The Importance of the ¹³C(α,n)¹⁶O Reaction in Asymptotic Giant Branch Stars

As it is widely accepted, the main and strong components of the s-process (nuclei heavier than A ∼ 90) found in the solar system material have been produced by relatively low-mass asymptotic giant branch (AGB) stars (1.2 < M/M_⊙ ≤ 4.0), already extinct before the birth of the Sun. Their structures consist of three layers: a degenerate C-O core, a thin He-rich mantel (named He intershell), and a loose and largely convective H-rich envelope. They undergo recurrent He-shell flashes, called thermal pulses (TPs), separated by relatively long interpulse periods, during which a quiescent shell-H burning balances the energy lost by radiation from the stellar surface. In those objects, the most important neutron source is the ¹³C(α,n)¹⁶O reaction, which is active in the He intershell. According to the current paradigm, a ¹³C pocket forms at the beginning of each interpulse period in a small layer characterized by a variable H-abundance profile. Then, as this region contracts and warms up to ≈90–100 MK, ¹³C starts capturing α particles and, as a consequence, releases neutrons. A second neutron burst, more rapid than the first but only marginally contributing to the s-process nucleosynthesis, occurs during a TP, when the maximum temperature in the convective zone powered by the He flash exceeds 300 MK. In this case, neutrons are released through the ²²Ne(α,n)²⁵Mg reaction. This neutron source dominates the s-process nucleosynthesis of more massive AGB stars (4.0 < M/M_⊙ < 9.0).

Starting in the early 1980s, different physical processes were proposed as responsible for the synthesis of the ¹³C needed to reproduce observations. In particular, a thin transition zone containing a small amount of protons (only 10⁻⁶ M_⊙ of H) is needed between the He-rich mantel and the H-rich envelope at its deepest penetration during a third dredge up (TDU) episode. Then, at H reignition, a ¹³C pocket rapidly forms. Such a ¹³C, which was originally thought to be engulfed in the convective shell triggered by the following TP (see, e.g., Iben & Renzini 1983), burns in radiative conditions during the long interpulse phase (Straniero et al. 1995). As a matter of fact, the radiative s-process timescale is quite long (a few 10⁴ yr), and low neutron densities (namely n_n < 10⁸ cm⁻³) are attained. Note that in spite of the low neutron density, the long timescale ensures a high enough neutron exposure to synthesize substantial s-process nuclei belonging to the main component. In the late 1990s, Gallino et al. (1998) carried out a large number of nucleosynthesis post-process calculations. However, the mass and profile of ¹³C within the pocket were treated as free parameters. In particular, the standard case, corresponding to about 4 × 10⁻⁶ M_⊙ of ¹³C, provided the best reproduction of the main s-process component in the solar system. Subsequent papers confirmed the above described scheme (e.g., Goriely & Mowlavi 2000; Lugaro et al. 2003).

The question of what physical mechanism leads to the formation of the ¹³C pocket is still open. In the last 20 yr, several hypotheses have been advanced. Herwig et al. (1997) (and also Herwig 2000) first proposed a convective overshoot operating during the TDU. Based on prescriptions derived from 2D hydrodynamical calculations of stellar convection (Freytag et al. 1996), they assumed that the convective mixing velocity decreases exponentially below the convective border. In this way, they obtained ¹³C pockets with masses of the order of (2–4) × 10⁻⁷ M_⊙ (thus, 10–20 times smaller than the standard case defined by Gallino et al. 1998). Later, Langer et al. (1999) investigated the possibility of rotational induced mixing, while Denissenkov & Tout (2003) analyzed the effect of a weak turbulence induced by gravity waves (see also Battino et al. 2016). It should be noted that all these models adopt a diffusion scheme to treat the mixing of protons into the He-rich and C-rich zone, independently on the engine of this mixing.

Straniero et al. (2006; see also Cristallo et al. 2009a) proposed a different algorithm, in which the degree of mixing scales linearly with the mixing velocity instead of quadratically as in the diffusion scheme. Then, by adopting an exponential decrease of the convective velocity, they showed that it is possible to produce ¹³C pockets larger than that obtained in previous studies and able to provide the required production of main-component s-process isotopes. Such a velocity profile drops as

$$\begin{eqnarray}&&v={v}_{\mathrm{CE}}\exp \left(-\displaystyle \frac{{\rm{\Delta }}r}{\beta {H}_{P}}\right),\end{eqnarray} \tag{ 1 }$$

where Δr is the distance from the convective boundary defined by the Schwarzschild criterion, v_CE is the velocity at the formal convective boundary, H_P is the pressure scale height, and β is a free parameter (calibrated to β = 0.1; see Cristallo et al. 2011). We refer to this mixing scheme as exponential velocity profile (EVEP) mixing. Later, Piersanti et al. (2013) investigated the effects induced by rotation on the evolution of the ¹³C pocket and the related s-process nucleosynthesis. Note that the adoption of the EVEP scheme facilitates the penetration of the convective envelope during TDU episodes. As a consequence, the model experiences TDUs when the mass of its H-exhausted core is lower (with respect to models without EVEP mixing). Then, models may become C-rich (C/O > 1) at lower surface luminosities. Guandalini & Cristallo (2013) demonstrated that such a result well fits with the observational luminosity function of C stars.

More recently, Nucci & Busso (2014) advanced a new hypothesis about the formation of the ¹³C pocket (see also Trippella et al. 2016; Palmerini et al. 2018), suggesting that dynamo-produced buoyancy of magnetized materials could provide the necessary physical mechanisms to transport protons from the H-rich envelope into the He-C rich mantel. Magnetic instabilities also supply a sufficient transport rate capable of explaining the formation of a quite large zone with low ¹³C concentration. In this scenario, the original poloidal field of a rotating star generates a toroidal field of similar strength developing various instabilities (Parker 1960; Spruit 1999), among which is the buoyancy of magnetized structures (Schuessler 1977). The dynamics of buoyant magnetized domains is strongly dependent on the physics of the stellar environment, and it is in general very complex. However, Nucci & Busso (2014) showed that below the convective envelopes of AGB stars, special conditions are held, and the fully MHD equations can be solved exactly. Then, a simple formula for the radial component of the buoyancy velocity can be obtained, and it describes a fast transport mechanism (see the original paper for details).

Besides the difficulties related to the adopted physical recipe, models also deal with the uncertainties related to many input quantities, among which are nuclear reaction rates. The one affecting the main neutron source, i.e., ¹³C(α,n)¹⁶O, plays a relevant role. Cristallo et al. (2009a, 2011) found that in some low-mass AGB models, ¹³C may not be fully consumed during the interpulse. When this happens, the residual ¹³C is engulfed into the convective zone powered by the incoming TP and burns at a higher temperature. This additional neutron burst affects the compositions of isotopes in the neighborhood of critical branchings. Those authors showed that this process provides an excess of some radioactive nuclei, such as ⁶⁰Fe, which has been proved to be alive in the early solar system (Mostefaoui et al. 2005; Tang & Dauphas 2012). Although such an occurrence is limited to the early TP-AGB phase of low-mass, high-metallicity models, a variation of the ¹³C(α,n)¹⁶O reaction rate may enhance or suppress such a process. The ¹³C(α,n)¹⁶O reaction may also play an important role during proton ingestion episodes (PIEs). These peculiar events, possibly occurring in the early TP-AGB phase of low-mass, low-metallicity stars, are characterized by proton engulfment in the convective shell triggered by the first fully developed TP. As a consequence of a PIE, on-flight H-burning occurs, with important consequences for the energetic stellar budget and the following (rich) s-process nucleosynthesis (Cristallo et al. 2009b). During this event, the energy provided by the ¹³C(α,n)¹⁶O reaction (plus the additional contribution from the relative neutron capture) plays a key role. Note that a similar nucleosynthesis may develop during a (very) late He-shell flash or an AGB final TP (PG 1159 spectral class or Sakurai's objects; see, e.g., Werner & Herwig 2007; Herwig et al. 2011).

In the case of radiative ¹³C burning, the Gamow peak energy of the ¹³C(α,n)¹⁶O reaction for the relevant AGB temperature is about 200 keV, which is well below the lower limit so far reached by direct measurements. In Section 2, we report the state of the art relative to its low-energy cross section that serves as input for a sensitivity study of the ¹³C(α,n)¹⁶O reaction and the related s-process nucleosynthesis in low-mass AGB stars. This is presented in Section 3, where two different hypotheses about the formation of the ¹³C pocket, namely the EVEP mixing and magnetic mixing, have been taken under consideration. In Section 4, we describe planned experiments that strive to improve our current knowledge. Finally, our conclusions are presented in Section 5.

Owing to its astrophysical importance, the ¹³C(α,n)¹⁶O S-factor¹² has been the subject of many studies aiming at the direct determination of its cross section or focusing on specific ¹⁷O states. A schematic diagram of the level scheme of ¹⁷O is shown in Figure 1. The ${}^{13}{\rm{C}}+\alpha$ entrance channel and the two competing exit channels ¹⁶O + n and ¹⁷O + γ are also represented by black arrows. The ${}^{17}{\rm{O}}$ levels near and above the α-threshold of interest for AGB nucleosynthesis are marked in red.

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Focusing on direct measurements, the most recent work by Heil et al. (2008) pointed out the substantial scatter of existing data, showing a broad (up to a factor of 2) range of absolute values for the astrophysical S-factor. On the other hand, the trend of the astrophysical S-factor as a function of energy is consistent among different data sets (Davids 1968; Bair & Haas 1973; Kellogg et al. 1989; Drotleff et al. 1993; Harissopulos et al. 2005). The lowest-energy data point sits at a center-of-mass energy of about 280 keV (Drotleff et al. 1993), slightly above the Gamow window for α-induced ¹³C burning in radiative conditions. Therefore, extrapolation has proved necessary to supply a value of the reaction rate at the temperatures of astrophysical interest. The understanding of the low-energy behavior of the astrophysical S-factor is complicated by the interplay between the rise in the S-factor due to the excited state of ¹⁷O at E_x = 6.356 MeV (ENDFS 2017) or E_x = 6.363 MeV (Faestermann et al. 2015) with spin parity J^π = 1/2⁺ (see Figure 1) and the enhancement produced by the electron screening effect (Bracci et al. 1990). Moreover, such measurements are extremely challenging, since at ∼300 keV, the cross section is already as low as ∼10⁻¹⁰ b, and the neutron detection efficiency of about 30% further reduces the signal-to-noise ratio.

Therefore, indirect measurements turned out to be very useful to constrain the ¹⁷O 6.356 MeV level contribution. These were essentially spectroscopic measurements of the resonance energy (Faestermann et al. 2015), its squared Coulomb-modified asymptotic normalization coefficient (ANC; Johnson et al. 2006; Avila et al. 2015), and the corresponding spectroscopic factor (Keeley et al. 2003; Kubono et al. 2003; Pellegriti et al. 2008; Guo et al. 2012; Mezhevych et al. 2017), which were used to calculate the low-energy astrophysical S-factor and the ${}^{13}{\rm{C}}(\alpha$,n)¹⁶O reaction rate. Concerning these last measurements, aside from two conflicting cases (Kubono et al. 2003; Johnson et al. 2006), very different experiments and analyses supplied compatible values of ANCs, suggesting a minor contribution of systematic errors at odds with the present status of direct measurements.

2.1. Exploring the Threshold Region with the Trojan Horse Method

Previous papers devoted to the analysis of the ¹³C(α,n)¹⁶O reaction mainly focused on the effects induced by the adoption of rates proposed by different authors. On the other hand, a sensitivity study is well suited to evaluate the expected variations on the s-process, because it allows the determination of relative distributions and, thus, the corresponding derivatives with respect to variations of the rate itself. Being the most recent direct measurement, we assume as a reference the rate proposed by Heil et al. (2008), and we vary it uniformly over the whole energy range by a factor of 1.5. In such a way, we cover most of the rates presented in the literature so far. The two most extreme cases, i.e., those proposed by Kubono et al. (2003) and Caughlan & Fowler (1988), are taken into account by varying the reference rate by a factor of 2.

As already highlighted in Section 1, the identification of the physical mechanism leading to the formation of the main ¹³C reservoir in AGB stars (the so-called ¹³C pocket) is a scientific issue debated since the end of the 1980s. No consensus has been reached on this topic to date, with different processes proposed so far: opacity EVEP mixing (Straniero et al. 2006; Cristallo et al. 2009a), magnetic buoyancy (Nucci & Busso 2014; Trippella et al. 2016; Palmerini et al. 2018), and Kelvin–Helmholtz instability coupled to gravity waves (Battino et al. 2016). These approaches lead to similar (at first glance) s-process distributions; however, their comparison is not the main goal of this paper.

Hereafter, we concentrate on s-process nucleosynthesis variations induced by the adoption of a modified ¹³C(α,n)¹⁶O rate in models with various masses and metallicities, computed with different codes. We analyze two sets of AGB models.

• 1.  
  FUNS¹³ evolutionary models: 1.5 and 3.0 M_⊙ with [Fe/H] = −0.15 (corresponding to Z = 10⁻², slightly lower than the initial solar metallicity Z_⊙ = 1.38 ×10⁻²), 4.0 M_⊙ with [Fe/H] = −2.15 (Z = 2.45 × 10⁻⁴, considering an initial enrichment of α-elements [α/Fe] =0.5), and 1.3 M_⊙ with [Fe/H] = −2.85 (Z = 4.9 ×10⁻⁵, considering an initial enrichment of α-elements [α/Fe] = 0.5).
• 2.  
  NEWTON post-process calculations on a 2.0 M_⊙ with [Fe/H] = −0.15 using AGB stellar structures computed with the FRANEC¹⁴ code (i.e., adopting a pure Schwarzschild criterion for the identification of convective borders). Models of this kind were illustrated by, e.g., Straniero et al. (2003).

3.1. FUNS Models

A detailed description of FUNS models can be found in Cristallo et al. (2016) and references therein. FUNS is derived from the FRANEC code (Chieffi & Straniero 1989). Major improvements with respect to previous versions of the code are the mass-loss law, the use of a full nuclear network (from hydrogen to bismuth) directly coupled to the physical evolution of the model, and the use of an exponentially decaying profile of convective velocities at the inner border of the envelope (see Straniero et al. 2006 for details). As already reported in Section 1, the introduction of such an algorithm has deep consequences for the physical and chemical evolution of the model: it makes the border stable against perturbations, increases the efficiency of TDU episodes, and allows the formation of a tiny ¹³C-rich region (the ¹³C pocket) at the base of the envelope after each TDU. Note that the external region of the ¹³C pocket is ¹⁴N-rich, due to the larger number of available protons at the H-shell reignition. Such an isotope acts as a major neutron poison via the ¹⁴N(n,p)¹⁴C reaction, thus reducing the number of neutrons available for the synthesis of heavy elements.

3.1.1. Solar-like Metallicity

In this section, we discuss the effects induced by a change of the ¹³C(α,n)¹⁶O reaction rate in models of low mass (1.5 and 3.0 M_⊙) and [Fe/H] = −0.15. This metallicity is representative of the environment where pre-solar SiC grains formed. Those μm-sized particles are relics of the nucleosynthesis in the interiors of already extinct AGB stars, whose material polluted the solar system before its formation (see, e.g., Busso et al. 1999). The isotopic anomalies detected in those grains are, in fact, clearly connected to physical conditions not available in the solar system at the epoch of its formation. In the upper panel of Figure 2, we report the heavy-element (Z > 25) surface composition of a star with M = 1.5 M_⊙ and [Fe/H] = −0.15.¹⁵ Symbols refer to the different adopted rates for the ¹³C(α,n)¹⁶O reaction: reference case (REF; Heil et al. 2008; black circles), reference case divided by a factor of 2 (D2; red squares), reference case divided by a factor of 1.5 (D1p5; blue triangles), reference case multiplied by a factor of 1.5 (P1p5; green diamonds), and reference case multiplied by a factor of 2 (P2; cyan stars). The corresponding isotopic surface distributions are plotted in Figure 3. An inspection of the surface elemental composition of the REF model reveals an almost flat overproduction of the three s-process peaks (upper panel of Figure 2), namely the ls component (Sr–Y–Zr), the hs component (Ba–La–Ce–Pr–Nd), and lead. An increase of the ¹³C(α,n)¹⁶O reaction leads to a larger production of hs elements (∼15%) and an even slightly higher synthesis of lead (∼20%).¹⁶ On the contrary, elements lighter than strontium are slightly underproduced. At first glance, this result could appear in contrast to the fact that the ¹³C in the pocket fully burns in radiative conditions between two TPs (Straniero et al. 1995). Actually, this is true for most of the AGB models, but it does not hold for low-mass stars (M < 3.0 M_⊙) at solar-like metallicities ([Fe/H] ≥ −0.15). In fact, Cristallo et al. (2009a) demonstrated that, for those models, part of the ¹³C in the first pockets is engulfed in the convective shell generated by the following TP (see also Karakas et al. 2010). This derives from the fact that the ¹³C in the pocket does not have enough time to fully burn in a radiative environment. Instead, AGB models with larger initial masses have high enough internal temperatures to guarantee a complete radiative ¹³C burning before the onset of the following TP. Such a behavior is easily understood by inspecting the masses of the H-exhausted cores (M_H) at the beginning of the TP-AGB phase. At solar-like metallicities, stars with M < 3 M_⊙ have almost the same M_H, while in more massive stars, M_H linearly grows with the stellar mass (see, e.g., Figure 2 of Cristallo et al. 2015). The larger the M_H, the larger the temperature in the He-intershell region. In addition, the lower the metallicity, the larger the M_H. Therefore, the convective ¹³C burning disappears with increasing the stellar mass and/or decreasing the initial metal content. When some ¹³C is ingested in the TP, it burns at a definitely higher temperature, producing rather large neutron densities ($\sim {10}^{11}$  cm⁻³). During that episode, the production of heavier elements is disfavored by convection, which does not allow isotopes to locally pile up. Moreover, the abundant ¹⁴N in the upper region of the ¹³C pocket acts as a neutron poison (via the ¹⁴N(n,p)¹⁴C reaction), thus further decreasing the number of neutrons available for the nucleosynthesis of elements heavier than iron. Another interesting feature of convective ¹³C burning is that the production of neutron-rich isotopes close to s-process branchings largely increases with respect to standard radiative ¹³C burning. This is the case, for instance, for ⁶⁰Fe (the largest variation, almost a factor 20, is found for this isotope), ⁸⁶Kr, ⁸⁷Rb, and ⁹⁶Zr (see Figure 3). On the contrary, other isotopes are overproduced with a large ¹³C(α,n)¹⁶O rate. For example, the production of ¹⁵²Gd increases by more than 50% when the rate is multiplied by a factor of 2. This is due to the fact that the nucleosynthesis of such an isotope strongly depends on the ¹⁵¹Sm branching, which is open during standard radiative ¹³C burning (thus, ¹⁵²Gd is partly fed by the main s-process flow). On the contrary, this branching is closed at high temperatures, as it occurs during convective ¹³C burning.¹⁷ The increased production of ⁸⁷Rb characterizing the D1p5 and D2 cases leads to the largest element surface variation, which is found for rubidium (+30%). In principle, the observed Rb/Sr ratio in C-rich stars belonging to the Galactic disk could be used to constrain the efficiency of the ¹³C(α,n)¹⁶O reaction. Unfortunately, the mass of those stars is poorly determined; moreover, current observational uncertainties are of the same order of magnitude of the theoretical differences just described. Thus, from this analysis, only extremely low values for the ¹³C(α,n)¹⁶O reaction can be safely discarded. More precise hints on the efficiency of the ¹³C(α,n)¹⁶O reaction can be derived from the analysis of isotopic ratios in pre-solar SiC grains. In Figure 4, we compare FRUITY models with available laboratory measurements of selected key elements¹⁸ (Liu et al. 2014a, 2014b, 2015; Trappitsch et al. 2018). Note that we are evaluating the effects induced by a change of the ¹³C(α,n)¹⁶O rate and not the capability of the model to reproduce SiC data. The largest variations are found for ⁶⁴Ni and ⁹⁶Zr. This is somewhat expected, as both are neutron-rich isotopes whose production depends on an s-process branching (at ⁶³Ni and ⁹⁵Zr, respectively).

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Finally, we highlight that the D2 and D1p5 models show particularly large ⁸⁶Kr/⁸²Kr ratios (∼1) with respect to the REF case (0.8), possibly giving hints on the implantation energy of the ⁸⁶Kr-rich component (see Raut et al. 2013). In Table 1, we report the ⁸⁶Kr/⁸²Kr and ⁹⁶Zr/⁹⁴Zr number ratios of the computed models, together with their percentage variations with respect to the reference case.

Table 1.  ⁸⁶Kr/⁸²Kr and ⁹⁶Zr/⁹⁴Zr Number Ratios of the 1.5 M_⊙ Model with [Fe/H] = −0.15 for Different ¹³C(α,n)¹⁶O Rates, Together with Their Percentage Variations with Respect to the Reference Case

Case	86Kr/82Kr	96Zr/94Zr
D2	1.046 (+34%)	0.999 (+8%)
D1p5	0.966 (+24%)	0.107 (+13%)
REF	0.777	0.093
P1p5	0.669 (−14%)	0.061 (−35%)
P2	0.633 (−19%)	0.041 (−56%)

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In Figures 5–7, we performed the same analysis already described but for a star with initial mass M = 3 M_⊙ and the same metallicity ([Fe/H] = −0.15). The heavy-element distributions are almost indistinguishable, apart from the D2 case, which is characterized by a small (5%) underproduction of the heaviest elements (Z > 50). The same holds for the isotopic composition, where the only noticeable differences are found for ⁶⁰Fe and ¹⁵²Gd (in any case below 10%). The theoretical isotopic δ curves do not evidence any appreciable deviation from the reference case. This behavior confirms that, in more massive AGBs, the temperature of the He-intershell region is always large enough to allow a complete ¹³C burning during the radiative interpulse phase. As highlighted before, this is a consequence of the larger core mass of the 3.0 M_⊙ model at the first TP followed by TDU (M_H ∼ 0.59 M_⊙), with respect to the 1.5 M_⊙ model (M_H ∼ 0.56 M_⊙).

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Straniero et al. (2014) first demonstrated that the s-process-enriched ejecta of an early generation of massive AGBs may be responsible for the s-rich distributions observed in samples of red giant stars belonging to the globular clusters M4 and M22 (see also Shingles et al. 2014). Those authors found that a satisfactory match to observations can be obtained, with some noticeable exceptions. In particular, theoretical models produce too much lead with respect to observations. Massimi et al. (2017) highlighted that a variation of the ²²Ne(α,n)²⁵Mg rate may soften the current disagreement between theory and observations. Here we check if the variation of the ¹³C(α,n)¹⁶O rate may have some effects on lead (mainly ²⁰⁸Pb) production. To that purpose, we calculate an AGB model with initial mass M = 4 M_⊙ and [Fe/H] = −2.15. Objects like this are thought to be the typical polluters for those stellar systems. We find that the model is not sensitive to any variation of the rate (see Figure 8). This result confirms the trend already highlighted for the M = 3 M_⊙ model at higher metallicity (note that for this model, M_H ∼ 0.86 M_⊙).

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The situation is definitely different if protons are mixed within the convective shell generated by a TP. Those types of events may occur at the first fully developed TP of low-mass, low-metallicity stars or during a very late TP near the tip of the AGB (Sakurai's objects). As representative of this class of events, we analyze a low-mass (M = 1.3 M_⊙) model at very low metallicity (e.g., [Fe/H] = −2.85). At this Z, the H-shell entropy barrier is weaker and, as a consequence, some hydrogen may be engulfed in the growing convective shell triggered by a TP. Due to the large temperatures, protons are captured by the abundant ¹²C while they are mixed. On the contrary, the corresponding products (¹³C and, eventually, ¹⁴N), reach the base of the convective shell, where T ∼ 230 MK. Thus, ¹³C is exposed to a temperature well beyond the threshold activation for the ¹³C(α,n)¹⁶O reaction (∼100 MK). As a consequence, an efficient s-process takes place. After some months, the energy deposited by the on-flight burning of protons leads to the splitting of the convective shell. From that moment on, the upper shell is triggered by incomplete CNO burning, while the lower shell is sustained by the 3α and ¹³C(α,n)¹⁶O reactions. Cristallo et al. (2009b) demonstrated that, in order to properly model such a PIE, the physical evolution of the model must be coupled to a full nuclear network, including all chemical species up to lead (see also Cristallo et al. 2016). This derives from the fact that, before the splitting (and the full development of the TP), the ¹³C(α,n)¹⁶O reaction provides a substantial fraction of the local energy budget. This reaction, in fact, directly releases about 2.2 MeV. Furthermore, an additional energy contribution comes from the following neutron capture (5 MeV, on average). It is therefore obvious that any model aiming at following a PIE cannot overlook the proper calculation of the energetics provided by neutron captures. The number of available neutron depends on two factors: the mixing efficiency and the burning efficiency. In this paper, we test the latter by modifying the rate of the ¹³C(α,n)¹⁶O reaction.

However, before discussing our results, some important remarks about mixing have to be highlighted. First, it has to be stressed that the simulation of a three-dimensional hydrodynamic event, such as a PIE, in a one-dimensional hydrostatic code intrinsically implies the adoption of approximations. A key quantity in stellar evolution is the Damköhler number Da = τ_mix/τ_burn between the characteristic timescales of convective mixing (τ_mix) and nuclear burning (τ_burn). In the large majority of steady burning phases of a stellar evolution, Da ≪ 1 inside convective zones. During a PIE, instead, ${Da}\to 1$; i.e., nuclear burning occurs "on the fly" (at least for hydrogen). As a first consequence, the model timestep (Δt) needs to be reduced to follow the in-flight burning. In our models, we limit the timestep to 50% of the mixing turnover timescale of the convective shell (τ_mix), in order to avoid the unrealistic full homogenization of that region. All isotopes are mixed within the convective region, apart from protons, which are mixed down to the mass coordinate where τ_burn = 1/3 · Δt (see Cristallo et al. 2009b for details). At deeper coordinates, Da ≫ 1; i.e., the proton reduction (via nuclear burning) is much more rapid than the supply (via convective mixing). As a consequence, below this point, it is unlikely for protons to survive. Other approaches, however, may be implemented, possibly leading to different results. For instance, Campbell & Lattanzio (2008) followed a diffusive approach in their calculation of PIEs. Diffusion being a very efficient mixing mechanism, the adoption of a diffusion equation in their code leads to an early splitting of the convective region. This has strong consequences for the following s-process nucleosynthesis (strongly hampering it).

Second, we stress that in our code, the rate of mixing is calculated basing on the mixing-length theory (MLT). This is a very crude approximation, since the MLT is a local theory and cannot capture the three-dimensional hydrodynamic nature of a PIE. In particular, the inhomogeneities in the flow and their effect on the three-dimensional distribution of burning cannot be taken into account. Moreover, the MLT cannot treat macroscopic motions, such as the Global Non-spherical Oscillations (GOSH) for H-ingestion flashes described by Herwig et al. (2014). Due to the huge computer time requested, those simulations can be used to constrain one-dimensional hydrostatic calculations, but they cannot substitute them yet. Then, we reckon that explorations such as the one presented here still herald precious hints on the physics of PIEs, and, thus, we proceed in describing our result.

In Figure 9, we report the heavy-element surface distributions of a 1.3 M_⊙ model with [Fe/H] = −2.85. A dichotomy clearly emerges. The three cases with a high ¹³C(α,n)¹⁶O rate (P2, P1p5, and REF) show similar distributions, even if they have important exceptions (see, e.g., lead and bismuth). On the contrary, other cases (D1p5 and D2) display a completely different behavior, with a definitely lower production of heavy elements (more than a factor of 50). Thus, it looks like the variation of the ¹³C(α,n)¹⁶O rate induces a sort of threshold effect in the s-process nucleosynthesis of this model. This is well understood in the framework of the PIE mechanism. Once the shell has split, the nucleosynthesis of the two shells follows separate evolutions. The lower one consumes its ¹³C reservoir, for a while in a convective environment and later on, when convection has switched off, in a radiative way. The upper shell instead continuously ingests protons, producing a large amount of ¹³C, as well as ¹⁴N (which acts as a poison). As a consequence, the production of heavy elements in the upper shell almost freezes after the splitting. As soon as the H-burning switches off, the convective envelope penetrates inward up to the splitting coordinate, carrying to the surface the material processed in the upper shell. On the contrary, the s-process-rich material of the lower shell is diluted in the following TP, and it is mixed to the surface at the epoch of the following TDU episode. Later, the model follows a standard AGB evolution. The two cases with a reduced ¹³C(α,n)¹⁶O rate ingest protons as well, but the splitting occurs before enough ¹³C has been mixed to the bottom of the convective shell (and, thus, an efficient s-process nucleosynthesis has developed). In Figure 10, we report the maximum neutron densities attained at the base of the convective shell for the different adopted rates as a function of the time since the beginning of the PIE. It is evident that the D1p5 and D2 cases attain lower neutron densities (∼10¹³ cm⁻³ compared to ∼10¹⁵ cm⁻³) and, consequently, the following s-process enhancement is lower. The corresponding s-process indexes [hs/ls]¹⁹ and [Pb/hs] are reported in Table 2 (see discussion in Section 5).

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Table 2.  s-process Indexes of the 1.3 M_⊙ Model with [Fe/H] = −2.85 for Different Values of the ¹³C(α,n)¹⁶O Reaction Rate

Case	[hs/ls]	[Pb/hs]
D2	−0.33	0.63
D1p5	−0.49	0.66
REF	0.13	0.84
P1p5	0.23	0.63
P2	0.21	0.29

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3.2. NEWTON Models

The NEWTON code calculates s-process nucleosynthesis for low-mass AGB stars considering the contributions of both the ¹³C(α,n)¹⁶O neutron source and the ²²Ne(α,n)²⁵Mg one. It uses a network with 404 nuclei (from hydrogen to bismuth). As already highlighted in the Introduction, the formation of the ¹³C pocket is assumed to be induced by the stellar dynamo. By using the approach proposed by Nucci & Busso (2014), the proton penetration into the He-rich layers results in the advection to the envelope of the magnetized material from the inner radiative layers: as a consequence, the ¹³C reservoir formed is large (up to $5\times {10}^{-3}\,{M}_{\odot }$) with an almost flat profile. Such a profile is quite different from the exponentially decreasing trend assumed in several other models (see, e.g., the EVEP mixing), while the ¹⁴N abundance is high in just a very thin layer adjacent to the envelope. Trippella et al. (2016) first showed that the neutron source ¹³C formed by MHD processes can account for the abundances of the main s-component nuclei in solar proportions, as well as the abundance distribution of post-AGB objects. In the same way, Palmerini et al. (2018) demonstrated that the magnetic mixing model, applied to a set of low-mass AGB stars (with masses from 1.5 to 3 ${M}_{\odot }$ and metallicities from 1/3 to 1 ${Z}_{\odot }$), provides a satisfactory explanation for the s-element isotopic mix measured in pre-solar SiC grains.

As in previous sections, we discuss here the effect induced on NEWTON nucleosynthesis predictions by a change of the ¹³C(α,n)¹⁶O reaction rate on a model of 2.0 M_⊙ and [Fe/H] =−0.15.²⁰ The reaction rates adopted for the ¹³C(α,n)¹⁶O reaction (Heil et al. 2008 divided and multiplied for factors of 1.5 and 2) and the notation are the same as in the previous sections, as are as the SiC grain data used for comparison.

In Figures 11 and 12, the heavy-element surface composition and corresponding isotopic distribution of the 2.0 M_⊙ AGB star considered are reported for the five different choices of the ¹³C(α,n)¹⁶O reaction rate. Both the surface elemental composition and the isotopic one reveal no sensitivity to the ¹³C+α cross section adopted in the calculations except for a few nuclei, namely ⁶⁰Fe, ⁸¹Kr, ⁸⁶Kr, ⁸⁷Rb, and ⁹⁶Zr, whose abundance variations in any case range between 5% and 20%. The lower sensitivity of NEWTON to the ¹³C(α,n)¹⁶O reaction rate with respect to FUNS models is due to the fact that the ¹³C in the pocket is entirely consumed during the interpulse period. Thus, none is left to be burned in the convective shell triggered by the following TP for any choice of reaction rate. This is ascribed to the different stellar structure adopted to compute the post-process calculation with respect to the FUNS models described in the previous section. NEWTON models, in fact, use physical inputs computed with the FRANEC code (Straniero et al. 2003), in which the exponentially decaying profile of convective velocities was not implemented. Therefore, the core mass M_H at the first TP followed by TDU is larger for any computed model. For instance, in the 2.0 M_⊙ studied here, M_H ∼ 0.61, to be compared with the typical values characterizing FUNS models (${M}_{{\rm{H}}}\sim 0.56;$ see previous section). In Figure 13, we repeat the comparison between isotopic ratios found in pre-solar SiC grains and the theoretical predictions of the NEWTON code for the s-process nucleosynthesis of the 2.0 M_⊙ AGB model. No remarkable changes are yielded by the ¹³C(α,n)¹⁶O reaction rate adopted in the calculation. However, we highlight the different trends of the curves in this figure and the ones of Figures 4 and 7, which do not depend on the nuclear physics input but on the physical mechanism responsible for the ¹³C pocket formation.

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The theoretical results presented in the previous section, as well as the discussion of the experimental status of the ¹³C(α,n)${}^{16}{\rm{O}}$ reaction reported in Section 2, clearly call for further direct measurements of its S-factor within the Gamow window corresponding to a radiative ¹³C burning. Owing to the vanishingly small cross section, it is then very likely that only underground measurements can help to sort out the discrepancies affecting direct data on the ¹³C(α,n)¹⁶O reaction, provided that the neutron detection efficiency of the setup is accurately known. The application of underground facilities to its investigation is detailed in Section 4.1.

Alternative approaches would also help to clarify the contradictory results brought by direct and indirect methods. In particular, the use of the detailed balance theorem (Section 4.2) might be of great help as an independent indirect technique, since the ¹³C(α,n)¹⁶O cross section could be obtained with no need of neutron detectors, reducing possible sources of systematic errors linked to the detection of neutrons.

4.1. The LUNA Experiment

4.2. The n_TOF Experiment

The sensitivity study performed in this work heralds interesting results that further strengthen the need of more detailed nuclear data concerning the ¹³C(α,n)¹⁶O cross section. In order to evaluate the effect on s-process nucleosynthesis induced by the use of any of the ¹³C(α,n)¹⁶O rates currently available in the literature, our study focused on a theoretical range well beyond the most recent uncertainty estimates. First, we carefully scrutinized the pool of experimental data regarding the ¹³C(α,n)¹⁶O cross section to find the uncertainty affecting the corresponding reaction rate. Owing to the large scatter in direct measurements, discrepancies as large as a factor of 2 are apparent in the absolute value of the cross section and then in the calculated reaction rate (see Section 2), while an average error of  25% has been calculated by taking into account all available data sets (see Trippella & La Cognata 2017). Therefore, we varied by a factor of 1.5 and 2 (upward and downward) the rate proposed by Heil et al. (2008), assumed as a reference case, to include potential systematic errors. The most interesting results are as follows:

• 1.  
  A variation of the ¹³C(α,n)¹⁶O rate does not appreciably affect s-process distributions for masses above 3 M_⊙ at any metallicity. The results obtained with the NETWON post-process code and the FUNS evolutionary code are consistent among them. Apart from a few isotopes, the differences derived from a variation of the ¹³C(α,n)¹⁶O reaction are always below 5%. This basically confirms the previous finding by Guo et al. (2012) and Trippella & La Cognata (2017).
• 2.  
  The situation is completely different if the standard paradigm of the s-process (i.e., that all ¹³C within the pockets burns radiatively) is violated. This occurs in FUNS models of low mass (M < 3 M_⊙) at solar-like metallicities (Cristallo et al. 2009a; see also Karakas et al. 2010). In such a case, a change of the ¹³C(α,n)¹⁶O reaction rate leads to nonnegligible variations in both the elemental and isotopic composition of the model. On average, the element surface distribution differs by about 10% with respect to the reference model, the heavier species showing the largest variations. A peak of 30% is obtained for rubidium: this is a consequence of the large overproduction (underproduction) of ⁸⁷Rb for slower (faster) reaction rates. Such a trend is also found for other neutron-rich isotopes, such as ⁶⁰Fe (a factor of 20). Unfortunately, the typical uncertainties affecting the spectra of s-process-rich stars are beyond the expected theoretical variations. Our analysis can safely discard extremely low values for the ¹³C(α,n)¹⁶O rate only. More precise constraints can be extracted from pre-solar SiC grain data. A change of the ¹³C(α,n)¹⁶O reaction rate within the explored range produces substantial differences in the isotopic surface ratios, in particular for the zirconium isotopes. However, the intrinsic spread characterizing SiC grains does not allow us to draw any firm conclusions. Larger ¹³C(α,n)¹⁶O rates also produce larger surface ⁸⁶Kr/⁸²Kr isotopic ratios. Even if this result is going in the right direction, the extreme ratios (∼3) found by Verchovsky et al. (2004) cannot be matched. Therefore, we confirm the results by Guo et al. (2012).
• 3.  
  The by far more interesting result comes from low-mass, low-metallicity FUNS models (M = 1.3 M_⊙ with [Fe/H] =−2.85). In this case, during the first fully developed TP, some protons are engulfed in the underlying convective shell and burn on the fly (PIE). During this peculiar phase, characterized by an extraordinarily rich nucleosynthesis, the energy budget results from the balance between H-burning and He-burning. The latter receives an important energetic contribution from the ¹³C(α,n)¹⁶O reaction. Depending on the adopted rate, completely different results are attained. In particular, the surface abundances of the heavier elements may decrease by more than a factor of 50 when slow rates for the ¹³C(α,n)¹⁶O reaction are used. Our results are substantially different from those obtained by Guo et al. (2012). The reason lies in the different approaches to the modeling of proton ingestions. In Guo et al. (2012), a post-process technique has been applied to follow the ongoing s-process nucleosynthesis. In such a case, any energetic feedback from the ¹³C(α,n)¹⁶O reaction and, most important, from the following neutron capture is lost. It has been demonstrated in the past that the physical evolution of the model strongly depends on that. The difference in the results follows consequently. Unfortunately, also in this case, observations cannot help in constraining the ¹³C(α,n)¹⁶O rate. In fact, halo stars showing s-process-enhanced distributions owe their surface compositions to pollution events from an already extinct AGB. From the pollution episode up to now, other mixing processes may have modified the observed distribution (such as gravitational settling; see, e.g., Stancliffe et al. 2007). Thus, we cannot take advantage of absolute abundances to derive firm conclusions. In principle, this problem can be circumvented by looking at the relative enhancement of the three s-process peaks. Observations indicate that, on average, [hs/ls] > 0.3 (see, e.g., Figure 5 in Cristallo et al. 2016). These values are consistent with the numbers obtained with the highest ¹³C(α,n)¹⁶O reaction rates (REF, P1p5, and P2; see Table 2). On the other hand, there are some isolated stars showing very low [hs/ls] values (∼−0.5), which would be consistent with low rates (cases D1p5 and D2; see Table 2). Thus, if we base our reasoning on a mere probabilistic criterion, we should exclude the lowest rates. However, it has to be taken into account that the calculation of a PIE in a one-dimensional hydrostatic code involves many approximations, which are hard to verify (see, e.g., Herwig et al. 2014). Moreover, the occurrence itself of this peculiar type of mixing in low-mass, low-metallicity stars is still a matter of debate, since it strongly depends on the initial composition (and, in particular, on the initial enrichment of α-elements). In conclusion, regardless of the robustness of the obtained results, we cannot get any hints on the efficiency of the ¹³C(α,n)¹⁶O reaction in those stars.

Current observational and laboratory uncertainties cause our attempt to constrain the ¹³C(α,n)¹⁶O rate by means of stellar models to be almost fruitless. In the future, however, very high-resolution spectrographs mounted on next-generation telescopes (e.g., HIRES on the E-ELT telescope; Marconi et al. 2016), as well as advanced resonant ionization mass spectrometers (e.g., CHARISMA; Savina et al. 2003), will provide extremely precise data (reducing the related errors and disentangling an eventual contamination from solar system material, respectively). In the meantime, any improvement in our knowledge of the ¹³C(α,n)¹⁶O rate will be useful. The ongoing experimental effort illustrated in this work is expected to produce in the next years a new set of direct and indirect data on the ¹³C(α,n)¹⁶O reaction. The determination of its reaction rate will benefit from the crucial information from the direct measurement and take advantage of the additional constraints from indirect measurements (e.g., the strong sensitivity of THM to the ¹⁷O level near the α-threshold). In addition, the combination of the direct ¹³C(α,n)¹⁶O data from LUNA and the inverse ¹⁶O(n, α)¹³C data from n_TOF can be used to reduce any possible source of systematic uncertainties. In summary, an accurate characterization of the resonant structures in the ¹³C(α,n)¹⁶O reaction cross section in the energy region of interest can be attained. It will eventually constraint the reaction rate to a conclusive level, well below the limits considered in the present study.

S.P. acknowledges the support of Fondazione Cassa di Risparmio di Perugia.