A&A 469, 1013-1025 (2007)
A. Bonacic Marinovic - R. G. Izzard - M. Lugaro - O. R. Pols
Sterrekundig Instituut Utrecht (SIU), Universiteit Utrecht, PO Box 80000, 3508 TA Utrecht, The Netherlands
Received 2 December 2006 / Accepted 5 March 2007
Context. Thermally pulsating asymptotic giant branch (AGB) stars are the main producers of slow neutron capture (s-) process elements, but there are still large uncertainties associated with the formation of the main neutron source, 13C, and with the physics of these stars in general. Observations of s-process element enhancements in stars can be used as constraints on theoretical models.
Aims. For the first time we have applied stellar population synthesis to the problem of s-process nucleosynthesis in AGB stars, in order to derive constraints on free parameters describing the physics behind the third dredge-up and the properties of the neutron source.
Methods. We utilize a rapid evolution and nucleosynthesis code to synthesize different populations of s-enhanced stars, and compare them to their observational counterparts to find out which values of the free parameters in the code produce synthetic populations that fit the observed populations best. These free parameters are the amount of third dredge-up, the minimum core mass for third dredge-up, the effectiveness of 13C as a source of neutrons, and the size in mass of the 13C pocket.
Results. We find that galactic disk objects are reproduced by a spread of a factor of two in the effectiveness of the 13C neutron source. Lower metallicity objects can be reproduced only by lowering the average value of the effectiveness of the 13C neutron source needed for the galactic disk objects by at least a factor of 3. Using observations of s-process elements in post-AGB stars as constraints we find that dredge-up has to start at a lower core mass than predicted by current theoretical models, that it has to be substantial ( ) in stars with mass , and that the mass of the 13C pocket must be about 1/40 that of the intershell region.
Key words: stars: AGB and post-AGB - stars: abundances - nuclear reactions, nucleosynthesis, abundances
The synthesis of s-process elements requires a source of free neutrons that can be captured by iron seeds to build up heavier nuclei along the isotopic stability valley. During the period of quiescent hydrogen-shell burning in between thermal pulses, the inter-pulse period, the conditions in the intershell (temperature, density and helium abundance) favour the reaction. For this reaction to be an effective source of free neutrons, a 13C-rich region or pocket in the intershell is needed. The production of 13C takes place by means of the reaction. It is assumed that the protons needed for this reaction penetrate the top of the intershell after TDU has occurred, as a result of a mixing process of an as yet uncertain nature: semi-convection (Hollowell & Iben 1988), hydrodynamical overshooting (Herwig et al. 1997), rotation (Herwig et al. 2003; Langer et al. 1999), and gravity waves (Denissenkov & Tout 2003) have all been proposed. As a consequence large uncertainties exist in the mass size of the 13C pocket and the 13C abundance profile inside it. In addition, due to the existence of several neutron poisons, especially 14N, not all free neutrons released by the reaction participate in the s-process, so the effectiveness of 13C as a neutron source for the s-process is also very uncertain. Another neutron burst is also released in the convective intershell during the thermal pulses due to the reaction, but the activation of this reaction is only marginal, which makes its contribution to the neutron exposure smaller than that of the reaction (see Busso et al. 1999, for a review of s-process nucleosynthesis in AGB stars).
Among stars that are enhanced in s-process elements (s-enhanced stars), one distinguishes intrinsic and extrinsic objects. Intrinsic s-enhanced stars are typically late-type giants of spectral classes S and C that show over-abundances of s-process elements (see, e.g., Smith & Lambert 1990; Abia et al. 2002), including the radioactive element Tc (Merrill 1952). With a half-life of a few million years, the presence of Tc lines indicates that the synthesis of s-process elements happened recently and in situ. Therefore intrinsic s-enhanced stars must be objects in the TP-AGB phase or post-AGB phase, which have recently experienced thermal pulses and TDU. The study of elemental abundances in these stars gives clues to the uncertain physical phenomena that take place in their interiors. Extrinsic s-enhanced stars also show over-abundances of s-process elements, but no Tc lines are present (Jorissen et al. 1993). This indicates that Tc has decayed since the s-process elements were synthesized. In addition, extrinsic stars are often observed to be in an evolution phase earlier than the TP-AGB; i.e., they are (sub)giants or main sequence stars. This implies that the s-process elements were not produced in situ, but that they have been accreted from a more massive TP-AGB companion star. These stars can therefore act as probes for studying mass transfer processes in binaries and for tracing the nucleosynthesis that occurred in their companion stars.
Busso et al. (2001) carried out a detailed comparison between observations of s-process enhanced stars and model predictions for the s-process in single AGB stars of different metallicities. They found that a large spread (a factor of 20) in the effectiveness (see Sect. 2.3 for a definition of this parameter) of the main neutron source 13C was needed to match spectroscopic observations at a given metallicity. Their analysis is based on a small set of initial stellar masses, and they have taken only the final abundances of their stars into consideration. In this paper we improve on their analysis by comparing observations of different types of stars belonging to the AGB family to stellar population synthesis models computed with the inclusion of the s-process. Our use of a rapid synthetic evolution code allows us to study a large set of finely spaced initial masses and metallicities and to trace the complete AGB evolution of each star. Using this method we are able to put much tighter constraints on the effectiveness of 13C as a neutron source. In addition, by comparing our models to post-AGB observations, we also put constraints on the minimum core mass for TDU, the TDU efficiency, and the size of the 13C pocket.
The AGB stellar models and the associated free parameters are described in Sect. 2. Our stellar population synthesis method is described in Sect. 3 and the results are presented in Sect. 4. In Sect. 5 we discuss our results and draw our conclusions.
Our TP-AGB star models are calculated with a modified version of the rapid evolutionary code by Hurley et al. (2000) and Izzard et al. (2004). We limit ourselves to a brief overview of the ingredients of this code and a discussion of the most important free parameters. A detailed description of our modifications and improvements can be found in Appendix A. Stellar evolution and nucleosynthesis are modelled synthetically, i.e., by means of analytical fits to the detailed evolutionary models of Karakas et al. (2002) and Stancliffe et al. (2004), and in some cases by tabular interpolation of these detailed models. Fits and interpolations depend on global stellar parameters, such as metallicity, mass, and core mass. This synthetic approach makes the calculation of the evolution very fast. The s-process element nucleosynthesis is calculated by interpolating results based on detailed models by Gallino et al. (1998), as described below. With these models we are able to follow the chemical abundances of stars as a function of time, from their initial abundances at the beginning of the main sequence, for which we use those of Anders & Grevesse (1989), to their chemically enhanced abundances of the latest evolutionary TP-AGB and post-AGB phases.
Many uncertainties exist in the detailed evolutionary models, which we treat as free parameters. In the following subsections we discuss the free parameters that are most relevant for stellar chemical evolution.
Detailed evolution models disagree on the amount of third
dredge-up that occurs in a star of a particular mass and metallicity,
i.e., how deep the convective envelope penetrates the intershell
(see, e.g., Frost & Lattanzio 1996;
Mowlavi 1999, for discussion).
The amount of dredge-up can be measured by the
third dredge-up efficiency
Comparing the absolute elemental abundances produced by s-process models directly to observations is difficult because they depend on the dilution of material from the intershell into the envelope, which in turn depends on several uncertain factors such as the size in mass of the pocket, the amount of TDU, mass loss, and mass accretion. However, the s-process element abundance ratios mostly remain unaffected by these processes, so they provide direct constraints on (see Sect. 4).
Note that with this procedure we do not properly account for the 22Ne neutron source. However, this neutron source does not contribute to defining the overall s-process distribution, at least for stellar models of mass , which are believed to be the counterparts of the observed s-process enhanced stars that we will discuss in Sect. 4 (e.g., Abia et al. 2001).
We assume that the 13C pocket mass is
a fixed fraction,
mass of the intershell
at the moment just before TDU occurs,
which in turn is approximated well by the maximum mass
of the convective intershell pocket during the thermal pulse.
We employ the relation proposed by
Iben (1977) for the maximum convective
The mass-loss rate during the giant branch phase affects the envelope mass at the moment when a star begins its TP-AGB phase. We have modelled mass-loss from giant branch stars with the widely used prescription by Reimers (1975),
|Figure 1: Range of initial masses, , of our population synthesis grid as a function of metallicity, depicted by the shaded area. For a given metallicity this shows the initial-mass range of stars which can be observed in the TP-AGB phase, with a maximum mass of 8 .|
We make use of our synthetic evolutionary models to carry out population synthesis of single stars, on a grid of 50 logarithmically spaced metallicities ranging between and 500 logarithmically spaced initial masses, whose range depends on the metallicity and on whether we are modelling a population of intrinsic or extrinsic s-enhanced stars.
In the case of intrinsic stars, for each metallicity we
apply the initial mass ranges shown in Fig. 1, with 8
as a maximum.
We derive these ranges from combining our stellar evolution models
with the age-metallicity relation (AMR)
for the galactic disk of Pont & Eyer (2004)
Given that we employ single stellar models, we treat the case of extrinsic stars indirectly. These stars have acquired their s-process enhancements by accreting part of the mass ejected by an initially more massive binary companion. Therefore we assume a simple model in which the initial mass of the companion is assumed to be distributed according to the single-star IMF. Hence the probability of an extrinsic star to show a given elemental abundance is weighted by the amount of mass which is lost by its companion in the form of that element and by the IMF weight of this companion. Because mass transfer could have occurred at any time since the formation of the system, we consider all stars whose evolution is beyond the TP-AGB phase, which in each metallicity bin corresponds to initial masses greater than the lower limit in Fig. 1 and smaller than 8 . For the models with [Fe/H] <-1 we consider 0.8 to be the lower initial mass limit.
The asymptotic envelope ratio is therefore only determined by (see Sect. 2.3), but not by other free parameters in our model, such as the size of the pocket ( ). Figure 2 shows the envelope [hs/ls] ratios obtained from the models of Busso et al. (2001), which in each model reach their corresponding intershell asymptotic value, but take a number of pulses to do so. Hence, considering stars that have experienced only a few TDU episodes, either because they are in an early stage of the TP-AGB phase or because they have masses of , results in a natural spread of [hs/ls] values. Only in AGB stars with is the amount of dredged-up material is so small that the surface [hs/ls] ratios differ from those of the intershell.
|Figure 2: Envelope [hs/ls] during the evolution of 1.5 and 3 star models of different metallicity, as indicated in the panel, and with 13C .|
To compare these abundance ratios with the observed data we have binned the resulting number distribution with a resolution of 0.1 dex. Whenever a star has a certain abundance ratio, its probability is added to the corresponding bin. This is done separately for the different metallicity bins and in each we normalize the resulting probability distribution by dividing it by its highest value.
|Figure 3: Intrinsic MS/S stellar population synthesis results compared to the observations. The panels show our results calculated with four increasing values as indicated above the panels. The grey scale is a logarithmic measure of the normalized number distribution of stars over [hs/ls]. The crosses are the observational data gathered by Busso et al. (2001), which have an average error given by the size of the cross in the upper right of each plot.|
We select those stars to be s-enhanced that have [ls/Fe] and/or [hs/Fe] larger than 0.1 dex. First we analyse the [hs/ls] ratios of intrinsic s-enhanced stars, which only have a strong dependence on the value. In all the runs we use , , and , based on our attempts to fit the post-AGB star observations of [Zr/Fe] ratios and carbon abundances, as explained in detail in Sect. 4.1.4. In Sect. 4.2 we study the [hs/ls] ratios, but this time of extrinsic s-enhanced stars, which give clues on at metallicities lower than that of the galactic disk.
|Figure 4: As Fig. 3, but for intrinsic SC and C stars. The horizontal crosses are observational data of intrinsic SC/C star gathered by Busso et al. (2001), complemented with data of Abia et al. (2002). The squares are SC/C stars in which the presence of Tc is doubtful (Abia et al. 2002) and the diagonal crosses are halo C stars from Busso et al. (2001). The observations have an average error given by the size of the cross in the upper right of each plot.|
|Figure 5: As Fig. 3, but for post-AGB stars The horizontal crosses are the observational data (see references in the text), which have an average error given by the size of the cross in the upper right of each plot.|
Post-AGB stars can also be used to provide constraints on the minimum core mass for TDU, the TDU efficiency, and the size of the 13C pocket. The observed values of [Zr/Fe] show a bimodal distribution: and , which suggests that some post-AGB stars did not experience TDU, while others did and suffered a strong enhancement of Zr (van Winckel 2003). This can be explained when we consider low mass stars ( ) at sub-solar metallicities. They have low-mass envelopes which easily become strongly s-enhanced. As shown in Fig. 6, only one TDU episode after the intershell material has become s-enriched is enough to raise their envelope abundance ratios by almost 1 dex. We compare the observational [Zr/Fe] ratios of van Winckel (2003), Reyniers et al. (2004) and Giridhar & Arellano Ferro (2005) to our models, from which we consider all post-AGB stars, both s-enhanced and not s-enhanced.
|Figure 6: Enhancement of Zr in TP-AGB stars as function of time. The models shown are for [Fe/H] =-0.5, with initial masses as indicated in the figure. When the stars reach the post-AGB phase, i.e., by the end of their evolution, there is a strong Zr enhancement (1 dex) as long as at least one s-enriched dredge-up episode has taken place. If no dredge-up occurs there is no enhancement at all, as seen in the case of the 1 star.|
|Figure 7: Ratio of the number of s-enhanced post-AGB stars which have ( ) to the number of s-enhanced post-AGB stars (Ns) from our synthetic models. The curves show the ratios calculated with different values of , as indicated in the figure. The straight dash-dotted and double-dotted lines show the upper limit to the ratio at [Fe/H , with a confidence of 68% and 95%, respectively, given the distribution of the observed data.|
Intrinsically s-enhanced post-AGB stars are observed at metallicities as low as (van Winckel 2003). When taking into account the AMR, this implies that at metallicities [Fe/Hstars with initial masses above 0.9 must experience TDU. At [Fe/H, the observed number of s-enhanced stars (which experienced TDU) is comparable to that of stars with (which did not undergo TDU). Thus, taking into account the IMF and the initial mass range given by the AMR at low metallicities, we estimate the minimum initial mass at which TDU must take place so that the number of s-enhanced stars is similar to that of non-enhanced stars. Applying this in our models, we find the corresponding minimum core mass for TDU as a function of metallicity. The resulting relation between minimum core mass and metallicity is best modelled by the fitting formula of Karakas et al. (2002), with an offset . This outcome depends somewhat on the adopted age-metallicity relation that sets the initial mass range of TP-AGB stars at each metallicity (Fig. 1), but the sensitivity of the mass of a low-mass star to its lifetime is only slight. Our result depends more strongly on the choice of mass loss. To test this we applied different mass-loss rates on the TP-AGB with the prescription of Reimers (1975), Eq. (5), by varying the coefficient from 0.5 to 3.0, and find that . This result is consistent with the findings of, e.g., Groenewegen & de Jong (1993), Marigo et al. (1999), Izzard et al. (2004), and Stancliffe et al. (2005) when studying the carbon star luminosity functions of the Small and Large Magellanic Clouds.
We now address the question of how much intershell material is dredged up to the surface. The surface s-process enhancement in our models is controlled by the dredge-up efficiency () and by the fractional mass of the 13C pocket relative to the intershell ( ). The carbon enrichment in the envelope depends only on the amount of dredge-up, independent of the size of the 13C pocket. The observations show that below [Fe/H all s-enhanced post-AGB stars have C/O >1 and that the non-enhanced ones have C/O <1 (van Winckel 2003). We use this information to break the degeneracy between and . Let be the observed number of s-enriched post-AGB stars that have C/O <1 and Ns be total number of s-enriched post-AGB objects. Figure 7 illustrates the ratio as a function of metallicity calculated with our models for different values of . With a simple application of Bayesian statistics we calculate that, given the observed data distribution, and Ns=11, there is a 32% probability that and only a 5% probability that . The latter sets the 95% confidence limit , as shown in Fig. 7. With this lower limit we calibrate to fit the Zr enhancement of post-AGB stars. Figure 8 shows our population synthesis results for the [Zr/Fe] ratio calculated with different values of and , compared to the observations. The upper left panel shows our results for . With this choice there are no s-enhanced post-AGB stars at low metallicities ( ) since these would all be low-mass stars due to the AMR and they experience negligible dredge-up. This is in contradiction to the observations. Moreover, as shown above (Fig. 7), a choice of is not compatible with the number of carbon enhanced stars. If we choose (upper right panel) then low-metallicity s-enhanced post-AGB stars are indeed produced. The large spread in the s-enhancement at higher metallicities is due to stars with mass . However, the choice , which roughly corresponds to that used by Gallino et al. (1998) for their detailed nucleosynthesis calculations, gives too much Zr for many of the s-enhanced post-AGB stars with . The lower right panel shows that most of the observations can be reproduced with and , with two exceptions which are extremely s-enhanced. From the constraint that all s-enhanced post-AGB stars are carbon stars, there is a 68% confidence lower limit on of 0.4. This implies that is needed to fit the post-AGB observations (lower right panel of Fig. 8). However, this choice of gives synthesized [ls/Fe] and [hs/Fe] ratios which are too small to reproduce those observed in MS/S, SC, and C stars. Figure 9 shows a comparison of the observed [ls/Fe] and [hs/Fe] of MS/S, SC, and C stars to our models, calculated with , , and . Most of the MS/S star abundances are fitted within the errors, with two high-metallicity exceptions that are only matched with a larger choice of or . The s-process abundances of most of the SC and C stars in the galactic disk are also matched well using the same choice of free parameters, except for three low metallicity objects that need a smaller .
From the combined evidence of MS/S, SC, C, and post-AGB stars, we find that with a few exceptions, all observations of intrinsic s-enhanced stars can be matched by models with the following set of parameters:
|Figure 8: Post-AGB stellar population synthesis results compared to the observations. The grey scale is a logarithmic measure of the normalized number distribution of stars in the [Zr/Fe]-[Fe/H] plane for . As indicated above the panels, plots are presented for different values of and . The crosses are the observational data (see references in the text), which have an average error given by the size of the cross in the upper right of each plot.|
|Figure 9: Stellar population synthesis results of MS/S stars ( top panels) and SC/C stars ( bottom panels) compared to the observations. The grey scale is a logarithmic measure of the normalized number distribution of stars in the [ls/Fe]-[Fe/H] plane ( left panels) and the [hs/Fe]-[Fe/H] plane ( right panels), using (as derived in Sects. 4.1.1 and 4.1.2), and . The observational data are from the same references and with the same coding as those of Figs. 3 and 4, which have an average error given by the size of the cross in the upper right of each plot.|
|Figure 10: Population synthesis results for single-star yields compared to observations of extrinsic s-enhanced stars for different values of 13C . The grey scale is a normalized logarithmic distribution of [hs/ls] for a population of stars as a function of metallicity. The symbols indicate observed data from different stellar types and references as follows: + MS/S stars; Ba II giants; CH sub-giants; CH giants; C giant; halo CH giant; halo yellow symbiotic; halo C-rich giant; halo C-rich sub-giant; halo N-rich dwarf, all previous gathered by Busso et al. (2001); C stars from Abia et al. (2002); lead stars gathered by Bisterzo et al. (2006). The observed data have an average error given by the size of the upper right cross in each plot.|
Extrinsic s-enhanced stars can be in any evolutionary stage prior to the TP-AGB phase, but they are mainly giants and main sequence stars. Their absolute s-process abundances cannot be studied without complete binary evolution models, but the abundance ratios of s-process elements are not substantially affected by binary processes. The [hs/ls] ratio represents that of the former TP-AGB companion which produced the s-process elements and it can be reasonably approximated by taking into account the yields of single stellar evolution models. Figure 10 shows a comparison of our synthetic [hs/ls] ratios to the extrinsic star data from Busso et al. (2001), Aoki et al. (2002), Abia et al. (2002) and van Eck et al. (2003). Similar to our results for intrinsic stars, the spread observed in these results is due to the range of initial masses that we use, although, in this case the mass range is wider and includes AGB stars up to 7-8 . An AGB star of has a very thin intershell and a massive envelope, so its surface [hs/ls] ratio varies smoothly between its value in the intershell and the solar ratio.
We focus first on the galactic disk ( ) stars. Figure 10 shows that no single choice of value can make our results fit all the observational data, but almost all observations can be fitted with a range of values . This is consistent with what we found for the intrinsic s-enhanced stars (see Sect. 4.1).
For , our synthetic [hs/ls] ratios are relatively insensitive to changes in because the hs-element synthesis saturates, opening the way to the synthesis of lead. There is some indication that a smaller value of is needed to reproduce objects with [Fe/H which have a high [hs/ls], but even when using small values [hs/ls] does not exceed about 1 dex. Thus we are unable to explain the extreme low-metallicity objects with . Our synthetic calculations do not extend below [Fe/H] <-2.3 because of the lack of detailed models with metallicity below this value.
As stated above our calculations do not extend below , but we notice that lead stars with lower metallicity could also be explained if the trend shown by our models continues at lower metallicities. This seems likely because [Pb/hs] does depend mostly on and not on other uncertain parameters in our models, however, explaining those stars at which show is still a problem.
Finally, we note that Cui & Zhang (2006) also presented a possible solution to explain the spread of [Pb/hs] observed in low-metallicity stars. These authors obtained a spread of neutron exposures (and hence [Pb/hs]) in low-metallicity AGB stars due to variations of the inter-pulse period in stars of different masses. Their results are based on the assumption that the neutron irradiation time is proportional to the inter-pulse period. However, this assumption is incorrect since the neutron irradiation time depends instead on the timescale at which the reactions occur. At the temperature of 108 K that Cui & Zhang (2006) use in their models, the timescale for captures is of the order of 300 yr (see Fig. 2 of Gallino et al. 1998), and this is independent of the inter-pulse period. In our models, instead, the spread in [Pb/hs] derives naturally from the fact that the neutron exposure changes with time, especially at low metallicities.
Based on the results of our population synthesis study, we find that it is not possible to reproduce the observed heavy-element abundances of s-enhanced stars with the standard set of values for the free parameters in our AGB models. The minimum core mass for TDU has to be reduced by at least 10 per cent with respect to what theoretical models predict; low-mass stars must have a significant TDU efficiency and the 13C pocket mass must be somewhat lower than what is usually considered standard. We also find that the observations can be matched with a narrow range of values and that the mean value in this range apparently decreases with metallicity.
The observations in the metallicity range are fitted well by , i.e., a spread of a factor of 2, while Busso et al. (2001) needed a factor of 20. This is a consequence of the natural spread that we find in the s-process element ratios caused mainly by the truncation of AGB evolution due to the total loss of the stellar envelope. Stars with initial mass start the TP-AGB phase with a relatively similar core mass, but different envelope mass. Consequently, a star with a certain low mass will experience fewer thermal pulses than one with a slightly higher mass, which affects its yields and composition during its life as a TP-AGB star. A different choice of mass loss in our models, either during the red giant branch or the AGB phase, would move the mass range in which this effect takes place, but not eliminate it.
a spread in [Pb/hs] arises because of
the shift in the neutron exposure.
Observations at these lower metallicities need a somewhat
perhaps down to 1/12.
This result may be due to the increasingly important
effect of -enhanced light neutron poisons at low
metallicities (Busso et al. 1999), which needs to be
investigated in more detail.
|Figure 12: [Pb/hs] results from our population synthesis of lead stars, shown in the same way as in Fig. 10 and compared to the data gathered by Bisterzo et al. (2006). The upper right cross shows the average error of the data points.|
It is reassuring that our conclusion of a small spread in is reached independently by the study of isotopic s-process signatures in pre-solar silicon carbide (SiC) grains from carbon stars. While Lugaro et al. (2003) needed a spread of 24 in the values of to cover the SiC data, by eliminating the effect of contamination of solar material, more recent laboratory studies (Marhas et al. 2007; Barzyk et al. 2007) have also reduced the spread of the values needed to cover the grain data to exactly the same range we found here. Although pre-solar grains can only give us information on the s-process taking place in stars with , such an independent check also gives us confidence in our results at lower metallicities.
Now the question arises as to whether the small spread that we find is compatible with the possible mechanisms currently proposed for the proton diffusion leading to the formation of the 13C pocket. We note that semi-convection (Hollowell & Iben 1988), hydrodynamical overshooting (Herwig et al. 1997), and gravity waves (Denissenkov & Tout 2003), which are three out of the four proposed mechanisms, produce a proton profile with the number of protons varying continuously from the envelope value to zero. The possible shapes of the proton profile were discussed by Goriely & Mowlavi (2000, see their Fig. 10), who concluded that "the s-process predictions are only weakly dependent on the shape of the H-profile''. Hence we would not expect a wide range of neutron exposures to occur in the current scenario. Future work should quantitatively link the shape of the proton profile to the range of we found in this study. The fourth process proposed for the formation of the neutron source is rotation (Herwig et al. 2003; Langer et al. 1999). Also in this case the formation of the pocket starts from a continuous proton profile; however, current models find that further mixing during the inter-pulse period completely inhibits the s-process (Herwig et al. 2003; Siess et al. 2004). Future work is needed to ascertain this point.
To explain the existence of Galactic post-AGB stars that are s-process enhanced at [Fe/H, we find that the minimum core mass for TDU must be 0.055-0.075 smaller than that found in the models of Karakas et al. (2002) and Stancliffe et al. (2004). This is consistent with the findings of several authors who studied the carbon-star luminosity functions of the SMC and LMC (e.g., Groenewegen & de Jong 1993; Marigo et al. 1999; Izzard et al. 2004; Stancliffe et al. 2005) and sets a strong constraint on future detailed evolutionary model results.
We find that to account for the fact that all 11 Zr-enhanced post-AGB stars observed so far are also C enhanced, in stars of mass . The precise value of this lower limit may depend somewhat on our adopted choices for mass loss, the AMR, and the minimum core mass for TDU. We leave a detailed analysis of this dependence for future study. However, it is clear from our results that a value of cannot reproduce the observations, and we are therefore confident in concluding that stars of mass must experience a significant amount of TDU.
The amount of Zr enhancement in post-AGB stars is determined by the amount of third dredge-up and the size of the 13C pocket. Assuming indicates that in our models the 13C pocket must have a mass of 1/40 of that of the intershell to fit the observed data from Zr-enhanced post-AGB stars. This is somewhat smaller than the size of the 13C pocket assumed in the models of Gallino et al. (1998), which is about 1/20 of the mass of the intershell.
Future detailed modelling of AGB stars will need to address the constraints we have set here. We are currently extending our analysis of extrinsic s-enhanced stars by means of binary population synthesis, i.e., by explicitly following the evolution and interaction processes of populations of binary stars.
The authors are deeply indebted to Roberto Gallino for providing the data from his detailed nucleosynthesis models and for insightful discussions and comments on this work. A.B.M. thanks Maarten Reyniers and Hans van Winckel for useful discussions and for providing data on post-AGB stars. A.B.M. also thanks Selma de Mink for refreshing his knowledge of statistical analysis. R.G.I. is supported by NWO. M.L. is supported by NWO (VENI fellow).
Hurley et al. (2000, hereafter H00) have developed a comprehensive synthetic evolution code for single stellar evolution, based on detailed models of Pols et al. (1998, hereafter P98). The P98 models include convective core overshooting, but do not undergo thermal pulses during the AGB phase. To model TP-AGB stars, H00 apply the same luminosity-core mass relation as in the early (E-) AGB phase. The TP-AGB section of the H00 code was improved by Izzard et al. (2004, hereafter I04) with a synthetic code based on the detailed model calculations by Karakas et al. (2002, hereafter K02) that undergo thermal pulses. The latest version (Izzard et al. 2006) also follows the nucleosynthesis of many s-process isotopes based on the models of Gallino et al. (1998).
However, unlike the P98 detailed models, the K02 detailed models do not include convective overshooting. This means that the transition from the E-AGB to the TP-AGB phase is not self-consistent in the I04 code, leading to discontinuities in the evolution of the stellar core mass, of the luminosity, and of the radius of the AGB star. To overcome these problems, we have made modifications to the I04 code, which are explained in detail below.
For an AGB star to enter the TP-AGB phase, its helium intershell mass must be small enough to undergo a flash. This occurs at the end of the E-AGB phase when the helium-burning shell almost reaches the hydrogen-burning shell. In our code the H-depleted core mass at the beginning of the TP-AGB phase, , is given by the mass of the H-depleted core at the base of the AGB, . However, stars with and larger undergo second dredge-up by the end of the E-AGB phase, which makes .
When convective overshooting is taken into account, the value of for a star with a given initial mass is higher than that of a star with the same initial mass without overshooting. The effects of overshooting become important in AGB stars of , having a direct impact on the subsequent evolution.
We have improved the linear relation between
from H00 (denoted as
in that paper) with a
quadratic one, based on data from the P98 models (hereafter OV):
|Figure A.1: Luminosity-core mass relation of a 3 star with metallicity Z=0.01. The arrows show the way in which the core evolves in time. During the E-AGB phase, the hydrogen-exhausted core mass remains constant at 0.716 . In our model the core mass at the start of the TP-AGB is the same as in the H02 model, which is greater than in the I04 model.|
|Figure A.2: Evolution of luminosity in time of the same star as in Fig. A.1. The E-AGB evolution is slightly modified in our model and the discontinuity in luminosity is fixed.|
For the beginning we use Eq. (56) from H00 for the luminosity
at the base of the AGB and the following relation
at the base of the AGB
For a smooth transition to the TP-AGB phase, we require
|Figure A.3: Examples of our analytic fit (solid lines) compared to the data from the detailed models (dashed lines) from Stancliffe et al. (2004) for different stellar masses and metallicities, as indicated in the panels.|
Following Paczynski (1970), it has become the usual
practice to apply a linear luminosity-core mass relation in
synthetic AGB models.
However, this relation fails in the presence of dredge-up and hot
bottom burning. This becomes a problem when comparing models
that include overshooting to models that do not: two TP-AGB stars with
the same core mass, but different envelope mass, do not behave in
the same way. To overcome this we designed
a luminosity relation that depends on both the
core mass and the envelope mass of the star. It
successfully reproduces both the P98 models that include
overshooting and models calculated with the same code used by P98,
which do not include overshooting.
Stancliffe et al. (2004, hereafter S04) employed a modified
version of the code used by P98 and computed detailed models that do
not include overshooting, but do undergo thermal pulses.
In an attempt to make our rapid TP-AGB
models more self-consistent, we have modified our luminosity
relation to fit the S04 models, assuming that with the inclusion
of overshooting the S04 models are modified in the same way
as those of P98. In addition to this, we find that in
the presence of third dredge-up the results of
the detailed stellar evolution models are better represented by
relating the luminosity to
the core mass
as it would be in the absence of dredge-up:
The S04 models have no mass loss, so we have to use a different set
of models to fit the
behaviour of the luminosity in terms of the loss of the envelope mass.
Based on the K02 models, the contribution of the envelope
to the luminosity can be modelled as
We use the H00 radius-luminosity relation, given that
it accurately fits the S04 models, but we have modified
the interpolation procedure slightly.
The H00 radius-luminosity relation (Eq. (74) in that paper) was
designed to fit the P98
models at seven different metallicities in the following way: