THE s-PROCESS IN ASYMPTOTIC GIANT BRANCH STARS OF LOW METALLICITY AND THE COMPOSITION OF CARBON-ENHANCED METAL-POOR STARS

From center to surface an AGB star consists of: a C–O degenerate core, an He-burning shell, an He-rich intershell, an H-burning shell, and an extended convective envelope, which is constantly eroded by strong stellar winds. When the whole envelope is lost the hot stellar core is left to evolve into the post-AGB phase, then it becomes the central star of a planetary nebula and eventually a cooling white dwarf. For a detailed review on AGB evolution we refer the reader to Herwig (2005). A peculiarity of the AGB evolution is that the H- and the He-burning shells are activated alternately. Most of the time the H-burning shell is active and freshly made He increases the mass of the intershell for periods of time of around 10³–10⁵ yr, depending on the stellar mass, with shorter periods associated with more massive AGB stars. The increasing mass of the intershell leads to an increase in the temperature at its base, until He burning is suddenly triggered resulting in a thermal pulse (TP). The energy released in this first phase of He-shell burning cannot be carried out radiatively and a convective region develops in the intershell, which lasts a few hundred years (the exact duration depends on mass) and normally extends up to just below the H-burning shell. In low-mass and low-metallicity AGB stars this convective region can extend into the H-rich layer and peculiar nucleosynthesis is expected to occur there (Campbell & Lattanzio 2008), including s-process nucleosynthesis (Cristallo et al. 2009a). Mild proton-ingestion events of this type are found in our low-mass models. They have a strong impact on the s-process distributions and will be discussed in detail in Section 3.

A TP causes the star to expand, which in turn causes the H-burning shell to be quenched. At this stage, the convective envelope can penetrate into the underlying layers and carry to the stellar surface material processed by H and He burning (a process referred to as the third dredge-up, or TDU for short), while He burning carries on radiatively. Eventually, He burning is also quenched and the whole cycle of H burning, TP, and TDU starts again. This cycle is repeated a few to hundreds of times depending on the time it takes for the mass loss to erode the envelope, which in turn depends on the initial stellar mass, and possibly the metallicity. The best evidence that TPs and the TDU occur in AGB stars is the formation of carbon stars, which show C > O at their surface as well as the presence of the radioactive element Tc (Wallerstein & Knapp 1998). This observed composition is explained by the combined effect of partial He burning during TPs (which produces more C than O) and the s-process (which produces Tc), along with the TDU, which carries a fraction of the intershell to the stellar envelope. As mentioned above, HBB is present in AGB stars with masses higher than ≃3 M_☉ at the metallicity considered here, converting C into N.

The s-process occurs in the intershell of AGB stars where He is abundant and (α, n) reactions can be efficiently activated. For a detailed review on the s-process in AGB stars we refer the reader to Busso et al. (1999). There are two possible neutron sources: the ¹³C(α, n)¹⁶O and the ²²Ne(α, n)²⁵Mg reactions. The ²²Ne neutron source is activated at temperatures higher than roughly 300 × 10⁶ K (hereafter MK). These high temperatures can only be reached inside the convective TPs and mainly in AGB stars more massive than ≃3 M_☉ (Truran & Iben 1977). The ²²Ne is produced via double α captures on ¹⁴N when the H-burning ashes are engulfed in the TP. The resulting neutron flux occurs over a timescale of the order of a few years with relatively high neutron densities, up to 10¹⁴ n cm⁻³ (Gallino et al. 1998; Lugaro et al. 2003; van Raai et al. 2011). The ¹³C neutron source is instead activated at temperatures as low as 90 MK (Cameron 1955), but there is not enough ¹³C in the H-burning ashes of AGB stars to make it an efficient neutron source. Without this low-temperature neutron source it is not possible to explain the s-process enhancements which have been observed in AGB stars since the 1950s (Merrill 1952; Smith & Lambert 1986). This is because most of the observed s-process-enriched AGB stars are of relatively low mass (<4 M_☉) where the temperature is not high enough to activate the ²²Ne neutron source. Thus, it has become standard practice to artificially include the ¹³C neutron source in AGB models of the s-process.

The most likely way to form the required ¹³C is to allow some extra mixing of protons to occur following each TDU episode at the deepest extent of the convective envelope, where a sharp composition discontinuity forms at the interface between the H-rich envelope and the He-rich intershell. This extra mixing may be physically explained by the effect of semi-convection (Hollowell & Iben 1988), hydrodynamical overshoot (Herwig 2000; Cristallo et al. 2009b), or gravity waves (Denissenkov & Tout 2003). However, there is no agreement yet on which of these proposed mechanisms is responsible for the extra mixing and thus we do not know what its detailed features are. The protons are captured by the abundant ¹²C, producing a layer rich in ¹³C and ¹⁴N (the ¹³C pocket, Hollowell & Iben 1988; Gallino et al. 1998; Goriely & Mowlavi 2000; Lugaro et al. 2003; Cristallo et al. 2009b). The resulting ¹³C usually burns under radiative conditions via ¹³C(α, n)¹⁶O before the onset of the following TP (Straniero et al. 1995; Gallino et al. 1998). This burning releases a large number of free neutrons on a long timescale (around 10⁴ yr) and with low neutron densities <10⁸ n cm⁻³. In the region of the pocket where ¹⁴N is more abundant than ¹³C, no s-process nucleosynthesis occurs owing to the dominance of the ¹⁴N(n, p)¹⁴C reaction over neutron captures by Fe seed nuclei and their progeny. In the intermediate-mass AGB stars that experience HBB, the formation of a ¹³C pocket may be inhibited by proton captures occurring at the hot base of the convective envelope during the TDU, which produce ¹⁴N and not ¹³C (Goriely & Siess 2004). Extremely deep TDU may also inhibit the activation of the ¹³C(α, n)¹⁶O reaction by penetrating into regions of the stellar core with a low abundance of ⁴He (Herwig 2004a). While these processes do need further investigation, on first sight a dichotomy appears in models of the s-process in AGB stars where the ¹³C neutron source is activated in the low-mass models, while the ²²Ne neutron source is activated in the higher-mass models. At the metallicity we consider here the mass at which the importance of the two neutron sources switches is ≃3 M_☉ (Goriely & Siess 2004).

An interesting feature of the ¹³C neutron source is that it is a primary neutron source because it is produced from the H and He originally present in the stars. Since the total time-integrated neutron flux τ from the ¹³C source is proportional to ¹³C/Z, the s-process predictions have a strong dependence on the metallicity of the star (Clayton 1988; Busso et al. 2001). Specifically, with τ increasing when decreasing Z, heavier elements are produced at lower metallicities. This property allowed Gallino et al. (1998) to predict the existence of low-metallicity Pb-rich stars, which was confirmed by observations of Pb-rich CEMP stars (e.g., Van Eck et al. 2003). On the other hand, the ²²Ne neutron source is traditionally considered to be a secondary neutron source, since it relies on the original CNO nuclei present in the star. However, at the low metallicities considered here ²²Ne also has an important primary component because the N abundance in the H-burning ashes increases significantly from its initial value. This is due to the effect of TDU carrying primary ¹²C to the envelope, which is then converted into N by H burning.

The ¹³C and the ²²Ne neutron sources produce neutron fluxes and neutron-capture nucleosynthesis with different features. In summary, it has been shown that the ¹³C source is responsible for the production of the bulk of the s-process elements in low-mass AGB stars reaching, as mentioned above, up to Pb at low metallicities (Gallino et al. 1998). Branching points on the s-process path are mostly closed during this neutron flux. The ²²Ne source instead produces smaller abundances of the s-process elements but it efficiently activates branching points along the s-process path (Abia et al. 2001; van Raai et al. 2011). Note that at low metallicity ²²Ne is both a neutron source and a neutron poison via (n, γ) reactions, leading to the production of light elements such as Na and Mg (Herwig 2004b; Karakas 2010; Bisterzo et al. 2010). Based on our models in Section 3 we revise the different neutron-capture scenarios for AGB stars of low metallicity.

To study the nucleosynthesis of the elements up to Bi we have used a post-processing code that takes as input the stellar evolutionary sequences described above. The stellar inputs are the temperature, density, and convective velocities (for convective regions) at each mass shell in the star as a function of time during the life of the star. Convective velocities are needed because the code includes both the changes due to nuclear reactions and those due to mixing in the equations to be solved for the abundances. An implicit method is used to solve N equations, N being the number of species included in the nuclear network, by inverting a large matrix containing N² elements. The code has been previously described in detail by, e.g., Cannon (1993), Karakas & Lattanzio (2007), and Lugaro et al. (2004).

Given that the number of species N determines how large the matrix to be solved is, in the interest of making the computation relatively fast (from a few days to a few weeks, depending on the initial stellar mass), we used for this work a basic s-process network, where we included a total of 320 species mostly along the valley of β stability and 2336 reactions. We based our nuclear reaction network on the JINA REACLIB database, as of 2009 May. From this database, we updated the rates for the neutron source reactions: we took the ¹³C(α, n)¹⁶O rate from Heil et al. (2008) and the ²²Ne(α, n)²⁵Mg from Karakas et al. (2006).

For the initial composition of most models we used the solar distribution of abundances from Asplund et al. (2009) scaled down to [Fe/H] = −2.3. Solar abundances of C, N, O, Ne, Mg, Si, S, Ar, and Fe are the pre-solar nebula values from Table 5 of Asplund et al. (2009); F is the meteoritic value of log  (F)_☉ = 4.42 from Table 1 of the same paper (chosen because it has a lower uncertainty), and for many of the elements heavier than Fe we use the meteoritic values for the solar abundances (e.g., Sr, Eu, and Pb).

We also computed some models by varying the initial composition to: (1) the values predicted by galactic chemical evolution models for solar neighborhood stars of metallicity [Fe/H] = −2.3 from Kobayashi et al. (2011)—this case affects only the initial composition of the elements up to Zn; (2) the values obtained assuming that the star was born with an r-process enrichment and in some cases also an s-process enhancement. Enhancements in the initial r-process abundances may have arisen from the pollution from one or a few primordial SNeII. Enhancements in the initial s-process abundances may have arisen from pollution from primordial massive stars producing s-process elements during core He and shell C burning (Pignatari et al. 2008), or via the still unidentified "light element primary process" (Travaglio et al. 2004; Montes et al. 2007). To compute these r- and s-process enhancements we used the s and r contributions given in Sneden et al. (2008).

The methodology for the inclusion of the ¹³C pocket deserves specific attention as the formation of this neutron source is still the main uncertainty in the s-process models. For reasons discussed above (Section 1.1), we introduced a ¹³C pocket for models of masses <4 M_☉, while we did not introduce it for the higher masses, except in one test case, where we introduced a ¹³C pocket in the Stromlo 5.5 M_☉ model. Also the 3 M_☉ model was computed with and without the inclusion of the ¹³C neutron source to illustrate in detail the different effects of the two neutron sources. This mass lies where the two regimes switch: the s-process dominated by the ¹³C neutron source and the s-process dominated by the ²²Ne neutron source.

The inclusion of the ¹³C neutron source was performed artificially during the post-processing by forcing the code to mix a small amount of protons from the envelope into the intershell at the deepest extent of each TDU episode. All the studies to date investigating the formation of the ¹³C pocket by various mechanisms have found that the proton abundance in the intershell decreases monotonically. Hence, we apply the basic assumption that the proton abundance in the intershell decreases monotonically from the envelope value of ≃0.7 to a minimum value of 10⁻⁴ at a given point in mass located at "M_mix" below the base of the envelope. This method is described in some more detail in Lugaro et al. (2004) and is very similar to that used by Goriely & Mowlavi (2000). We made the further choice that the proton abundance decreases exponentially, i.e., linearly in a logarithmic scale. Studies of the formation of the ¹³C pocket have found profiles that can slightly differ from this basic assumption, as well as from each other. For example, compare the profile shown in Figure 4 of Herwig (2000) to the profiles shown in Figures 1–4 of Cristallo et al. (2009b). Goriely & Mowlavi (2000) tested the effect of using a "slow" or a "fast" profile (see their Figure 10) and found no major differences in the s-process distribution.

We tested different values of our parameter M_mix, representing the extent in mass of the proton profile. The extent in mass of the resulting ¹³C pocket is <M_mix (Goriely & Mowlavi 2000; Lugaro et al. 2003; Cristallo et al. 2009a). For some models of mass <3 M_☉ we tested M_mix = 0, 0.0006 M_☉, 0.001 M_☉, 0.002 M_☉, and 0.004 M_☉. For the 3 M_☉ models we used M_mix = 0, 0.0005 M_☉ and 0.001 M_☉. For the 5.5 M_☉ model, we tested M_mix = 0 and 0.0001 M_☉. To keep the standard assumption that the ¹³C pocket should be a minor fraction (≃1/10–1/20) of the intershell mass (Gallino et al. 1998; Goriely & Mowlavi 2000) we had to decrease the value of M_mix with the stellar mass because the mass of the intershell is at least an order of magnitude smaller in the intermediate-mass AGB stars than in the lower-mass models: around 10⁻² M_☉ in a 1.5 M_☉ model compared to around 10⁻³ M_☉ in a 6 M_☉ model.

The C, N, O, and F abundances in low-metallicity AGB stars have been already extensively presented by Herwig (2004b), Karakas & Lattanzio (2007), Lugaro et al. (2008), Stancliffe (2009), and Karakas (2010) so here we only discuss them briefly. In Figure 3 we present the final [X/Fe] for the elements lighter than Fe for the same selected models of Figure 2. In Table 5 we present the final [X/Fe] for C, N, O, F, and Ne for all our models and the indicated choices of M_mix. These light elements mostly do not change when varying this free parameter: the only exceptions are for F and Ne. For example, F decreases by 0.41 dex and Ne by 0.45 dex in the Stromlo 1 M_☉ model when M_mix = 0. All of our models are very rich in carbon and relatively poor in nitrogen, with typical [C/N] varying from ≃1 dex, for the 0.9 and 1 M_☉ models, to ≃2 dex for models up to 3 M_☉ (Figure 3). From this stellar mass and above, HBB is activated and the nitrogen abundance increases and quickly overcomes the abundance of carbon. Oxygen is produced in our models, ranging from 0.42 to 1.34 dex. A typical product of AGB nucleosynthesis is fluorine, which we find to be strongly produced in our models: [F/Fe] is comparable to [C/Fe] up to the stellar mass where HBB is efficient. For higher masses fluorine is destroyed by proton captures. Neon is also a typical product of AGB stars, in the form of ²²Ne, and its production factor varies in our models from ∼+1 to ∼+3 dex (Karakas & Lattanzio 2003). This element is observable in planetary nebulae and can be used as a further probe of AGB nucleosynthesis. The Ne abundance was derived for the halo planetary nebula BoBn 1, which is also rich in C, N, and F, and can be explained by low-metallicity AGB models of ≈1.5 M_☉ (Otsuka et al. 2010). The production of F and Ne is primary in low-metallicity AGB stars because it is driven by the TDU of ¹²C. Our 2 M_☉ models produce very similar results to the model of Cristallo et al. (2009b).

The light elements Na and Mg require specific discussion because they depend on the abundance of ²²Ne—which in turn depends on the amount of TDU and therefore on the stellar mass—on the formation of the ¹³C pocket, and on the overall neutron flux. Our predictions for these elements are shown in Figure 3 for selected models and listed in Tables 3 and 4 for all the models. Na is strongly produced in our models due to proton and neutron captures on ²²Ne. For models without HBB, the increasing efficiency of the TDU with stellar mass means that the amount of primary ²²Ne and Na in the envelope also increases with mass up to ≃2 M_☉; this is illustrated in Figure 3. In these low-mass models the ²³Na is synthesized by proton captures on ²²Ne during both the interpulse and convective TP (see also Goriely & Mowlavi 2000; Cristallo et al. 2009b). Neutron captures on ²²Ne occur any time there are neutrons available given the large abundance of ²²Ne present in the intershell (see also the discussions in Karakas 2010; Bisterzo et al. 2010). In models of masses higher than around 2 M_☉, less Na is produced because the total amount of TDU as well as M_mix are smaller and because Na production by HBB is sensitive to mass. Na production requires higher temperatures than CNO cycling and takes place most efficiently at ≈4 M_☉; at higher masses the temperature at the base of the envelope is so high that the Na is effectively destroyed by proton captures. At the other end of our mass range, in the lowest-mass models 0.9 and 1 M_☉, we also obtain a range of Na abundances, depending on the features of the proton ingestion and the inclusion of the ¹³C pocket. The 0.9 M_☉ and 1.0 M_☉ Stromlo models are the only models to have [Na/Fe] significantly lower than [Ba/Fe]. In all the other models [Na/Fe] is comparable and even higher than [Ba/Fe].

Magnesium is also significantly produced in our models when the mass is greater than ≃1.25 M_☉. In the models with mass lower than ≃2.5 M_☉ all the isotopes of Mg are produced via neutron captures in the ¹³C pocket starting on ²³Na. For this reason the abundance of Mg follows the abundance of Na, keeping roughly to one order of magnitude lower than Na. In the models with mass higher than ≃2.5 M_☉, the heavier isotopes of Mg, ²⁵Mg and ²⁶Mg, are also produced by α-capture during the TP via the ²²Ne + α reactions. HBB in the most massive models leads to activation of the Mg–Al chains, which produces ²⁶Mg and ²⁶Al and alters the composition of the He-shell prior to each TP (Karakas & Lattanzio 2003). In these higher mass models the abundance of Mg becomes comparable to, and even larger than, that of Na.

In Figure 4, we show the Mg isotopic ratios at the stellar surface at the end of the computed evolution. These ratios vary by up to three orders of magnitude with the stellar mass. The derivation of the Mg isotopic ratios from the spectra of CEMP stars would represent very strong constraints on the mass of the donor AGB star. We note that the models shown in Figure 4 were computed using the solar ratios of ²⁴Mg/²⁵Mg = 7.9 and ²⁴Mg/²⁶Mg = 7.2. In selected models with mass ⩾1.5 M_☉, we tested the effect of starting with ratios derived from the detailed Galactic chemical evolution models of Kobayashi et al. (2011), i.e., ²⁴Mg/²⁵Mg = 158 and ²⁴Mg/²⁶Mg = 179. While this did not have any impact on the final ratios for the masses considered, the final compositions of stars with masses lower than 1.5 M_☉ may be more strongly effected by the initial composition. This is because these models experience less dredge-up and for this reason, the final Mg isotopic ratios are likely to be higher than given here.

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In Table 6, we present a comparison of our results with those of Cristallo et al. (2009b) and Bisterzo et al. (2010) for the same element ratios as shown in Tables 3 and 4. The results from the stars and Cristallo's 2 M_☉ models show very good agreement, both in the light (see Table 5) and in the heavy elements. Both these models were computed with the inclusion of molecular opacities. The Stromlo 2 M_☉ models both with and without inclusion of realistic molecular opacities produces higher abundances of both light and heavy elements in AGB models, with respect to Fe. Varying the mass of the ¹³C pocket can also lead to significant changes in the predicted abundances of both light and heavy elements in AGB models. The model by Cristallo et al. (2009b) includes the ¹³C pocket using a self-consistent approach based on convective overshooting motivated by hydrodynamical simulation of convective/radiative boundaries. These authors found that the extent in mass of the pocket decreases with the TP number. In our models, instead we kept M_mix constant with the TP number. This explains why the abundances with respect to Fe by Cristallo et al. (2009b) are closer to those computed with the smaller M_mix value in our models. In any case, the ratios [Ba/Sr], [Ba/Eu], and [Pb/Ba] are very similar in all three sets of models. This comparison indicates that the s-process distributions obtained using our artificial treatment of the ¹³C pocket are in agreement with those obtained using the self-consistent treatment of Cristallo et al. (2009b).

Bisterzo et al. (2010) treated the neutron production in the ¹³C pocket as a free parameter and varied its efficiency with respect to their standard (ST) case, which reproduces the solar system s-process abundances at Z = 0.01. This approach results in a larger allowed range of variations of the resulting abundances. The results from the Stromlo, stars, and Cristallo et al. models are typically included within the range predicted by Bisterzo et al. (2010).

Overall, Table 6 presents a relatively consistent set of results for neutron-capture nucleosynthesis in AGB stars of low metallicity. The main features of these models are the production of s-process elements up to +3 dex and a variety of s-process distributions, typically with 0 < [Ba/Sr] < 1.0, [Ba/Eu] ≃ 0.9, and 0 < [Pb/Ba] < 2.0. [Ba/Eu] reaches lower values only if there is not a large s-process enhancement, i.e., [Ba/Fe] < 1.0. The "radiative + convective" model of Goriely & Siess (2005) produces a heavy-element distribution in the range displayed in Table 6, with [Ba/Sr] ≃ 0.9, [Ba/Eu] ≃ 0.8, and [Pb/Ba] ≃ 1.0; see their Figure 6. The production of Na and Mg always increases with the stellar mass, something also seen in the Bisterzo et al. (2010) models. Their 1.3 M_☉ models produce lower Na than the 1.25 M_☉ models presented here probably because these models do not include production of Na via proton captures.

Finally we find that, in spite of the fact that the mixing scheme strongly affects the proton-ingestion episodes, our results for the s-process are qualitatively the same as the results presented by Cristallo et al. (2009a) for a 1.5 M_☉ star of metallicity half of that used in our models (Z = 5 × 10⁻⁵) which also experience a proton-ingestion episode. From Figure 6 of Cristallo et al. (2009a) we see that after the first TDU episode, which records the effect of the proton-ingestion event, the s-process favors the production of the lighter elements, just like our models. The final [Ba/Sr] ≃ 0.40, [Ba/Eu] ≃ 0.95, and [Pb/Ba] ≃ 0.85 are close to those obtained in our 1.25 M_☉ and 1.5 M_☉ models.

Figure 5 presents an overview of the observational data for CEMP stars and carbon-normal metal-poor stars of the rI and rII subclasses following the data and classification of Masseron et al. (2010). In this and all the following figures we used different symbols for stars in different metallicity ranges, as indicated in the caption. There are no noticeable differences in the distribution of any of the plotted abundances when changing the metallicity, the only noticeable properties being that there are more CEMP-s than CEMP-s/r stars with [Fe/H] >−2, and more CEMP-no stars with [Fe/H] <−3 than in the other CEMP star classes. The different classes of CEMP stars appear to be fairly well distinguished in this plot. Most of the CEMP-no Eu data are upper limits, and the only CEMP-rII star, CS 22892-052, appears indistinguishable from the rest of the rII group in terms of its heavy element composition.

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The two groups of CEMP-s and CEMP-s/r appear to be distinct, with CEMP-s/r having higher [Eu/Fe] ratio in absolute terms, but also higher [Ba/Fe] on average. In Figure 5, we plot the correlation lines of the two groups. The similar slope, around 1, indicates that for both classes the linear dependence would be preserved in a non-logarithmic scale. What counts here is the y-intercept, which corresponds to [Ba/Eu] and thus defines the slope of the linear correlation between the two variables in a non-logarithmic scale: in the case of CEMP-s stars the process responsible for their compositions produces on average around 400 times more Ba than Eu, with a spread of a factor of about two; in the case of CEMP-s/r stars, instead, the process responsible for their composition produces on average around 200 times more Ba than Eu, with a spread of a factor of about two. Not surprisingly, the r stars also sit on a correlation line with a slope of around 1 and y-intercept of around −0.6, corresponding to the solar-system r-process Ba/Eu ratio of about 10. In the case of CEMP-s the [Ba/Eu] value of +0.89 agrees very well with the typical [Ba/Eu] ratios reported in Tables 3, 4, and 6, while in the case of CEMP-s/r, the value of +0.6 does not agree with any AGB models producing [Ba/Fe] > 1.

The three CEMP-s/r with the lowest Ba abundance in their group may belong instead to the CEMP-s group. This seems likely for BS 16080-175, with [Ba/Fe] = 1.55 and [Eu/Fe] = 1.05, which can be reclassified as CEMP-s within the 0.07 dex uncertainties. The other two outliers BS 17436-058, with [Ba/Fe] = 1.60 and [Eu/Fe] = 1.17, and HD 187861, with [Ba/Fe] = 1.39 and [Eu/Fe] = 1.34, may be reclassified as CEMP-s within 2σ of their 0.11 and 0.20 dex [Eu/Fe] uncertainties. However, we note that [La/Fe] = 1.73 ± 0.2 in HD 187861 indicates that it should be classified as a CEMP-s/r star. More observations and analysis of these stars will help to clarify their status.

Another interesting case is that of CS 30322-023 ([Ba/Fe] = 0.52 and [Eu/Fe] = −0.63), which has been previously classified as a CEMP-s (Aoki et al. 2007) and was included in the CEMP-low-s class by Masseron et al. (2010). This star has the highest [N/C] ≃ +2 and the lowest, and only negative, [Eu/Fe] ratio in the sample, as well as [Fe/H] ≃ −3.3, at the lower end of the CEMP-s range. Another three stars have [N/C] >1: CS 22949-037, CS 22960-053, and CS 29528-041, which all have [Fe/H] similar to CS 30322-023 and are classified as CEMP-no by Masseron et al. (2010). However, CS 22960-053 and CS 29528-041 have [Ba/Fe] ≃ 0.9 but no Eu observations, so they may be belong to the CEMP-low-s class (as defined by Masseron et al. 2010), together with CS 30322-023. In fact, Aoki et al. (2007) classified both CS 22960-053 and CS 30322-023 as CEMP-s stars following Ryan et al. (2005).

Figure 6 includes a selection of our model predictions together with the data of Figure 5. In this as in all the following figures the solid lines are a visual aid connecting the predicted envelope composition after each TDU episode. These prediction lines are not straight because the composition of the s-process material in the He intershell evolves with time and because the variables are shown on a logarithmic scale. The process of transferring and mixing AGB material onto the companion star would result in mixing lines connecting the AGB composition with the initial composition. These mixing lines closely follow the plotted AGB evolution lines and are not shown in the figure. We include in the figure the same stellar models plotted in Figures 2 and 3, representing cases where the neutron-capture nucleosynthesis occurs in the different regimes described in Section 3. All the plotted models start from a solar composition, [Ba/Fe] = [Eu/Fe] = 0, except for the three 2 M_☉ models from the Stromlo set with different initial composition: [r/Fe] = +0.4, [r/Fe] = +1.0, and [r/Fe] = +2.0 included in the lower panel of Figure 6.

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All the models produce both Ba and Eu. The predictions lines follow the trend of the CEMP-s group. A small spread in the initial r-process composition up to [r/Fe] = ±0.4 dex, which is within the composition observed in field stars of the corresponding metallicity (see, e.g., Figure 2 of Bisterzo et al. 2011b), is enough to cover the data. CS 30322-023 indicates an initial negative [r/Fe] ≈−0.6, which would also be within the observed range at the corresponding metallicity [Fe/H] = −3.3. On the other hand, AGB models do not produce the high [Eu/Fe] observed in CEMP-s/r stars.

One solution to this problem may be to increase the initial [r/Fe] abundance by up to +2 dex under the assumption that the r-process enrichment is primordial in these systems (bottom panel of Figure 6). This primordial enrichment can arise by assuming that the molecular clouds where CEMP-s/r stars formed were polluted by r-process elements from a nearby SNII. This scenario provides a solution for the highest measured [Eu/Fe] abundances in CEMP-s/r stars as discussed in detail by Bisterzo et al. (2011b). However, it comes with three problems, some of which have been discussed previously (see, e.g., Jonsell et al. 2006; Lugaro et al. 2009). (1) The initial [r/Fe] value does not affect the final [Ba/Fe] value, which remains constant; hence the linear correlation observed between the Eu and Ba enhancements in the CEMP-s/r sample is not reproduced. (2) The number of rII stars (around 10) is smaller than the number of CEMP-s/r stars (about 30), so it seems unlikely that CEMP-s/r stars evolved from rII stars in binary systems because these should make up a fraction of all rII stars. This does not appear to be an observational bias since stellar surveys such as the Hamburg/ESO R-process-Enhanced Stars have specifically targeted rI and rII stars. It is also difficult to invoke supernova triggering as the cause for the formation of binaries: though early work indicated this link possible (Vanhala & Cameron 1998), it is now understood that formation of binaries occurs very naturally whenever a turbulent velocity field is present in a molecular cloud (Bate 2009). Furthermore the whole concept of supernova-triggered star formation is still controversial, both observationally and theoretically (Leão et al. 2009; Elmegreen 2011). (3) The metallicity distribution is different for the two samples: [Fe/H] is centered at ≃ −2.5 for CEMP-s/r stars and at ≃ −2.8 for rII stars.

In this and the following subsections we compare our model predictions to the observations using "intrinsic" indicators, i.e., elemental ratios that include only elements that are produced in AGB stars. Different from the [Ba/Fe] and [Eu/Fe] ratios considered above, where Fe is not produced in AGB models, the "intrinsic" ratios move away from their initial solar values toward their s-process values already after a small number of TPs (though this is less true for the 5 M_☉ model where the dilution in the envelope is much larger than in the other models). For the CEMP stars, this means that the intrinsic ratios are close to the AGB values even for relatively large dilutions of the AGB material onto the CEMP star. Hence, intrinsic ratios are, in a first approximation, independent of the TDU, the mass loss, the stellar lifetime, and the accretion and mixing processes on the binary companion. Instead, they mostly constrain the nucleosynthesis occurring in the deep layers of the star, and in our context, the neutron source and the neutron flux for the s-process.

We also note that we consider here ratios of elements that are all significantly produced in AGB stars, so that variations in the initial composition of these elements due to the chemical evolution of the Galaxy do not have a significant effect on the AGB predictions. Using the initial composition from Kobayashi et al. (2011) (i.e., [F/Fe], [Na/Fe], and [Mg/Fe] equal to −1.5, −0.2, and +0.3 dex, respectively) did not lead to modifications of the results plotted and discussed here in selected models with mass ⩾1.5 M_☉. Only the [Mg/Fe] ratio of stars with masses lower than 1.5 M_☉ may be more strongly effected by the initial composition because these models experience less dredge-up. For these models, this effect would at most shift the [Mg/hs] ratios by +0.3 dex.

Figure 7 shows the [ls/hs] and [Mg/hs] observed in CEMP-s and CEMP-s/r stars together with our model predictions. The dispersion of the data is quite large, however, it clearly shows the presence of a linear correlation with slope ≃ 1. The y-intercept ≃ 0 indicates that this correlation is most likely due to variations in the amount of the hs elements, Ba, La, and Ce. As noted in previous studies (see, e.g., Figure 9 and Figure 6 of Jonsell et al. 2006; Bisterzo et al. 2011b,, respectively), the CEMP-s/r stars are separated from the CEMP-s in that they show [ls/hs] lower by at least 0.5 dex. Here we show that CEMP-s/r stars also show lower [Mg/hs] than CEMP-s. It seems obvious to ascribe these differences to the fact that CEMP-s/r stars have, on average, higher hs abundances than CEMP-s (Figure 6).

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All our models produce [ls/hs] > −1. This derives from the basic way in which the s-process operates (see also Busso et al. 2001). As the neutron flux increases, first the ls elements, then the hs elements, and finally Pb are produced. As the flux goes through these peaks the abundances reach their equilibrium values, above which it is not possible to increase them. This is similar to the situation of buckets connected to each other. When water is poured into the first bucket, the bucket fills up. Once it is full, the water starts moving into the second bucket and so on. The relative amount of water in the two buckets is only determined by their size (the neutron-capture cross sections in the case of the ls and hs elements) and not by how much more water is poured into the system.⁹ Hence, the minimum [ls/hs] ratio is the value expected when the s-process is in equilibrium through the first and the second s-process peaks (with σ_AN_A = constant). This [ls/hs] ratio is roughly equal to −1 because the neutron-capture cross section and the solar abundance of, e.g., ¹³⁸Ba are about 1.5 and 6 times lower (respectively) than those of, e.g., ⁸⁸Sr. On the other hand, the [Pb/hs] ratio cannot reach an equilibrium value because Pb is at the end of the s-process chain. The Pb abundance can continue to grow indefinitely as more neutrons are made available.

Unless we ascribe the overall mismatch to the large error bars, which still would not explain why CEMP-s/r stars are separated from CEMP-s stars in terms of both [ls/hs] and [Mg/hs], it appears that a fraction of CEMP-s/r stars do not carry a typical pure s-process signature in the relative abundances of their s-process elements. Together with Figure 6, Figure 7 demonstrates that (1) CEMP-s/r stars have the highest abundances of hs elements, and thus the lowest [ls/hs] and [Mg/hs] down to −2,¹⁰ and (2) our s-process models can match the observations well for CEMP-s stars, but cannot match those of CEMP-s/r stars, not only in terms of their Eu abundance, but also in terms of their ls and hs abundances.

Focusing on the CEMP-s stars only, they appear to be overall better matched by the models with the ¹³C neutron source burning radiatively. The 5 M_☉ model, which could not be ruled out as a match to the observations on the basis of the Ba and Eu enhancements (Figure 6), produces too much Mg and ls with respect to hs to be able to match any of the CEMP-s and CEMP-s/r data. Inspection of the results for the 5.5 M_☉ model in Table 4 shows that inclusion of the ¹³C pocket does not solve this problem. On the other hand, the composition of CS 30322-023, the only CEMP-s with [Mg/hs] > 0, is compatible with the models of 3 M_☉ and 5.5 M_☉ computed with the inclusion of the ¹³C pocket.

Figure 8 presents the [Pb/hs] and [Mg/hs] observed in CEMP-s and CEMP-s/r stars together with our model predictions. The dispersion of the data is large and there is no obvious correlation between the two ratios. While the two classes are different in all the other abundance ratios considered so far, the distribution of [Pb/hs] in CEMP-s and CEMP-s/r stars covers the same range. The models with the ¹³C neutron source burning radiatively provide a good match to CEMP-s stars, where the spread in the [Pb/hs] ratio can be explained by varying the TDU number or, equivalently, the initial stellar mass (see also Bonačić Marinović et al. 2007). The models where the ¹³C neutron source burns convectively can provide a match for the lower end of the [Pb/hs] ratios observed in CEMP-s/r stars, however, as shown in the previous section, they do not match the [ls/hs] values of CEMP-s/r. Even though the error bars are large, CEMP-s/r stars with high [Pb/hs] ratios are not matched by any models.

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We note that the 5 M_☉ model does not cover any of the data points. As in the case of the [ls/hs] versus [Mg/hs] the composition of CS 30322-023 can be matched by intermediate-mass AGB models (e.g., 3 M_☉ and 5.5 M_☉) if a ¹³C pocket is included.

Figures 9 and 10 present the [Na/hs] and [F/hs] ratios plotted versus the intrinsic s-process indicators, [ls/hs] and [Pb/hs]. There are not enough data to be able to determine if CEMP-s and CEMP-s/r stars are distinguished in their [Na/hs] and [F/hs] abundance ratios. The models where the ¹³C neutron source burns radiatively, which provide a good match to the overall composition of CEMP-s stars in terms of their [Mg/hs], [ls/hs], and [Pb/hs] compositions, typically produce too much Na and F with respect to the hs elements to match any of the observational data. One exception is the case of CS 30322-023, the only CEMP-s with [Na/hs] > 0: this star is again distinguished from the rest of the CEMP-s stars and as mentioned above may carry the signature of a more massive AGB star. A better overall match to [Na/hs] and [F/hs] is provided by the low-mass Stromlo models where the ¹³C neutron source burns convectively; however, these models do not cover the corresponding [Pb/hs] (and [ls/hs]) ratios.

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As discussed in Section 3.1 the predicted abundance of Na is made by both neutron and proton captures on ²²Ne. The fraction made by neutron captures is linked to the production of Mg, and the models where the ¹³C neutron source burns radiatively provide a good match to the [Mg/hs] observed in CEMP-s. The fraction made via proton captures, on the other hand, is disconnected from the production of Mg, but depends on the details of the proton-ingestion episodes, the profile of protons introduced to make the ¹³C pocket, and the proton-capture reaction rates of ²²Ne and ²³Na (which are uncertain, see Iliadis et al. 2010). All these uncertainties need to be tested to see if it is possible to match the observed [Na/hs] and [Mg/hs] ratios, which indicate that the production of Na via proton captures should be minimized. During the proton diffusion leading to the formation of the ¹³C pocket, a proton profile weighed toward a low proton abundance (e.g., the "fast" profile in Figure 10 of Goriely & Mowlavi 2000) may contribute to the solution of this problem. This possibility will be investigated in detail in a forthcoming study and has the potential to provide a strong constraint on the formation of the ¹³C pocket. A lower value of the ²²Ne(p, γ)²³Na reaction rate or a higher value of the ²³Na(p, α)²⁰Ne reaction rate could also help lower the predicted Na abundances.

The predicted [F/hs] ratios are also higher than the observed upper limits. The operation of some form of deep mixing of the envelope material of the AGB companion star down to regions where H burning occurs, which is also invoked to explain the observed high N enhancements and low ¹²C/¹³C ratios (see, e.g., Stancliffe 2010) may contribute to decreasing the F abundance in these stars. Further investigations are required to test this idea. Note that in the 5 M_☉ model with HBB, F is strongly depleted by proton captures.