties of a model in which a generic URCA pair A (N, M) is efficient.Upper left panel: abundance in mass fraction as a function of the interior mass of the two interacting nuclei N and M. Upper right panel: density as a function of the interior mass.Lower left panel: number of captures/decays per unit time in seconds (λ; black lines, right y-axis), number of reactions occurring per unit mass and unit time in grams per second (r; green lines, right y-axis), and nuclear energy of the URCA pair in MeV (E; red lines, right y-axis); the solid lines refer to the EC, while the dashed lines refer to the beta decay.Lower right panel: nuclear energy of the URCA pair in ergs per unit mass and unit time (e =n rE ); the black line refers to the EC, the red line to the beta decay, and the orange line to the total of the URCA pair.In all of the panels, the gray vertical dashed line marks the URCA shell.

Figure 18 .
Figure 18.Selected central quantities as a function of the central density of the 9.2 M e model during the activation of the 25 (Mg, Na) pair: temperature (red line, left y-axis), adimensional entropy per baryon (green line, right y-axis), and nuclear (purple), gravitational (blue), and neutrino (magenta) energies (right y-axis).
the 23 (Na, Ne) URCA pair is reached.During the TIR, the central 25 Mg mass fraction increases by ∼3 orders of magnitude, i.e., from -

Figure 20 .
Figure 20.As Figure 17, in the case of a 9.20 M e model during the phase in which a convective core induced by the 25 (Mg, Na) URCA pair is formed.The vertical gray dashed line marks the 25 (Mg, Na) URCA shell.

Figure 21 .
Figure 21.Same as Figure 20 but during a phase in which a convective core, driven by the 23 (Na, Ne) URCA pair, and a convective shell, induced by the 25 (Mg, Na) URCA pair, are formed.The gray and green vertical dashed lines mark the 25 (Mg, Na) and 23 (Na, Ne) URCA shells, respectively.

Figure 22 .
Figure 22.Same as Figure 21 but zoomed in on the inner 0.03 M e .

Figure 23 .
Figure 23.Selected quantities of the 9.20 M e model during the breathing pulses induced by the 23 (Na, Ne) URCA pair: the central 23 Na mass fraction (green line, left y-axis), the mass of the convective core (blue line, right y-axis), and the central temperature (red line, right y-axis).

Figure 24 .
Figure24.Convective (green shaded areas) and chemical (color codes reported in the color bar) internal history during the phase when the URCA processes are active.The x-axis reports the logarithm of the time until the end of the evolution (t fin − t) in units of years.

Figure 25 .
Figure 25.Evolution of the central density as a function of time for three models of the initial mass 9.20 M e prior the onset of the TIR (see text), computed with the following assumptions: URCA processes and convective mixing taken into account (reference model; red line), URCA processes taken into and convective mixing artificially suppressed (blue line), and URCA processes neglected (green

Figure 27 .
Figure 27.Same as Figure 23 but for the 9.15 M e model and during the TIR (see text) induced by the 25 (Mg, Na) URCA pair.

Figure 28 .
Figure 28.Same as Figure 27 but for the 9.10 M e model.

Figure 30 .
Figure30.Final CO and ONeMg core masses obtained with the "extrapolated evolution" based on a linear regression (see text).Also shown is the CO core mass corresponding to the threshold central density for the activation of the EC on 24 Mg.

Figure 31 .
Figure 31.CO (dashed line) and ONeMg (solid line) core mass as a function of the initial mass at various evolutionary stages: core He depletion (red line and dots), first thermal pulse (blue line and dots), and Ne ignition (magenta line and dots).

Figure 32 .
Figure 32.Convective (green shaded areas) and chemical (color codes reported in the color bar) internal history during off-center Ne burning for the 9.25 M e (left panel) and 9.50 M e (right panel) models.The x-axis reports the logarithm of the time until the end of the evolution (t fin − t) in units of years.

Figure 33 .
Figure 33.Selected chemical and physical properties (see the legend) of the 9.50 M e model during the off-center neon burning.

Figure 36 .
Figure 36.Selected chemical and physical properties (see the legend) of the 9.22 M e star during the off-center Si-S burning.

Figure 37 .
Figure 37. Selected chemical and physical properties (see the legend) of the 9.22 M e star when the Si-S burning front has reached the center.

Figure 38 .
Figure 38.Selected chemical and physical properties (see the legend) of the 9.22 M e star at the presupernova stage.

Figure 39 .
Figure 39.Selected chemical and physical properties (see the legend) of the 11.0 M e model during the shell O-burning phase.

Figure 34 .
Figure 34.Selected chemical and physical properties (see the legend) of the 9.22 M e (left panel) and 12.0 M e models when the ONe burning front has reached the center.

Figure 35 .
Figure 35.Selected chemical and physical properties (see the legend) of the 9.22 M e star at off-center Si-S ignition.

Table 1
Nuclear Network Adopted in the Present Calculations Generally speaking, if the heat transfer is efficient enough, one expects a burning front continuously propagating toward progressively more internal zones.On the contrary, if the heat transfer is not efficient enough, the propagation of the burning front toward the center, e.g., ONe burning, has been found to occur by compressional heating(Figures 26 and 27 in

Table 3
Main Properties of the Computed Models Evolution toward the TP-AGB Phase: Stars with Initial Mass M 7.0 M e (M CO 0.69 M e )

Table 4
Main Properties of the TP Phase of the 7.00 M e Model