Issue 
A&A
Volume 609, January 2018



Article Number  A29  
Number of page(s)  10  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201731427  
Published online  22 December 2017 
Sensitivity of the sprocess nucleosynthesis in AGB stars to the overshoot model
Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles (ULB), CP 226, 1050 Brussels, Belgium
email: sgoriely@astro.ulb.ac.be
Received: 23 June 2017
Accepted: 22 August 2017
Context. Sprocess elements are observed at the surface of low and intermediatemass stars. These observations can be explained empirically by the socalled partial mixing of protons scenario leading to the incomplete operation of the CN cycle and a significant primary production of the ^{13}C neutron source. This scenario has been successful in qualitatively explaining the sprocess enrichment in AGB stars. Even so, it remains difficult to describe both physically and numerically the mixing mechanisms taking place at the time of the third dredgedup between the convective envelope and the underlying Crich radiative layer
Aims. We aim to present new calculations of the sprocess nucleosynthesis in AGB stars testing two different numerical implementations of chemical transport. These are based on a diffusion equation which depends on the second derivative of the composition and on a numerical algorithm where the transport of species depends linearly on the chemical gradient.
Methods. The sprocess nucleosynthesis resulting from these different mixing schemes is calculated with our stellar evolution code STAREVOL which has been upgraded to include an extended sprocess network of 411 nuclei. Our investigation focuses on a fiducial 2 M_{⊙}, [Fe/H] = −0.5 model star, but also includes four additional stars of different masses and metallicities.
Results. We show that for the same set of parameters, the linear mixing approach produces a much larger ^{13}Cpocket and consequently a substantially higher surface sprocess enrichment compared to the diffusive prescription. Within the diffusive model, a quite extreme choice of parameters is required to account for surface sprocess enrichment of 1–2 dex. These extreme conditions can not, however, be excluded at this stage.
Conclusions. Both the diffusive and linear prescriptions of the overshoot mixing are suited to describe the sprocess nucleosynthesis in AGB stars provided the profile of the diffusion coefficient below the convective envelope is carefully chosen. Both schemes give rise to relatively similar distributions of sprocess elements, but depending on the parameters adopted, some differences may be obtained. These differences are in the element distribution, and most of all in the level of surface enrichment.
Key words: nuclear reactions, nucleosynthesis, abundances / stars: AGB and postAGB
© ESO, 2017
1. Introduction
The determination of the surface composition of evolved stars provides important clues to the evolution of their internal structure and their role in the cosmic cycle. In that respect, asymptotic giant branch (AGB) stars form an important class of objects for several reasons. They correspond to the late evolutionary phase of stars with masses between about 1 and 8 M_{⊙}; they exhibit peculiar chemical patterns at their surface as compared to other red giant stars; and many of them are characterized by strong mass loss (up to 10^{4}M_{⊙} yr^{1}) that ejects the surface material into the interstellar medium, contributing thereby to the galactic chemical evolution.
The abundance peculiarities observed at the surface of AGB stars are the consequence of (i) Hburning at the base of the convective envelope during the socalled hot bottom burning that can lead to a temporary production of and affects the CNO isotopes, or and (ii) the action of the third dredgeup (denoted hereafter 3DUP) that allows elements synthesized in the Heburning layers to reach the surface. The signatures of the 3DUP are multiple and can explain, for example, the formation of carbon stars, and the surface enrichment in , Mg isotopes or heavy elements such as Ba or Pb by the socalled slow neutron capture process (for a recent review, see Karakas & Lattanzio 2014). The reaction is currently believed to be the major neutron source in lowmass AGB stars leading to the production of sprocess elements (Busso et al. 1999). In more massive AGB stars, neutrons can also be released at the bottom of the thermal pulse by the reaction. This production requires high temperatures (T ≳ 3.2 × 10^{8} K) achieved in stars with M ≳ 3−4M_{⊙}, depending on the metallicity. The canonical scenario to produce fresh is to invoke the mixing of protons from the envelope into the rich layers during a 3DUP event (e.g., Iben & Renzini 1982), followed by the incomplete operation of the CN cycle. The region over which protons are transported is referred to as the partial mixing (PM) zone. Unfortunately, AGB models are still subject to large uncertainties concerning the consistent prediction of both the 3DUP and transport processes. In particular, the 3DUP and PM properties are sensitive to the stellar characteristics (such as stellar mass, metallicity or mass loss rate) and strongly depend on the numerical and physical treatment of the convective boundaries and more specifically on the prescriptions used to account for extramixing at the base of the envelope. These additional transport processes are generally attributed to convective overshooting (Herwig et al. 1997), rotationally induced mixing (Herwig et al. 2003; Siess et al. 2004; Piersanti et al. 2013), gravity waves (Denissenkov & Tout 2003) or mixing driven by magnetic buoyancy (Nucci & Busso 2014) but remain parametric and poorly constrained. Recent attempts to perform two and threedimensional hydrodynamical simulation of the He shell convective zone below the AGB envelope (Herwig et al. 2006, 2014; Stancliffe et al. 2011) provide some hints on the nature of the mixing but the resolution and short time scales of the simulations still limit their predictive power.
This paper investigates the impact on the sprocess nucleosynthesis of the numerical treatment of overshoot mixing at the base of the convective envelope in AGB star. In Sect. 2, we describe the different prescriptions adopted for the implementation of the chemical transport and the nuclear ingredients for the sprocess reaction network included in our stellar evolution code STAREVOL. The nucleosynthesis of the heavy elements is analyzed in details for a fiducial 2 M_{⊙}[Fe/H] = −0.5 model star up to the end of the AGB phase. In Sect. 3, we study the abundance redistribution of heavy elements (A ≳ 40) due to the neutron captures taking place during the core Heburning phase. In Sect. 4, we pay special attention to the sprocess taking place during the AGB phase with a detailed study of its sensitivity with respect to different overshoot models and different choices of their parameters. We also discuss the final surface abundances for stars of different masses and metallicities in Sect. 5. We finally draw our conclusions in Sect. 6.
2. Mixing and nucleosynthesis modeling
The present calculations are based on the stellar evolution code STAREVOL, the description of which can be found in Siess et al. (2000), Siess (2006) and references therein. In our computations, we have used the standard mixing length theory (MLT) with α = 1.75 and take into account the change in opacity due to the formation of molecules when the star becomes carbon rich, as prescribed by Marigo (2002). The reference solar composition is given by Asplund et al. (2009) which corresponds to a metallicity Z = 0.0134. For the mass loss rate, we used the Reimers (1975) prescription with η_{R} = 0.4 from the main sequence up to the beginning of the AGB and then switched to the Vassiliadis & Wood (1993) rate. The following sections describe in more detail our extended treatments of chemical transport and nuclear burning that have now been consistently included in STAREVOL. We stress that in the present study, we have not considered extramixing at the boundaries of the thermal pulse, but only at the base of the convective envelope.
2.1. An empirical description of the convective overshoot
The temperature gradient in the convective regions is usually computed by means of the MLT where the radial fluid velocity (v_{conv}) depends on the local thermodynamical properties. From this velocity, a diffusion coefficient is defined: (1)where Λ = αH_{p} is the mixing length, α the MLT parameter and H_{p} the pressure scale height. This local timeindependent theory predicts that at the convective boundary the acceleration of the convective bubbles (but not its velocity) vanishes. However, pushed by their inertia, nothing can prevent the convective cells to overshoot beyond the Schwarzschild limit. This extension of the mixing is usually referred to as overshooting.
In stellar evolution calculations, essentially two standard prescriptions have been proposed to describe convective overshooting. The first and oldest one simply assumes that the convective zone extends over of fraction beyond the Schwarzschild limits, where is the pressure scale height at the convective boundary. The value of d was constrained by fitting the width of massive main sequence stars (e.g., Maeder & Meynet 1987) and varies between ≈0.1−0.25. The other alternative was proposed by Herwig et al. (1997; see also Herwig 2000) based on the twodimensional hydrodynamical simulations of Freytag et al. (1996) and assumes a diffusion coefficient in the overshoot region of the form (2)where D_{cb} is the value of the diffusion coefficient at the base of the convective envelope (as defined by the Schwarzschild criterion), z is the distance from the formal convective boundary and f_{over} a free parameter describing the efficiency of the diffusive mixing. In a more recent paper, Battino et al. (2016) upgraded this prescription by implementing a double exponential where a second flatter exponential term takes over when the diffusion coefficient has fallen below a threshold value D_{2} ≈ 10^{5} cm^{2}/ s. This second term was fitted on a 3 M_{⊙} model to account for the mixing induced by gravity waves. Their formulation corresponds to (3)where the switch between the two regimes occurs at the depth z_{2}, determined by imposing that (4)In this prescription, z is the distance from the convective boundary and the parameters are f_{1},f_{2} and D_{2}. During the AGB phase, Battino et al. (2016) use the default values of f_{1} = 0.014, D_{2} = 10^{11} cm^{2}/ s and f_{2} = 0.25 at the bottom of the convective envelope.
Essentially, two different approaches can be found in the literature regarding the numerical implementation of the chemical transport. The diffusive mixing scheme (referred to as DM) describes the transport of the nuclear species by a diffusion equation (5)where D = D_{conv} + D_{over} is the sum of the convective and overshoot diffusion coefficients, respectively. However, a diffusive approach of this process may not be very realistic as convection is in essence the advection of convective bubbles.
An alternative algorithm was devised by Sparks & Endal (1980) and later improved by Chieffi et al. (2001) and Straniero et al. (2006) where mixing is now described by a linear equation coupling the mixing between neighboring layers moving at a velocity v over a time step Δt. This formulation, which we refer to as linear mixing (LM), is given by (6)In this expression the summation runs overs the whole zone affected by mixing and the superscript 0 refers to abundances prior to the mixing. ΔM_{k} is the mass of the layer k and the factor (7)where the mixing turnover time between mesh points j and k is given by (8)where Δr_{i} = r_{i + 1}−r_{i} is the size of the cell. The velocity v_{i} is set to v_{conv} in the convective layers and below the envelope it decreases exponentially with where v_{cb} is the velocity at the base of the convective envelope and β a free parameter. We note that this formulation is equivalent to the expression for D_{over} given by Eq. (2).
In our computation, the total mass of the mixing zone was calculated as M_{mix} = ∑ _{k}ΔM_{k} × f_{k} with f_{k} = Δt/τ_{k} = Δt × v_{k}/ Δr_{k} if Δ_{t}<τ_{k}, or 1 otherwise. In comparison with the standard expression provided by Straniero et al. (2006) where M_{mix} = ∑ _{k}ΔM_{k}, our formulation presents the advantage of being independent of the limit in the summation, since in the layers where mixing is not present v_{k} and f_{k} = 0.
Fig. 1
Profile of the overshoot diffusion coefficient D_{over} as a function of the reduced depth z/z^{∗} for different values of p and assuming D_{cb} = 10^{16} cm^{2}/ s, D_{min} = 10^{3} cm^{2}/ s. 
In the present study, we have considered both the DM (Eq. (5)) and LM (Eq. (6)) prescriptions and in each case, the adopted mathematical profile for the overshoot diffusion coefficient reads (9)where the characteristic length z^{∗} is given by z^{∗} = f_{over} × H_{p} × ln(D_{cb})/2, D_{min} is the value of the diffusion coefficient at the boundary z = z^{∗} and p is an additional free parameters defining the slope of the exponential decrease of D_{over} with respect to z. Some examples of the zdependence of the overshoot diffusion coefficient are illustrated in Fig. 1. Note that below D_{min} we have assumed that D_{over} drops to zero, in contrast to Straniero et al. (2006) who do not consider a lower limit for v_{over}, hence for D_{over}. So, with our prescription, the overshoot mixing is only taking place over the radial region defined by 0 ≤ z ≤ z^{∗}. We note that Eq. (9) is just an extension of Eq. (2) from the original formulation of Herwig et al. (1997) which is recovered by setting p = 1 and f_{over} = 0.02 and D_{min} = 1 cm^{2}/ s. Similarly, the default parameters used by Straniero et al. (2006) in their simulations correspond to p = 1 and f_{over} = 0.2 and D_{min} = 1 cm^{2}/ s. We also point out that for a given value of D_{cb}, different parameters f_{over}, D_{min} and p can lead to similar profiles of D_{over} for z ≤ z^{∗}. For example for D_{cb} = 10^{16} cm^{2}/ s and p = 1, the same profile is obtained using (D_{min} = 10^{4} cm^{2}/ s; f_{over} = 0.25) or (D_{min} = 1 cm^{2}/ s; f_{over} = 0.2) with the only difference that, in the former case, the diffusion coefficient drops to zero earlier, that is, when D_{over} reaches 10^{4} cm^{2}/ s.
We also stress that at the time of maximum deepening of the convective envelope during the 3DUP, the size of the CO core is ≈0.3–0.4 R_{⊙} for our 2 M_{⊙} model. This number slightly varies with the mixing parameters (which impact the efficiency of the 3DUP) but remains always of the same order of magnitude as R_{⊙}. As a consequence, using unrealistically large values of f_{over} can lead to proton mixing well below the convective envelope and induce a substantial reduction of the CO core size eventually leading to convergence issues. In practice, values of f_{over} larger than typically 0.1 can hardly be considered in our model star.
In light of the previous remark, we find that using the double exponential prescriptions of Battino et al. (2016) leads to a complete disappearance of the sprocess nucleosynthesis. In our stellar models, during the interpulse phase the second exponential term (obtained with the large value of f_{2} = 0.25) was responsible for the transport of the neutron poison ^{14}N in the region where ^{13}C is burnt. As for rotation (Herwig et al. 2003; Siess et al. 2004), this pollution of the pocket induced by this sustained chemical transport inhibits the production of sprocess nuclei. We do not understand why this effect does not show up in the computations of Battino et al. (2016).
2.2. Sprocess reaction network
A full nuclear reaction network including 411 nuclei between H and Po with some 734 nuclear (n, p, and αcaptures), weak (electron captures, βdecays) and electromagnetic interactions has been implemented in the STAREVOL code. Nuclear reaction rates were taken from the Nuclear Astrophysics Library of the Brussels University^{1} (Arnould & Goriely 2006), and include the latest experimental and theoretical cross sections through the interface tool NETGEN (Xu et al. 2013a). In particular, all the chargedparticleinduced reaction rates of relevance in the H and Heburning calculations were taken from the NACRE and NACREII evaluations (Angulo et al. 1999; Xu et al. 2013b) as well as STARLIB library (Sallaska et al. 2013). When not available experimentally, the cross sections were calculated within the statistical HauserFeshbach model with the TALYS reaction code (Goriely et al. 2008). The TALYS calculations were also used systematically to deduce from the laboratory neutron capture cross sections the stellar rates by allowing for the possible thermalization of lowlying states in the target nuclei. We note that at low temperature, the nonthermalization of the isomeric state of ^{26}Al, ^{85}Kr, ^{115}In, ^{176}Lu and ^{180}Ta is introduced explicitly in the reaction network (Käppeler et al. 1989; Nemeth et al. 1994). The temperature and densitydependent βdecay and electron capture rates in stellar conditions were taken from Takahashi & Yokoi (1987) with the update of Goriely (1999). The (n,α) reactions and αdecays are also introduced when relevant, in particular for the Bi and Po isotopes.
3. Nucleosynthesis process during core Heburning
Fig. 2
Upper panel: Kippenhahn diagram of our 2 M_{⊙}[Fe/H] = −0.5 during the first dredgeup and core He burning phases; the green and blue dotted lines delineate the region of maximum nuclear energy production by H and Heburning, respectively. Middle panel: evolution of the central neutron density N_{n}, central ^{13}C mass fraction (×10^{6}) and neutron exposure τ. Lower panel: evolution of the central temperature T_{c} and density ρ_{c}. 
When the star leaves the main sequence and the central temperature reaches ≈5 × 10^{7} K (for a density ρ_{c} ≈ 3 × 10^{4} g cm^{3}), the burning of the CNO byproduct ^{13}C via produces a first burst of neutrons (Fig. 2). This occurs during the first dredgeup of our 2 M_{⊙}[Fe/H] = −0.5 model star and leads to a maximum neutron density of N_{n} = 13 cm^{3} and a timeintegrated neutron exposure of , where v_{T} is the most probable relative neutronnucleus velocity at the temperature T. The main neutron emission lasts until is depleted at the center and then proceeds at a much lower rate in the contracting shells surrounding the free core. Eventually helium ignites offcenter and after a series of shell flashes, convection reaches the center at an age 9.57 × 10^{8} yr. The temperature in the core reaches 1.2 × 10^{8} K and can trigger the production of a small amount of neutrons by the activation of the reaction. The neutron exposure increases up to τ = 3.7 × 10^{4} mb^{1} but remains too small to generate an efficient sprocess nucleosynthesis. Instead, it is responsible for the rearrangements of the abundances of specific heavy nuclei. As shown in Fig. 3, elements such as ^{40}K, ^{108}Cd, ^{128,130}Xe, ^{154}Gd, ^{160}Dy, ^{170}Yb or ^{198}Hg can be significantly produced with abundances increased by more than ~0.4 dex with respect to their initial value and others partially destroyed, like ^{107}Ag, ^{149}Sm or ^{151}Eu. At the temperatures encountered in the stellar center during core He burning, we notice that ^{123}Te, ^{176}Lu, ^{180}Ta and ^{187}Os become βunstable and fully decay (Takahashi & Yokoi 1987). When helium is exhausted in the core, the chemical profiles in the central 0.4–0.5 M_{⊙} are frozen and the abundances of these heavy nuclei will be part of the future white dwarf. We note that the distributions of overabundances of elements heavier than typically A = 40 show very small differences between our 2 and 3 M_{⊙} stellar models, almost independently of the metallicity.
Fig. 3
Final overproduction factors [X/Fe] of all stable nuclei with A ≥ 30 (full squares) at the center of our 2 M_{⊙}[Fe/H] = −0.5 model star at the end of core helium burning. The sonly nuclei are shown by red diamonds connected by a solid line. The vertical line depicts the total decay of the sonly nucleus ^{123}Te. 
4. Sprocess nucleosynthesis during the AGB phase of a 2 M_{⊙} [Fe/H] = –0.5 model star
Mixing of the nuclear species beyond the convective boundary in the LM and DM models essentially depends on three parameters: D_{min}, p and f_{over}. While f_{over} describes the radial extent over which mixing takes place, the other two parameters D_{min} and p modify the profile of the diffusion coefficient and thus the distribution of protons in the Crich region and incidentally the sprocess nucleosynthesis. In Sect. 4.1 we first study the proton profiles resulting from the overshooting within the LM and DM models and then analyze the production of sonly nuclei during a given interpulse phase (Sect. 4.2) and in the convective pulse (Sect. 4.3). The surface enrichment resulting from the overshoot mixing since the beginning of the AGB phase is discussed in Sect. 4.4.
4.1. Proton profiles resulting from the treatment of overshoot mixing
To compare the effects of the two mixing schemes, we began our calculations with the same initial model computed without any extramixing to avoid any chemical contamination resulting from previous evolution. We recall that we have not considered overshooting at the convective boundaries of the thermal pulse. The initial model is taken at the time when the convective envelope approaches its deepest extent during the 3DUP following the 8th thermal pulse of our 2 M_{⊙} [Fe/H]=−0.5 model star. We note that the mass coordinate reached by the convective envelope during the 3DUP depends on the mixing scheme.
We show in Fig. 4 the diffusion coefficients and proton mass fractions below the convective envelope obtained with the LM formalism for different values of the parameters D_{min} and p. As expected, a more efficient transport resulting for example from a higher value of D_{min} or a flatter profile obtained with a lower p leads to a deeper 3DUP. Following Goriely & Mowlavi (2000), we defined the extent of the PM zone ΔM_{pmz} as the mass range over which the hydrogen mass fraction decreases from X(H) = 0.5 (close to its envelope value) to X(H) = 10^{5}. As shown in Table 1, ΔM_{pmz} varies between ≈6 × 10^{5}M_{⊙} (for p = 1, D_{min} = 10^{9} cm^{2}/ s) to almost 7 × 10^{4}M_{⊙} (for p = 1, D_{min} = 10^{6} cm^{2}/ s). We also note that with increasing values of D_{min}, the H mass fraction remains relatively high in the mixing region and then drops rapidly.
Fig. 4
Left panels: proton mass fraction (bottom panel) and diffusion coefficient (top panel) profiles below the convective envelope resulting from the LM scheme with overshoot parameters f_{over} = 0.1, D_{min} = 10^{3} cm^{2}/ s and 3 values of p = 1/2, 1, 5. Right panels: same but for f_{over} = 0.1, p = 1 and 4 values of D_{min} = 10^{1.9} cm^{2}/ s. The profiles were taken when the convective envelopes reaches its deepest inward extent. For display convenience, the D_{min} = 10^{9} profiles have been shifted outward by 0.00085 M_{⊙}. 
Extent in mass (in unit of 10^{4}M_{⊙}) of the PM zone (ΔM_{pmz}) and sprocess region (ΔM_{spro}) for the LM and DM models and the different adopted values of the parameters p and D_{min} (in cm^{2}/ s). In all cases, f_{over} = 0.1.
In the DM case (Fig. 5), the extent of the PM zone is systematically smaller by a factor of between three and eight compared with the values obtained with the LM prescription, except in the rather extreme case of D_{min} = 10^{9}. Although this study has been restricted to a specific interpulse phase, the derived properties (extent of the mixing, H profiles) remain general and weakly dependent on mass or composition as long as the background stellar stratification is not significantly affected (Herwig et al. 2007).
4.2. Sprocess nucleosynthesis during an interpulse
In this section we analyze the overproduction factors [X/Fe] of the 28 sonly nuclei at the end of the eighth interpulse of our 2 M_{⊙}[Fe/H] = −0.5 model star resulting from the two different mixing schemes (DM and LM). The abundances are determined by averaging the chemical composition over a mass range of 0.01 M_{⊙} between M_{r} = 0.595 and 0.605M_{⊙}. We recall that the same initial stellar structure is considered in all cases (the fully consistent stellar evolution sequences are presented in Sects. 4.4 and 5).
The results of our exploration of the mixing parameter space are displayed in Figs. 6 and 7 for the LM and DM schemes, respectively. The first obvious observation is the similarity of the distribution of selements between all the models, except may be for the extreme case p = 1, D_{min} = 10^{9} cm^{2}/ s. The main difference is the level of overproduction.
During the interpulse phase, the sprocess nucleosynthesis is known to take place in a relatively small radiative region where the protonto^{12}C ratio ranges between 0.06 and 0.6, although in more metal poor stars with [Fe/H] ≲ −1, layers with ratios as low as 0.002 also contribute (Goriely & Mowlavi 2000). For an He intershell carbon mass fraction X(^{12}C) = 0.2, the neutron production will mainly occur in layers where the proton mass fraction X(H) ≃ 10^{3}−10^{2}, while for larger values of X(H) ≃ 10^{2}−10^{1} an efficient production of ^{19}F and ^{23}Na is expected. As discussed in Goriely & Mowlavi (2000), the detailed profile of protons in this region dictates the subsequent sprocess nucleosynthesis (see in particular their Fig. 3), so the extent of the PM zone Δ_{pmz} as defined in Sect. 4.1 is only a qualitative and approximate indicator of the sprocess efficiency. For example, the value of Δ_{pmz} picked in Table 1 for the DM scheme is similar between the models D_{min} = 1 and D_{min} = 10^{9} but the resulting sprocess enrichment (Fig. 7) is significantly different. For this reason, we defined a new indicator ΔM_{spro} corresponding to the mass extent where 10^{3} ≤ X(H) ≤ 10^{2}. An inspection of the numbers presented in Table 1 indicates that ΔM_{pmz} and ΔM_{spro} are quite different but strongly correlated. There are, however, two exceptions, one is related to the extreme case D_{min} = 10^{9} cm^{2}/ s mentioned previously and the second to the LM cases p = 1/2 and p = 1 where the nucleosynthesis is very similar. Comparing Figs. 6, 7, and Table 1 indicates that the highest overproduction factors are obtained for the largest values of ΔM_{spro}. In the LM case, ΔM_{spro} reaches a maximum of ~3 × 10^{4}M_{⊙} for p = 1/2−1 and D_{min} ≈ 10^{3}−10^{6} cm^{2}/ s, leading to a significant production of sprocess nuclei (Fig. 6). In contrast, in the DM case, ΔM_{spro} never exceeds 3 × 10^{5}M_{⊙}, that is, about an order of magnitude lower than in the LM case. As shown in Fig. 7, the overall overproduction factors are indeed a factor of between five and ten lower than in the LM model. This significant enrichment in sprocess elements by the LM scheme was also reported by Cristallo et al. (2009).
Fig. 6
Overabundance distributions [X/Fe] for the 28 sonly nuclei inside a region of 0.01 M_{⊙} around M_{r} = 0.595−0.605M_{⊙} at the end of the 8th interpulse of our 2 M_{⊙}[Fe/H] = −0.5 model star. The results are shown for different LM conditions, i.e. different values of p (upper panel) or D_{min} (in cm^{2}/s; lower panel); in all cases, f_{over} = 0.1 is assumed. 
4.3. Convective nucleosynthesis in the thermal pulse
The distribution of selements resulting from the radiative interpulse nucleosynthesis, as described in Sect. 4.2, may still be affected by the second neutron irradiation taking place in the pulsedriven convective region, at least if the temperature T_{p} at its base is high enough to activate the reaction. T_{p} depends on the core mass and is known to increase with increasing pulse number, stellar mass or decreasing metallicities and ranges from 1.5 × 10^{8} K to above 3.5 × 10^{8} K. In our 2 M_{⊙}[Fe/H] = −0.5 model star, T_{p} reaches 3 × 10^{8} K in the last thermal pulses, so only few neutrons are produced. As seen in Fig. 8, the impact of the convective pulse nucleosynthesis on the abundance distribution of sonly nuclei is negligible. However, locally, the abundances of some nuclei, such as ^{93}Nb, ^{113}In, ^{176}Lu or ^{187}Os may be modified through the impact of Tdependent sprocess branching during the neutron irradiation (Takahashi & Yokoi 1987). In the mass and metallicity range considered in this study, none of the ^{13}C produced by the PM of protons survives the interpulse phase. Therefore, it remains to be checked if, in lowmass metalpoor stars (M ≲ 1.5M_{⊙}, [Fe/H] ≲ −2), fresh ^{13}C is ingested in the convective pulse, as found in previous studies (Lugaro et al. 2012).
Fig. 8
Ratio of the nuclei mass fraction after (X_{conv}) and before (X_{rad}) the 9th thermal pulse of our 2 M_{⊙}[Fe/H] = −0.5 model star. Open and filled squares represent stable and sonly nuclei, respectively. 
4.4. Nucleosynthetic yields
In this section, we present the nucleosynthesis yields of a 2 M_{⊙}[Fe/H] = −0.5 model computed with the LM and DM mixing formalisms. For these simulations, the same mixing parameters are used, namely D_{min} = 10^{3} cm^{2}/ s, p = 1 and f_{over} = 0.1, and applied since the beginning of the AGB phase. The thermal pulseAGB evolution is similar in both cases and leads to the occurrence of ten thermal pulses and 8 3DUP. The surface overabundance of sonly nuclei obtained at the end of the computation is presented in Fig. 9. Although the two distributions are relatively similar, the overproduction obtained with the DM is about eight times smaller than with the LM prescription. Again, this has to be ascribed to the smaller mass of the PM zone in the DM formulation. While the LM prescription gives rather satisfactory overproduction factors of the order of 1.5 dex, they do not exceed 0.9 dex in the DM case. The latter would consequently face problems explaining the large surface abundances of some intrinsic sprocessrich Cstars that show for example [Ba/Fe] as high as 1.65 (Zamora et al. 2009).
In the specific DM case, we have investigated the impact of the mixing parameters on the final surface enrichment. The results for nine sets of mixing parameters, that is, D_{min} = 10^{3},10^{6}, and 10^{9} cm^{2}/ s, p = 1/5,1,5 are shown in Fig. 10. In all these models, f_{over} = 0.1 and overshooting is activated at the beginning of the AGB phase. Most cases lead to a rather low surface enrichment, except for the case D_{min} = 10^{6} cm^{2}/ s, p = 1 where an enrichment of the order of 1 dex is achieved and for the set D_{min} = 10^{9} cm^{2}/ s, p = 5 which gives rise to an overproduction of about 2 dex for the heaviest sonly nuclei comparable to the one obtained with the LM prescription. In this latter quite extreme case, a relatively flat proton profile with X(H) ≃ 10^{3}−10^{2} is achieved over a large fraction of the PM zone, leading to a significant production of sprocess nuclei during the interpulse phase.
Fig. 9
Surface overabundance distributions [X/Fe] for the 28 sonly nuclei at the end of the AGB phase of a 2 M_{⊙}[Fe/H] = −0.5 model star resulting from the LM and DM schemes. In both cases, D_{min} = 10^{3} cm^{2}/ s, p = 1 and f_{over} = 0.1. 
Fig. 10
Surface overabundance distributions [X/Fe] for the 28 sonly nuclei at the end of the AGB phase of a 2 M_{⊙}[Fe/H] = −0.5 model star computed with the DM overshoot formalism and 9 different sets of mixing parameters, i.e. D_{min} = 10^{3},10^{6}, and 10^{9} cm^{2}/ s, p = 1/5,1,5 and f_{over} = 0.1. 
5. Sprocess in stars of different masses and metallicities
As shown in Sect. 4, the abundance distribution of snuclei is rather insensitive to the mixing scheme and parametrisation but the production factors strongly depend on its mathematical formulation. The LM model consistently produces significant surface enrichment while the DM approach requires a fine tuning of the underlying parameters to achieve a similar result. Assuming the production of selements is similar, it remains to be seen if the overall abundance distributions obtained using both models are comparable for stars with different initial masses and metallicities. The parameters used are D_{min} = 10^{3} cm^{2}/ s, p = 1 for the LM scheme and D_{min} = 10^{9} cm^{2}/ s, p = 5 for the DM (f_{over} = 0.1 is taken in both cases). The high value of D_{min} = 10^{9} cm^{2}/ s was chosen, so that similar surface enrichments are produced between both schemes, making the comparison easier. We are aware that such a high value is difficult to justify on physical grounds but considering a value lower than D_{min} = 10^{7} cm^{2}/ s would reduce the PM zone significantly, leading to a negligible sprocess production. Figure 11 shows the mass extent of the PM zone ΔM_{pmz}, sprocess region ΔM_{spro}, as defined in Sect. 4.2, and the mass ΔM_{3DUP} of 3DUP material as a function of the pulse number, for a 2 M_{⊙} model computed with three different initial metallicities [Fe/H] = 0, −0.5 and −1. The decline of the PM and sprocess zones with the pulse number is a consequence of the growing core mass which produces a strong gravitational pull and thus a higher compression of the layers located below the convective envelope (see also Cristallo et al. 2009). While the LM models give rise to a rather smooth and constant decrease of the PM and sprocess zones, irrespective of the metallicity, significantly larger variations are found with the DM prescription. Again, one can see the correlation between the size of the sprocess zone (Fig. 11) and the surface overproduction factors (Fig. 12), except to some extent for the [Fe/H] = −1 model star where the 10^{4} ≲ X(H) ≲ 10^{3} mass range, neglected in the definition of ΔM_{spro}, may contribute to the sprocess nucleosynthesis (Goriely & Mowlavi 2000). As far as the dredgeup mass is concerned, both prescriptions yield rather similar values, at least when the 3DUP efficiency has reached its asymptotic value, that is, after the first three to five thermal pulses depending on the stellar metallicity (Fig. 11).
Fig. 11
Upper panel: extent of the PM zone ΔM_{pmz} in a 2 M_{⊙} model as a function of the pulse number for 3 different initial metallicities; [Fe/H] = 0, −0.5 and −1. The LM scheme uses D_{min} = 10^{3} cm^{2}/ s, p = 1 and f_{over} = 0.1 and the DM case D_{min} = 10^{9} cm^{2}/ s, p = 5 and f_{over} = 0.1. Middle panel: same but for the mass of the sprocess zone ΔM_{spro}. Lower panel: same but for the dredgedup mass ΔM_{3DUP} 
The sensitivity of the sprocess production on the mixing scheme and initial composition is illustrated in Fig. 12. As can be seen, the final surface enrichment between the two mixing schemes mainly differs in terms of overproduction factors, but also in terms of isotopic and elemental distributions. With our specific choice of parameters, the DM scheme yields similar or even higher sprocess overproduction factors than the standard LM parametrization. It remains, however, difficult to reach surface enrichments larger than typically 2 dex. At solar metallicity the sprocess surface enrichment is relatively weak, despite the fact that the extent of the sprocess zone (Fig. 11) is relatively similar to the other model stars with lower metallicities. This is essentially due to the smaller number and relatively shallower 3DUP episodes found in solar metallicity stars. While six 3DUP are predicted in our 2 M_{⊙}[Fe/H] = 0 model star, nine are found for a metallicity of [Fe/H] = −1. In addition, in the last five thermal pulseinterpulse sequences, the 3DUP of the solar metallicity star drags from 5% to 25% of the thermal pulse material into the envelope, while the 3DUP efficiency reaches more than 35% in the [Fe/H] = −1 model star. This pattern is found to be qualitatively independent of the adopted overshoot model, although quantitatively some minor differences are found.
Some differences in the element distribution can also be found. In particular, when compared with the LM scheme, the DM models tends to give rise to a lower production of light selements with respect to the heavy selements, as seen in Figs. 12 and 13, and in Table 2. While this difference is marginal in the solar metallicity star, this is particularly clear in the 2 M_{⊙}, [Fe/H] = −1 model star, where a lower 90 ≲ A ≲ 140 distribution by almost 0.5 dex is found with the DM model compared to the LM formulation. The corresponding observational [hs/ls] index, where hs describes the heavy selements (Ba, La and/or Nd) abundance and ls the light selements (Sr, Y and/or Zr) abundance, can differ by up to 0.2 dex (Table 2). The low production of light selements with the DM model is due to the specific profile of the diffusion coefficient obtained with the large p = 5 and D_{min} = 10^{9} cm^{2}/ s values (see Figs. 1 and 5). With these parameters, more weight is given to the layers with proton abundances close to X(H) ≃ 10^{3} at the expense of X(H) ≃ 10^{2}, and as a consequence the production of light elements is reduced with respect to heavy ones (see in particular Fig. 3 of Goriely & Mowlavi 2000).
Fig. 12
Dependence on stellar metallicity of the final surface overabundance distributions [X/Fe] of the 28 sonly nuclei at the surface of a 2 M_{⊙} model resulting from the LM scheme with D_{min} = 10^{3} cm^{2}/ s, p = 1 and the DM formulation with D_{min} = 10^{9} cm^{2}/ s, p = 5. In both cases, f_{over} = 0.1 is taken. Four metallicities [Fe/H] = 0, −0.5, −1 and −2 have been considered. 
Fig. 13
Top panel: as Fig. 12 for the dependence on the stellar mass of the final surface overabundance distribution of sonly nuclei. Stars of 2 and 3 M_{⊙} with a metallicity [Fe/H] = −0.5 are considered. Lower panel: same as the left panel for the final surface elemental distribution of all elements above Ge. 
We also notice that with decreasing metallicity, the production of heavier elements is favored as a consequence of the lower abundance of iron seed nuclei, hence a larger neutrontoseed ratio. We note the systematic overproduction of Pb at [Fe/H] ≤ −1 which accounts for Pbstars (Van Eck et al. 2001), but none of the models studied here can explain some sprocess enriched Cenhanced metalpoor stars (Bisterzo et al. 2011; Piersanti et al. 2013) as well as lowmetallicity postAGB stars (e.g., De Smedt et al. 2014, 2016) that are poorly enriched in Pb.
Finally, the effect of varying the initial mass is illustrated in Fig. 13. As confirmed by Table 2, these calculations indicate that, with increasing stellar mass, the [hs/ls] ratio decreases, essentially because of some additional production of light selements in hotter thermal pulses. While in our 2 M_{⊙}[Fe/H] = −0.5 model star, T_{p} does not exceed 3 × 10^{8} K, in the 3 M_{⊙} model it reaches 3.5 × 10^{8} K, allowing for the activation of the ^{22}Ne(α, n)^{25}Mg reaction in the convective thermal pulse. The [hs/ls] index is found to be smaller by 0.3 (0.2) dex in the 3 M_{⊙} star with respect to the 2 M_{⊙} star when adopting the LM (DM) model. Future calculations will explore stellar models within a wider range of masses and metallicities.
Independently of this work, Buntain et al. (2017) investigated the sensitivity of the sprocess nucleosynthesis to the proton profile and mass extent of the PM zone. Their approach differs from ours as they perform postprocessing calculations in which they impose the proton profile at the time of the 3DUP and the extent of the PM zone ΔM_{pmz}. These authors find that the resulting sprocess abundance distribution remains weakly (by more or less 0.2 dex) dependent on the shape of the mixing function, except at low metallicity (Z = 0.0001) where the Pb abundance can be significantly affected. These results are consistent with those presented here, except for the Pb production in metal poorstars that always remain relatively high in our calculations.
6. Conclusions
Surface [hs/ls] index deduced from the [La/Y] ratio at the end of the evolution of the five model stars studied here and for both LM and DM schemes.
Assuming overshooting below the convective envelope of an AGB stars can be described by an exponential decrease of the diffusion coefficient or equivalently of the convective velocity, we analyzed the impact of two different numerical modeling of the chemical transport on the resulting sprocess nucleosynthesis. We have shown that for the same set of parameters, both descriptions give a rather similar abundance distribution of sonly nuclei (though some differences are found in the relative production of light to heavy selements), but the surface enrichment can differ drastically. As found by Cristallo et al. (2009), the linear algorithm leads to the formation of a PM zone that can easily account for large surface sprocess enrichments. On the other hand, the DM approach requires a quite extreme set of parameters producing diffusion profiles that sharply drop to zero below a threshold value of D_{min} ~ 10^{7}−10^{9} cm^{2}/s. At this stage, it is however not possible to exclude this set of parameters, nor to favor one prescription over the other. Both schemes are able to phenomenologically simulate the PM of protons inside the Crich layers. Our exploration of the parameters attests of the difficulty to find a choice of parameters that leads to surface enrichments significantly larger than typically 2 dex for the bulk of selements. However, it remains unclear if stars of different masses and metallicities are characterized by similar sets of parameters and if these parameters are constant during the evolution. Furthermore, additional processes induced by gravity waves or rotational mixing can interfere with the sprocess nucleosynthesis. Differences in the sprocess distributions between the two schemes may be more perceptible in higher mass stars with the LM scheme producing more light selements. We also report that, contrary to Battino et al. (2016), our simulations do not show evidence for a strong sprocess nucleosynthesis when using their double exponential overshooting prescription. The reason is ascribed to the fact that with this formulation, mixing is still active in the PM zone during the interpulse phase and allows pollution of the pocket by the neutron poison.
Available at http://wwwastro.ulb.ac.be/Bruslib
Acknowledgments
L.S. and S.G. are FRSF.N.R.S. research associates.
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All Tables
Extent in mass (in unit of 10^{4}M_{⊙}) of the PM zone (ΔM_{pmz}) and sprocess region (ΔM_{spro}) for the LM and DM models and the different adopted values of the parameters p and D_{min} (in cm^{2}/ s). In all cases, f_{over} = 0.1.
Surface [hs/ls] index deduced from the [La/Y] ratio at the end of the evolution of the five model stars studied here and for both LM and DM schemes.
All Figures
Fig. 1
Profile of the overshoot diffusion coefficient D_{over} as a function of the reduced depth z/z^{∗} for different values of p and assuming D_{cb} = 10^{16} cm^{2}/ s, D_{min} = 10^{3} cm^{2}/ s. 

In the text 
Fig. 2
Upper panel: Kippenhahn diagram of our 2 M_{⊙}[Fe/H] = −0.5 during the first dredgeup and core He burning phases; the green and blue dotted lines delineate the region of maximum nuclear energy production by H and Heburning, respectively. Middle panel: evolution of the central neutron density N_{n}, central ^{13}C mass fraction (×10^{6}) and neutron exposure τ. Lower panel: evolution of the central temperature T_{c} and density ρ_{c}. 

In the text 
Fig. 3
Final overproduction factors [X/Fe] of all stable nuclei with A ≥ 30 (full squares) at the center of our 2 M_{⊙}[Fe/H] = −0.5 model star at the end of core helium burning. The sonly nuclei are shown by red diamonds connected by a solid line. The vertical line depicts the total decay of the sonly nucleus ^{123}Te. 

In the text 
Fig. 4
Left panels: proton mass fraction (bottom panel) and diffusion coefficient (top panel) profiles below the convective envelope resulting from the LM scheme with overshoot parameters f_{over} = 0.1, D_{min} = 10^{3} cm^{2}/ s and 3 values of p = 1/2, 1, 5. Right panels: same but for f_{over} = 0.1, p = 1 and 4 values of D_{min} = 10^{1.9} cm^{2}/ s. The profiles were taken when the convective envelopes reaches its deepest inward extent. For display convenience, the D_{min} = 10^{9} profiles have been shifted outward by 0.00085 M_{⊙}. 

In the text 
Fig. 5
As Fig. 4 but for the diffusive treatment (DM) of overshooting. 

In the text 
Fig. 6
Overabundance distributions [X/Fe] for the 28 sonly nuclei inside a region of 0.01 M_{⊙} around M_{r} = 0.595−0.605M_{⊙} at the end of the 8th interpulse of our 2 M_{⊙}[Fe/H] = −0.5 model star. The results are shown for different LM conditions, i.e. different values of p (upper panel) or D_{min} (in cm^{2}/s; lower panel); in all cases, f_{over} = 0.1 is assumed. 

In the text 
Fig. 7
As Fig. 6 for the sprocess distributions obtained with the DM model. 

In the text 
Fig. 8
Ratio of the nuclei mass fraction after (X_{conv}) and before (X_{rad}) the 9th thermal pulse of our 2 M_{⊙}[Fe/H] = −0.5 model star. Open and filled squares represent stable and sonly nuclei, respectively. 

In the text 
Fig. 9
Surface overabundance distributions [X/Fe] for the 28 sonly nuclei at the end of the AGB phase of a 2 M_{⊙}[Fe/H] = −0.5 model star resulting from the LM and DM schemes. In both cases, D_{min} = 10^{3} cm^{2}/ s, p = 1 and f_{over} = 0.1. 

In the text 
Fig. 10
Surface overabundance distributions [X/Fe] for the 28 sonly nuclei at the end of the AGB phase of a 2 M_{⊙}[Fe/H] = −0.5 model star computed with the DM overshoot formalism and 9 different sets of mixing parameters, i.e. D_{min} = 10^{3},10^{6}, and 10^{9} cm^{2}/ s, p = 1/5,1,5 and f_{over} = 0.1. 

In the text 
Fig. 11
Upper panel: extent of the PM zone ΔM_{pmz} in a 2 M_{⊙} model as a function of the pulse number for 3 different initial metallicities; [Fe/H] = 0, −0.5 and −1. The LM scheme uses D_{min} = 10^{3} cm^{2}/ s, p = 1 and f_{over} = 0.1 and the DM case D_{min} = 10^{9} cm^{2}/ s, p = 5 and f_{over} = 0.1. Middle panel: same but for the mass of the sprocess zone ΔM_{spro}. Lower panel: same but for the dredgedup mass ΔM_{3DUP} 

In the text 
Fig. 12
Dependence on stellar metallicity of the final surface overabundance distributions [X/Fe] of the 28 sonly nuclei at the surface of a 2 M_{⊙} model resulting from the LM scheme with D_{min} = 10^{3} cm^{2}/ s, p = 1 and the DM formulation with D_{min} = 10^{9} cm^{2}/ s, p = 5. In both cases, f_{over} = 0.1 is taken. Four metallicities [Fe/H] = 0, −0.5, −1 and −2 have been considered. 

In the text 
Fig. 13
Top panel: as Fig. 12 for the dependence on the stellar mass of the final surface overabundance distribution of sonly nuclei. Stars of 2 and 3 M_{⊙} with a metallicity [Fe/H] = −0.5 are considered. Lower panel: same as the left panel for the final surface elemental distribution of all elements above Ge. 

In the text 
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