THE ¹²C + ¹²C REACTION AND THE IMPACT ON NUCLEOSYNTHESIS IN MASSIVE STARS

During the evolution of massive stars (initial mass ≳8 M_☉), ¹²C fusion occurs in the stellar core and, during more advanced phases, in convective shells. The lifetime of central carbon burning is governed by the balance between the energy generated by the fusion reaction and the energy lost by neutrinos (e.g., Arnett & Thielemann 1985; El Eid et al. 2004). Carbon fusion has three relevant channels:

$${}^{12}{\rm C}+{}^{12}{\rm C} \rightarrow {}^{24}{\rm Mg}^{*} \rightarrow {}^{20}{\rm Ne} + {}^{4}{\rm He} (Q = + 4.616\,{\rm MeV}),$$

$${}^{12}{\rm C}+{}^{12}{\rm C} \rightarrow {}^{24}{\rm Mg}^{*} \rightarrow {}^{23}{\rm Na} + p (Q = + 2.238\,{\rm MeV}),$$

$${}^{12}{\rm C}+{}^{12}{\rm C} \rightarrow {}^{24}{\rm Mg}^{*} \rightarrow {}^{23}{\rm Mg} + n (Q = - 2.605\,{\rm MeV}).$$

The first two channels have approximately the same probability, whereas the probability of the neutron channel is about two orders of magnitude lower (Dayras et al. 1977). However, it is the uncertainty of the relative strength of these channels at stellar energies (see next section for details), where no direct measurements are available, that is of particular relevance.

Despite considerable experimental efforts, the total ¹²C+¹²C fusion reaction rate remains uncertain at stellar temperatures. Recent heavy ion fusion systematic studies have indicated that the fusion cross section may be hindered at low energies (Jiang et al. 2007). This would result in a lower rate than commonly used (Caughlan & Fowler 1988, hereafter CF88). On the other hand, recent low-energy experiments (Spillane et al. 2007) suggest an enhancement of the rate due to the presence of resonant structure effects at lower energies than previously considered (Gasques et al. 2005; Jiang et al. 2007; Gasques et al. 2007). This could result in a much higher ¹²C+¹²C fusion rate close to the Gamow peak temperature. Therefore, the present uncertainty of the ¹²C+¹²C rate still covers orders of magnitude at stellar temperatures.

Because both protons and α particles are products of ¹²C+¹²C fusion, C-burning is the first phase during massive star evolution where proton- and α-capture reactions can be efficiently activated at the same time. Leftover ²²Ne, from the previous convective He-burning core, efficiently captures α particles via the reaction ²²Ne(α, n)²⁵Mg, driving neutron-capture nucleosynthesis on the stellar material (e.g., Raiteri et al. 1991a). The ²²Ne(α, n)²⁵Mg reaction has been recognized as the dominant source of neutrons for the s-process in massive stars. Indeed, the reaction is active in the convective core He-burning and subsequent C-burning phases, and is responsible for the majority of the s-process species between iron and strontium observed in the solar system (Peters 1968; Lamb et al. 1977; Couch et al. 1974; Käppeler et al. 1989; Prantzos et al. 1990a; Raiteri et al. 1993; The et al. 2007; Pignatari et al. 2010). A large fraction of pre-supernova material is affected by convective carbon shell nucleosynthesis. On the other hand, material processed by C-burning in the core is usually modified and processed by later evolutionary burning phases (e.g., Woosley & Weaver 1995; Limongi et al. 2000; Rauscher et al. 2002; Woosley et al. 2002). This scenario may be modified however by uncertainties in the ¹²C+¹²C reaction rate.

The impact of the ¹²C+¹²C uncertainty in massive star simulations has been studied previously by, e.g., Gasques et al. (2007), where the consequences arising from a lower rate were considered, and by Bennett et al. (2012; hereafter Paper I), where a higher rate was used for stars of various initial masses at solar metallicity. The main results obtained in Paper I are the following: (1) For all stellar masses considered (M = 15–60 M_☉), using the upper limit ¹²C+¹²C rate (CU rate, see also the next sections) results in ¹³C(α, n)¹⁶O becoming the predominant neutron source, compared with the established source ²²Ne(α, n)²⁵Mg and increases the s-process stellar yields by up to ∼2 orders of magnitude. (2) When an intermediate rate (CI, Paper I) is used, s-process-rich material from the C core is only ejected for the 20 M_☉ mass. For this one case, the stellar model shows similar results to that of the 20 M_☉ star using the CU rate, whereas all other models show results comparable to the standard case using the CF88 ¹²C+¹²C rate. (3) For all of the masses considered, the lifetime of central C-burning increases with increasing the ¹²C+¹²C rate. The density and temperature of C-burning decrease with increasing ¹²C+¹²C rate. No significant variation trends are observed for more advanced phases. (4) Point (1) suggests that the CU rate is likely too high, since it would be hard to reconcile the predicted yields with observations, such as the solar s-process distribution.

In this paper, for one 25 M_☉ star we consider the complete range of the present ¹²C+¹²C uncertainty, in agreement with Gasques et al. (2007) and Paper I. We discuss the importance of the rate uncertainty on the s-process and on explosive p-process yields, which are driven by photodisintegration on s-process-rich material during core collapse (e.g., Arnould & Goriely 2003, and reference therein). We consider the implications for the s-process arising from the large uncertainty regarding the relative strengths of the three carbon fusion channels, an effect that was not included in the two works mentioned above. The impact of the neutron source ¹³C(α, n)¹⁶O is also considered for non-rotating and fast-rotating massive stars, with halo metallicities. Finally, the production of unstable long-lived species ²⁶Al and ⁶⁰Fe in the pre-explosive phase is revised within the ¹²C+¹²C uncertainty (e.g., Arnett & Truran 1969; Timmes et al. 1995b; Limongi & Chieffi 2006).

The paper is organized as follows. The present uncertainty of the ¹²C+¹²C reaction rate and its nucleosynthesis channels is discussed in Section 2. Presented in Section 3 are the main features of the stellar models used in this work. In Sections 4 and 5, the s-process in the convective He-burning core and in C-burning conditions is described. The impact on nucleosynthesis of different ¹²C+¹²C channels is discussed in Sections 6 and 7. The p-process nucleosynthesis is analyzed in Section 8, and in Section 9 the importance of the ¹²C+¹²C uncertainty is explored at low metallicity, for non-rotating and fast-rotating stars. Finally, conclusions are presented in Section 10.

The ¹²C+¹²C reaction is one of the most studied heavy ion fusion processes. Considerable effort has been expended over the last decades on the measurement of the ¹²C+¹²C fusion cross section at very low energies (Patterson et al. 1969; Mazarakis & Stephens 1973; High & Čujec 1977; Kettner et al. 1980; Becker et al. 1981; Rosales et al. 2003).

Despite these dedicated efforts, the low-energy reaction rate still carries considerable uncertainties due to pronounced resonance structures that are thought to be associated with molecular ¹²C+¹²C configurations in the ²⁴Mg* system (Bromley et al. 1960). While this observation is significant for the understanding and interpretation of nuclear structure configurations (Freer 2007), it may also have a significant impact on the actual reaction rate of ¹²C+¹²C fusion in stellar burning environments. The stellar reaction rate depends critically on the S-factor in the stellar energy regime, which so far has been inaccessible to laboratory studies. The low, sub-picobarn cross sections and the substantial beam-induced background events on target impurities make the identification and measurement of the reaction products difficult (Strieder 2010), and generally the expanding of the measurement of the ¹²C+¹²C fusion cross section to very low energies (Zickefoose et al. 2011). In particular, the resonance configurations are characterized by pronounced structures associated with the ²⁴Mg* compound nucleus in the corresponding excitation range, and should be strongly populated in the fusion process except for the hindrance by Coulomb and orbital momentum barrier. The experimental data indicate that these resonances have relatively narrow total widths, Γ  ≈  100 keV. Since these resonance levels are highly unbound with respect to α and proton decay, the α cluster and proton single-particle component in the wave function is relatively small, corresponding to spectroscopic factors of ⩽0.01 for both decay channels. This is also reflected in the results of elastic ²⁰Ne(α, α) (Abegg & Davis 1991; Davis 1981) and ²³Na(p, p) (Vanhoy et al. 1987) scattering experiments, which indicate no pronounced α or single-particle structure in this particular excitation range of ²⁴Mg. Despite considerable theoretical effort, the exact nature and structure of these molecular resonance levels is not fully understood and low-energy reaction cross section extrapolations are mostly based on averaging the cross section over the resonance structure (Fowler et al. 1975). A recent potential model analysis of the fusion cross section (Gasques et al. 2005) was based on the Sao Paulo model and resulted in a theoretical prediction of the cross section, describing well the experimental data and allowing extrapolation to lower energies. The results were in good agreement with the previous phenomenological approach by Fowler et al. (1975).

More recently, however, it has been argued on the basis of heavy ion fusion systematics that the low-energy cross section of fusion reactions declines faster with decreasing energy than projected by the potential model. This has been modeled by introducing an additional term in the potential that is related to the incompressibility of nuclear matter (Mişicu & Esbensen 2006). This term translates into a hindrance effect for the fusion process, suggesting a significant reduction of the overall cross section toward low energies. This effect has been confirmed by a number of low-energy fusion reactions with positive Q-values (Jiang et al. 2008), and it is argued that the hindrance effect may also reduce the low-energy S-factor of the ¹²C+¹²C fusion process significantly (Jiang et al. 2007). The impact of such a reduction on a number of stellar burning scenarios was discussed in a previous study (Gasques et al. 2007).

On the other hand, recent low-energy experiments (Spillane et al. 2007) indicate the existence of strong molecular resonances at even lower energies than previously considered (Gasques et al. 2005, 2007; Jiang et al. 2007). These resonances may contribute significantly to the low-energy cross section and enhance the reaction rate substantially. One pronounced ¹²C+¹²C resonance structure has been reported in both the proton and the α channel, at E_cm = 2.138 MeV (Spillane et al. 2007), and another resonance at E_cm  ≈  1.5 MeV has been projected on the basis of a theoretical evaluation of recent ¹²C+¹²C scattering data, which seems to indicate the existence of a pronounced resonance also at 1.5 MeV (Perez-Torres et al. 2006). The total strength of the resonance at 2.14 MeV is projected to be ωγ ⩽ 0.13 MeV. This is remarkably large and translates into a pronounced ¹²C+¹²C cluster configuration with a spectroscopic factor of C²S$_{\bf ^{12}C}$ ≈ 1.¹¹ Adopting this value for the resonance at 1.5 MeV, we obtain an upper limit for the total resonance strength of ωγ ⩽ 0.01 μeV. A thorough search was performed to identify this resonance at low ¹²C+¹²C in the proton channel ¹²C(¹²C, p)²³Na. Unfortunately, the measurements were affected by the beam-induced proton background from the resonant ¹²C(d, p)¹³C reaction, triggered by deuteron recoils from elastic ¹²C beam scattering on low-level deuteron impurities in the ¹²C target (Zickefoose et al. 2011). The yield of this two-step background process presently sets the upper limit for the resonance strength of ¹²C(¹²C, p)²³Na below E_cm = 2.5 MeV. Recent unpublished experimental data with ultrapure ¹²C targets and greatly reduced deuterium contamination did not show any indications for lower energy resonances (Zickefoose 2011). While this background also affects the proposed strong ¹²C(¹²C, p)²³Na resonance at E_cm = 2.138 MeV (Spillane et al. 2007), the possibility of a ¹²C+¹²C cluster resonance reflected in the ¹²C(¹²C, α)²⁰Ne reaction channel cannot be excluded, since the α channel was not observed in this particular experiment.

Complementary information may come from α-scattering, α-capture and α-emission data because a ¹²C cluster configuration may also show an enhanced α-cluster signature. Indeed, the ²⁰Ne(α, α₀) elastic scattering data by Abegg & Davis (1991) indicate a number of strong natural parity states in the excitation range of interest. No radiative capture data are available for this energy range (Endt 1990), but several levels have been observed in the ²³Na(p, α)²⁰Ne reaction. Overall, the complementary information is sparse, since not many experiments have systematically probed the level structure of ²⁴Mg above 15 MeV.

Evaluating the available information, most notable are the two states at 15.44 MeV (J^π = 0⁺, T = 2) and 16.07 MeV (J^π = 6⁺, T = 0), which match the energies of the two postulated low-energy resonances in the ¹²C+¹²C channel at E_cm = 1.5 MeV (Perez-Torres et al. 2006) and E_cm = 2.14 MeV (Spillane et al. 2007). The 15.44 MeV level has been observed in a number of scattering and reaction studies, and was identified as a T = 2 isospin state. The total width, as well as the proton and α partial widths, is known or can be deduced from the available experimental information such as ²³Na(p, α) and ²³Na(p, p) (McDonald et al. 1978; Endt 1990).

The critical parameters that need to be defined in order to determine the resonance strengths in the fusion process are the ¹²C partial widths for these levels. The level at 16.07 MeV has only been observed in high spin transfer reactions (Ford et al. 1974), and the proton and α partial widths cannot be directly evaluated. If these two levels correspond to the observed ¹²C+¹²C resonance states, it would require a pronounced ¹²C+¹²C cluster configuration for both levels. Some information can be extracted on the basis of this assumption. The population of a pure T = 2 level is isospin forbidden; it is only possible with strong isospin mixing. In the case of the 15.44 MeV state, the existence of isospin mixing is confirmed by the observation of the α emission (McDonald et al. 1978) and the α decay of this level (McGrath et al. 1970). The isospin mixing can actually be deduced from the strength of the α decay branch to be ≈0.01, adopting a pronounced ²⁰Ne+⁴He cluster configuration. This result is relatively insensitive to the choice of the radius of the system. On the other hand, a pronounced ¹²C+¹²C cluster structure is associated with a large interaction radius of R = 8.5 fm. The ¹²C partial width of the 16.07 MeV 6⁺ state is limited by the ℓ = 6 orbital momentum barrier. This level has not been observed in any of the proton or α induced scattering or reaction processes. The available information about the proton and α partial widths relies on the recent measurement of the resonance strengths for both branches in the ¹²C+¹²C fusion reaction (Spillane et al. 2007). The measured resonance strengths are consistent with the large interaction radius of R = 8.5 fm, anticipated for a cluster configuration of this level.

On the basis of these considerations, we have re-evaluated the reaction rate for the ¹²C+¹²C fusion reaction as well as the rates for the two most important reaction channels, ¹²C(¹²C, α)²⁰Ne and ¹²C(¹²C, p)²³Na. While these two channels are responsible for the energy production by carbon fusion and for the production of seed material for the next Ne-burning stage, the third channel, ¹²C(¹²C, n)²³Mg, may contribute as the neutron source for s-process nucleosynthesis at higher temperature environments. This possibility will be discussed in Section 7. Figure 1 shows the total ¹²C+¹²C rate with the evaluated uncertainties. Note that in the figure the neutron channel is neglected; the reaction is endothermic with a negative Q-value of −2.6 MeV and contributes less than 4% at higher energies to the total cross section. The "lower limit" (CL) of the rate corresponds to the one suggested on the basis of an extrapolation of the averaged data, using a potential model approach that includes a hindrance term for low-energy fusion processes (Gasques et al. 2007). The recommended rate is based on the classical extrapolation of the averaged S-factor data using a standard potential model (Gasques et al. 2005) plus the contribution of a single resonance observed at E_cm = 2.13 MeV, and is quite similar to the standard CF88 rate. The "upper limit" (CU) includes an additional term resulting from a possible strong ¹²C+¹²C cluster resonance at E_cm = 1.5 MeV. The resonance parameters have been estimated as outlined above. The total rates are listed in Table 1.

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In order to assess the importance of the ¹²C+¹²C rate for the s-process in massive stars, five 25 M_☉ stellar models were calculated using different ¹²C+¹²C rates. Apart from the ¹²C+¹²C rate, all of the models were calculated with identical input physics, and were therefore identical at the end of He-burning. Calculations were performed using the Geneva stellar evolution code (GENEC), described in Hirschi et al. (2004) and Eggenberger et al. (2008). We recall here the main input physics used.

The initial metal content was Z = 0.01, where the solar abundances were given by Anders & Grevesse (1989) for the total elemental abundances, and by Lodders (2003) for the relative isotopic abundances. The corresponding OPAL opacity tables are taken from Rogers et al. (1996), except at low temperatures, where instead opacities from Ferguson et al. (2005) were used. The convective boundaries were defined according to the Schwarzschild criterion. Overshooting was applied for core H- and core He-burning only, using an overshooting parameter, α = 0.2 H_P, as in Maeder (1992). Effects from rotation and magnetic fields were not included in the calculation, but this does not detract from the main focus of the present investigation. For log T_eff > 3.9, O-type star mass-loss rates were adopted from Vink et al. (2001), and from de Jager et al. (1988) otherwise. With a few exceptions, reaction rates were taken from the NACRE compilation (Angulo et al. 1999). For temperatures below 0.1 GK, the ¹⁴N(p, γ)¹⁵O rate is taken from Mukhamedzhanov et al. (2003). Above 0.1 GK, the lower limit NACRE rate was used. This combined rate is very similar to the more recent LUNA rate (Imbriani et al. 2005), which is used in the post-processing at relevant temperatures. The 3α rate adopted is from Fynbo et al. (2005), and the ¹²C(α, γ)¹⁶O rate from Kunz et al. (2002). For the ²²Ne(α, n)²⁵Mg reaction, we used the rate of Jaeger et al. (2001) for temperatures of 1 GK and below. The NACRE rate was used for higher temperatures. The ²²Ne(α, n)²⁵Mg rate competes with ²²Ne(α, γ)²⁶Mg, where the NACRE rate is used.

In Figure 2, we present the Kippenhahn evolution diagram for the five stellar models, where the ¹²C+¹²C rate used in the calculations are, respectively, (1) the upper limit provided in Section 2 (M25CU model, upper left panel), (2) the CF88 rate multiplied by a factor of 10 (M25CF88t10 model, upper right panel), (3) the CF88 rate (M25CF88 model, central panel), (4) the CF88 rate divided by a factor of 10 (M25CF88d10 model, lower left panel), and (5) the lower limit provided in Section 2 (M25CL model, lower right panel). The rates used for the different cases are shown in Figure 1.

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All models were evolved beyond the end of central O-burning. The central temperature and density evolution are shown in Figure 3. As mentioned, the evolutionary curve in Figure 3 is the same for all the models until central He exhaustion. Models calculated using the higher ¹²C+¹²C rates generally show the signature of central C ignition at lower temperature and density conditions (upward kink in the curve in the range log(ρ_c) = 4–6 g cm⁻³). After central C-burning, no significant variations may be noticed in the T_c–ρ_c diagram in the following evolutionary phases (see below for more details), as already noted in Paper I. The main parameters of the stellar models presented in Figure 2 and 3 are summarized in Tables 2 and 3. Similar data are provided for a grid of masses and at solar metallicity in Paper I, which is similar to the case discussed here at lower metallicity. However, the range of ¹²C+¹²C rate explored in this work is larger than in Paper I, including the possibility of a rate lower than CF88. At T ∼ 0.7–0.8 GK, the CL rate varies from the CF88 rate by less than a factor of 10. However, there is a factor of 10 difference between CF88d10 and CF88 over the same temperature range (Figure 1). Therefore, because the CL rate rapidly increases with temperature, CF88d10 is actually the lowest carbon fusion rate we consider during central C-burning conditions. Note that the CF88d10 and CF88t10 rates are given by applying a temperature invariant correction factor to the CF88 rate over the entire temperature range. Though such constant variation is unlikely, the present models calculated with these rates are still useful as a guide, and can provide qualitative insights about the impact of the ¹²C+¹²C rate on stellar evolution, including later stages.

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The time elapsed between He exhaustion and core C-burning activation decreases with increasing ¹²C+¹²C rate. However, the early C ignition is compensated by a longer core burning phase (see Figure 6 and Table 3). In Figures 4–6, the central temperature and density at C ignition and C-burning lifetimes¹² are compared. The trend observed in Paper I is also confirmed for ¹²C+¹²C rates lower than CF88: the temperature and density of central C ignition decreases with increasing carbon fusion rate, whereas the C-burning lifetime shows the opposite behavior. The M25CL and M25CF88 models show similar conditions for central C-burning. Indeed, at the point where central carbon ignition occurs in these models, the CL rate and the CF88 rate are quite similar, despite the fact that for lower temperatures the CL rate drops quickly compared with the standard rate. On the other hand, the M25CF88d10 and M25CF88t10 models show significantly different behaviors; reducing the ¹²C+¹²C rate has a larger impact compared with the CF88 case. Finally, because of the high upper limit, the M25CU model (which is about 50,000 larger than the CF88 rate, Figure 1) shows a large effect in the stellar conditions (see Tables 2 and 3), in agreement with calculations presented in Paper I.

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Note that the behavior of C-burning temperature with the ¹²C+¹²C rate can simply be derived analytically, if we consider the temperature dependence of the C-burning rate, where $\lambda _{^{12}{\rm C}^{12}{\rm C}} \propto {\rm T9}^{29}$ (where T9 is temperature in GK units; see Woosley et al. 2002). Indeed, if we simply consider that the amount of energy required for the stellar structure is the same, regardless of the C-burning rate, we obtain for the upper limit a burning temperature of T9 = 0.50, and for the model M25CF88d10 T9 = 0.79. These values are only about 10% lower than the activation temperatures shown in Table 2. Such small differences are likely due to minor effects considered in the full stellar model, as for instance the neutrino feedback.

Figure 2 shows that carbon core burning is radiative for the models M25CL, M25CF88, and M25CF88t10. For the lowest rate, CF88d10, core C-burning induces a tiny convective core. The CU case, on the other hand, shows an extended convective core (up to 4.12 M_☉ in size). Generally, the standard 25 M_☉ model is close to the lower mass limit for radiative core carbon burning, which is around 22 M_☉ (Hirschi et al. 2004). Core C-burning becomes convective once the energy produced by ¹²C+¹²C is higher than the energy lost by neutrinos (El Eid et al. 2004). In the CF88d10 case, the lower rate leads to a delayed C ignition at higher temperatures, where the energy generation slightly overtakes the neutrino losses. In the M25CU model, the formation of an extended convective core is due to the early activation of C-burning, where the lower temperature makes the neutrino energy loss much less efficient (Itoh et al. 1996; see also discussion in Paper I).

After central carbon exhaustion, C-burning develops in outer convective shells. Models M25CF88d10, M25CL, M25CF88, and M25CU show one extended convective shell, whereas M25CF88t10 develops two convective shells. In this last case, the first convective shell overlaps with the second and final convective shell by about 2/3 of its total extent. In a similar way, the M25CU model shows a similar overlap between the central convective C core and the subsequent convective C shell. Such overlap has a strong impact on the composition of the final stellar ejecta. Looking at Table 3, we can see that contrary to carbon burning, all stellar models are quite similar during the subsequent burning stages (neon and oxygen burning stages; see also Paper I).

As mentioned before, the ¹²C+¹²C rate is the only reaction relevant for energy generation tested in this work. Note however that the uncertainty of other reactions such as ¹²C(α, γ)¹⁶O (e.g., Imbriani et al. 2001; El Eid et al. 2004, 2009; Woosley et al. 2002) or ¹⁶O+¹⁶O may affect stellar structure. In particular, the ¹²C(α, γ)¹⁶O affects the amount of ¹²C left after He-burning and therefore the ensuing C-burning and Ne-burning stages. In principle, using a different ¹²C(α, γ)¹⁶O rate could affect the results discussed in the following sections. However, after having explored the impact of the ¹²C + ¹²C rate for stars of different masses (and therefore with different ¹²C and ¹⁶O composition at the onset of central C ignition) in Paper I, we believe that our present results are robust and cannot be significantly changed by using different ¹²C(α, γ)¹⁶O rates.

Stellar physics uncertainties (mixing-length theory approximations, prescriptions adopted for convective boundary mixing, mass loss, impact of rotation, and magnetic field; see e.g. Limongi et al. 2000; Woosley et al. 2002; Meynet et al. 2006) may also affect stellar models. Furthermore, multi-dimensional simulations for advanced stages of massive stars showed several features that cannot be properly reproduced by basic one-dimensional models (e.g., Arnett & Meakin 2011). However, as also discussed in Paper I, all those uncertainties do not modify the general conclusions of this work on the impact of the ¹²C+¹²C rate uncertainty, for both the stellar structure and the nucleosynthesis.

In massive stars, central He-burning evolves in a convective core. The nuclear reactions driving the energy generation are the 3α reaction, converting three ⁴He nuclei into ¹²C, and the ¹²C(α, γ)¹⁶O reaction. The most abundant species at the end of core He-burning are ¹²C and ¹⁶O (e.g., Arnett & Thielemann 1985). The ¹⁴N left from the CNO cycle during the previous H-burning is fully converted to ²²Ne at the beginning of core He-burning, via the reaction chain ¹⁴N(α, γ)¹⁸F(β⁺ν)¹⁸O(α, γ)²²Ne. Therefore, the amount of ²²Ne produced depends on the initial abundance of the CNO species in the star (e.g., Prantzos et al. 1990a). The main source of neutrons for the s-process is ²²Ne, via the reaction ²²Ne(α, n)²⁵Mg. This occurs at the end of the He core when the He abundance drops below 10%, in the last ∼10⁴ yr, the central temperature becomes larger than 2.5 × 10⁸ K, and α-capture on ²²Ne is efficiently activated (e.g., Raiteri et al. 1991b; The et al. 2007; Pignatari et al. 2010). As mentioned in Section 3, all models treated share the same evolution and nucleosynthesis history until central He exhaustion.

The complete calculations for the s-process were performed using the parallel version of the multizone driver of the post-processing code PPN, developed by the NuGrid research platform (http://www.astro.keele.ac.uk/nugrid) and described by Herwig et al. (2008; MPPNP). Reference sources for basic charged particle reactions, 3α and ¹²C(α, γ)¹⁶O, have been selected in agreement with the stellar code (Fynbo et al. 2005; Kunz et al. 2002, respectively). As outlined in Section 3, the ²²Ne(α, n)²⁵Mg and ²²Ne(α, γ)²⁶Mg rates are from Jaeger et al. (2001) and Angulo et al. (1999), respectively. Neutron-capture reaction rates for stable isotopes, and also for unstable isotopes if available, are provided by the KADoNIS Web source (Dillmann et al. 2006). For neutron-capture rates not included in KADoNIS, we refer to the Basel REACLIB database, revision 20090121 (Rauscher & Thielemann 2000). The β-decay rates are provided by Oda et al. (1994) and Fuller et al. (1985) for light species, and by Aikawa et al. (2005) for species heavier than iron. Exceptions are ²⁶Al and ⁸⁵Kr β-decay rates, where the isomeric state and the ground state must be considered as separate species at He-burning temperatures and terrestrial rates must be used (e.g., Pignatari et al. 2010, and reference therein).

The isotopic and elemental distributions between Fe and Mo at the end of core helium burning are presented in Figure 7. For elements between iron and strontium, most of the s-process abundances in the solar system have been produced by massive stars (weak s-process component; e.g., Käppeler et al. 1982, 1994; Rauscher et al. 2002; The et al. 2007; Pignatari et al. 2010).

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The overabundance of ¹⁶O is reported in Figure 7, as a reference. ¹⁶O is a primary isotope; therefore, its production does not change with the initial metallicity of the star. The ¹⁶O observed in the solar system today is mainly produced in massive stars, in the same region of the star where the s-process yields are synthesized. Unlike primary isotopes, however, s-process yields in massive stars show a direct dependence on the initial stellar metal content, which is closer to a secondary-like nucleosynthesis (see, e.g., Tinsley 1980; Raiteri et al. 1992, for a definition of secondary nucleosynthesis and for the analysis of the metallicity dependence for the s-process in massive stars). According to Tinsley (1980), isotopes that are fully produced by the s-process in massive stars are expected to show an overabundance a factor of two higher than ¹⁶O at solar metallicity. Actually, the weak s-process is not a pure secondary process (e.g., Raiteri et al. 1992; Baraffe et al. 1992; Pignatari & Gallino 2008; Nomoto et al. 2006). However, we may still use the ¹⁶O overabundance as a guidance for the s-process efficiency. From Figure 7 (left panel), Cu and Ga are the elements with the highest s-process production. In the right panel, the isotopes show an odd–even pattern typical of the s-process, due to their neutron-capture cross sections behavior (see, e.g., Käppeler et al. 1982). Among the others, the isotopes ⁵⁸Fe and ^{63, 65}Cu show the highest overabundances, just above the iron seeds. As expected, the s-process efficiency drops beyond the Sr–Y–Zr neutron magic peak (N = 50), due to the total amount of neutrons produced. In Table 4, we report the abundances of the main He core products ¹²C and ¹⁶O, of the main neutron source ²²Ne, of the s-process seed ⁵⁶Fe, of the s-only isotopes between Fe and Sr, and of neutron magic ⁸⁸Sr at the He exhaustion. Note that the amount of ²²Ne available before the s-process starts is about 1% by mass (given mostly by the initial CNO) and that 36% of it is still available for more s-processing during the following C-burning phase.

The abundance of ⁵⁴Fe is also reported in Table 4, together with the neutron exposure. The neutron exposure τ_n is calculated according to the formula proposed by Woosley & Weaver (1995) and it is given by the ⁵⁴Fe depletion via neutron capture (note that ⁵⁴Fe is destroyed by neutron capture):

$$\tau _{\rm n} = - ({\rm MACS}_{^{54}{\rm Fe}(n,\gamma)})^{-1}\times \ln (X(54)/X(54)_{\rm ini}) \,({\rm mbarn}^{-1}),$$

where the MACS$_{^{54}{\rm Fe}(n,\gamma)}$ is the Maxwellian-averaged neutron-capture cross section on ⁵⁴Fe (for simplicity hereinafter in this formula, we choose for every temperature the rate at 30 keV by Coquard et al. 2006, 30.03 mbarn⁻¹), X(54) is the present ⁵⁴Fe mass fraction and X(54)_ini is its initial abundance (X(54)_ini = 5.339 × 10⁻⁵). The total neutron exposure at the end of the He core is reported in Table 4. Final mass fraction abundances of ¹²C and ¹⁶O are 0.27 and 0.71, respectively.

As discussed in Section 3, increasing the ¹²C + ¹²C rate results in an early central C ignition. This is because increasing the rate affects the equilibrium between the energy generation due to C fusion reaction and the energy lost by neutrinos. Therefore, the total lifetime of the central C-burning is affected and more massive stars may develop a convective C core (see Paper I). On the other hand, a lower ¹²C + ¹²C rate delays the C-burning activation to higher temperature and density conditions (see, e.g., Gasques et al. 2007). In the following section, we analyze the impact of these features on s-process nucleosynthesis.

5.1. The s-process Nucleosynthesis in C-burning Conditions, and the cs-component

In Figure 8, we present the abundances for a sample of species at the end of central O-burning as a function of the mass coordinate, for the models CU, CF88t10, CF88, CF88d10, and CL. Together with light isotopes indicative of different burning regions (¹H, ⁴He, ¹²C, ¹⁶O, and ²⁰Ne), we report the abundance profile of the s-only isotopes ⁷⁰Ge, ^{80, 82}Kr, r-only isotope ⁷⁰Zn (which can also be produced by direct neutron capture via the ⁶⁹Zn branching and is therefore a good tracer for neutron densities higher than ∼10¹¹ cm⁻³), and the isotopes ⁸⁸Sr, ¹³⁸Ba, and ²⁰⁸Pb. These last three nuclides belong to the three neutron magic peaks along the neutron-capture path (N = 50, 82, and 126, respectively), and are mostly of s-process origin. However, the bulk of their s-process abundance in the solar system distribution was produced by low-mass asymptotic giant branch (AGB) stars, which are responsible for the main and strong s-process components (see Gallino et al. 1998, and reference therein). By comparing the abundances of these isotopes in the stellar envelope (which roughly coincides with the initial abundance) with the abundances in the O-rich zone, the s-process efficiency producing these species can be deduced. In all cases, the production of ⁸⁸Sr is significantly higher than ¹³⁸Ba and ²⁰⁸Pb, as expected from the neutron exposure regime typical for the s-process in massive stars (e.g., Raiteri et al. 1991a). The further s-process contribution during C-burning over the He core ashes may be observed in Figure 8 by comparing the abundances in the He core window with the deeper profiles in the convective C shell region. Production of ⁷⁰Zn, as well as the isotopic ratio ⁸⁰Kr/⁸²Kr, provides an indication of the neutron densities reached during the convective He-burning core, in the He core window, and in the C shell material. This is due to the effect of neutron-capture branching points at ⁶⁹Zn and ⁷⁹Se (e.g., Pignatari et al. 2010, and reference therein). In particular, ⁷⁰Zn and ⁸⁰Se both increase with increasing the neutron density. In Table 5, we report the mass-fraction abundances for a sample of species in the C shell (including the ones listed above) and for the cases reported in Figure 8.

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Table 5. Pre-explosive Abundances in the Convective C Shell Region of the Models M25CF88d10, M25CL, M25CF88, M25CF88t10, and M25CU (Columns 2–6, Respectively) for a Sample of Selected Species

Xi	M25CF88d10	M25CL	M25CF88	M25CF88t10	M25CU
C 12	4.546E−02	2.626E−02	2.589E−02	5.783E−03	1.651E−02
O 16	6.011E−01	5.814E−01	5.998E−01	5.724E−01	6.168E−01
Ne 20	2.861E−01	3.332E−01	3.234E−01	3.571E−01	2.951E−01
Ne 22	9.342E−05	1.498E−04	1.132E−04	9.748E−05	2.022E−03
Na 23	5.716E−03	8.421E−03	8.234E−03	7.744E−03	2.517E−02
Fe 54	7.761E−08	8.764E−08	7.571E−08	8.567E−08	6.951E−08
Fe 56	1.100E−04	1.175E−04	1.113E−04	1.140E−04	6.898E−05
Zn 70	9.146E−08	1.693E−08	1.836E−08	4.729E−08	2.071E−08
Ge 70	4.332E−06	3.881E−06	4.366E−06	4.062E−06	1.345E−05
Se 76	7.816E−07	6.063E−07	6.862E−07	6.223E−07	6.505E−06
Kr 80	5.154E−08	1.616E−07	1.605E−07	2.580E−08	3.372E−06
Kr 82	4.377E−07	4.336E−07	4.767E−07	3.387E−07	1.266E−05
Sr 88	4.196E−07	4.370E−07	4.660E−07	5.576E−07	7.740E−05
Y 89	9.578E−08	8.965E−08	9.595E−08	1.046E−07	1.709E−05
Zr 96	7.669E−09	2.214E−09	2.732E−09	2.178E−09	4.552E−08
Te124	9.139E−10	1.005E−09	1.009E−09	7.768E−10	6.765E−08
Xe130	1.087E−09	1.223E−09	1.211E−09	1.131E−09	4.824E−08
Xe134	9.884E−10	1.893E−10	2.164E−10	5.124E−10	7.992E−10
Ba138	3.196E−08	2.981E−08	3.026E−08	3.021E−08	1.322E−07
Al 26	1.690E−09	8.470E−09	6.728E−10	2.274E−06	1.367E−08
Fe 60	4.605E−06	1.132E−06	2.450E−06	3.326E−06	9.021E−08

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The abundances of the main products of C-burning, ²⁰Ne and ²³Na, do not show a linear correlation with the C-burning rate. In particular, the ²⁰Ne/²³Na isotopic ratio ranges from about 12 (model M25CU) to 50 (model M25CF88d10). The maximum abundance of ²⁰Ne is obtained in M25CF88t10, whereas the largest ²³Na amount is in M25CU, since in this last case the abundances are dominated by the C-burning at lower central temperatures (Arnett 1972). The distribution of light species in the C-burning region is important for the s-process, since 90% or more of the neutrons produced are captured by light neutron poisons (e.g., Pignatari et al. 2010). Therefore, the abundances of light nuclides need to be considered to preserve the consistency of nucleosynthesis calculations. In particular, at metallicities close to solar, the main neutron poison is ²⁵Mg, with an important contribution from ^{24, 26}Mg, ^{20, 22}Ne, and ²³Na.

From Table 5, we see that for model CF88d10 the C shell material is exposed to the highest neutron density. This is because of the late C-burning activation at higher temperature and density (see later in this section and Gasques et al. 2007). On the other hand, the s-process is more efficient in the CU model; compared with the standard case, the ⁸⁸Sr abundance is boosted by more than two orders of magnitude. Such a result has already been mentioned (Bennett et al. 2010) and is discussed in more detail in Paper I.

Interestingly, all models show an overproduction of ⁸⁸Sr in the central zone, at a mass coordinate lower than 2 M_☉, compared with more external C-burning regions. Indeed, during both Ne-burning and early O-burning conditions, ⁸⁸Sr is still produced, or at least not efficiently depleted, in conditions of partial photodisintegration. Also, typical p-rich species, e.g., ⁷⁸Kr or ⁸⁴Sr, are produced in this phase with different efficiencies. Such nuclear species will be further created or destroyed by photodisintegrations in later O-burning and Si-burning evolutionary stages (e.g., Thielemann & Arnett 1985). However, as pointed out by Rauscher et al. (2002), such pre-explosive abundances could be mixed by the merging of different shells into more external regions of the star, where they can survive and provide a significant contribution to p-process species yields, together with the explosive p-process abundances. Such contributions can also be relevant for non-p-rich isotopes like ⁸⁸Sr. Along similar lines, Arnett & Meakin (2011) showed how multi-dimensional effects may affect the stellar structure before the supernova (SN) explosion, allowing a more efficient exchange of material between different burning zones deep in the star compared with standard one-dimensional models. Therefore, caution must be maintained concerning nucleosynthesis abundances in the later evolutionary stages of massive stars.

The CF88d10 model is characterized by a lower production of neutrons compared with the CF88 model (⁸⁸Sr is lower by 11%), but it has a higher neutron density (higher ⁷⁰Zn amount, lower ⁸⁰Kr/⁸²Kr ratio; see Table 5). This effect was discussed by Gasques et al. (2007). The CL and CF88 models show similar results, because of the similarities between the two stellar models (Section 3). The CF88t10 model shows an increase of ⁸⁸Sr by only a factor of 1.2 in the final convective C shell compared to CF88 (Table 5), confirming earlier results presented by Bennett et al. (2010). In the CU case, ⁸⁸Sr increases by a factor of 166 relative to the standard case. Major differences are observed for the whole s-process distribution. The reason for such a large overproduction in the CU case is the strong s-process efficiency reached in the convective C-burning core. For CU, in the C core the dominant neutron source is ¹³C(α, n)¹⁶O, not ²²Ne(α, n)²⁵Mg. When the burning temperature is lower than ∼0.70–0.75 GK, ¹³C is efficiently produced via the ¹²C(p, γ)¹³N(β⁺ν)¹³C channel. The ashes of the C core are later mixed in the convective C shell and ejected in the interstellar medium by the final SN explosion (Figure 2). For the CU ¹²C+¹²C rate, the overlap between the convective core and the final shell acts to drastically increase the s-process efficiency, as explained in Paper I.

The activation of the ¹³C(α, n)¹⁶O reaction in the C-burning core was previously discussed by Chieffi et al. (1998). However, in stellar models using the standard ¹²C+¹²C rate, the central C-burning is not convective. Even in the eventuality that a small convective core is developed, the enriched material is further processed by the more advanced burning phases and eventually collapses to form the neutron star. On the other hand, when the temperature is higher than ∼0.70–0.75 GK, ¹³N(γ, p)¹²C reaches equilibrium with its direct reaction (Clayton 1968). In these conditions, the only way to produce ¹³C is via ¹²C(n, γ)¹³C, and so the ¹³C(α, n)¹⁶O reaction only recycles neutrons captured by ¹²C. The neutrons have to be produced by other reactions, e.g., by ²²Ne(α, n)²⁵Mg.

In Figure 9 (left panel), the final s-process elemental distribution between Fe and La is shown for all models considered. The first four cases present similar s-process abundance distributions. In Table 6, the overproduction factors for elements between C and Mo are reported for a detailed comparison. The CU model produces an s-process distribution that is peaked in the region of Sr–Y–Zr–Nb (with an overproduction factor larger than 1000 for Sr and Y). The neutron magic peak at N = 50 is bypassed by the neutron-capture path efficiently and a relevant production of s elements (with an overproduction factor larger than 100) is observed up to Sn (Z = 50). Negligible abundances are observed above Ba–La (Z = 56, 57), compared with the Sr–Y–Zr–Nb bulk. In Figure 9 (right panel), the respective isotopic distributions between Fe and Mo are presented. We recall again that the different distributions shown in Figure 9 share the same neutron-capture process history until the end of the core He-burning. After core He exhaustion, the heavy isotope distribution in the C shell ejecta is driven by different neutron exposure histories.

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Table 6. Pre-explosive Element Overabundances in the Convective C Shell Region from C to Mo for the Models M25CF88d10, M25CL, M25CF88, M25CF88t10, and M25CU (Columns 2–6, Respectively)

Xel/X☉	M25CF88d10	M25CL	M25CF88	M25CF88t10	M25CU
C	1.311E+01	7.575E+00	7.470E+00	1.668E+00	4.763E+00
N	5.139E−04	8.662E−03	9.398E−03	5.553E−03	3.202E−02
O	6.230E+01	6.025E+01	6.215E+01	5.932E+01	6.392E+01
F	1.369E−02	2.631E−02	2.518E−02	3.063E−02	2.652E−01
Ne	1.454E+02	1.694E+02	1.644E+02	1.816E+02	1.520E+02
Na	1.429E+02	2.105E+02	2.059E+02	1.936E+02	6.293E+02
Mg	7.433E+01	6.250E+01	5.587E+01	7.081E+01	5.192E+01
Al	2.613E+01	2.974E+01	2.849E+01	2.782E+01	5.220E+01
Si	6.938E+00	3.961E+00	3.366E+00	4.117E+00	1.979E+00
P	1.146E+01	8.229E+00	8.065E+00	7.423E+00	1.240E+01
S	3.506E−01	3.216E−01	3.161E−01	3.153E−01	2.560E−01
Cl	2.498E+00	2.814E+00	2.818E+00	2.901E+00	3.259E+00
Ar	3.586E−01	3.568E−01	3.575E−01	3.552E−01	3.280E−01
K	2.368E+00	2.137E+00	2.172E+00	2.068E+00	2.111E+00
Ca	2.273E−01	2.315E−01	2.282E−01	2.292E−01	1.836E−01
Sc	1.154E+01	9.397E+00	9.780E+00	9.859E+00	1.180E+01
Ti	9.323E−01	9.359E−01	9.576E−01	9.415E−01	1.798E+00
V	5.069E−01	4.622E−01	4.823E−01	4.784E−01	1.069E+00
Cr	3.439E−01	3.521E−01	3.474E−01	3.494E−01	2.950E−01
Mn	1.032E−01	1.028E−01	1.013E−01	1.030E−01	7.756E−02
Fe	2.260E−01	2.372E−01	2.292E−01	2.328E−01	1.335E−01
Co	2.741E+01	2.589E+01	2.634E+01	2.521E+01	1.079E+01
Ni	2.759E+00	2.740E+00	2.752E+00	2.715E+00	2.069E+00
Cu	7.410E+01	7.078E+01	7.259E+01	6.934E+01	6.346E+01
Zn	2.602E+01	2.623E+01	2.946E+01	3.074E+01	5.052E+01
Ga	6.785E+01	6.604E+01	7.312E+01	6.709E+01	2.436E+02
Ge	5.763E+01	4.726E+01	5.341E+01	5.323E+01	2.692E+02
As	4.441E+01	3.353E+01	3.787E+01	3.913E+01	2.659E+02
Se	2.852E+01	1.896E+01	2.179E+01	2.088E+01	2.248E+02
Br	2.629E+01	1.554E+01	1.814E+01	1.876E+01	2.049E+02
Kr	1.455E+01	1.308E+01	1.441E+01	1.344E+01	5.039E+02
Rb	2.022E+01	1.065E+01	1.257E+01	1.875E+01	4.854E+02
Sr	9.261E+00	1.182E+01	1.249E+01	1.365E+01	1.739E+03
Y	7.791E+00	7.293E+00	7.805E+00	8.507E+00	1.390E+03
Zr	3.394E+00	3.103E+00	3.327E+00	3.258E+00	6.575E+02
Nb	5.012E+00	4.053E+00	4.407E+00	4.401E+00	8.198E+02
Mo	1.224E+00	1.245E+00	1.316E+00	1.179E+00	2.866E+02

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In the CF88 model, any ²²Ne left in the He core ashes is burnt with a peak neutron density of ∼10¹¹ cm⁻³ (e.g., Raiteri et al. 1991a) and a neutron exposure of τ ∼ 0.05 mbarn⁻¹. The final convective C-burning shell episode is switched off by the onset of central O-burning and is not active in the last evolutionary phase before the SN explosion. The final distribution in Figure 9 is peaked in the Cu–Ga–Ge region. In the CU model, the dominant neutron exposure contribution is provided during the convective C-burning core, driven by the ¹³C(α, n)¹⁶O reaction. During the final convective C shell, ²²Ne drives the neutron flux. The component built in the C core and peaked in the Sr–Y–Zr–Nb region is dominant in the final s-process rich ejecta because of the larger neutron exposure and the large overlapping factor between the core and shell (∼2/3 of the mass of the convective C shell is made of C-burning core ashes). Similar results were found in Paper I for a large range of masses using the CU rate, and for the specific case of the 20 M_☉ star model and a rate intermediate between CF88 and CU (CI rate; Paper I). In cases where the convective C core overlaps with the final convective C shell, the material from the core dominates the final s-process ejecta. The resulting s-process yields are anomalous compared with the classical weak s-process component.

If all, or most, of the s-process-rich material lost to the interstellar medium by core collapse SNe is similar to the CU ejecta, then the predicted abundances cannot be reconciled with the solar system distribution. In other words, if all massive stars were to eject material exposed to ¹³C(α, n)¹⁶O in the central C-burning conditions observed for CU, an anomalous abundance distribution would be observed. This may indicate that the upper limit ¹²C+¹²C rate is overestimated, a conclusion already reached in Paper I. We will discuss in Section 6 the impact of the ¹²C(¹²C, α)²⁰Ne and ¹²C(¹²C, p)²³Na strength uncertainty on these results. Tackling the discussion from a more speculative point of view, two s-process components may be identified: the standard weak s-process component, dominated by ²²Ne(α, n)²⁵Mg, and a stronger s-process component coming from ¹³C(α, n)¹⁶O, triggered when the temperature is low enough to prevent the ¹³N(γ, p)¹²C reaction from becoming efficient. We refer to the latter component as the "cold" C-burning component or "cs-component" hereafter.

5.2. Possible Implications of the cs-component for the Lighter Element Primary Process

5.3. Pre-explosive Production of ²⁶Al and ⁶⁰Fe

The ¹²C+¹²C reaction has two main channels with comparable strength, producing respectively one α particle and one ²⁰Ne nucleus, or one proton and one ²³Na nucleus. However, the predicted C-burning yields consist of more than 20% ²⁰Ne, and only few per cent of ²³Na. This is due to the efficient activation of ²³Na(p, α)²⁰Ne, which processes ²³Na in ²⁰Ne (e.g., Arnett & Thielemann 1985; Chieffi et al. 1998). As an example, in Figure 10, we show the isotopic chart with the main reaction fluxes responsible for productive and destructive nucleosynthesis of ²³Na, at typical shell C-burning conditions. Most of the ²³Na produced via ¹²C(¹²C, p)²³Na is depleted by the ²³Na(p, α)²⁰Ne reaction. Therefore, the final ²³Na abundance is mostly given by the equilibrium between those two reactions. The other significant flux in Figure 10 is due to the ²³Na(p, γ)²⁴Mg reaction, which competes with the (p, α) channel (with a typical flux ratio of (p, α)/(p, γ) ∼ 4; e.g., NACRE). Note that most of the sodium observed today in the solar system is produced in massive stars in the conditions described.

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At stellar temperatures, the uncertainty in the relative strength between the ²⁴Mg* α-decay and proton-decay channels has a relevant impact on nucleosynthesis calculations. In the present work, we use the ratio R_α/R_p = 0.65/0.35, keeping in mind that we also properly consider the small contribution from the neutron channel (see Section 7 for a more detailed discussion about the neutron channel). However, this ratio becomes more uncertain at temperature conditions closer to the Gamow peak energies and typical C core conditions.

In order to test the impact of this uncertainty, we have performed two sets of calculations using a simple single-zone post-processing method: using the CF88 ¹²C+¹²C rate (set1, CF88 rate; T = 1.0 GK, ρ = 10⁵ g cm⁻³) for typical shell C-burning conditions and the CU rate for conditions dominated by core C-burning (set2, CU rate; T = 0.65 GK, ρ = 10⁴ g cm⁻³), the ratio R_α/R_p has been modified within the range 0.95/0.05 ≲ R_α/R_p ≲ 0.05/0.95. In Figure 11, the C-burning yields from the most extreme cases are compared with the yields obtained using the standard ratio 0.65/0.35. Results for set1 and set2 are presented in the left and right panels, respectively. For a more direct comparison, in Table 7 the abundances for selected species corresponding to the cases presented in Figure 11 are reported.

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Concerning set1, despite the large difference in the R_α/R_p ratio, the variation is below a factor of two for most species. A larger departure from the standard isotopic distribution is related to p-process species (e.g., ⁹²Mo), which are only depleted by neutron captures and have in all cases negligible abundances. Increasing the number of α particles compared with protons has the effect of increasing the number of neutrons available, therefore increasing the s-process efficiency. In particular, the s-process species between Fe and Zr are boosted (see Table 7). On the other hand, increasing the number of protons reduces the s-process efficiency, even if protons are mostly converted to α particles via ²³Na(p, α)²⁰Ne. The main reason for this behavior is that in C-burning conditions the ²²Ne(α, n)²⁵Mg nucleosynthesis channel competes with ²²Ne(p, γ)²³Na. This is shown in Figure 12 for the standard case set1, where the fluxes producing and depleting ²²Ne are presented. Therefore, variation of the R_α/R_p ratio also affects ²²Ne nucleosynthesis. Note that during C-burning the ²²Ne(α, γ)²⁶Mg rate is marginal compared to the (α, n) rate.

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In Figure 11 (right panel), set2 calculations show a larger variation than set1. In particular, a stronger ¹²C+¹²C proton channel does not significantly modify the s-process yields and the results discussed in Section 5 are confirmed. Conversely, a stronger α channel reduces the s-process yields by more than a factor of 10 at the Sr–Y–Zr peak. For instance, ⁸⁸Sr (neutron magic number, N = 50) in the a95p05* case (where α-channel probability = 95%, proton-channel probability = 5%) is reduced by a factor of ∼30 compared with the st* case (α-channel probability = 65%, proton-channel probability = 45%; see Table 7). This effect is the opposite to that in set1, where a stronger α channel leads to a more efficient s-process production. The main neutron source in set2 (and the s-process ejecta when using the CU ¹²C+¹²C rate) is the ¹³C(α, n)¹⁶O, not ²²Ne(α, n)²⁵Mg. In the temperature conditions of set2, the photodisintegration timescale of ¹³N via ¹³N(γ, p)¹²C is comparable with the β-decay of ¹³N to ¹³C (see Section 5). As a result, the available ¹³C abundance depends on the amount of protons available to activate the ¹²C(p, γ)¹³N reaction. As such, increasing the R_α/R_p ratio with respect to the standard value reduces the proton density and consequently the ¹³C neutron source. Calculations in set2 also show significant variations for light species; a stronger α-channel leads to a significant decrease of ¹⁶O and ²³Na and an increase of ²⁰Ne (see details in Table 7). Conclusions obtained in Section 3 need to be viewed cautiously since the CU model, as well as the other four models, was calculated using a constant value of R_α/R_p = 0.65/0.45. If the ¹²C+¹²C rate is indeed larger than CF88, then the impact of the R_α/R_p ratio uncertainty on stellar calculations becomes more important and may not be negligible with regard to the main features of subsequent evolutionary stages.

In summary, the total ¹²C+¹²C fusion rate is not the only uncertainty to consider when comparing different nucleosynthesis calculations. In this section, we showed that the R_α/R_p ratio must also be considered and that more experimental efforts are required to constrain the relative strength of the primary decay channels.

Besides the two main nucleosynthesis channels, feeding ²⁰Ne and ²³Na, respectively, the ¹²C+¹²C reaction can also result in the production of neutrons via the ¹²C(¹²C, n)²³Mg channel. The isotope ²³Mg is unstable, decaying rapidly (τ_1/2 = 11.3 s) into ²³Na. The reaction is endothermic with a Q-value of −2.6 MeV and will contribute only at higher temperatures. Yet, even at high energies the contribution of the reaction branch is small. An early analysis by Dayras et al. (1977, D77 hereafter) indicates 2% at 3 MeV gradually increasing up to 10%. This is confirmed by more recent measurement (X. Tang 2012, private communication). For the present calculations, the ¹²C(¹²C, n)²³Mg rate of D77 has been used. In simulations where the total ¹²C+¹²C fusion rate corresponds to the CF88 rate, the neutron flux produced by the ¹²C(¹²C, n)²³Mg branch is weaker than the ²²Ne(α, n)²⁵Mg, at least for solar-like metallicities. Note, however, that the ²²Ne(α, n)²⁵Mg neutron source is secondary and its efficiency depends on the initial metal content, whereas ¹²C(¹²C, n)²³Mg is a primary source. In comparison with the third potential neutron source ¹³C(α, n)¹⁶O, the ¹²C(¹²C, n)²³Mg probability is marginal for models using the CU rate as will be discussed below.

The experimental uncertainty in the ¹²C(¹²C, n)²³Mg data is approximately 50% at energies above 4 MeV, and toward lower energies it increases up to 80%. The cross section is expected to decline rapidly since the neutron channel closes at 2.6 MeV, yet the existence of molecular resonance structures near the threshold cannot be excluded, because such features are visible in the proton and α-channels at the corresponding energy range. For low temperature environments typical for C shell burning near 1 GK, we therefore consider an uncertainty range of approximately a factor of 10; for higher temperatures as found at the bottom of the C shell during the later evolutionary stages prior to core collapse, we reduce the uncertainty range to a factor of five since the Gamow window extends into the excitation range well covered by experiment. We do not consider the reaction for typical core C-burning conditions, since the neutron channel of C-burning is endoergic and therefore suppressed toward the low temperatures associated with the core burning environment.

In order to test the impact of the neutron channel, and taking into account its uncertainty, we have performed two sets of calculations using different ¹²C(¹²C, n)²³Mg rates (D77, and D77 multiplied by a factor of 2, 5, and 10, which are called D77t2, D77t5, and D77t10, respectively): set3, where typical C shell conditions with constant temperature and density are considered (CF88 ¹²C+¹²C rate, T_set3 = 1.1 GK and ρ = 10⁵ g cm⁻³, the ¹²C(¹²C, n)²³Mg probability is ∼0.12% of the total rate), and set4, where realistic temperature and density profiles are extracted at the bottom of the final convective carbon shell of the CF88t10 model (using the CF88t10 rate). Note that cases with rates lower than D77 are not considered here, since the ¹²C(¹²C, n)²³Mg contribution becomes negligible. The temperature and density profiles used for set4 calculations are reported in Figure 13 and are characterized by an increase in the final temperature and density, up to T = 1.36 GK and ρ = 1.33 × 10⁵ g cm⁻³. During core collapse, and shortly before the SN explosion, thermal instabilities occur in O-burning layers and temperature and density may rapidly increase in the outer stellar material (Arnett 1974; Arnett & Wefel 1978). We choose the CF88t10 model because it shows a convective C shell lasting until the SN explosion. In all of the other models considered in this work, the convective phase terminates at an earlier point in the evolution. In these cases, stellar structure modifications triggered by O ignition in the core prohibits convective shell C-burning from continuing (e.g., Hirschi et al. 2004) and the yields are not affected by the temperature increase shown in Figure 13. In the CF88t10 model, the final temperature and density rise causes a boost in the neutron density, resulting in distinctive effects in the s-process distribution. For instance, all the isotopes considered as r-only between iron and strontium may be significantly produced (⁷⁰Zn, ⁷⁶Ge, ^{80, 82}Se), ⁷⁵As overproduction might be comparable to neighbor s-only isotopes (e.g., ⁷⁶Se), and, in general, s-process branchings are significantly affected (e.g., Pignatari et al. 2010).

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During this final phase, the ¹²C(¹²C, n)²³Mg channel may also become more efficient, since its strength compared with α and the proton channel increases with temperature (Dayras et al. 1977). The main purpose of the two sets presented in this section is to test the impact of the ¹²C(¹²C, n)²³Mg channel in typical C shell conditions (set3) and in the final temperature-rise phase (set4).

The relative isotopic distributions for the D77t2, D77t5, and D77t10 cases are shown for set3 in Figure 14. The right panel shows the same calculations for set4. Abundances for selected species are reported in Tables 8 and 9 for the most extreme cases, compared to D77. In both sets of calculations presented in Figure 14, the D77t2 case shows small departures from the standard case within 10%. On the other hand, in the D77t5 and D77t10 cases, the s-only species between iron and strontium (⁷⁰Ge, ⁷⁶Se, ^{80, 82}Kr and ^{86, 87}Sr) show an increase in their average mass fraction of 1.30 and 1.77, respectively (Table 8). This means that the average neutron exposure has increased (see also ⁸⁸Sr in Table 8). Interestingly, an increase in the availability of neutrons due to a more efficient ¹²C(¹²C, n)²³Mg channel causes a propagation effect over the s-process species up to Sr, with significant differences along the neutron capture path. Illustrating this point, the variation of ^{80, 82}Kr is larger than lighter or heavier s-only species (e.g., in the D77t10 case compared to the D77 case, the ⁸²Kr abundance increases by a factor of 2.1, compared with 1.4 for ⁷⁰Ge; Table 8). The r-only isotopes like ⁷⁰Zn or ⁷⁶Ge are enhanced for the cases D77t5 and D77t10 compared with the standard case, with a production comparable to the s-process species (see Table 8).

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The evolution history of the stellar structure has a relevant impact on the C shell nucleosynthesis, affecting for example the amount of ¹²C available when C ignites in a convective shell (e.g., Imbriani et al. 2001; El Eid et al. 2009). The set4 calculations presented in Figure 14 use the initial abundances and the temperature–density evolution of the last convective C shell in the CF88t10 model. Compared to set3, set4 models show less ¹²C fuel available (the ¹²C abundance is ∼0.075, compared to 0.27 in set3). This is because of the large overlap with the previous convective shell, where most of the carbon fuel was consumed.

Therefore, the relatively low amount of ¹²C significantly reduces the impact of a higher ¹²C(¹²C, n)²³Mg rate compared with set3, and the average mass fraction is only marginally modified. On the other hand, in the D77t5 case, most of the r-only isotopes are increased by a factor of ∼1.2 compared with the standard case, and, at the same time, the p-only isotopes are reduced by a similar factor. In fact, such modifications are triggered by local neutron captures due to the final short neutron burst, where r-only isotopes are fed from local abundant s-process isotopes and the p-rich isotopes are more depleted via neutron capture. The impact of a final neutron burst, arising from a higher ¹²C(¹²C, n)²³Mg rate, may be more severe than the case considered in set4 since its relevance is directly related to the amount of carbon fuel that is still available when the temperature starts to rise. As such, the estimation regarding the impact of the ¹²C(¹²C, n)²³Mg uncertainty on C shell yields may change according to the ¹²C+¹²C rate used and to the stellar model considered.

Summing up, present s-process abundance predictions may be affected locally (at the different branching points producing more r-only species, less ⁸⁰Kr, etc.), and/or more generally over the entire s-process distribution, by the uncertainty associated with the ¹²C+¹²C neutron channel. Figure 14 shows the two different effects that can be obtained, depending on the stellar model. The ¹²C(¹²C, n)²³Mg probability needs to be known at stellar carbon burning conditions with a precision significantly better than a factor of five. We have shown that a factor of two precision would have an impact on the s-process predictions within an acceptable 10%.

In this section, we explore the impact of the ¹²C+¹²C rate uncertainty on p-process nucleosynthesis. In the solar abundance distribution, there are 35 p-rich stable nuclei, from ⁷⁴Se to ¹⁹⁶Hg, that are called p-only isotopes since they cannot be produced by either the s-process or the r-process. In the solar system, the p-only nuclei are about two orders of magnitude less abundant when compared with the other stable isotopes of the same element fed by the s- and r-processes. Exceptions are the p-nuclei ^{92, 94}Mo and ^{96, 98}Ru, which represent respectively 14.84, 9.25% and 5.52, 1.88% of the total abundance of the respective element.

Different astrophysical sources have been proposed to either reproduce or contribute to the solar p-process distribution (see, e.g., Arnould & Goriely 2003; Fröhlich et al. 2006; Travaglio et al. 2011). At present, the most well-established astrophysical site for p-process nucleosynthesis is the O/Ne-rich layers in massive stars, before and after the SN explosion (e.g., Arnould 1976; Woosley & Howard 1978). The pre-explosive p-process component is mostly re-processed by the explosive nucleosynthesis triggered by the SN shock wave. Its effective impact on the total p-process yields depends on the explosion mechanism and on the stellar structure behavior in the last days before core collapse. For instance, extensive convection and mixing between different burning shells (e.g., convective O shell mixing with convective C shell; Rauscher et al. 2002) may trigger the enrichment of SN O-rich ejecta by pre-explosive p-process nuclei (see also Arnett & Meakin 2011).

Explosive p-process nucleosynthesis in standard SNe Type II explosions shows a weak dependence on the initial mass of the star (e.g., Rayet et al. 1995). The initial abundances for the p-process are provided by the s-process distribution, built in the previous evolutionary phases. This implies that the p-process efficiency depends on the initial metallicity of the star, showing a secondary nature similar to the weak s-process (Rayet et al. 1995; Arnould & Goriely 2003).

In order to investigate the impact of the ¹²C+¹²C rate on the p-process abundances, we have performed a set of one-zone explosive nucleosynthesis simulations. We have used the same explosive trajectories used by Rapp et al. (2006). Initial abundances are given by the pre-explosive composition calculated in the models presented in this paper. In Figure 15 (left panel), we compare the p-process abundance distribution obtained using the abundance seeds from the CF88t10 and CU models (pCF88t10 and pCU, respectively). The other models discussed in the previous sections are similar to pCF88t10 (Section 3). The abundance distribution of pCF88t10 is consistent with previous analyses of the p-process nucleosynthesis in explosive O/Ne layers in massive stars (e.g., Prantzos et al. 1990b; Rayet et al. 1995). About 60% of the p-process isotopes are produced in comparable amounts (e.g., Rayet et al. 1990, 1995). A sample of isotopes is underproduced compared to the average production factor F₀. In all present models, the nuclei ^{92, 94}Mo and ^{96, 98}Ru are systematically underproduced by an order of magnitude or more in all nucleosynthesis scenarios using realistic massive stars conditions (Arnould & Goriely 2003). This problem cannot be resolved by nuclear physics uncertainties (Rauscher 2006; Rapp et al. 2006). A possible solution was proposed by Arnould et al. (1992), showing that artificially increasing s-process abundances for A ⩾ 90 would solve the Mo–Ru puzzle. It was argued by Costa et al. (2000) that using a larger ²²Ne(α, n)²⁵Mg rate (within the upper limit of Angulo et al. 1999) could increase s-process yields in massive stars at the Sr peak and beyond. However, not only is this finding unconfirmed by other results (Heger et al. 2002), such a high rate would also result in disagreement between present weak and main s-process component predictions in the solar system abundance distribution. Furthermore, the capability of the present s-process stellar AGB models to reproduce the s-process signature measured in carbonaceous presolar grains would be called into question (see also discussion in Pignatari et al. 2010).

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Alternatively, other processes in massive stars different from the classical p-process have been proposed to contribute to the missing Mo–Ru p abundances, e.g., in α-rich freezout conditions during the SNII explosion (e.g., Hoffman et al. 1996) or the νp process in proton-rich neutrino-wind conditions (Fröhlich et al. 2006). The p-process underproduces ¹¹³In and ¹¹⁵Sn within present nuclear uncertainties, despite the potential contribution from the r-process (Rapp et al. 2006; Dillmann et al. 2008, and references therein). On the other hand, ¹⁵²Gd and ¹⁶⁴Er receive a significant contribution from the s-process (e.g., Arlandini et al. 1999; Abbondanno et al. 2004; Best et al. 2001). Finally, ¹³⁸La might receive a significant contribution from neutrino capture on ¹³⁸Ba (Goriely et al. 2001), which is not considered in these calculations.

For the pCF88t10 case, we obtain an overproduction of ¹³⁰Ba, ¹⁸⁰Ta, and ¹⁹⁶Hg. The overproduction is acceptable within the nuclear uncertainties for ¹³⁰Ba. The p-process production of ¹⁸⁰Ta is affected by the pre-explosive s-process abundance distribution. The pCF88t10 yields presented in Figure 15 are characterized by a general underproduction compared with ¹⁶O. According to Rayet et al. (1995), the total production factors for p-nuclei and ¹⁶O in the ejected SN material should be comparable, assuming that massive stars are their main astrophysical site. For instance, Rayet et al. (1995) estimated such underproduction to be at least a factor of two for a 25 M_☉ star, considering uncertainties associated with both the stellar model and nuclear input, such as the ¹²C(α, γ)¹⁶O rate. In the pCF88t10 case, the average overproduction factor is F₀ = 42.3.

Considering a mass correction factor of f_m = 0.2, the p/O yield ratio is ∼0.12. This takes into account an ¹⁶O overproduction factor compared with the solar overabundance (∼70) and the amount of p-rich and O-rich mass ejected by a standard 25 M_☉ core collapse SN model (see for instance Table 3 of Rayet et al. 1995, where f_m ranges from 0.19 to 0.23 depending on the different model prescriptions).

The average overproduction factor of the pCF88t10 case (F₀ = 42.3) is consistent with the value provided by Rayet et al. (1995) for a 25 M_☉ model at solar metallicity and for the initial mass function weighted p-process distribution (F₀ = 130 and 100, respectively), considering that the classic p-process in massive stars is a secondary process. Indeed, its efficiency critically depends on the s-process abundance seeds from the previous stellar phases (Arnould & Goriely 2003). It is also important to note that according to Tinsley (1980), at solar metallicity secondary species (in this case, the p-process yields) should be produced twice as much as primary species (e.g., ¹⁶O). Consequently, the underproduction of the p-process in the SNII scenario is likely to be more severe than mentioned by Rayet et al. (1995); the yield ratio required to reproduce the p-process inventory should be p/O = 2 at solar metallicity, twice as much as previously estimated.

In Figure 15 (left panel), the pCU case shows significant differences compared with the pCF88t10 case. First, F₀ is more than a factor of eight higher than in the standard case (F₀ = 348). Second, a good fit is obtained for ^{92, 94}Mo and ⁹⁸Ru within a factor of two uncertainty and only a factor of three underproduction for ⁹⁶Ru. The higher s-process efficiency of the cs-component in the CU model at the Sr peak and beyond provides the required seed distribution to feed this mass region. It is important to mention that for the pCU case the lighter p-process species are mostly produced by the coldest (and more external) trajectories, whereas in standard p-process conditions they are mostly produced by the hottest (and deepest) particles due to photodisintegration flows destroying heavier species. Therefore, in this case, local photodisintegration channels are mainly responsible for p-nuclei yields, and previous impact studies of nuclear uncertainties (e.g., Rauscher 2006; Rapp et al. 2006) should be reconsidered. The underproduction of ¹¹³In, ¹³⁸La, ¹⁵²Gd, and ¹⁶⁴Er is also confirmed for the pCU case. The problem of ¹³⁰Ba, ¹⁸⁰Ta, and ¹⁹⁶Hg overproduction, seen in pCF88t10, is also solved due to the increase of F₀. However, ¹⁹⁰Pt is now underproduced.

As a verification test, we compare the pCU p-process abundances with the results obtained by the code used in Rapp et al. (2006), using the same initial distribution and the same trajectories (Figure 15). Considering that the two codes used different nuclear networks, the final compositions are consistent, with maximum yield variation within a factor of two. We also tested how the present results may be affected by using different trajectories. Only marginal variations in the final p-process distribution were obtained by either changing the freezout temperature in the explosive particles, using different power-law prescriptions, or assuming an exponential decrease. Therefore, the present conclusions are not compromised.

From the present calculations, for the higher (CU) ¹²C+¹²C rate, the overlap between the convective C core and the convective C shell provides a higher s-process efficiency at the Sr peak and beyond, forming the cs-component (Section 5). Such a distribution feeds enhanced p-process yields, also reproducing the solar p-process abundances in the Mo–Ru region within a factor of 2–3 (Figure 15). Therefore, the cs-component obtained from the CU model provides a set of s-process seeds that may efficiently feed the final abundances of the puzzling Mo and Ru p-only species, and provide a satisfactory distribution of heavier p-nuclei.

We have seen that the cs-component is difficult to reconcile with the s-process abundances observed in the solar system, unless only a few per cent of the total s-process-rich mass ejected by massive stars can carry it. By extension, the p-process-rich distribution obtained from CU (cp-component, hereafter) has to follow the same restrictions as discussed in Section 5. Indeed, we have shown that if about 2% of the s-process-rich mass ejected into the interstellar medium has the enhanced component, then this could provide a possible scenario to solve the LEPP puzzle in the solar system distribution. Assuming that 2% of the ejected p-process-rich mass carries the cp-component (with F₀ = 348 in the pCU case), only about 2% of the average p-process abundance in the solar system distribution could be reproduced by the cp-component (348 × f_m × 0.02/70, where f_m = 0.2 is the mass correction factor described above and 70 is the ¹⁶O overproduction). The F₀ from the pCF88t10 case, assuming that the remaining ejected mass shows a standard s-process and p-process signature, would correspond to ∼12% of the average p-process solar distribution. The low p-process efficiency in massive stars simulations cannot therefore be solved by considering the cp-component in the p-process inventory, and it is even worse than previously estimated if we also consider the secondary nature of the classic p-process. An amount of only a few per cent of solar Mo–Ru p-nuclei could be produced. If such speculations will be confirmed, another astrophysical source for the cosmic abundances of the p-process nuclei is needed. In this direction, thermonuclear supernovae (SNeIa) might efficiently produce p-process abundances in their ejecta (Howard et al. 1991; Howard & Meyer 1993; Kusakabe et al. 2011; Travaglio et al. 2011). In particular, Travaglio et al. (2011) provide SN Ia simulations where a relevant fraction of the p-process abundances in the solar system inventory are reproduced.

These final considerations however are based on only one stellar mass, and the trajectories used for the p-process calculations are based on one-zone SN explosion trajectories (Rapp et al. 2006). Recent calculations have shown that fall-back in core-collapse SNe could actually drastically reduce the amount of O-rich mass ejected for stars with masses heavier than ∼20 M_☉ (Fryer et al. 2009). The impact of this on final p-process yields still needs to be estimated in detail, and for different masses. One possible implication is that p-process yields may depend on which SN engine is considered. Nevertheless, the intriguing scenario where part of the missing LEPP component may be related to the missing p-process abundances of Mo and Ru, and more in general with a new component for the p-process inventory, can be constrained or ruled out by an experimental measurement of the ¹²C+¹²C rate at stellar temperatures.

In previous sections, the impact of the ¹²C+¹²C rate uncertainty on s- and p-process nucleosynthesis near solar metallicity has been analyzed. In this section, we discuss the implications for the s-process at low metallicity. In Section 5, we showed that the cs-process may at least partially contribute to the LEPP component in the solar system. Travaglio et al. (2004) proposed that the solar LEPP component is due to a primary process already active in the early universe, leaving a signature that has been observed in the Galactic halo for Sr, Y, and Zr (e.g., Truran et al. 2002). Whether the "solar" LEPP and the "stellar" LEPP (see definitions and discussion in Montes et al. 2007) are either the same process or due to a combination of different processes is still matter of debate. More observations involving Sr–Y–Zr and other elements in the LEPP mass region (from the Sr peak up to Te–Xe) are required in order to shed more light into the origin of LEPP (Farouqi et al. 2009; Chiappini et al. 2011; Hansen & Primas 2011). Following the analysis of the previous sections, one possible question is: Can the cs-process efficiently contribute to the so-called stellar LEPP at low metallicity?

Another efficient s-process component has been proposed, occurring in fast rotating stars of low metallicity (Pignatari et al. 2008; Frischknecht et al. 2012). This component is triggered by the production of primary ²²Ne and the consequent activation of ²²Ne(α, n)²⁵Mg. Since, in principle, the cs-process may also be produced and ejected by fast rotating massive stars, what is the impact on the s-process ejecta of these two combined components? Stellar evolution models including the effects of rotation are successful in reproducing several observations, like nitrogen chemical enrichment in the early Galaxy (Heger & Langer 2000; Meynet & Maeder 2000) or the ratio of Wolf–Rayet to O-type stars (Vázquez et al. 2007; Frischknecht et al. 2010, and reference therein). There are, however, some unresolved issues in rotating models. A more efficient breaking mechanism is necessary to explain the rotation rates of white dwarfs and millisecond pulsars. Although magnetic fields help, it is not clear yet whether their effect is sufficient (Suijs et al. 2008). Rotation-induced shear mixing at the envelope–core interface in AGB stars may be too strong and thus disable the s-process via radiative ¹³C-burning in the interpulse phase of low-mass AGB stars (Herwig et al. 2003; Siess et al. 2004). Keeping this in mind, the cs-process behavior with metallicity does not depend on rotation-induced mixing, and the s-process component in fast rotators mentioned above is a realistic scenario to consider within the uncertainties of the present models.

We have performed a set of calculations at [Fe/H] = −3. Stars with this metallicity showing the LEPP component are observed in the Galactic halo (e.g., HD 122563 at [Fe/H] = −2.7; Honda et al. 2006). A set of different initial abundances were taken from Pignatari et al. (2008), at the end of convective core He-burning, considering a large spread of primary ²²Ne available in the He core due to rotational mixing. Two different C-burning trajectories were used in the simulations: a set1 trajectory (Section 6, T = 1.0 GK, ρ = 10⁵ g cm⁻³, using the CF88 ¹²C+¹²C rate, hereafter set1b), which reproduced C shell conditions without the cs-process component, and a set2 trajectory (Section 6, T = 0.65 GK, ρ = 10⁴ g cm⁻³, using the CU ¹²C+¹²C rate, hereinafter set2b), which did include the cs-process component. For complete fast rotating stellar model calculations at different metallicities, we refer to Frischknecht et al. (2012). In Figure 16 (left panel), the production of elements located at the various neutron magic peaks is shown for set1b for different initial abundance distributions, defined by the amount of ²²Ne that is consumed by ²²Ne(α, n)²⁵Mg in the previous He core phase (²²Ne_burnt). The first case at ²²Ne_burnt ∼ 4 × 10⁻⁵ uses an abundance distribution from a non-rotating star, where the available ²²Ne comes only from the initial CNO abundances at [Fe/H] = −3, considering also the α-enhancement at this metallicity. The other cases have initial abundances with an enhanced s-process distribution, due to the primary ²²Ne neutron source being activated in fast rotating massive stars (Pignatari et al. 2008). In particular, the test at ²²Ne_burnt ∼ 2 × 10⁻³ corresponds to the standard case considered at [Fe/H] = −3 in Pignatari et al. (2008, with the primary ²²Ne around 1% by mass in the He core). We also used as initial distributions unpublished results obtained from the same study, where different amounts of primary ²²Ne were available in the He core. The cases with ²²Ne_burnt ∼ 4 × 10⁻³ correspond to ∼2% of available primary ²²Ne. Finally, we also included an extreme case, where about 1% of ²²Ne is burnt by the ²²Ne(α, n)²⁵Mg in the He core.

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As already mentioned by Pignatari et al. (2008), in set1b most of the s-process yields are produced in the core He-burning phase, and only a partial modification occurs in the later shell C-burning phase. Production of the Sr–Y–Zr peak is robust over a large range of primary ²²Ne_burnt (∼7 × 10⁻⁴–3 × 10⁻³). Above 3 × 10⁻³, material at the Sr peak starts to be depleted, feeding heavier elements along the s-process path. For ²²Ne_burnt ∼ 2 × 10⁻³ Ba starts rising, as well as Pb for ²²Ne_burnt ∼ 3 × 10⁻³. At ²²Ne_burnt ∼ 5 × 10⁻³, Ba and Pb productions reach their largest abundance. Increasing the amount of ²²Ne_burnt to ∼10⁻² causes the Sr peak material to rise again, whereas Ba and Pb decrease. To burn 1% of ²²Ne via ²²Ne(α, n)²⁵Mg requires that the amount of primary ²²Ne is 5% or more. Such a large abundance would make ²²Ne acting mostly as a neutron poison, consuming the neutrons produced. Indeed, the production of Ba and Pb is limited not only by the amount of iron seeds available, but also by the dual neutron source/neutron poison nature of ²²Ne. Note that, however, such an amount of ²²Ne is a significant overestimate from present predictions of fast rotating massive stars (Meynet et al. 2006; Hirschi et al. 2008; Frischknecht et al. 2012). Interestingly, this last case is different from AGB stars at low metallicity, where ²²Ne becomes both a seed and a poison for s-process nucleosynthesis up to lead. In the AGB case, the main neutron source is the ¹³C(α, n)¹⁶O reaction and the neutron exposure is high enough to produce heavy elements starting from light species (Gallino et al. 2006).

In Figure 16 (right panel), we show the same calculations for set2b, where the cs-process contribution has also been considered, due to the activation of ¹³C(α, n)¹⁶O. Compared with the abundance at the He core exhaustion for the non-rotating case, the Sr production factor compared with solar increases by about 40 times, up to 3.9. Production factors of Ba and Pb increase by about 5 and 2 times, up to 0.028 and 0.006, respectively. At both solar-like and [Fe/H] = −3 metallicities, the bulk of s-process production is found at the Sr peak, with minor contributions to heavier elements. If we compare the Sr production factor obtained here and the one obtained with the M25CU model and half-metallicity (Table 6), a secondary-like trend with the initial composition is observed, while light neutron poisons are mostly primary. Consequently, the iron seeds force the nature of this s-process component to be secondary, just like the s-process in fast rotators.

Concerning fast rotating massive stars, the production of Sr–Y–Zr is not affected much by the cs-process component. In fact, the production of Sr has almost been saturated during the previous core He-burning phase, since most of the Fe seeds have already been consumed (Pignatari et al. 2008; Frischknecht et al. 2012). As a result, the Sr–Y–Zr s-process yields from fast rotating massive stars are robust over a large range of ²²Ne_burnt (i.e., the spread of initial stellar mass and the efficiency of rotation-induced mixing), within nuclear uncertainties of critical reaction rates like ²²Ne+α (Pignatari et al. 2008), ¹⁷O(α, γ)²¹Ne (Hirschi et al. 2008), and ¹²C+¹²C. However, relevant uncertainties may be associated with other nuclear reactions not considered here, and with the treatment of rotation in stars.

The production of Ba and Pb is boosted by the cs-process, which eventually feeds the Pb peak more efficiently for larger values of ²²Ne_burnt. Indeed comparing results from set1b and set2b, Figure 16, shows that the maximum production factors of Sr–Y–Zr and Ba are similar, whereas Pb is produced by a factor of two more in set2b. Additionally, in set2b calculations, ²²Ne is only partially destroyed and also behaves like an efficient neutron poison. Therefore, the more primary ²²Ne (and ^{25, 26}Mg, direct products via α capture on ²²Ne) there is available in the He core ashes, the less efficiently the cs-process feeds the heavier elements. In general, the maximum elemental ratio Pb/Ba that can be produced in the present calculations for fast rotating massive stars is ∼3, including the uncertainty related to the amount of primary ²²Ne available and/or consumed by the ²²Ne(α, n)²⁵Mg, as well as the uncertainty in the ¹²C+¹²C rate. Taking into account the present ²²Ne(α, n)²⁵Mg uncertainty, such conditions can be obtained when the primary ²²Ne abundance in the He core is about 2% for a 25 M_☉ star (about a factor of two more than predicted by Hirschi et al. 2008), or for stars with larger initial mass, where ²²Ne depletion via α-capture is more efficient. According to stellar model calculations by Hirschi et al. (2008), the production factors of Sr, Ba, and Pb are 7.5, 12.0, and 0.08, respectively, becoming 5.9, 31.9, and 0.9 when the contribution from the cs-process is included (Figure 16). In this case, therefore, s-process elements are efficiently produced only up to Ba, also taking into account the ¹²C+¹²C uncertainty.

In Pignatari et al. (2008), it was shown that the s-process in fast rotating massive stars due to primary ²²Ne is a secondary process. This was confirmed by similar results obtained for a complete set of fast rotating stellar models (Frischknecht et al. 2012). As discussed before, the cs-process is also secondary-like. In Section 5, it was shown that in principle the cs-component is a possible scenario for explaining the LEPP in the solar system without failing the weak s-process distribution. Assuming that the same constraints discussed in Section 5 are also valid for low metallicity, only a few massive stars would eject such a component at [Fe/H] ≲ −3.

The LEPP signature could have been observed in stars such as HE 1327−2326, where [Fe/H] = −5.45 (Frebel et al. 2005), for which the efficiency of the cs-process compared with other primary explosive processes in massive stars proposed to feed the Sr–Y–Zr peak is supposed to be lower (Woosley & Hoffman 1992; Qian & Wasserburg 2007; Farouqi et al. 2008; Arcones & Montes 2011). However, because at present only the upper limits of Ba and Eu are available for this star, whether or not HE 1327−2326 is an LEPP star still requires confirmation. A detailed comparison between stellar yields and spectroscopic observations of metal-poor LEPP stars is beyond the scope of this paper, since complete massive stellar calculations at low metallicity would be required.

In fast rotating massive stars, the occurrence of the cs-process would not affect the conclusions of Pignatari et al. (2008) concerning nucleosynthesis at the Sr–Y–Zr peak. If such a component does exist however, a more efficient production of Sr–Y–Zr is obtained also at lower rotation efficiencies, and for fast rotators Ba peak elements, and possibly Pb, could be more abundant in the final s-process yields. The present calculations explored the possible impact of the ¹²C+¹²C rate uncertainty on the s-process at low metallicity, in non-rotating and fast rotating stars. The analysis presented in this section is based on one-zone nucleosynthesis calculations and not on full stellar models. However, we did use realistic stellar conditions and explored the impact of various uncertainties on theoretical abundance predictions, which provides some guidance about s-process nucleosynthesis in massive stars at low metallicity. In the case that future nuclear physics experiments suggest a higher ¹²C+¹²C rate than the CF88 rate for central C-burning conditions, a new set of complete stellar models at low metallicity and different rotation efficiencies would be required in order to revise the present estimates of s-process nucleosynthesis stellar yields in the early universe.

The aim of the present work was to explore the impact of the ¹²C+¹²C rate uncertainty in massive stars, on pre-SN nucleosynthesis, and on the explosive p-process. Our analysis was focused on one mass and metallicity (25 M_☉ and half solar metallicity) and considered a large variation of the ¹²C+¹²C rate: the lower rate limit took into account possible hindrance effects in the carbon fusion reaction, whereas the upper limit considered contributions from potential subthreshold resonances at low temperature.

In all, we considered five different ¹²C+¹²C rates: the lower limit (CL), the standard rate of CF88, CF88 divided and multiplied by a factor of 10 (CF88d10 and CF88t10, respectively), and the upper limit (CU). Other reactions critical for the energy generation during stellar evolution were left unchanged. Up to central He exhaustion, the five stellar models obtained with these rates (M25CL, M25CF88d10, M25CF88, M25CF88t10, and M25CU) share the same conditions.

With the lower ¹²C+¹²C rates, CL and CF88d10, the s-process during convective shell C-burning shows a slightly lower neutron exposure, coupled with a higher neutron density peak, causing local effects at the branching points along the s-process path (e.g., ⁷⁹Se and ⁸⁵Kr). The production of long-lived isotopes ²⁶Al and ⁶⁰Fe during the pre-explosive phase is also affected, as discussed by Gasques et al. (2007). In particular, we confirm that ⁶⁰Fe tends to be produced more with lower ¹²C+¹²C rates. However, results for ²⁶Al are strongly affected by the stellar model used. In fact, if the convective C shell is switched off before the core collapse starts, all of the ²⁶Al has the time to decay. As a result, the final stellar yields only contain explosive ²⁶Al. The model with the highest pre-explosive ²⁶Al abundance in the C shell material is M25CF88t10, since it is the only one to keep a convective C shell until core collapse. Because the pre-explosive component of ²⁶Al is model dependent, the final ²⁶Al/⁶⁰Fe in SN ejecta needs to be regarded cautiously.

For the higher ¹²C+¹²C rates, CF88t10 and CU, the final s-process yields tend to increase toward the Sr peak with respect to lighter s-process elements. In the M25CF88t10 model, such an increase is within 20%, depending on the isotope considered, and is due to the overlap between the first shell and the second convective carbon shell (Bennett et al. 2010). In the M25CU model, the activation of ¹³C(α, n)¹⁶O in the C core causes the formation of the cs-process component, which will also dominate the final C shell ejecta. In agreement with Paper I, the s-process yields at the Sr peak increase by about two orders of magnitude. It was shown that if only ∼2% of the s-process-rich material from massive stars carries the cs-process signature, the weak s-process component from other stars would not be affected and the cs-process could reproduce at least part of the LEPP component in the solar system.

We have provided a series of tests in which the strengths to the two dominant nucleosynthesis channels of the carbon-fusion reactions ¹²C(¹²C, α)²⁰Ne and ¹²C(¹²C, p)²³Na have been changed. The impact of using a different R_α/R_p ratio changes according to the total ¹²C+¹²C rate. For the calculations using CF88, the amount of neutrons available (and the s-process efficiency) decreases with increasing proton channel strength compared to the α channel. Indeed, ²²Ne depletion in C-burning conditions is dominated by the (α, n) and (p, γ) channels, which are respectively affected by the amount of α and protons available. The cs-process, on the other hand, drastically loses its efficiency with increasing α channel strength, because less protons are available to produce ¹³C via ¹²C(p, γ)¹³N(β⁺)¹³C. For instance, using a ratio R_α/R_p = 0.95/0.05 (10 times higher than the standard R_α/R_p = 0.65/0.35), the Sr abundance decreases by about a factor of 30. Small variations are obtained from reducing the α channel strength. Critical light species are also affected by R_α/R_p using the CU ¹²C+¹²C rate: ¹⁶O and ²³Na decrease with a high R_α/R_p ratio, while ²⁰Ne increases. Therefore, uncertainty in the R_α/R_p ratio may affect abundances of species that are the main fuel for following evolutionary phases.

We also tested the impact of the uncertainty of the channel ¹²C(¹²C, n)²³Mg on the s-process calculations, assuming a range of probability between the standard rate (D77) and D77 multiplied by 10 (D77t10). First of all, the impact of the neutron channel increases with increasing amounts of ¹²C available at the onset of shell C-burning. This in turn may depend on the ¹²C(α, γ)¹⁶O rate during the previous core He-burning and convection prescription, as well as the total ¹²C+¹²C rate. In the simulations reproducing the typical C shell conditions using the CF88 ¹²C+¹²C rate, it was seen that the neutron channel uncertainty causes a propagation effect in the s-process distribution, peaked at the s-only isotopes ^{80, 82}Kr, with an increase of more than a factor of two. On the other hand, in the last convective C shell of the M25CF88t10 model, uncertainty in the neutron channel only becomes important in the final phase, associated with a temperature and density rise, when the higher neutron density (being driven mostly by ¹²C(¹²C, n)²³Mg) increases the abundance of the r-only isotopes (e.g., ⁷⁰Zn and ⁷⁶Se) by 20%–30%.

We have studied the impact of the ¹²C+¹²C rate uncertainty on the p-process yields, using standard p-process trajectories from SN explosion (Rapp et al. 2006). In particular, with the exception of the M25CU yields, the p-process calculations show comparable results. The cs-process component feeds the production of extremely abundant p-process yields (cp-process), with average p-only abundance 8.3 times higher than the standard p-process, and also reproducing the Mo and Ru p-only abundances. GCE calculations based on stellar models of different masses and using updated SN explosions would be required to study the impact of the cp-process on the chemical inventory of the p nuclei.

Finally, we have explored the possible impact of the ¹²C+¹²C rate for non-rotating and fast-rotating massive stars with low metallicity ([Fe/H] = −3). It was found that at the considered metallicity the cs-process is secondary, despite the primary neutron source ¹³C(α, n)¹⁶O and mainly primary neutron poisons. The possible existence of the of the cs-component does not modify previous conclusions regarding the s-process in fast rotating massive stars, due to the primary ²²Ne neutron source. The main effects of the cs-process are a relevant abundance production at the Sr peak, even without rotation, and the production of elements heavier than Sr–Y–Zr, up to the Ba peak and eventually up to Pb, in fast rotating massive stars. However, an efficient production of lead would occur for concentrations of primary ²²Ne in the He core that are a factor of two higher than have previously been considered (see for more details Pignatari et al. 2008). Considering the cs-process contribution or, in other words, the ¹²C+¹²C rate uncertainty, a maximum ratio of Pb/Ba ∼ 3 was estimated for the s-process yields. The production of more Pb compared with other lighter elements is limited by the amount of primary ²²Ne. If there is too much of it available in the He core, then its neutron poison efficiency reduces the s-process beyond iron, or alternatively a ²²Ne(α, n)²⁵Mg rate higher than the present uncertainty would be required.

NuGrid acknowledges significant support from NSF grants PHY 02-16783 and PHY 09-22648 (Joint Institute for Nuclear Astrophysics, JINA) and EU MIRG-CT-2006-046520. The continued work on codes and in disseminating data is made possible through funding from STFC (R.H., UK), an NSERC Discovery grant (F.H., Canada), and an Ambizione grant of the SNSF (M.P., Switzerland). NuGrid computations were performed at the Arizona Advanced Computing Center (USA), the high-performance computer KHAOS at EPSAM Institute at Keele University (UK) as well as CFI (Canada) funded computing resources at the Department of Physics and Astronomy at the University of Victoria and through Computing Time Resource Allocation through the Compute Canada WestGrid consortium. M.P. also thanks the support from Core project Eurogenesis.