2.2.1. An important aside on convective borders

It is well known that convective flows may lead to mixing beyond the formal Schwarzschild boundary, which is the position where the buoyancy force is zero. This is often referred to as ‘convective overshoot’. Historically, this term was used for the case where momentum carried a hypothetical fluid element beyond the Schwarzschild border (where the acceleration is zero). The true border is where the velocity reaches zero.

There are two common approaches to modelling this overshoot. In the simplest case, one simply extends the convective region by a chosen amount, typically a multiple α_over times the local pressure scale-height H _p. Another approach is due to Herwig et al. (Reference Herwig, Bloecker, Schoenberner and El Eid1997), which is based on the study of Freytag, Ludwig, & Steffen (Reference Freytag, Ludwig and Steffen1996). Here, one introduces a diffusion coefficient D _OV, which is used beyond the formal convective border. Herwig et al. (Reference Herwig, Bloecker, Schoenberner and El Eid1997) proposed

(1) $$\begin{equation} D_{\rm OV} = D_{0} \text{exp} {\left(\frac{-2z}{f_{\rm {over}} H_{\rm p}}\right),} \end{equation}$$

where D ₀ is the diffusion coefficient near the convective boundary, z is the radial distance from the edge of the convective zone and H _p is the pressure scale height at the convective edge.

Typically, the amount of overshooting used in stellar calculations is based on calibrations to a variety of observations such as the width of the main sequence in stellar clusters, eclipsing binaries (e.g. Schaller et al. Reference Schaller, Schaerer, Meynet and Maeder1992; Schroder, Pols, & Eggleton Reference Schroder, Pols and Eggleton1997; Claret Reference Claret2007; Stancliffe et al. Reference Stancliffe, Fossati, Passy and Schneider2015) and more recently on asteroseismic observations (e.g. Montalbán et al. Reference Montalbán, Miglio, Noels, Dupret, Scuflaire and Ventura2013; Aerts Reference Aerts, Pavlovski, Tkachenko and Torres2013). In the case of the exponential diffusive approach, the value of f _over = 0.016 was shown to reproduce the observed width of the main sequence (Herwig Reference Herwig2000). We note that α_over ≈ 10 × f _over (Herwig et al. Reference Herwig, Bloecker, Schoenberner and El Eid1997; Stancliffe et al. Reference Stancliffe, Fossati, Passy and Schneider2015), with α_over values from the literature typically in the range 0.1–0.4 (e.g. Schaller et al. Reference Schaller, Schaerer, Meynet and Maeder1992; Stancliffe et al. Reference Stancliffe, Fossati, Passy and Schneider2015; Claret & Torres Reference Claret and Torres2016).

However, we note that it is possible, even likely, that other mechanisms may be involved, such as gravity waves, or other hydrodynamical phenomena. In lieu of a complete theory to describe the behaviour of the fluid at the convective boundary, we shall instead refer to ‘convective boundary mixing’ (hereinafter CBM) as the generic process(es) acting at this location. CBM is usually specified via some numerical algorithm, which must suffice until we have a better description of the behaviour of the fluid at convective boundaries.

2.2.2. Incomplete C-burning—hybrid CO–Ne cores

In stars with initial mass slightly above M _up, carbon ignites in the very outer regions of the core. In some cases, after the primary flash has occurred, no further carbon burning takes place. These aborted carbon ignition models have an interior comprised of a very large central CO region surrounded by a thin ONe layer and a further outer CO region (e.g. Doherty et al. Reference Doherty, Siess, Lattanzio and Gil-Pons2010; Ventura & D’Antona Reference Ventura and D’Antona2011). The initial mass range for creation of this class of hybrid CO–Ne cores is very narrow, being at most about 0.1 M_⊙. CO–Ne cores can also be formed when the carbon burning flame stalls on its journey towards the centre.

Due to off-centre (convective) carbon burning in super-AGB stars, a molecular weight inversion is created, which can drive thermohaline mixing (Siess Reference Siess2009). The resultant mixing transports carbon from the inner region towards the burning flame, replacing it with the heavier products of carbon burning. Thermohaline mixing below the C-convective shell is thus able to decrease significantly the C content in the zones ahead of the C-burning flame, and deprived of fuel, this causes the extinction of the flame before it reaches the centre. The result is a hybrid degenerate core, comprising an inner zone of unburnt (but depleted) CO and an outer ONe zone. In such a case, in contrast to calculations that do not include thermohaline mixing, the ONe core is left with a larger amount of unburnt ¹²C, between 2% and 5%, in the centre.

Later, using downward revised values of the thermohaline mixing coefficient based on multi-dimensional hydrodynamic simulations, Denissenkov et al. (Reference Denissenkov, Herwig, Truran and Paxton2013) discounted the ability of thermohaline mixing to stall the carbon burning flame, suggesting that the mechanism was too inefficient. Some CBM was included by Denissenkov et al. (Reference Denissenkov, Herwig, Truran and Paxton2013) at the base of the C-burning convective shell of super-AGB stars, where the C-burning flame was indeed deprived of fuel and stalled. This resulted in hybrid CO–Ne cores being formed.

For determining the composition of stellar material, the main difference between thermohaline mixing and CBM concerns the extent of the induced mixing. Thermohaline mixing connects the entire interior to the region with the higher molecular weight that is being produced by the burning flame. With CBM, the mixing only occurs directly below the convection over a region whose width is determined by some model or algorithm. In contrast to thermohaline mixing, this leaves a pristine CO core interior to the maximum extent of the CBM. Thus, CBM also produces hybrid CO–Ne cores, but these hybrids can be produced with a range of configurations and quite widely varying widths of the ONe shell. This is unlike the structures produced from the lower-mass super-AGB stars that have thin ONe shells in the far outer core.

Chen et al. (Reference Chen, Herwig, Denissenkov and Paxton2014) investigated the effects of CBM and the quite uncertain ¹²C reaction rates on hybrid core creation. They found that varying the efficiency of CBM in addition to the ¹²C reaction rates resulted in the formation of hybrid CO–Ne cores over a wide range of core masses from ≈0.93–1.30 M_⊙. This corresponds to an initial mass range for hybrid CO–Ne core creation (defined as $\Delta M_{\rm {CO\text{--}Ne}}$ ) of up to 1 M_⊙, which would make the CO–Ne cores very common. This wide channel and very large core mass of 1.30 M_⊙ for hybrid CO–Ne WDs could have important implications for the rate of Type 1a SNe due to an increased initial-mass range for progenitors, a reduction of core growth required prior to explosion, and to a reduced delay time between star formation bursts and the occurrence of Type 1a SNe (e.g. Meng & Podsiadlowski Reference Meng and Podsiadlowski2014; Wang et al. Reference Wang, Meng, Liu, Liu and Han2014; Liu et al. Reference Liu, Stancliffe, Abate and Wang2015; Denissenkov et al. Reference Denissenkov, Herwig, Truran and Paxton2013).

Farmer et al. (Reference Farmer, Fields and Timmes2015) studied carbon ignition within intermediate/massive stars with an extensive grid of models looking at the effects of rotation, convective overshooting (using the formalism of Herwig et al. (Reference Herwig, Bloecker, Schoenberner and El Eid1997) with f _over in the range 0–0.02), thermohaline mixing and combinations of these processes. In their study, they found that a substantial number of stars which ignited carbon off-centre went on to form CO–Ne cores. In particular, in agreement with Denissenkov et al. (Reference Denissenkov, Herwig, Truran and Paxton2013) and Chen et al. (Reference Chen, Herwig, Denissenkov and Paxton2014), models with efficient overshooting at the base of the convective carbon burning region led to a very wide initial-mass range for hybrid CO–Ne WDs.

Recently, a work by Lecoanet et al. (Reference Lecoanet2016) using 3D hydrodynamic simulations has suggested that convective mixing cannot stall the carbon burning flame due to the large buoyancy barrier that needs to be crossed to reach the radiative burning front, and hence formation of CO–Ne WDs would not be typical. Irrespective of whether CO–Ne cores could actually form, Brooks et al. (Reference Brooks, Schwab, Bildsten, Quataert and Paxton2017) showed that a structure composed of a higher density ONe mantle above a CO core would be unstable to rapid mixing shortly after the onset of the WD cooling sequence. Thus, the actual occurrence of hybrid cores and their possibility to remain unmixed throughout the latest stages of stellar lives is still a matter of debate.

One of the main interests in hybrid cores is related to the potential eventual fates of SNIa. The amount of available C would probably be high enough so that, if the degenerate core were able to increase in mass up to M _Ch, a thermonuclear (single) SN explosion would result (Poelarends et al. Reference Poelarends, Herwig, Langer and Heger2008). Alternatively, if the super-AGB star were the primary component of a close binary system with specific initial orbital parameters, SNIa explosions might occur.

This possibility was explored by Bravo et al. (Reference Bravo, Gil-Pons, Gutiérrez and Doherty2016), who computed the hydrodynamical explosion of white dwarfs hosting hybrid cores, under different conditions (size of the hybrid cores, and ignition by deflagration or detonation). These authors showed that SNIa harbouring hybrid cores would be characterised by lower kinetic energies and lower amounts of ejected ⁵⁶Ni than their pure CO WD counterparts. Explosions of these hybrid cores may be the theoretical counterparts of the sub-luminous class of SN2002 cx-like SN or SNIax. Denissenkov et al. (Reference Denissenkov, Truran, Herwig, Jones, Paxton, Nomoto, Suzuki and Toki2015) also pointed out the fact that hybrid CO–Ne cores might be one possible reason for the inhomogeneity of observed SNIa. The multi-dimensional hydrodynamical simulations by Kromer et al. (Reference Kromer2015) and Willcox et al. (Reference Willcox, Townsley, Calder, Denissenkov and Herwig2016) also reproduced the same trend, that is, their SNIa models hosting hybrid degenerate cores also produce less ⁵⁶Ni and release less kinetic energy.

2.2.3. Incomplete Ne-burning – failed massive stars

The ability of the Ne burning flame to propagate to the centre is of crucial importance in determining if the star ends its life as an EC-SN or an FeCC-SN. As mentioned in Section 2.2, if the ONe core mass exceeds 1.37 M_⊙, it is assumed that Ne shall ignite and the star will follow the massive star channel. However, there are slight complications to this standard picture. The behaviour of Ne burning is very similar to that seen during the earlier phase of C burning. Efficient neutrino cooling causes the temperature maximum to move away from the centre, resulting in Ne ignition occurring further off-centre for lower masses. Akin to the aborted carbon ignition models described in Section 2.2.2, if neon is ignited at the very outer edge of the core, there will be a brief neon flash but neither subsequent burning nor flame propagation (Timmes et al. Reference Timmes, Woosley and Taam1994; Ritossa et al. Reference Ritossa, García-Berro and Iben1999; Eldridge & Tout Reference Eldridge and Tout2004). It is expected that these stars with core masses so close to the Chandrasekhar mass will end life as EC-SNe after a very brief thermally pulsing phase. Doherty et al. (Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015) proposed a new nomenclature for models that undergo only very slight off-centre neon burning and then later reach the thermally pulsing super-AGB phase, calling them ‘hyper-AGB’ stars.

In addition to the super-AGB evolution towards an EC-SN, a second possible single star EC-SN channel exists, that of ‘failed massive stars’ (Jones et al. Reference Jones2013; Jones, Hirschi, & Nomoto Reference Jones, Hirschi and Nomoto2014). A failed massive star is formed in stars with ONe core masses slightly above the value for Ne ignition. If Ne is ignited far enough off-centre and convective boundary mixing is employed at the base of the Ne burning shell, then instead of a flame progressing smoothly towards the centre, the Ne burning can be stalled and undergo multiple flashes. After each flash, there is a period of contraction that, given enough time, can ultimately result in the core reaching sufficient densities for the Urca processes to be activated and the star to subsequently reach conditions for an EC-SN prior to the Ne flame being able to reach the centre. However, if no CBM is employed and the strict Schwarzschild boundary is used at the base of the Ne convective region, as can be seen in Jones et al. (Reference Jones, Hirschi and Nomoto2014), then the class of failed massive stars ceases to exist and the Ne flame is free to propagate inward towards the centre with the star most likely becoming an FeCC-SN. The exact contribution from failed massive stars to the EC-SN channel is highly uncertain, but if this class of star only occurs for models in which the H-exhausted core has been reduced to precisely the Chandrasekhar mass, (refer to next Section 2.3), then we expect a narrow channel.

2.3.1. Second dredge-up

Due to the gravitational contraction of the core after CHeB, the envelope expands and cools, with convection penetrating inwards into the H exhausted core (Becker & Iben Reference Becker and Iben1979). The SDU event occurs at different stages of the C-burning phase for stars of different initial masses. For low-mass super-AGB stars, it takes place prior the first C-flash. Stars of higher initial mass evolve faster and thus ignite C earlier. Normally, SDU only brings to the surface material that has undergone H burning. However, the more massive stars can experience what is called a ‘corrosive” SDU episode (Gil-Pons et al. Reference Gil-Pons, Doherty, Lau, Campbell, Suda, Guilani, Gutiérrez and Lattanzio2013; Doherty et al. Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b): in these cases the bottom of the convective envelope is able not only to reach below the former HBS, but also to reach deeper, to where the products of the HeBS reside. Corrosive SDU enriches the surface with substantial amounts of primarily ¹²C (and ¹⁸O, e.g. Becker & Iben Reference Becker and Iben1979; Herwig Reference Herwig2004), while in the more massive models corrosive SDU also enriches the surface with substantial amounts of ¹⁶O. The masses of the stellar cores for which corrosive SDU occurs vary between studies, with about 1.15–1.28 M_⊙ in Doherty et al. (Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b), down to 1.03 M_⊙ in Herwig (Reference Herwig2004). These values are generally lower for lower metallicity models due to their broader residual He shells. The degree of metal enrichment from SDU in the envelope of low metallicity stars is critical for their future evolution.

2.3.2. The dredge-out episode

Ritossa et al. (Reference Ritossa, García-Berro and Iben1999) first named, described and provided an extensive analysis of the phenomenon known as dredge-out. It was later reported by Siess (Reference Siess2007), Poelarends et al. (Reference Poelarends, Herwig, Langer and Heger2008), Takahashi et al. (Reference Takahashi, Yoshida and Umeda2013), Gil-Pons et al. (Reference Gil-Pons, Doherty, Lau, Campbell, Suda, Guilani, Gutiérrez and Lattanzio2013), Doherty et al. (Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015) and Jones et al. (Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a) using different evolutionary codes and different input physics (in particular the treatment of mixing and treatment of convective borders). This phenomenon occurs for massive super-AGB stars regardless of their metallicity (e.g. Gil-Pons et al. Reference Gil-Pons, Doherty, Lau, Campbell, Suda, Guilani, Gutiérrez and Lattanzio2013) and occurs for stars in the upper ≈ 0.3 M_⊙ range of super-AGB stars.

Figure 4 shows the evolution during C-burning and dredge-out phase for a 9.5 M_⊙ Z = 0.001 star from Siess (Reference Siess2007). Nearing the end of the carbon burning phase, a convective He shell develops near the upper boundary of the partially degenerate core. This shell is initially separated from the base of the convective envelope by a relatively extended radiative region (about 1 M_⊙) and a thin semiconvective region near the He–H interface. As described in Ritossa et al. (Reference Ritossa, García-Berro and Iben1999), the He convective shell is initially sustained mainly by C-burning, and gravothermal energy, but He-burning powers its final approach towards the base of the convective envelope. Eventually, these convective regions meet and protons are ingested into very high temperature (≳ 10⁸ K) He- and C-rich regions. These ingested protons rapidly undergo the ¹²C(p,γ)¹³N reaction leading to a H-flash with peak luminosities of L _H ~ 10⁹ L_⊙. The associated total energy release from the H-flash is vast, and is generated in a very small region. According to estimates by Jones et al. (Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a), the energy released by this process represents about 11% of the internal energy and about 8% of the binding energy of the combustive flame layer. It is likely that this has hydrodynamical consequences and the assumption of hydrostatic equilibrium should be doubted. At the very least, it is likely that time-dependent convection is required (Herwig et al. Reference Herwig, Pignatari, Woodward, Porter, Rockefeller, Fryer, Bennett and Hirschi2011). Jones et al. (Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a) suggest that this dredge-out may provoke a phenomena similar to the global oscillation of shell-H ingestion (GOSH) event (Herwig et al. Reference Herwig, Woodward, Lin, Knox and Fryer2014), which could potentially drive more powerful and non-radial hydrodynamic events leading to mass ejection.

Figure 4. Kippenhahn and luminosity diagram during the carbon burning phase and dredge-out episode for a 9.5 M_⊙ Z = 0.001 model from Siess (Reference Siess2007). Time is counted backwards from the last computed model.

With an abundant supply of ¹³C now in a high-temperature, helium-rich region, the ¹³C(α,n)¹⁶O reaction is expected to take place at a rapid pace and produce a substantial number of free neutrons (Doherty et al. Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015; Jones et al. Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a).

A dredge-out event is expected to produce neutron densities of the order N _n ≈10¹⁵ n cm⁻³ corresponding to the intermediate n-capture regime (known as the ‘i-process’, see Cowan & Rose Reference Cowan and Rose1977). This process in super-AGB stars was suggested by Jones et al. (Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a) to be responsible for the occurrence of some carbon-enhanced metal poor stars enriched in s- and r-process elements (the CEMP s/r stars, see Beers & Christlieb Reference Beers and Christlieb2005). However, based on an IMF argument, the relatively few super-AGB stars seem unlikely to be a major source of pollution of the CEMP s/r stars (Abate, Stancliffe, & Liu Reference Abate, Stancliffe and Liu2016).

Besides the possibility of ejection of heavier-than-iron elements, the dredge-out process also alters surface abundances of light elements, in particular He and the He-burning product ¹²C (Ritossa et al. Reference Ritossa, García-Berro and Iben1999). This results in the most massive super-AGB stars becoming carbon stars. We discuss the importance of the surface composition during the super-AGB phase in Section 2.4.

In models of slightly lower mass than those that undergo dredge-out, near the end of carbon burning and prior to the SDU, there is also the formation of a convective He region. However, this convective zone decays before the convective envelope penetrates inwards, and therefore does not merge with the proton-rich region. This material will be highly enriched in ¹²C and will later also be dredged up to the surface (e.g. Herwig et al. Reference Herwig, VandenBerg, Navarro, Ferguson and Paxton2012).

3.1.1. The effect of the initial composition

Decreasing the stellar metallicity means decreasing the number of CNO nuclei and hence the efficiency of the CNO cycles. To maintain the required energy production for fewer CNO seeds, the stars attain higher central temperatures and luminosities during the main sequence phase of evolution. This results in a larger He core mass for the same initial mass and also more massive cores during CHeB, and hence results in more massive CO cores. Due to this, the M _up values are seen to decrease with decreasing metallicity until reaching a plateau (or minimum e.g. Cassisi & Castellani Reference Cassisi and Castellani1993; Bono et al. Reference Bono, Caputo, Cassisi, Marconi, Piersanti and Tornambe2000) at about Z = 0.001 to 10⁻⁴. As can be seen in Figure 6, the behaviour of M _mas with metallicity echoes that of M _up albeit with an offset of about 1.5–2.1 M_⊙ to higher initial masses.

Models that employ the strict Schwarzschild criterion for convective boundaries such as those from Siess (Reference Siess2007) typically produce the smallest HeB core, and hence represent a reasonable upper limit to the values of M _up and M _mas. For near solar composition (Z = 0.02) and using the strict Schwarzschild criterion, the M _up and M _mas values are ~9 and 11 M_⊙, respectively (Garcia-Berro & Iben Reference Garcia-Berro and Iben1994; Ritossa et al. Reference Ritossa, García-Berro and Iben1999; Siess Reference Siess2007; Doherty et al. Reference Doherty, Siess, Lattanzio and Gil-Pons2010; Takahashi et al. Reference Takahashi, Yoshida and Umeda2013).

The values of M _up and M _mas also show large variations due to the He content (e.g. Becker & Iben Reference Becker and Iben1979; Bono et al. Reference Bono, Caputo, Cassisi, Marconi, Piersanti and Tornambe2000). Stars with larger initial He contents are more luminous and develop more massive convective cores during CHB. Their CHB lifetime is also substantially shorter due to both the reduced amount of H fuel and the hotter, larger cores that burn the fuel more efficiently. This larger core follows through to the CHeB phase resulting in a larger CO core that leads to a reduction of ~1.6–2 M_⊙ in M _up and M _mas when enrichments of Y ~ 0.08–0.15 are used (e.g. Bono et al. Reference Bono, Caputo, Cassisi, Marconi, Piersanti and Tornambe2000; Shingles et al. Reference Shingles, Doherty, Karakas, Stancliffe, Lattanzio and Lugaro2015).

3.1.2. The effect of overshooting

Convective overshooting during the core H and He burning phases acts to mix additional fuel into the core. This increases the lifetime of these phases. It also increases the maximum size of the convective cores with this effect being more prominent during CHeB. With commonly used values of core overshoot, e.g. f _over = 0.016, the M _up and M _mas values are generally reduced by about 2–2.5 M_⊙ (e.g. Bertelli, Bressan, & Chiosi Reference Bertelli, Bressan and Chiosi1985; Siess Reference Siess2007; Gil-Pons, Gutiérrez, & García-Berro Reference Gil-Pons, Gutiérrez and García-Berro2007; Poelarends et al. Reference Poelarends, Herwig, Langer and Heger2008; Farmer et al. Reference Farmer, Fields and Timmes2015). This is highlighted in Figure 6 by comparing the large/small open square values from Siess (Reference Siess2007), which are for models with/without overshoot, respectively. We note that values of M _mas from the recent literature that include convective overshooting are in reasonable agreement with these results. For example, for Z = 0.015, Woosley & Heger (Reference Woosley and Heger2015) find 9 M_⊙, while Jones et al. (Reference Jones2013) find 8.8–9.5 M_⊙ for Z = 0.02. However, even assuming the same convective approach during the pre-carbon burning phases, the M _up values vary considerably between studies. For example, Girardi et al. (Reference Girardi, Bressan, Bertelli and Chiosi2000) find very low values of about 4.5–5 M_⊙ at Z = 0.02. The causes of the differences are hard to attribute in some cases, as discussed in Siess (Reference Siess2007).

3.1.3. The effect of rotation

Stellar rotation has an impact similar to that of overshooting, in that it increases both the duration and the size of the convective core during CHB (e.g. Maeder & Meynet Reference Maeder and Meynet2000; Ekström et al. Reference Ekström2012). This larger core is then inherited by the CHeB phase, and hence we expect a larger CO core and presumably this would lead to a reduction in the initial mass for carbon ignition with increasing rotation rate. However, in their grid of intermediate-mass Z = 0.02 metallicity models with overshooting, Farmer et al. (Reference Farmer, Fields and Timmes2015) found that for a given initial mass, the CO core mass at carbon ignition was practically the same, irrespective of the initial rotation rate that ranged Ω/Ω_crit = 0–0.5 (their Figure 15). In this case, it seems the rotation does not impact the M _up and M _mas values provided that overshoot is efficient enough, and that rotation velocities are not close to their critical values.

3.1.4. Reaction rates

The rate of the ¹²C + ¹²C reactions can either hasten or delay the onset of C burning and hence alter the contraction time between He burning and C ignition. The CO core grows considerably during this phase and the impact of this should not be overlooked. The carbon burning ¹²C + ¹²C reaction rates can also modify the M _up value. It has been suggested that there exists a possible unknown/unmeasured resonance (Spillane et al. Reference Spillane2007) that would increase the reaction rate above that currently recommended (Caughlan & Fowler Reference Caughlan and Fowler1988, hereafter CF88) and lead to a reduction in the core mass which ignites carbon, and hence reduces M _up. Straniero, Piersanti, & Cristallo (Reference Straniero, Piersanti and Cristallo2016) examined the impact of including a narrow resonance at 1.45 MeV in the standard carbon burning rate. They found that the minimum CO core mass for carbon burning was shifted from ~1.06 M_⊙ down to 0.95 M_⊙, and this resulted in a uniform decrease of M _up by about 2 M_⊙ for their study over metallicities Z = 10⁻⁴−0.03. This can be seen in Figure 6 by the extent of the arrows representing models with the modified carbon burning rate. This reduction in M _up with increased carbon reaction rate is in agreement with the results of Chen et al. (Reference Chen, Herwig, Denissenkov and Paxton2014). However, with their maximum rate (1 000 × CF88), they find a lesser decrease, at about 1.3 M_⊙ (to M _up ~ 5.3 M_⊙ at metallicity Z = 0.01). In Fraser et al. (Reference Fraser2011), the M _mas value was seen to decrease by 1 M_⊙ (to 7 M_⊙ at metallicity Z = 0.02) when the ¹²C + ¹²C reaction rates were enhanced by a factor of 10⁵ compared to the standard CF88 rates.

Given the shape of the IMF and the decrease in M _mas with decreasing metallicity, we expect that the SN rate was higher in the past. In summary, the two limiting masses M _up and M _mas are very uncertain and even with ‘reasonable’ choices of input physics their values may vary by over 3 M_⊙. For example, at close to solar metallicity (Z = 0.02), M _up can vary between about 5.5 and 9 M_⊙. In Section 5, we discuss the observational probes that are being used to aid in constraining these important mass boundaries.

3.2.1. Core growth

The core growth rate during the super-AGB phase is dictated by the outward movement of the H burning shell and progresses at ~10⁻⁶ M_⊙ yr⁻¹ (Ritossa et al. Reference Ritossa, García-Berro and Iben1999; Poelarends et al. Reference Poelarends, Herwig, Langer and Heger2008; Siess Reference Siess2010; Doherty et al. Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015) with typically faster growth rates in the more massive and/or metal-rich models.

An important factor that influences the effective core growth rate is the TDU. Unfortunately, whilst the efficiency of TDU is a very important quantity, it is also one of the most poorly constrained aspects of AGB modelling, especially at larger core masses. It depends on many factors such as the resolution (Straniero et al. Reference Straniero, Chieffi, Limongi, Busso, Gallino and Arlandini1997), numerics (Stancliffe, Tout, & Pols Reference Stancliffe, Tout and Pols2004; Stancliffe Reference Stancliffe2006), and treatment of convective boundaries (Frost & Lattanzio Reference Frost and Lattanzio1996; Herwig et al. Reference Herwig, Bloecker, Schoenberner and El Eid1997; Mowlavi Reference Mowlavi1999; Jones et al. Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a). As mentioned in Section 2.4, in super-AGB stars, the efficiency of TDU varies considerably between studies and ranges from λ = 0 to λ > 1, typically being smaller for more massive models. There is some evidence for TDU in massive AGB stars of high metallicities. This is in the form of Rb over-abundances observed in bright O-rich AGB stars in the Galaxy and Magellanic Clouds (García-Hernández et al. Reference García-Hernández, García-Lario, Plez, D’Antona, Manchado and Trigo-Rodríguez2006, Reference García-Hernández2009). The existence of very luminous carbon rich stars in the Magellanic Clouds is suspected to arise from TDU events after the cessation of HBB (Frost et al. Reference Frost, Cannon, Lattanzio, Wood and Forestini1998; van Loon, Zijlstra, & Groenewegen Reference van Loon, Zijlstra and Groenewegen1999). Unfortunately, we are yet to unambiguously identify any super-AGB stars (see Sections 5.1 and 5.3), and there are no constraints from lower metallicity objects, because such stars have long since died. A standard tracer of AGB nucleosynthesis is ⁹⁹Tc, which is produced by neutron captures in the deep layers of the star. Technetium has no stable isotope, and its longest lived isotope is ⁹⁹Tc with a half-life of 0.21 Myr. Therefore, the detection of Tc in the stellar spectra is the signature of recent nucleosynthetic activity and the presence of TDU. Unfortunately, super-AGB stars are not expected to be able to produce large enough amounts of Tc for it to be observable (e.g. from the massive AGB stars study by García-Hernández et al. Reference García-Hernández, Zamora, Yagüe, Uttenthaler, Karakas, Lugaro, Ventura and Lambert2013).

3.2.2. Mass loss and mass ejections

Whilst the mass-loss rate is fundamental to determining the final fates of super-AGB stars, unfortunately it is also highly uncertain especially at lower metallicities. Mass loss in (super-)AGB stars is thought to be via pulsation aided dust-driven winds. First, large amplitude pulsations are responsible for forcing material to large enough radii and increasing the density enough for dust grains to form. The radiation pressure from the star is then able to accelerate these dust particles which are collisionally coupled to the gas, leading to quite efficient mass loss (Wood Reference Wood1979). Although there is no mass-loss rate derived specifically for super-AGB stars, it is common to use rates derived for lower mass AGB stars (e.g. Vassiliadis & Wood Reference Vassiliadis and Wood1993; Bloecker Reference Bloecker1995), or for rates taken from red super-giant and O-rich AGB stars (van Loon et al. Reference van Loon, Cioni, Zijlstra and Loup2005). Using these prescriptions, the mass-loss rates in super-AGB stars are of the order 10⁻⁴–10⁻⁵ M_⊙ yr⁻¹.

The mass-loss rate for super-AGB stars at low metallicity is unknown and we can only apply prescriptions that were derived using observations of solar metallicity, or moderately metal-poor stars. Due to their more compact structure, low metallicity stars are expected to have slower mass-loss rates. Kudritzki, Pauldrach, & Puls (Reference Kudritzki, Pauldrach and Puls1987) proposed a metallicity scaling proportional to $\sqrt{Z/Z_\odot }$ in an attempt to take this into account. We note however that this scaling was derived for radiative line-driven winds of hot luminous O stars whose conditions are quite unlike the cool super-AGB stars we are considering here.

Another important factor that determines the mass-loss rate in super-AGB stars is the envelope opacity. The use of low temperature molecular opacities that take into account the envelope composition variations is crucial for cases where the envelope molar abundance ratio C/O exceeds unity. The change in molecular chemistry when a star becomes carbon rich leads to an increase in opacity, which results in a cooler and more extended stellar envelope and a higher mass-loss rate (Marigo Reference Marigo2002; Cristallo et al. Reference Cristallo, Straniero, Lederer and Aringer2007; Ventura & Marigo Reference Ventura and Marigo2010; Constantino et al. Reference Constantino, Campbell, Gil-Pons and Lattanzio2014; Doherty et al. Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b). This effect may play an important role for low metallicity super-AGB stars. This is especially true for the most massive super-AGB stars, with post-SDU/dredge-out core masses closest to M _Ch. These stars have often had their surface enriched in C due to dredge-out events and are already carbon rich (with C/O > 10 in some cases) at the start of the TP-SAGB phase.

The final fates of metal poor super-AGB stars are also strongly influenced by the efficiency of SDU/dredge-out prior to the TP-SAGB phase. These processes are able to mix significant amounts of metals from the stellar interior leading to surface metallicities up to Z ≈ 0.001, with the amount of enrichment increasing with stellar mass (Gil-Pons et al. Reference Gil-Pons, Doherty, Lau, Campbell, Suda, Guilani, Gutiérrez and Lattanzio2013). Furthermore, the nucleosynthesis and mixing processes that occur during the TP-SAGB also alter their total surface metallicity. As a consequence, their envelope opacity values, surface luminosities and radii become very similar to their higher Z counterparts. Envelope metallicity during the TP-SAGB is critical in terms of the strength of stellar winds, as higher Z objects are thought to be able to drive higher mass-loss rates.

At low metallicities, if the mass-loss rate is sufficiently low, there is the possibility for stars with initial masses below M _up to be able to grow enough to reach M _Ch and explode as a Type 1.5 SN (Arnett Reference Arnett1969; Iben & Renzini Reference Jr and Renzini1983; Zijlstra Reference Zijlstra2004; Gil-Pons et al. Reference Gil-Pons, Gutiérrez and García-Berro2007; Lau, Stancliffe, & Tout Reference Lau, Stancliffe and Tout2008; Wood Reference Wood, Qain, Leung, Zhu and Kwok2011). This class of SN is suspected to occur when a degenerate CO core within an AGB star grows to the Chandrasekhar mass prior to removal of its envelope. Carbon is then ignited under conditions such that the thermonuclear runaway will lead to an explosion that will disrupt the entire star. Due to the massive H-rich envelope, the early light curve should resemble that of a Type II SN; however, due to the production of a substantial amount of radioactive Ni within the explosion, the resultant light curve will also have a late exponential luminosity decline reminiscent of a Type Ia SN.

In addition to standard super-wind mass loss from super-AGB stars, there have also been two suggested potential mass expulsions events, from either the Fe-peak instability (Lau et al. Reference Lau, Gil-Pons, Doherty and Lattanzio2012) or as a result of convective-reactive H-ingestion events (Jones et al. Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a). These phenomena are expected to occur at different evolutionary phases, with the Fe-instability generally seen near the end stages of the TP-(S)AGB when the envelope has reduced below about 3M_⊙, whilst convective-reactive H-ingestion episodes and potential mass ejection could occur prior to the start of the thermally pulsing phase during the dredge-out phase. These types of mass ejections make determining the final fates of super-AGB stars, in particular the most massive near the EC-SN boundary, quite problematic.

3.2.3. Structural models and/or population synthesis

Whilst the final fates of thermally pulsing super-AGB stars had been examined for individual models of Z = 0.02 in the pioneering series of papers by Garcia-Berro, Iben and Ritossa (Garcia-Berro & Iben Reference Garcia-Berro and Iben1994; Ritossa et al. Reference Ritossa, Garcia-Berro and Iben1996; Garcia-Berro, Ritossa, & Iben Reference Garcia-Berro, Ritossa and Iben1997; Iben, Ritossa, & Garcia-Berro Reference Iben, Ritossa and Garcia-Berro1997; Ritossa et al. Reference Ritossa, García-Berro and Iben1999), the global final fates problem for super-AGB stars was first tackled in a parametric fashion by Poelarends (Reference Poelarends2007), Poelarends et al. (Reference Poelarends, Herwig, Langer and Heger2008) and Siess (Reference Siess2007), with each of these studies using slightly different approaches.

In Poelarends (Reference Poelarends2007), a suite of synthetic models was computed exploring the rate of EC-SNe when using differing mass-loss rates (Vassiliadis & Wood Reference Vassiliadis and Wood1993; van Loon et al. Reference van Loon, Cioni, Zijlstra and Loup2005), efficiencies of TDU and a metallicity scaling included in the mass-loss rate. The final fate results from their best estimate, which included the mass loss prescription from van Loon et al. (Reference van Loon, Cioni, Zijlstra and Loup2005), parameterised TDU from Karakas, Lattanzio, & Pols (Reference Karakas, Lattanzio and Pols2002), and metallicity scaling of Kudritzki et al. (Reference Kudritzki, Pauldrach and Puls1987), are shown in the top panel of Figure 7 (which is adapted from Figure 12 in Langer Reference Langer2012). This figure includes the critical mass limits delineating different evolutionary fates. We define ΔM _EC-SN and ΔM _ONe as the range of initial masses that produces EC-SNe and ONe WDs respectively. For Z = 0.02, this EC-SN channel is narrow with $\Delta M_{\rm {EC\text{-}SN}}$ ~ 0.2 M_⊙, but at the lowest metallicity, all super-AGB stars would end life as EC-SNe, giving $\Delta M_{\rm {EC\text{-}SN}}$ ~ 1.8 M_⊙, and leaving no ONe WDs. Interestingly at Z = 10⁻⁵ even the most massive CO cores are able to grow to M _Ch and explode as Type 1.5 SN. The cause of this increase in the EC-SN rate with decreasing metallicity is primarily the application of the metallicity scaling on the mass-loss rate. In the case with no metallicity scaling applied to the mass loss (dashed line in Figure 7), they find $\Delta M_{\rm {EC\text{-}SN}} \sim 0.25$ –0.55 M_⊙ with the wider range at lower metallicity.

Figure 7. Final fates of intermediate-mass stars from Poelarends (Reference Poelarends2007) (top panel) and Doherty et al. (Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015) (bottom panel). Solid lines delineate M _up, M _n, and M _mas. The dashed line in the top panel represents the M _n value in the case where no metallicity factor is applied to the mass-loss rate. The hatched region represents the width of the EC-SNe channel. As mentioned in Section 3.1, there is a slight offset in the M _up and M _mas values between the two sets of models, with this due to the different method for treatment of convective boundaries during CHeB: Poelarends (Reference Poelarends2007) included convective overshooting via the method of Herwig et al. (Reference Herwig, Bloecker, Schoenberner and El Eid1997) with an overshoot parameter of f _over = 0.016, whilst Doherty et al. (Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015) used the search for convective neutrality approach of Lattanzio (Reference Lattanzio1986). We note that if no super-AGB stars become EC-SN, then M _n=M _mas. It is also worth noting that M _n will always be greater than M _up because even if explosions of CO cores occur (the Type 1.5 SNe), they are not expected to leave neutron star remnants. The lowest metallicity examined in Doherty et al. (Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015) was Z = 10⁻⁴, but here we present new models for Z = 10⁻⁵ calculated using the same methodology as in the previous work.

Also critical to the width of the EC-SN channel is the occurrence of TDU, with lower efficiencies resulting in a wider channel. However, we expect the impact of the metallicity scaling upon the mass-loss rate to be greater than the possible lack of TDU. In Poelarends et al. (Reference Poelarends, Herwig, Langer and Heger2008), the width of the EC-SN channel was explored for solar metallicity using a selection of commonly used mass-loss rates (Reimers Reference Reimers1975; Vassiliadis & Wood Reference Vassiliadis and Wood1993; Bloecker Reference Bloecker1995; van Loon et al. Reference van Loon, Cioni, Zijlstra and Loup2005) as well as the efficiency of TDU, finding $\Delta M_{\rm {EC\text{-}SN}}$ ranging from ~0.2–1.4 M_⊙. For a rough estimation, the order of increasing mass-loss rate is the following: Reimers (Reference Reimers1975) (with η = 5), van Loon et al. (Reference van Loon, Cioni, Zijlstra and Loup2005), Vassiliadis & Wood (Reference Vassiliadis and Wood1993) and Bloecker (Reference Bloecker1995) (with η = 0.1), with approximate values for a typical metal-rich super-AGB star being 0.2, 0.4, 0.8 and 3 × 10⁻⁴ M_⊙ yr⁻¹, respectively (see Figure 7 in Doherty et al. Reference Doherty, Gil-Pons, Lau, Lattanzio and Siess2014a).

In Siess (Reference Siess2007), post-SDU/dredge-out core masses were taken from detailed calculations and then the further evolution during the TP-SAGB phase was extrapolated based on the ratio of the average envelope mass-loss rates $\dot{M}_{\rm {env}}$ to average effective core growth rates $\dot{M}_{\rm {core}}$ characterised by a ζ parameter defined as $\zeta =\mid {\frac{\dot{M}_{\rm {env}}}{\dot{M}_{\rm {core}}}}\mid$ . With best estimate values of $\dot{M}_{\rm {core}} = 5 \times 10^{-7}$  M_⊙ yr⁻¹, TDU efficiency λ values between 0.3–0.9, and with $\dot{M}_{\rm {env}}$ = 5 × 10⁻⁵ M_⊙ yr⁻¹, the ζ values range from 140–1000. With these values, the $\Delta M_{\rm {EC\text{-}SN}}$ ranged from ~ 0.1 M_⊙ (ζ = 1000) to ~0.25–0.6 M_⊙ (ζ = 100) with these widths relatively constant over the entire metallicity range Z = 0.04–10⁻⁵. When the metallicity scaling of Kudritzki et al. (Reference Kudritzki, Pauldrach and Puls1987) was applied to the mass-loss rate, a very similar result to that of Poelarends (Reference Poelarends2007) was found with $\Delta M_{\rm {EC\text{-}SN}}$ increasing substantially at lower metallicity, and at Z = 10⁻⁵, the ONe WD channel disappears entirely.

Using detailed evolutionary calculations, Doherty et al. (Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015) confirmed the results of these earlier parametric studies. They found that when using reasonable mass-loss rates (Vassiliadis & Wood Reference Vassiliadis and Wood1993), without an explicit metallicity scaling but with efficient TDU, the width of the EC-SN channel from super-AGB stars is narrow with $\Delta M_{\rm {EC\text{-}SN}}$ ~0.1–0.2 M_⊙(bottom panel of Figure 7). This figure shows that the vast majority of stars that enter the TP-SAGB phase will end life as ONe WDs, in sharp contrast with the favoured set by Poelarends (Reference Poelarends2007).

By taking the mass range of the EC-SN channel and weighting it with an IMF, the importance and fractional contribution towards the overall core-collapse supernova rate can be determined. Using a Salpeter IMF and the EC-SN, width and mass limits from the synthetic calculations of Poelarends (Reference Poelarends2007) result in 5%, 17% and 38% of all Type II-P SNe coming from the EC-SN channel for metallicities Z = 0.02, 0.001 and 10⁻⁵, respectively. For our calculations, we assumed that the maximum mass for a Type II-P SN is 18M_⊙ based on the analysis of SN observations by Smartt (Reference Smartt2015).

In contrast to using the results from Doherty et al. (Reference Doherty, Gil-Pons, Siess, Lattanzio and Lau2015) over these same metallicities, we find a far smaller percentage, of about 2–5% of all Type II-P SNe will be EC-SNe. This large variation in frequency of EC-SNe highlights the importance of constraining the mass-loss rate at low metallicity.

We note here that models that undergo core overshooting will have far smaller envelopes (by as much as 2–3 M_⊙) to remove in order for a star to avoid the EC-SN channel. This will result in fewer SNe for models with larger values of overshooting.

4.3.1. Light elements

Super-AGB star yields for the production of elements lighter than Fe have only quite recently become available. There is now a variety of calculations from different research groups, which cover a large range of masses and metallicities including a wide variety of input physics. Grids of nucleosynthesis calculations of super-AGB stars have been produced from three main groups: those using starevol (Siess Reference Siess2010), aton (Ventura et al. Reference Ventura, Di Criscienzo, Carini and D’Antona2013; Di Criscienzo et al. Reference Di Criscienzo2016), and monstar/monsoon (Doherty et al. Reference Doherty, Gil-Pons, Lau, Lattanzio and Siess2014a, Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b). For full details of choices of stellar model parameters the reader should refer to these publications. Here, we briefly describe the major differences between the different set of models.

In contrast to the model grids using monstar, those using starevol and aton do not undergo TDU events and therefore show the abundance patterns of pure HBB. Another significant difference between models is that starevol and monstar use standard mixing-length theory with an alpha calibrated to produce solar radius. The aton models utilise the FST (Canuto & Mazzitelli Reference Canuto and Mazzitelli1991) model for convection, which produces higher temperatures during HBB, with the result that these models undergoing more advanced nucleosynthesis.

The mass-loss prescriptions also vary. The standard model sets for starevol and monstar both use the mass-loss prescription from Vassiliadis & Wood (Reference Vassiliadis and Wood1993), whilst aton use the more rapid Bloecker (Reference Bloecker1995) rate with an η value of 0.01. Slower mass-loss rates lead to more TDU enrichment (if TDU is occurring) and the longer duration on the super-AGB gives HBB more time to process material. This, together with higher temperatures at the bottom of the convective envelope, will lead to more advanced nucleosynthesis. Another important factor is the nuclear reaction rates, especially for the heavier proton capture reactions.

A common feature of all super-AGB models at all metallicities is the large production of HBB products ¹³C and ¹⁷O as well as ⁴He, with most coming from efficient SDU. Lithium can be either destroyed or produced in super-AGB stars dependent primarily on the mass-loss rate. When HBB begins we see production of Li, but once the ³He is all used, the destruction of Li dominates. So if the mass-loss is sufficiently high that the star ends its life early, before the ³He is all destroyed, then the star may be a nett producer of Li. If the mass-loss rate is lower, and the Li is destroyed before most of the mass is ejected, then the Li yield is negative (Ventura & D’Antona Reference Ventura and D’Antona2010; Doherty et al. Reference Doherty, Gil-Pons, Lau, Lattanzio and Siess2014a). ¹⁴N is increased at all dredge-up events and is also greatly increased through HBB. The TDU products ¹²C, ¹⁶O, and ²²Ne are also subsequently processed via HBB.

Models of very metal-poor super-AGB stars (Z ≈ 10⁻⁴) create similar isotopes to their metal rich counterparts. However, in low metallicity models that experience TDU, such as those by Doherty et al. (Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b), one find positive yields of the species ¹²C, ¹⁶O, ¹⁵N, ²⁸Si, which is not the case for the more metal-rich super-AGB stars.

In Figure 9, we compare yields of light elements from super-AGB stars (in [X/Fe])Footnote ⁹ for models with Z = 10⁻⁴ (bottom panel) and close to solar composition (top panel) in common between the studies by Doherty et al. (Reference Doherty, Gil-Pons, Lau, Lattanzio and Siess2014a, Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b) and Siess (Reference Siess2010). We also include the slightly higher metallicity (Z = 3 × 10⁻⁴) model from Ventura et al. (Reference Ventura, Di Criscienzo, Carini and D’Antona2013). Clearly seen is the close agreement found between studies for metal-rich super-AGB star yields. This is due to nucleosynthesis in these stars being primarily driven by HBB. However, as the metallicity decreases the differences between results from different groups begins to increase. This is due to a variety of factors, such as variations in the HBB temperatures and the choice of nuclear reaction rates but it is primarily related to the occurrence or not of TDU, with the relative TDU contribution being higher at lower metallicity. At the very low metallicity the yield of many major elements such as C, O, F, Ne, Na, and Mg differ so much between calculations that there is no consensus on whether these elements are produced or destroyed within super-AGB stars.

Figure 9. Comparison of a selection of light element yields for models of 9.0 M_⊙ Z = 0.02 and 7.5 M_⊙ Z = 10⁻⁴ (or Z = 3 × 10⁻⁴) from Doherty et al. (Reference Doherty, Gil-Pons, Lau, Lattanzio and Siess2014a, Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b), Siess (Reference Siess2010) and Ventura et al. (Reference Ventura, Di Criscienzo, Carini and D’Antona2013). Note the change of scale for the y-axis between panels.

We stress that although agreement between results from different groups at high metallicity is somewhat comforting, the principal test of the validity of our results will come when we can compare against observations, which requires us to positively identify super-AGB stars.

For details of nucleosynthesis in extremely metal-poor (Z ⩽10⁻⁵) and primordial super-AGB stars, we refer to the companion review by Gil-Pons et al. (in preparation).

4.3.2. Heavy elements

Due to both the numerical complexities and the time consuming nature of the calculations, currently published heavy element yield predictions of super-AGB stars are limited to a small selection of individual models in Fishlock et al. (Reference Fishlock, Karakas, Lugaro and Yong2014), Shingles et al. (Reference Shingles, Doherty, Karakas, Stancliffe, Lattanzio and Lugaro2015) and Karakas & Lugaro (Reference Karakas and Lugaro2016). There are also calculations for selected species in Karakas et al. (Reference Karakas, García-Hernández and Lugaro2012) and Lugaro et al. (Reference Lugaro2014). In these works, heavy element production is limited to predominantly Rb and in not large quantities. This nucleosynthesis is illustrated by Figure 10 in which we present a new set of yields of heavy elements (via s-process nucleosynthesis) for a range of metallicities Z = 0.02 − 0.001. These calculations are based on the evolutionary models from Doherty et al. (Reference Doherty, Gil-Pons, Lau, Lattanzio and Siess2014a, Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b) and use the 324 species nucleosynthetic network detailed in Lugaro et al. (Reference Lugaro2014). These models are chosen as representative of intermediate-mass super-AGB stars and all have very similar core masses (~1.15 M_⊙), thermally pulsing durations (~6 × 10⁴ yr), and number of thermal pulses (≈100). As mentioned previously, the high temperature within the helium burning shell during the convective thermal pulses leads to strong activation of the ²²Ne neutron source with very high peak neutron densities N _n ~10¹⁴ cm⁻³. However, with the very short thermal pulse duration, the integrated neutron exposure τ is not very high, at most ~ 0.04 mbarn⁻¹. This results in a large production of Kr (Z = 36), Rb (Z = 37) and light s-process elements Sr, Y, and Zr (Z = 38, 39, and 40, respectively), and a small synthesis of heavy s-process elements and Pb. The heavy element abundance patterns are similar between models of the different metallicities even with the associated large variation in availability of Fe-seeds. This is due to the low neutron exposure that inhibits formation of substantial amounts of elements past the light s-process peak as well as the small overlap factor between successive thermal pulses which leads to little build up of heavy elements to be subsequently reprocessed. Even with their large number of thermal pulses, heavy element production within super-AGB stars (at least for the moderate metallicities presented here), is reasonably modest, with the yield of no heavier than Fe element exceeding unity in [X/Fe]. This is mostly a consequence of the small mass contained within each convective thermal pulse (~5 × 10⁻⁴ M_⊙) and therefore considerable dilution of s-process enriched intershell material within a massive envelope.

Figure 10. Heavy element nucleosynthesis yields for super-AGB stars for a range of metallicities (in [X/Fe]) all scaled to the solar abundances of Asplund et al. (Reference Asplund, Grevesse, Sauval and Scott2009). The breaks in the distribution are for the elements Tc (Z = 43) and Pm (Z = 61) which have no stable isotopes. The shaded regions represent the elements used to represent the three s-process peaks ls, hs, and Pb. The maximum production is for the element Rb (Z = 37).

The super-AGB models of Jones et al. (Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a) found proton ingestion during TPs where protons from the envelope were making contact with the convective HeB during the thermal pulse. This leads to rapid production of ¹³C, which then undergoes the ¹³C(α,n)¹⁶O reaction. The neutrons produced during this event are available to then form heavy elements via the intermediate capture process. Jones et al. (Reference Jones, Ritter, Herwig, Fryer, Pignatari, Bertolli and Paxton2016a) estimated the potential i-process heavy element production within super-AGB stars to be of the order of 1–2 dex, comparable to the maximum 1–2.5 dex production from the ²²Ne source as estimated by Doherty et al. (Reference Doherty, Gil-Pons, Lau, Lattanzio and Siess2014a, Reference Doherty, Gil-Pons, Lau, Lattanzio, Siess and Campbell2014b).