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The evolution and nucleosynthesis of low-mass and intermediate-mass stars is significantly affected by numerical modelling uncertainties as well as uncertainties in the input physics. Here we review the main uncertainties affecting AGB model calculations including convection, which affects the occurrence and efficiency of TDU, and mass loss, which determines the AGB lifetime. We also comment on uncertainties affecting s-process element predictions.

4.1. Convection and the third dredge-up

Dealing with convection in stellar interiors is one of the major problems of stellar evolution modelling. There are two main ways that convection affects AGB evolution and nucleosynthesis. The first is through the way stellar evolution codes treat the interface between radiative and convective regions within a stellar model (Frost & Lattanzio 1996b; Mowlavi 1999a). The second is how the temperature gradient affects the energy transport in the convective regions.

4.1.1. Determining the borders of convection

We know from observations that AGB stars experience third dredge-up. We unambiguously observe C and Tc-rich AGB stars but we still (after 40 years) do not know at which initial stellar mass that dredge-up begins. Circumstantial evidence suggests that the minimum mass be about 1.5 M in the Galaxy (e.g., Wallerstein & Knapp 1998) but theoretical models mostly struggle to obtain enough TDU at this stellar mass without the inclusion of some form of convective overshoot (Herwig et al. 1997; Mowlavi 1999a; Herwig 2000; Cristallo et al. 2009; Karakas et al. 2010; Kamath et al. 2012).

Observations of external galaxies such as the LMC, SMC, and dwarf spheroidal galaxies show higher numbers of C-stars than in our Galaxy (e.g., Zijlstra et al. 2006; Sloan et al. 2008, 2012), which is evidence that it is easier to obtain TDU at lower metallicities, for a given mass. On the other hand, Boyer et al. (2013) found a lack of C stars in the inner metal-rich region of M31, indicating that there is a metallicity ceiling for the operation of TDU. Stellar evolution codes qualitatively agree with these observations.

The main problem for calculating TDU is how we determine the border between a convective and radiative region. Applying the Schwarzschild criterion for convection (∇rad > ∇ad) is too simplistic because while the accelerations of blobs at this border is zero, the velocities may be finite. This suggests that some overshoot is inevitable and to be expected. The question then is how much?

While the amount of convective overshoot can be constrained by considering various observations (Herwig 2000; Cristallo et al. 2009; Weiss & Ferguson 2009; Kamath et al. 2012), this provides little insight into the actual physics occurring in convective envelopes, given the numerical differences between the various stellar evolution codes. The fact that Kamath et al. (2012) found such a diversity in the amount of overshoot required to match observations may indicate that this is not the best way to characterise the depth of mixing required.

The problem of determining convective borders in stellar interiors is surely also linked to the question of 13C pocket formation, which can be formed by the inclusion of a partially mixed overshoot region, for example. Uncertainties not only include the formation mechanism but also the size of the pocket in the He intershell as well as the shape. When more protons are added the resultant 13C pocket is accompanied by a sizeable 14N pocket, which acts as an efficient neutron absorber and suppresses the s-process.

The way forward is to consider multi-dimensional simulations which unfortunately have not yet advanced sufficiently to answer the problem of overshoot or 13C pocket formation.

4.1.2. Structural changes from convection

The second important way that convection affects models is the substantial effect on the structure of AGB convective envelopes. The treatment of convection in stellar envelopes determines surface properties such as the luminosity and effective temperature and it has an impact on the efficiency of hot bottom burning (Ventura & D’Antona 2005a). The most commonly used treatment of convection is the MLT which has a free parameter, the mixing-length parameter α, which is usually set by calibrating a 1 M, Z = Z stellar model to the Sun’s present day radius. The parameter α is then assumed to remain constant throughout the star's evolution, with the same value used for all masses and metallicities.

However, AGB stars have very different envelope structures to the shallow convection zone found in our Sun. There is no good reason why the value of α required to fit a standard solar model is appropriate for AGB stars. Furthermore Lebzelter & Wood (2007) found evidence for α to increase with evolution along the AGB, which suggests that α need not be constant. Increasing α leads to shallower temperature gradients which produces higher luminosities and stronger burning in intermediate-mass AGB stars (Ventura & D’Antona 2005a). Convective models other than the MLT are also used, such as the Full Spectrum of Turbulence (Ventura et al. 1998; Mazzitelli, D’Antona, & Ventura 1999) applied by, e.g., Ventura & D’Antona (2005a).

4.2. Mass loss

Dealing with the extent and temporal variation of mass loss in AGB stars is one of the main uncertainties in stellar modelling (Blöcker 1995; Habing 1996; Ventura & D’Antona 2005b). This is because the mass-loss rates of AGB stars are very uncertain and difficult to determine from observations without a priori assumptions about dust mass and type, and/or radiative transfer modelling (e.g., Bedijn 1988; Vassiliadis & Wood 1993; Habing 1996; van Loon et al. 1999a; Groenewegen et al. 2009a, 2009b; Lagadec & Zijlstra 2008; Lagadec et al. 2010; Guandalini 2010; De Beck et al. 2010; Riebel et al. 2012). Mass loss on the AGB determines the AGB lifetime and the number of thermal pulses experienced by TP-AGB models. This then limits the number of TDU episodes and the duration of HBB; and hence determines the level of chemical enrichment expected from a population of AGB stellar models at a given metallicity.

In order to calculate stellar yields, mass loss has to be included in the calculation of AGB models. The available prescriptions are simple, parameterised formulae that result in mass being removed from the envelope smoothly in time, in contrast to observations which suggest that AGB mass loss in real stars is clumpy and asymmetric (e.g., Meixner et al. 1998; Dinh-V-Trung & Lim 2008; Olofsson et al. 2010; Wittkowski et al. 2011; Paladini et al. 2012; Lombaert et al. 2013).

The upper part of the AGB in particular is dominated by continuously increasing mass loss (Habing 1996). Observations indicate rates increase from 10-7 M year-1 for short period Mira variables to ≈ 10-4 M year-1 for luminous long-period variables including OH/IR stars and Miras (Groenewegen et al. 2009a; Justtanont et al. 2013). These winds are most likely dust and shock driven (Winters et al. 2003), which leads to the star becoming completely enshrouded by dust and visible predominantly in the infra-red (Habing 1996; Uttenthaler 2013).

One of the biggest uncertainties is the rate of mass loss from low-metallicity AGB stars. Based on theoretical calculations, Mattsson et al. (2008) conclude that low-metallicity C stars have similar mass-loss rates to their metal-rich counterparts. Observations showed that mass-loss rates in low metallicity C-rich AGB stars in nearby galaxies are of a similar magnitude to AGB stars in our Galaxy (e.g., Sloan et al. 2009; Lagadec et al. 2009).

The most widely used prescriptions in AGB evolutionary calculations include the Vassiliadis & Wood (1993) mass-loss law, based on empirical observations of mass-loss rates in C and O-rich AGB stars in the Galaxy and Magellanic Clouds; the Blöcker (1995) formula, based on dynamical calculations of the atmospheres of Mira-like stars; and the Reimer’s mass-loss prescription (Reimers 1975; Kudritzki & Reimers 1978), even though it was originally derived for first ascent giant stars and does not predict a superwind (Groenewegen 2012). Both the Blöcker (1995) and Reimers (1975) rates depend on an uncertain parameter η which typically takes values from 0.01 to 10 (Ventura et al. 2000; Straniero et al. 1997; Karakas 2010). Ventura et al. (2000) found η = 0.01 by calibrating their intermediate-mass AGB models to Li abundances in the LMC.

Other mass-loss prescriptions for AGB stars are available (e.g., Bedijn 1988; Arndt, Fleischer, & Sedlmayr 1997; Wachter et al. 2002; van Loon et al. 2005; Wachter et al. 2008; Mattsson, Wahlin, & Höfner 2010). Some of these prescriptions are specifically for C-rich stars e.g., Arndt et al. (1997) and Wachter et al. (2002) and are not appropriate for bright intermediate-mass AGB stars. The theoretical mass-loss rates from Mattsson et al. (2010) for solar-metallicity C stars are available as a FORTRAN routine that can be coupled to a stellar evolution code.

4.3. Extra mixing in AGB stars

A case has been made for some form of slow non-convective mixing to operate in AGB envelopes, in an analogous situation to the extra mixing operating in first giant branch envelopes (Sections 2.2.4 and 2.3).

The evidence for deep mixing on the AGB comes mainly from O and Al isotope ratios measured in pre-solar oxide grains, which support the existence of such extra mixing in low-mass (M ≲ 1.3 M) AGB stars (Busso et al. 2010; Palmerini et al. 2009, 2011). The C isotopic ratios measured in AGB stars span a large range, from very low 12C / 13C ratios of ≈ 4 to a maximum of about 100 (Lambert et al. 1986; Abia & Isern 1997). The sample by Lambert et al. (1986) has an average value of 58 (without the J-type C stars whose origin is unknown). The range of 12C / 13C ratios in AGB stars is similar to that measured in mainstream pre-solar silicon carbide grains, with values between 40 ≲ 12C / 13C ≲ 100 with an average 12C / 13C ≈ 60 (Zinner 1998). The lowest values of the 12C / 13C ratio in AGB stars suggest that some small fraction of Galactic disk C-rich AGB stars experience extra mixing. In Section 3.5.1 we summarise the C isotope predictions from AGB models, including the range expected when extra mixing occurs on the first gaint branch.

Carbon-enhanced metal-poor stars that are s-process rich presumably received their C, N and s-process enrichments from a previous AGB companion and their 12C / 13C ratios are therefore an indicator of extra mixing and nucleosynthesis in the AGB star. The 12C / 13C ratio has been measured in CEMP stars covering a range of evolutionary phase, from turn-off stars through to giants. Figure 7 from Stancliffe et al. (2009) illustrates the observed C isotopic ratios of unevolved (log g ≥ 3) CEMP stars are ≲ 10 (e.g., Cohen et al. 2004; Sivarani et al. 2006; Jonsell et al. 2006; Aoki et al. 2007; Beers et al. 2007; Lucatello et al. 2011). Such low observed ratios are difficult to reconcile with standard AGB nucleosynthesis models, which produce very high 12C / 13C ratios (> 103) at low metallicity (e.g., Karakas 2010; Cristallo et al. 2011; Lugaro et al. 2012). Such low 12C / 13C ratios could reveal a metallicity dependence to the extra-mixing occurring in AGB envelopes.

Nitrogen abundances of CEMP stars also show a spread that is not easily explained by canonical models as shown by the population synthesis study by Izzard et al. (2009). Stancliffe (2010) studied thermohaline mixing in low-metallicity AGB models and found it not strong enough to explain the low C isotopic ratios. Clearly additional mixing - whatever the mechanism - is required in low-metallicity AGB envelopes, but the need for extra mixing in solar-metallicity C-rich AGB stars is more ambiguous as discussed in detail by Karakas et al. (2010, but see Busso et al. 2010).

4.4. Low-temperature opacities

In recent years there has been considerable effort put into developing accurate low-temperature molecular opacity tables for stellar evolution calculations. The opacity tables of Alexander & Ferguson (1994) and later Ferguson et al. (2005) included the first detailed treatment of the inclusion of molecules to the total opacity at temperatures where T ≲ 104 K. These tables were only available for solar or scaled-solar abundance mixtures. As we have seen, AGB stars experience multiple mixing episodes that alter their envelope compositions, such that the stars may become C and N-rich and in some cases, the envelope C/O ratio can exceed unity.

Marigo (2002) showed that at the transition from C/O < 1 to C/O ≥ 1 the dominant source of molecular opacity changes from oxygen-bearing molecules to C-bearing molecules. In AGB stellar models, this change in opacity leads to a sudden decrease in the effective temperature and subsequent expansion in radius. These changes to the stellar structure cause an increase in the rate of mass loss. Marigo (2002) showed that this resulted in shorter AGB lifetimes and therefore smaller stellar yields. AGB models with HBB can also deplete C and O (while producing N), which causes changes to the stellar structure and nucleosynthesis (Weiss & Ferguson 2009; Ventura & Marigo 2009, 2010; Fishlock et al. 2014). Despite claims to the contrary, Constantino et al. (2014) showed that there was no threshold in [Fe/H] below which the composition dependent molecular opacities would not produce significant changes. It is therefore necessary to use low-temperature molecular opacity tables that follow the change in C, N, and C/O ratio with time at all masses and compositions.

Besides the scaled-solar tables from Ferguson et al. (2005), other tables currently available for stellar evolutionary calculations are: (1) Lederer & Aringer (2009), who account for an enhancement of C and N compared to the initial abundance for various metallicities, and (2) Marigo & Aringer (2009), who provide the ÆSOPUS on-line downloadable tables 5. These tables are available for essentially arbitrary variations in C, N, and C/O (including enhancements and depletions) for whatever metallicity desired and for various choices of the solar composition.

4.5. The s-process

While the formation mechanism of the 13C pocket is still the main uncertainty in the s-process models, there are a number of further problems associated with the s-process scenario discussed previously in Section 3.7.

Lugaro et al. (2012) identified four different regimes of neutron captures that can occur in theoretical AGB models including: (1) the 22Ne(α,n)25Mg source operates during convective thermal pulses, (2) the 13C(α,n)16O reaction burns under radiative conditions, with the 13C produced via the inclusion of a 13C pocket, (3) the 13C(α,n)16O reaction burns under convective conditions during a thermal pulse, with the 13C produced via the inclusion of a 13C pocket, and (4) the 13C(α,n)16O reaction operates under convective conditions with the 13C produced via the ingestion of a small number of protons from the tail of the H shell during the thermal pulse.

The mass range at which these regimes occur is model dependent as described in Lugaro et al. (2012). At the metallicity considered in that study (Z = 0.0001) proton ingestion (Regime 4) dominates the s-process abundance predictions at the lowest masses (M ≤ 1 M at Z = 0.0001) when no (or small) 13C pockets are present. Proton ingestion is expected to be more important at even lower metallicities, although the mass and metallicity range where proton ingestion occurs is still very uncertain and may occur at solar metallicities under specific conditions (e.g., Sakurai's Object). Again, the problem comes down to the treatment of convection and convective borders in stars. The first multi-dimensional studies are becoming available to guide the one-dimensional models (Stancliffe et al. 2011; Herwig et al. 2011, 2013).

For the first few pulses in low-mass stars, the temperature in the intershell is not large enough for efficient radiative burning of the 13C. Hence Regime 3, where all or most of the 13C is burnt under convective conditions during the next thermal pulse is possibly a common occurrence during the first few thermal pulses for all low-mass AGB stars of ≲ 2 M, regardless of Z (see also Cristallo et al. 2009; Lugaro et al. 2012). The main result of Regime 3 is that the overall neutron exposure (i.e., total number of free neutrons) decreases owing to ingestion of the neutron poison 14N alongside the 13C from the H-shell ashes. Also during Regime 3 the neutron density increases owing to the short timescale of thermal pulses (order 102 years) relative to radiative burning during the interpulse (≈ 104 years).

The occurrence of 13C ingestion during thermal pulses is strongly connected to the uncertainties related to the onset of TDU in the lowest-mass AGB stars (e.g., the initial stellar mass for the onset of the TDU at a given Z, the efficiency of TDU as a function of stellar mass, core mass, metallicity; Frost & Lattanzio 1996b; Straniero et al. 1997; Mowlavi 1999a; Karakas et al. 2002; Stancliffe & Jeffery 2007; Karakas et al. 2010).

If stars as low as ≈ 1.2 M experience TDU at solar metallicities then we expect Regime 3 to be dominant in these stars for the first few thermal pulses, before the He shell region has heated up to sufficient temperature to ignite 13C(α,n)16O under radiative conditions. If these stars only experience a few TDU episodes, then this regime will dominate the stellar yields of s-process elements.

All stars rotate but the effect of rotation on stellar structure in general, and the s-process in particular, is still poorly known. The angular velocity profile inside AGB stars may produce a strong shear instability between the contracting core and the expanding envelope. Unlike the (partial) mixing postulated to come from overshoot during TDU, this shear layer does not disappear at the end of TDU. This is expected to result in continuous mixing of protons into the top layers of the He intershell, resulting in the complete operation of the CN cycle and the production of a higher abundance of the neutron-poison 14N instead of 13C. This has been shown to lower the neutron exposure and suppress the formation of s-process elements (Herwig et al. 2003; Piersanti et al. 2013). The effect of magnetic fields (Suijs et al. 2008) and gravity waves may modify the angular momentum in the star and reduce the mixing between core and envelope but has not been considered in detailed stellar evolution models so far.

We have mentioned convective overshoot at the bottom of the thermal pulse in the context of O intershell abundances. Such overshoot can lead to increased temperatures and the activation of the 22Ne neutron source at lower initial stellar mass than canonical models with no overshoot. While it has been shown for a 3 M star that such overshoot produces Zr isotopic ratios inconsistent with those measured in stardust grains (Lugaro et al. 2003), other observations such as oxygen in post-AGB stars suggest that such overshoot occurs. While the first s-process yields from models with overshoot into the C-O core are becoming available (Pignatari et al. 2013) a comprehensive study on the effect of such overshoot on the s-process is still lacking.

4.6. Binary evolution

Most stars (roughly 60%) exist in binary or multiple systems (Duquennoy & Mayor 1991). Not all stars in binary systems (or higher order multiples) will be close enough to interact and hence they will evolve essentially as single stars. The fate of these stars and their contribution to the enrichment of the Galaxy is determined by their initial mass and metallicity. For binary stars that are close enough to interact, there are many more variables that determine the type of interaction including the orbital parameters of the system and the mass ratio between the two stars.

Possible interactions include mass transfer via Roche Lobe Overflow (RLOF), which can lead to a common envelope and possible stellar merger. For example, the warm R-type C stars are all single stars which has led various authors to propose that they must be the result of stellar mergers (McClure 1997b; Izzard et al. 2007). If the stars do not merge during the common envelope phase the orbital period will be dramatically shortened, allowing for later mass transfer. The details of common envelope evolution are complex and not well understood (e.g., Taam & Ricker 2010).

Regardless of the final outcome of the common envelope, the evolution of the two stars will be significantly altered from a single stellar evolution channel. Common envelopes may truncate the evolution of the more massive star on the first giant branch, which means it will never become an AGB star. Clearly the nucleosynthesis yields will be significantly altered from the expectations from single stellar evolution (Izzard 2004; Izzard et al. 2006).

Interactions can come in other forms. For example if one of the stars is on the AGB and has a strong wind, then some of that wind may be transferred to the companion. That wind may then contain the products of AGB nucleosynthesis, which will later be observed on the surface of the lower mass companion. Stellar wind accretion is thought to be the dominant mechanism to produce barium and CH-type stars (McClure 1983; McClure & Woodsworth 1990; McClure 1997a; Boffin & Jorissen 1988; Han et al. 1995; Karakas, Tout, & Lattanzio 2000; Izzard, Dermine, & Church 2010; Miszalski et al. 2013), as well as carbon and nitrogen-enhanced metal-poor stars (Beers & Christlieb 2005; Lucatello et al. 2005; Izzard et al. 2009; Pols et al. 2012; Abate et al. 2013).

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