IMPACT OF A REVISED ²⁵Mg(p, γ)²⁶Al REACTION RATE ON THE OPERATION OF THE Mg–Al CYCLE

The strength of a resonance is defined in terms of nuclear resonance parameters:

$$\begin{equation} \omega \gamma =\frac{2J+1}{(2j_1+1)(2j_2+1)}\frac{\Gamma _a\Gamma _b}{\Gamma } \end{equation} \tag{ 1 }$$

with J, j₁, and j₂ being the spins of resonance, projectile, and target nucleus, respectively, and Γ_a, Γ_b, and Γ being the partial widths for the entrance and exit channels and the total resonance width at the resonance energy, respectively. The resonance strength for narrow resonances as in the case of ²⁵Mg(p, γ)²⁶Al can be measured directly in the thick-target yield approximation (see Rolfs & Rodney 1988, for details). Alternatively, the resonance parameters, e.g., the proton partial width Γ_p of the entrance channel, can be obtained from indirect experiments (see below).

The determination of weak low-energy resonance strengths from direct measurements is usually extremely difficult. Small target contaminations, e.g., oxygen, as well as stoichiometry changes under heavy proton bombardment may have a large effect on the absolute determination. A measurement relative to a well-known resonance can often avoid such difficulties. In Strieder et al. (2012), the low-energy resonances have been normalized to the 304 keV resonance which, in turn, was precisely measured with several different experimental techniques (Limata et al. 2010). The resonance strength values used for the present reaction rate calculation are summarized in Table 1 and compared to NACRE (Angulo et al. 1999) and a more recent compilation by Iliadis et al. (2010b). Additionally, the ground state feeding probability and the electron screening correction for directly measured ωγ values are listed.

The NACRE (Angulo et al. 1999) rate at low temperatures (resonances below 130 keV) is mainly based on a reanalysis (Iliadis et al. 1996) of proton partial width values from older proton-stripping data (Betts et al. 1978; Champagne et al. 1989; Rollefson et al. 1990). The same source of information was used in Iliadis et al. (2010b).

The proton width of the 92 keV resonance calculated from the recent direct experiment, Γ_p = (5.6 ± 1.1) × 10⁻¹⁰ eV (Strieder et al. 2012), deviates from the value used in the compilations, Γ_p = (2.8 ± 1.1) × 10⁻¹⁰ eV (Iliadis et al. 2010a), by 1.8σ. Therefore, at the 90% confidence level the two values are incompatible, while the proton width of Strieder et al. (2012) is in good agreement with the original value of Rollefson et al. (1990), Γ_p = (5.2 ± 1.3) × 10⁻¹⁰ eV. In contrast to the 92 keV resonance where a large spread of the indirect data is obvious (see Table II in Iliadis et al. 1996), the proton width data for the 37 and 57 keV resonances from different experiments are in much better agreement and we used the value quoted in Iliadis et al. (2010a) for the present work. As a general rule we have used data from direct experiments whenever available and the results from indirect measurements were included only where no direct data exist.

The ²⁵Mg(p, γ)²⁶Al resonances decay through complex γ-ray cascades either to the 5⁺ ground state or the 0⁺ isomeric state at E_x = 228 keV. The ground state feeding is of particular relevance for astronomy since the ²⁶Al ground state decays into the first excited state of ²⁶Mg with subsequent γ-ray emission observed by satellite telescopes. The isomeric state of ²⁶Al decays (T_1/2 = 6.3 s) exclusively to the ground state of ²⁶Mg and does not lead to the emission of γ-rays. Therefore, a precise determination of the ground state feeding probability f₀ is important for the reaction rate calculation.

For the 189 and 304 keV resonances, this parameter could be reinvestigated experimentally in a high-resolution study using a high-purity germanium detector (Limata et al. 2010; Strieder et al. 2012). A high-precision determination for the low-energy resonances was impossible and the ground state feeding probabilities for these resonances rely mainly on previous literature information. The main source of information on the feeding probability is Endt & Rolfs (1987), which is to a large extent based on the experimental work published in Endt et al. (1988). For resonances at 37 and 57 keV the feeding probability seems to be well grounded while for the 92 keV resonance there is no experimental information from Endt et al. (1988). Unfortunately, the alternative literature information in the case of the 92 keV resonance is contradictory. A probability of 80% ± 15% was deduced from the experimental branching ratio determination measured in the ²⁴Mg(³He,pγ)²⁶Al reaction (Champagne et al. 1983a, 1983b). However, in Champagne et al. (1986), the same authors quote a value of 61%, while the compilation of Endt & Rolfs (1987) gives 85%. The origin of this large discrepancy is unknown, but may be attributed to different assumptions on the secondary branching ratios. Recent measurements (Strieder et al. 2012) suggested a stronger feeding of ²⁶Al states that predominately decay to the isomeric state reducing the ground state fraction. Therefore, a ground state feeding probability of 60⁺²⁰₋₁₀%, as recommended by Strieder et al. (2012), has been used in the present work for the 92 keV resonance. In general, the small uncertainty, e.g., 1%, quoted in Endt & Rolfs (1987) seems questionable due to the disagreement for certain resonances and a larger uncertainty has been assigned to these values (see Table 1).

In astrophysical environments, nuclear reactions usually take place at energies far below the Coulomb barrier where the probability for the incoming particle to overcome the repulsive force of the interacting partner decreases steeply with decreasing energy (Rolfs & Rodney 1988). In laboratory studies, the target nuclei are in most cases in the form of atoms or molecules while projectiles are usually in the form of positively charged ions. The atomic (or molecular) electron clouds surrounding the reacting nuclei act as a screening potential reducing the Coulomb barrier effectively seen by the penetrating particles. Thus, the penetration through a shielded Coulomb barrier at a given projectile energy E is equivalent to that of bare nuclei at energy E_eff = E + U_e. This so-called electron screening effect (Assenbaum et al. 1987) becomes very important for large nuclear charges at low energies.

In general, a resonance strength ωγ is proportional to the penetration probability through the Coulomb barrier, the penetrability P_l(E) of the orbital angular momentum l: ωγ∝Γ_p∝P_l(E). Thus, the enhancement factor f_es of the entrance channel can be expressed as

$$\begin{equation} {f_{{\rm es}}}=\frac{\omega \gamma _{\rm screen}}{\omega \gamma _{\rm bare}}=\frac{{P}_l({E+U_e})}{{P}_l({E})}, \end{equation} \tag{ 2 }$$

and for small l the approximation f_es ≈ exp (πηU_e/E) is valid (Assenbaum et al. 1987) where η is the Sommerfeld parameter (Rolfs & Rodney 1988). The screening potential U_e is usually calculated in the approximation that the projectile velocity is much smaller than the Bohr velocity of the electrons (Shoppa et al. 1993). This approximation represents the so-called adiabatic limit where the electrons remain in the lowest energy state of the combined projectile and target system with the same quantum numbers as the original system. Consequently, the screening potential is given by the difference in atomic binding energy between the original system and the single positively charged combined system.

The atomic binding energies can be found in the literature, e.g., Huang et al. (1976). In the case of ²⁵Mg(p, γ)²⁶Al in the adiabatic limit, a value of U_e = 1.14 keV was calculated leading to enhancement factors f_es quoted in Table 1. However, in most cases, experimental investigations of the electron screening potential resulted in larger values compared to the adiabatic limit (see, e.g., Strieder et al. 2001). This discrepancy is still far from being solved and certainly deserves further studies. It is worth noting that alternative approaches have been discussed in the literature (Liolios 2001, 2003) which lead to slightly different values for the screening potential. In order to account for this ambiguity in the theoretical calculation of the electron screening potential, we assign an uncertainty to the adiabatic limit enhancement factor equal to 30% of the difference between its value and unity.

Note that the electron screening effect is already sizeable for the 304 keV resonance but has been totally neglected in previous compilations, e.g., Angulo et al. (1999) and Iliadis et al. (2010b), when low-energy resonance parameters from direct studies were used.

Since the 1980s, the observations of the 1.809 MeV γ-ray line emitted in star-forming regions of the Milky Way have raised interesting questions about the origin of the galactic pollution of ²⁶Al (Mahoney et al. 1984; Diehl et al. 1995, 2006). Although it is commonly accepted that massive stars, i.e., those ending their life as core-collapse supernovae, are the main source of the galactic ²⁶Al, the precise nucleosynthesis scenario is still matter of debate. Favorable conditions are expected during the advanced phases of the evolution of such massive stars. In particular, a significant contribution should come from the pre-explosive as well as the explosive nucleosynthesis occurring in the C- and Ne-burning shells (Arnett & Wefel 1978; Woosley & Weaver 1980). Nevertheless, extant theoretical models show that an additional contribution may come from Wolf–Rayet (WR) stars (Dearborn & Blake 1985). In this case, the ²⁶Al is produced within the core of very massive main-sequence stars (M > 30 M_☉), where the temperature exceeds 50 MK. Since the main-sequence phase, these stars experience a huge mass loss. In such a way, even material located on the top of the H-convective core, which is enriched with the ashes of the Mg–Al cycle, may be ejected. The actual contribution of the WR stars to the galactic budget of ²⁶Al is rather controversial. While Palacios et al. (2005) find that these stars provide between 20% and 50% of all of the galactic ²⁶Al, Limongi & Chieffi (2006) conclude that the cumulative yield of WRs is negligible when compared to that from C- and Ne-burning shells.

Figure 4 illustrates the nucleosynthesis scenario for the core H-burning phase of a WR precursor. The burning timescales of ²⁴Mg, ²⁵Mg, and ²⁶Al^gs are reported as a function of the temperature and are defined as

$$\begin{equation} \tau _{i}=\frac{1}{X \rho N_A \langle \sigma v\rangle _i}, \end{equation} \tag{ 5 }$$

where i denotes ²⁴Mg, ²⁵Mg, and ²⁶Al^gs with the corresponding reaction rates for ²⁴Mg(p, γ)²⁵Al, ²⁵Mg(p, γ)²⁶Al^tot, and ²⁶Al^gs(p, γ)²⁷Si, respectively. We have assumed a hydrogen mass fraction of X = 0.1 and a density of ρ = 100 g cm⁻³. All the reaction rates are from the NACRE compilation except for ²⁵Mg(p, γ)²⁶Al where we have used both the present work and the Iliadis et al. (2010b) rates. The thick solid line represents the residual time for 80 M_☉ stellar models (from Limongi & Chieffi 2006), i.e., the fraction of the main-sequence lifetime during which the central temperature is larger than a given value.

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During most of the main-sequence lifetime, the ²⁵Mg(p, γ)²⁶Al is the fastest process of the Mg–Al cycle. As previously found by Limongi & Chieffi (see also Iliadis et al. 2011), the corresponding ²⁵Mg-burning timescale is sufficiently short to ensure that all the ²⁵Mg available in the convective core is converted into ²⁶Al. Note that only at the end of the main sequence, when the central H is close to complete exhaustion and the temperature is about 80 MK, the burning rate of ²⁴Mg becomes as short as that of ²⁵Mg. As a result, the burning of ²⁴Mg provides a negligible contribution to the ²⁵Mg abundance in the core, and, in turn, to the ²⁶Al production. The ²⁶Al^gs accumulated in the convective core is marginally depleted by the subsequent proton captures because its burning timescale is about two orders of magnitude larger than that of the ²⁵Mg. Finally, since the ²⁶Al^gs lifetime is comparable to the stellar lifetime, its radioactive decay must be considered.

In summary, the amount of ²⁶Al accumulated in the convective core of a massive star depends, essentially, on the original ²⁵Mg content and on the branching ratio between the two output channels of the ²⁵Mg(p, γ)²⁶Al reaction. Indeed, due to the competition between ²⁵Mg(p, γ)²⁶Al^gs and ²⁵Mg(p, γ)²⁶Al^m, only a fraction of the original ²⁵Mg is actually converted into ²⁶Al^gs. A comparison between the previous (Iliadis et al. 2010b) and the revised branching ratios shows that at temperatures of core H-burning, the new rates imply a substantial increase of the competitive channel, i.e., the isomeric state production, than previously assumed (Figure 2). As a consequence, the ²⁶Al^gs production in the convective core of H-burning massive stars is less efficient than was believed. Note that this finding does not necessarily imply that the contribution of WR stars to the galactic ²⁶Al is negligible. A reliable evaluation of this contribution still resides, for example, on the poorly known mass range of these stars, which is significantly affected by mass-loss uncertainties.

The chemical analysis of presolar grains, dust particles found in pristine meteorites with a size smaller than a few microns, reveals a variety of isotopic compositions. These presolar grains, e.g., mainstream SiC and O-rich grains (see Hoppe & Zinner 2000; Clayton & Nittler 2004, for a review), represent fossil records of the parent star atmospheres and provide unique information on stellar nucleosynthesis.

Mainstream SiC grains are believed to condense in the C-rich atmospheres surrounding low-mass (M < 3 M_☉) AGB stars of different metallicity (0.001 < Z < 0.04), which are very active nucleosynthesis sites (Iben & Renzini 1983; Busso et al. 1999; Straniero et al. 2006). Recursive dredge-up episodes powered by thermal pulses are responsible for the C enrichment of the atmosphere of these giant stars and SiC grains form when the C/O ratio becomes larger than 1, the so-called C-star phase. O-rich grains may also condense in AGB stars, but before the C-star phase is attained and C/O is still less than 1.

The ²⁶Mg excess observed in SiC as well as O-rich grains from AGB stars is interpreted as the signature of an in situ decay of ²⁶Al (Zinner et al. 1991; Nittler et al. 1994) and current theoretical models predict that low-mass AGB stars may deliver a substantial amount of ²⁶Al. The ²⁶Al is produced in the H-burning shell of an AGB star, accumulated in the H-exhausted region, and mixed via convection powered by thermal pulses to regions of higher temperatures. In the case where the maximum temperature remains below the threshold for the activation of the ²²Ne(α,n)²⁵Mg reaction (T < 300 MK), the ²⁶Al survives and, later on, may be dredged up to the stellar surface. By contrast, the ²⁶Al is destroyed by neutron captures occurring at the bottom of the convective zone and only ²⁶Al above this zone can be dredged up (Mowlavi & Meynet 2000; Cristallo et al. 2009). Based on full network stellar model calculations, Cristallo et al. (2011) found values of ²⁶Al/²⁷Al up to 5 × 10⁻³, in good agreement with those measured in mainstream SiC grains and several O-rich grains. However, in some O-rich grain values larger by up to one order of magnitude have been observed.

These extreme excesses of ²⁶Al are often explained by invoking an AGB extra-mixing which connects the bottom of the convective envelope to the hottest H-burning zone, where the Mg–Al cycle is at work. Note that the extra-mixing scenario provides a widely accepted explanation of the C and O isotopic ratios measured in the atmospheres of low-mass RGB stars (Boothroyd et al. 1994; Charbonnel 1995; Denissenkov & Weiss 1996) and O isotopic ratios found in a large sample of presolar grains suggest that extra-mixing should also be at work during the AGB phase (Nollett et al. 2003). Nevertheless, a reliable mechanism for such an extra-mixing has not yet been identified; possible candidates are rotational-induced instabilities, magnetic pipes, gravity waves, and thermohaline mixing.

The AGB extra-mixing hypothesis implies that parent stars of O-rich grains with large ²⁶Al excess never attain the C-star stage because otherwise one should also expect SiC grains showing similarly large values of ²⁶Al overabundance: this occurrence is considered a major drawback of the proposed scenario. However, as pointed out by Straniero et al. (2003), there is a lower limit for the mass of AGB stars with C/O > 1 and the larger the metallicity, the larger the minimum mass required, e.g., the C-star minimum mass is about 1.5 M_☉ at solar metallicity, while it is only 1.3 M_☉ at Z = 0.003. An extra-mixing in the AGB phase would increase the required minimum C-star mass because the dredged-up carbon is partially converted into nitrogen (by the CN cycle), so that the onset of the C-star stage is delayed. Thus, a very deep extra-mixing could prevent an AGB star from becoming a C-star and, at the same time, would allow for the development of high excesses of ²⁶Al.

In a recent work, Palmerini et al. (2011) showed that the O-rich grains with extreme ²⁶Mg excess can be explained by AGB stellar models with particularly deep extra-mixing, provided that (1) the initial mass is lower than 1.5 M_☉ and (2) the ²⁵Mg(p, γ)²⁶Al^gs reaction rate is enhanced by a factor of five with respect to the Iliadis et al. (2010b) rates.

The coupling between nuclear burning and mixing makes a quantitative analysis of the impact of the new rates on the isotopic composition of O- and C-rich presolar grains difficult and would require the computation of stellar models with an extended nuclear network. This effort is beyond the purpose of the present work, but some qualitative consideration may be drawn on the basis of bare network calculations. According to Palmerini et al. (2011), the maximum temperature attained by the extra-mixing is between 40 and 50 MK, corresponding to an energy range where the ²⁵Mg proton capture rate is dominated by the 57 keV resonance (see Figure 1). In Figure 5, we report the evolution of the ²⁶Al/²⁷Al ratio for material exposed to a constant temperature of 40 MK (lower panel) and 50 MK (upper panel), respectively. In this energy range the recommended new rate for the ²⁵Mg(p, γ)²⁶Al^gs is only about 10% larger with respect to Iliadis et al. (2010b), while the competing channel, ²⁵Mg(p, γ)²⁶Al^m, is about 40% larger. As a consequence, the resulting ²⁶Al/²⁷Al isotopic ratio at T = 50 MK is even lower than previously found, although the total rate is larger. Moreover, in spite of the large uncertainty of the dominant 57 keV resonance contribution, we can definitely exclude an increase of a factor of five for the ²⁵Mg(p, γ)²⁶Al^gs rate.

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In conclusion, AGB models without extra-mixing may account for ²⁶Al/²⁷Al up to 5× 10⁻³, values commonly found in mainstream SiC grains as well as in many O-rich grains from AGB stars. Larger values of ²⁶Al/²⁷Al may in part be explained by a deep extra-mixing (T ⩾ 50 MK), but even in the upper limit of the ²⁵Mg(p, γ)²⁶Al^gs rate, it is unlikely that the extra-mixing scenario could produce aluminum isotopic ratios with ²⁶Al/²⁷Al > 10⁻².

For many years, globular clusters (GCs) have been considered as simple stellar systems, made of nearly coeval stars and formed from a chemically homogeneous preexisting gas nebula. Nevertheless, a growing amount of photometric and spectroscopic observations indicates that many GCs actually harbor multiple stellar populations characterized by star-to-star chemical variations. Such chemical variations include the well-known O–Na and Mg–Al anticorrelations which are usually coupled to a nearly constant value of C+N+O. This chemical pattern is a characteristic signature for H burning, where the Ne–Na and the Mg–Al cycles are active. The first evidence of these "anomalies" was found in bright red giant stars (Kraft et al. 1997; Ivans et al. 1999). As it is well known, RGB stars have an extended convective envelope, but the innermost unstable layer does not reach the H-burning zone. Therefore, an extra-mixing was initially invoked to explain the observed anticorrelations. Nonetheless, this hypothesis is in contrast with the more recent discovery of O–Na and the Mg–Al anticorrelations in less evolved turnoff and sub-giant stars (Gratton et al. 2001; Yong et al. 2003). These observations definitely rule out the hypothesis that the anticorrelations are the result of an in situ physical process and prove that they were already present in the gas nebula from which these stars formed about 13 Gyr ago. Among the proposed alternative hypotheses, the pollution of the primordial gas by an early generation of massive AGB stars (perhaps super-AGB) appears promising (Cottrell & Da Costa 1981; Dantona et al. 1983; Ventura & D'Antona 2005). In these massive AGB stars, the convective envelope penetrates the regions where the H burning takes place: this phenomenon is usually called hot bottom burning (Renzini & Voli 1981). Then, the relatively low velocity wind of these stars ensures the required pollution of the intra-cluster medium. According to this scenario, stars with low Mg and high Al (or low O and high Na) would represent a second generation of cluster stars, formed after the intermediate-mass stars of the first generation passed through the AGB phase and polluted the intra-cluster gas with ashes of H burning.

However, the attempts made so far to simultaneously reproduce the observed O–Na and Mg–Al anticorrelations have produced controversial results (Fenner et al. 2004; Ventura & D'Antona 2005). Recently, Ventura et al. (2011) showed that an increase of the ²⁵Mg(p, γ)²⁶Al reaction rate by a factor of two, coupled with a more sophisticated treatment of the convective energy transport under super-adiabatic conditions, may reduce the discrepancy between the theoretical expectations and the observed cluster abundances of Mg and Al.

In order to illustrate the influence of the newly recommended reaction rates on the Mg–Al cycle operating in the H-burning shell of a massive AGB star, we have performed some bare network calculations. Values for the metallicity, the H mass fraction, the temperature, and the density representative of the innermost layers of the convective envelope of a massive AGB star have been selected, namely, Z = 0.001, X = 0.6, T = 100 MK, and ρ = 10 g cm⁻³. The result is shown in Figure 6, where the upper panel refers to the calculation obtained by means of the new reaction rates for the ²⁵Mg(p, γ)²⁶Al, while the bottom panel corresponds to the calculations obtained by adopting the Iliadis et al. (2010b) rates. The NACRE compilation has been used for all of the other reactions of the Mg–Al cycle, while for the ²⁶Al^gs decay rate we have assumed the terrestrial value (λ = 2.97 × 10⁻¹⁴ s⁻¹). The increase of the new total ²⁵Mg(p, γ)²⁶Al reaction rate by a factor of two with respect to Iliadis et al. (2010b) is indeed very close to the value found by Ventura et al. (2011) and supports the massive AGB self-pollution scenario. It should be noted that the largest variations in the evolution of the Mg and Al isotopic abundances are caused by the larger ²⁵Mg(p, γ)²⁶Al^m rate. This variation favors a prompt destruction of ²⁵Mg and a fast increase of the ²⁷Al production. In Figure 7, the Al (elemental) abundance is compared with the corresponding Mg abundance, for the T = 100 MK calculations. The new rate implies a steeper anticorrelation and a significant increase of the maximum Al abundance. Note the similarity of this figure to Figure 4 of Ventura et al. (2011) based on the result of AGB stellar models obtained under different assumptions for the ²⁵Mg(p, γ)²⁶Al rate.

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