Asymptotic Giant Branch Stars: Their Inﬂuence on Binary Systems and the Interstellar Medium

Asymptotic Giant Branch Stars: their inﬂuence on binary systems and the interstellar medium

Amanda I Karakas, BSc (Hons)

A thesis submitted for the degree of Doctor of Philosophy. School of Mathematics & Statistics, Monash University, Australia.

July, 2003 i

Quotes

Since September it’s just gotten colder and colder. There’s less daylight now, I’ve noticed too. This can only mean one thing – the Sun is going out. In a few more months the Earth will be a dark and lifeless ball of ice. Dad says the Sun isn’t going out. He says it’s colder because the Earth’s orbit is taking us farther from the Sun. He says winter will be here soon. Isn’t it sad how some people’s grip on their lives is so precarious that they’ll embrace any preposterous delusion rather than face an occasional bleak truth? “Calvin & Hobbes”, by Bill Watterson

The Cat's Eye Nebulae, NGC 6543, imaged by the Hubble Space Telescope. Planetary nebulae such as NGC 6543 are believed to consist of several tenths of a solar mass of gas and dust expelled during the latter stages of the asymptotic giant branch phase of stellar evolution. Image from http://www.astro.washington.edu/balick/WFPC2/. Contents

Abstract ...... vi Statement ...... vii Publications ...... viii Acknowledgments ...... x

1 Introduction 1 1.1 Evolution of Low and Intermediate Mass Stars ...... 2 1.2 Chapter Synopsis ...... 6 1.2.1 The Barium Stars ...... 6 1.2.2 Parameterizing the Third Dredge-Up ...... 7 1.2.3 Stellar Yields ...... 8

2 The Formation of the Barium Stars 10 2.1 Introduction ...... 10 2.2 Observations of the Barium Stars ...... 12 2.3 Description of the Model ...... 15 2.3.1 Mass Loss ...... 15 2.3.2 Stellar Wind Accretion ...... 16 2.3.3 Nucleosynthesis Model ...... 17 2.3.4 Monte-Carlo Simulation Parameters ...... 19 2.4 Tidal Evolution Model ...... 20 2.5 Results and Discussion ...... 22 2.5.1 Number of Ba stars to Red Giants ...... 24 2.5.2 Dependence of Results on Free Parameters ...... 25 2.5.3 Barium Over-abundances ...... 27 2.5.4 The Mass Distributions ...... 29 2.6 Summary and Further Work ...... 29

3 Pre–AGB Evolution and Nucleosynthesis 32 3.1 Numerical Details ...... 32 3.1.1 The Stellar Evolution Code ...... 32 3.1.2 The Nucleosynthesis Code ...... 36 3.2 Stellar Models ...... 41 3.3 Stellar Lifetimes ...... 41 3.3.1 Uncertainties Aﬀecting the Stellar Lifetimes ...... 42

ii CONTENTS iii

3.3.2 Model Results ...... 43 3.3.3 Comparison With Other Work ...... 47 3.4 First and Second Dredge-Up: the CNO Isotopes ...... 49 3.4.1 Conﬂict Between Theory and Observation ...... 50 3.4.2 Depth of the First and Second Dredge-Up ...... 51 3.4.3 Results from the FDU and SDU: Carbon ...... 53 3.4.4 Nitrogen ...... 54 3.4.5 Oxygen ...... 59 3.5 First and Second Dredge-Up: non-CNO Elements ...... 60 3.5.1 Lithium and Fluorine ...... 61 3.5.2 Neon and Sodium ...... 62 3.5.3 Magnesium and Aluminium ...... 63 3.6 The Core Helium Flash ...... 64 3.6.1 Results From the Core-helium Flash ...... 66 3.7 Summary ...... 68

4 AGB evolution 69 4.1 Evolution on the AGB ...... 69 4.1.1 The Third Dredge-Up ...... 70 4.2 Evolution During a Thermal Pulse ...... 75 4.2.1 Mass and Duration of the Convective Pocket ...... 75 4.2.2 Temperature in the Helium-burning Shell ...... 79 4.2.3 Secondary Convective Pockets ...... 82 4.3 Evolution During the Interpulse Phase ...... 84 4.3.1 The Interpulse Period ...... 85 4.3.2 Temperature at the Base of the Convective Envelope ...... 87 4.3.3 The Core–mass–luminosity Relationship ...... 91 4.4 Summary ...... 99

5 AGB nucleosynthesis 101 5.1 Introduction ...... 101 5.2 Nucleosynthesis Resulting From the Third Dredge-Up ...... 102 5.2.1 The Hydrogen-burning Shell ...... 103 5.2.2 The Helium-burning Shell ...... 106 5.3 Nucleosynthesis Resulting from Hot Bottom Burning ...... 110 5.3.1 The Creation of 7Li via HBB ...... 110 5.3.2 Nucleosynthesis from HBB: CNO Isotopes and Fluorine . . . . . 113 5.3.3 The Production of Primary 14N ...... 114 5.3.4 The Neon, Magnesium and Aluminium Isotopes ...... 117 5.4 Comparison of Model Predictions to Observations ...... 118 5.4.1 Observations of the Mg Isotopes in IRC+10216 ...... 119 5.4.2 Observations of the Mg Isotopes in Other Stars ...... 121 5.5 Neon in Planetary Nebulae ...... 126 5.5.1 The Observations ...... 127 CONTENTS iv

5.5.2 Model Results ...... 129 5.5.3 Discussion ...... 132 5.6 Summary and Further Work ...... 134

6 Parameterizing the Third Dredge-Up 138 6.1 Introduction ...... 138 6.2 Stellar Models Without Mass Loss ...... 140 6.2.1 Model Details ...... 140 6.2.2 Convection and the Third Dredge-Up ...... 140 6.2.3 Mass Loss ...... 141 6.2.4 Model Results ...... 141 min 6.3 The Fit for Mc(1) and Mc ...... 148 6.3.1 The Core Mass at the First Thermal Pulse ...... 148 6.3.2 The Core Mass at the First TDU Episode ...... 150 6.4 The Fit for λmax ...... 151 6.4.1 The Dredge-Up Parameter, λ as a Function of Time ...... 154 6.5 Discussion ...... 156 6.5.1 The Core Mass at the First Pulse ...... 156 min 6.5.2 The Third Dredge-Up: Mc and λmax ...... 157 6.5.3 The Carbon Star Luminosity Function ...... 158 6.6 Summary and Further Work ...... 159

7 Stellar Yields 160 7.1 Stellar Yield Calculation ...... 160 7.1.1 Estimating the Surface Enrichment from the Last Thermal Pulses 161 7.2 Stellar Yield Uncertainties ...... 164 7.2.1 Uncertainties From Mass Loss ...... 166 7.2.2 The Initial–ﬁnal Mass Relation ...... 167 7.2.3 Uncertainties in the Nuclear Reaction Rates ...... 168 7.3 Stellar Yields ...... 170 7.3.1 Dependence Upon the Initial Abundances ...... 171 7.4 Comparison to Other Authors ...... 171 7.4.1 Physical Assumptions ...... 172 7.5 Yield Results ...... 174 7.5.1 Hydrogen ...... 175 7.5.2 Helium: 4He ...... 178 7.5.3 Lithium: 7Li ...... 179 7.5.4 Carbon: 12C ...... 180 7.5.5 Carbon: 13C ...... 182 7.5.6 Nitrogen: 14N ...... 184 7.5.7 Nitrogen: 15N ...... 184 7.5.8 Oxygen: 16O ...... 186 7.5.9 Oxygen: 17O ...... 188 7.5.10 Oxygen: 18O ...... 189 CONTENTS v

7.5.11 Fluorine ...... 189 7.5.12 The Neon Isotopes ...... 191 7.5.13 Sodium ...... 193 7.5.14 The Magnesium Isotopes ...... 195 7.5.15 The Aluminium Isotopes ...... 196 7.5.16 The Silicon Isotopes ...... 199 7.6 Summary and Further Work ...... 200

8 Conclusion 203 8.1 Future Directions ...... 206

A HR Diagrams 207

B Surface Abundance Results From the First and Second Dredge-Up 211

C Surface Abundance Results for the Asymptotic Giant Branch 221

D Stellar Yields 261

Bibliography 272 CONTENTS vi

Abstract

In this thesis we investigate a diverse range of topics related to asymptotic giant branch (AGB) stars by using detailed stellar models with the latest input physics. We simulate the orbital parameters of the chemically-peculiar barium stars and their now extinct AGB companions using a rapid binary evolution algorithm with stellar-wind accretion, and including an improved treatment for the thermally-pulsing AGB phase of evolution. The orbital parameters of the barium stars are, on the whole, consistent with our model, but the best fit requires a mild tidally-enhanced stellar wind, a low accretion efficiency and much weaker tides than expected from either observation or theory. The model does not reproduce the orbital periods and eccentricities of the shortest-period barium stars, and tends to overproduce very-long period systems. To improve the stellar-wind accretion model, we require a better parameterization of the third dredge-up. We therefore generated a homogeneous set of low and intermediate mass AGB models with and without mass loss, that cover a larger range of mass and composition than any previously available. The effect of mass loss is negligible for M & 3M , but has a dramatic impact on the evolution of low-mass models causing their third dredge-up efficiency to be greatly reduced. The models without mass loss are used to parameterize the third dredge-up as a function of the stellar mass, metallicity and pulse number in a form suitable for synthetic evolutionary algorithms. The contribution made by AGB stars to the chemical enrichment of the interstellar medium is explored by performing detailed nucleosynthesis calculations for the mass-loss models. We investigate the stellar lifetimes and the surface abundance changes caused by the first and second dredge-up, and the structural features of the AGB models most relevant to nucleosynthesis. A comparison is made between our results and some recently published results in the literature. The nucleosynthesis that occurs during the thermally- pulsing AGB is reviewed in detail, with an emphasis on the surface abundance results for sodium and the neon, magnesium and aluminium isotopes. The intermediate mass stellar models efficiently produce the heavy magnesium isotopes, and models with M 3M produce the most 22Ne, owing to deep third dredge-up and low 22Ne destruction≈ rates. For completeness, we include the surface abundance results for all mass-loss models in Appendix C. We calculate the stellar yields of hydrogen through to 30Si in a form suitable for chemical evolution models, and discuss the main sources of uncertainty that affect the results. Our yields are compared to other calculations published in the literature, with the conclusion that the stellar yields are extremely sensitive to the details of the numerical model, in particular, the details of the third dredge-up and the mass-loss law used on the AGB. Our large set of AGB models by virtue of being detailed and self-consistent are expected to be valuable resource to which others in the field can compare. CONTENTS vii

Statement

This thesis contains no material which has been accepted for the award of any other degree or diploma in any university or other institution. To the best of my knowledge this thesis contains no material previously published or written by another person except where due reference is made in the text. The length of this thesis is less than 100,000 words, exclusive of ﬁgures, tables, appendices and bibliographies.

Amanda I Karakas CONTENTS viii

Publications

The following papers were published and/or prepared during the term of the author’s can- didature: Refereed Publications 1. The Chemical Evolution of Magnesium Isotopic Abundances in the Solar Neigh- bourhood submitted for publication in Publications of the Astronomical Society of Australia, 2003 Y. Fenner, B. K. Gibson, H.–C. Lee, A.I. Karakas, J. C. Lattanzio, A. Chieﬃ, M. Limongi and D. Yong

2. A New Synthetic Model for AGB Stars submitted for publication in Monthly Notices of the Royal Astronomical Society, 2003 R. G. Izzard, C. A. Tout, A. I. Karakas and O. R. Pols

3. The Production of Magnesium and Aluminium in Asymptotic Giant Branch Stars accepted for publication in Publications of the Astronomical Society of Australia, 2003 A. I. Karakas and J. C. Lattanzio 4. Chemical enrichment by Wolf-Rayet and AGB Stars published in Monthly Notices of the Royal Astronomical Society, 2002 Volume 338, Issue 4, Pages 973 – 989 L. Dray, C. A. Tout, A. I. Karakas and J. C. Lattanzio

5. The Eccentricity of the Barium Stars published in Monthly Notices of the Royal Astronomical Society, 2000 Volume 316, Issue 3, Pages 689 – 698 A. I. Karakas, C. A. Tout and J. C. Lattanzio

Conference Proceedings 1. Mg and Al Yields from Low and Intermediate Mass AGB stars in Carnegie Observatories Astrophysics Series, Vol. 4: Origin and Evolution of the Elements, 2003 ed. A. McWilliam and M. Rauch (Pasadena: Carnegie Observatories) http://www.ociw.edu/ociw/symposia/series/symposium4/proceedings.html A.I. Karakas and J.C. Lattanzio

2. Parameterizing the Third Dredge-up in Asymptotic Giant Branch Stars in Planetary Nebulae: Their Evolution and Role in the Universe, Proceedings of the 209th IAU Symposium, 2001, in press A. I. Karakas, J. C. Lattanzio and O. R. Pols CONTENTS ix

3. Thermal pulses and Dredge-up in AGB Stars in Evolution of Binary and Multiple Star Systems: A meeting in Celebration of Peter Eggleton’s 60th Birthday ASP Conference Series, Volume 229, Page 31, 2001 O. R. Pols, C. A. Tout, J. C. Lattanzio and A. I. Karakas

4. The Eccentricities of the Barium Stars in The Inﬂuence of Binaries on Stellar Populations Astrophysical and Space Science Library, Volume 264, Page 117, 2001 J. C. Lattanzio, A. I. Karakas, C. A. Tout

5. Evolution and Nucleosynthesis of AGB stars: current issues in Salting the Early Soup: trace nuclei from stars to the solar system Memorie della Societa Astronomica Italiana, Volume 72, Pages 255 – 264, 2001 J. C. Lattanzio and A. I. Karakas

6. The uncertainties of the 26Al(p,γ)27Si reaction rate and their impact on AGB nucleosynthesis calculations in Salting the Early Soup: trace nuclei from stars to the solar system Memorie della Societa Astronomica Italiana, Volume 72, Pages 319 – 328, 2001 M A. Lugaro, A. I. Karakas, A. Champagne, J. C. Lattanzio and R. C. Cannon

7. The Eﬀects of Binaries on Carbon Production published in Memorie della Societa Astronomica Italiana, 2000 Volume 71, Pages 797 – 804 C. A. Tout, J. C. Lattanzio, A. I. Karakas, J. R. Hurley and O. R. Pols

8. How Binary Stars aﬀect Galactic Chemical Evolution in Asymptotic Giant Branch Stars: Proceedings of the 191th IAU Symposium, 1998 Astronomical Society of the Paciﬁc, Page 447 C. A. Tout, A. I. Karakas, J. C. Lattanzio, J. R. Hurley and O. R. Pols

9. Eccentric Barium Stars and Wind Accretion in Nuclei in the Cosmos V, 1998 ed. by N. Prantzos and S. Harissopulos, Pages 235 – 238 A. I. Karakas, C. A. Tout and J. C. Lattanzio CONTENTS x

Acknowledgments

Foremost I would like to thank my supervisors, Dr. John Lattanzio of Monash University, and my co-supervisor, Dr. Chris Tout of Cambridge University for their guidance, support and friendship over the past 5 years. Extra special thanks to John – this thesis would not have been possible without his assistance in so many ways (port appreciation being one of them...). There are many people I would like to thank who have contributed to this work, including (in no particular order) Dr. Onno Pols of the Astronomical Institute Utrecht, for many useful discussions and guidance in analyzing the results; Dr. Robert Cannon of the University of Edinburgh, who provided the nucleosynthesis code and adapted it to our needs, and who helped find a few bugs at crucial times; Dr. Peter Wood of the Research School of Astronomy & Astrophysics at Mt.Stromlo, for providing the initial framework of the evolution code; Dr. Maria Lugaro of Cambridge University for updating the reaction network used in the nucleosynthesis code and many discussion about AGB stars; Rob Izzard of Cambridge University for providing results ahead of publication and many interesting discussions about synthetic AGB algorithms; Yeshe Fenner of Swinburne Uni- versity for providing results ahead of publication; Professor Brad Gibson of Swinburne University for access to the Swinburne Supercomputer; Dr. Alessandro Chieffi of the In- stitute of Space Astrophysics and Cosmic Physics for many lively discussions about stellar evolution (in particular with John) and Dr. Robin Humble of CITA and Swinburne Uni- versity, for helping with everything computer related and for proof-reading this thesis, and many excellent suggestions on the content and presentation. Further thanks go to some of the current and previous PhD students in the School of Mathematics & Statistics of Monash University. These students made life bearable at Monash and provided plenty of distractions when I’ve needed them including Dr. Robin Humble (of course), Dr. Maria Lugaro (life became less fun after you left), Kashif Rasul (ditto), Justin Freeman (thankfully there was Justin), Ashley Crouch (who’d have thought hey?), Lisa Elliott (for all those conference dinners), Simon Campbell (for putting up with me) and Mirko Velic (for the laughs). I would also like to thank some other friends including Dr. Katherine Dean (I’m so glad you were around), Matthew Nichol, Andre Peterson, Robert de Rozario, Bruce Shepherdley and Justine Taylor. Thanks to my family, especially my mother, for their encouragement and support to do what I really wanted. Lastly, to Robin, for his support and affection. Chapter 1

Introduction

Asymptotic giant branch (AGB) stars are important for many reasons. All stars in the mass range 0.8M to 8M will pass through the AGB phase of evolution, including our own Sun. Stars on∼the A GB phase are observed to be chemically diﬀerent from their less evolved counterparts, and are also observed to be losing mass rapidly. There are chemically-peculiar stars, such as the CH and barium-type stars, which we now believe obtained their unusual abundances by mass transfer from an AGB star. AGB stars are also the precursors of planetary nebulae (PNe), so the study of these evolved stars is necessary to provide insight into the interpretation of chemical abundances observed in PNe, and in turn the observations of PNe can improve our understanding of stellar structure and nucleosynthesis. Knowledge of AGB evolution is also vital to obtain an estimate of the contribution of low and intermediate mass stars to the chemical evolution of galaxies and globular clusters. Recently, the study of pre-solar grains has revolutionized the ﬁeld of stellar nucleosynthesis, by providing very precise measurements of isotopic ratios from material that con- densed in the circumstellar envelopes of stars (Zinner 1998). For example, the 20Ne/22Ne ratio can be directly measured in pre-solar grains (Lewis, Amari & Anders 1990) whilst only elemental neon can be measured in planetary nebulae. The isotopic ratios obtained from pre-solar grains cannot be measured by even the most determined spectroscopist. Many of the pre-solar grains are believed to originate in AGB stars (Amari, Lewis & An- ders 1994), making the need for detailed AGB models with up–to–date input physics more important than ever. In this thesis we are primarily interested in the changes to the surface composition of low and intermediate mass stars during the AGB. We begin with a brief overview of the evolution of low and intermediate mass stars from the zero-age main sequence to the AGB phase. We then discuss the aspects of AGB evolution that are covered in this work and how they are organised in this thesis.

1 CHAPTER 1. INTRODUCTION 2

Figure 1.1: Colour-magnitude diagram for M3, using data from Buonanno et al. (1994). The approximate positions of the main sequence, ®rst giant branch (FGB), horizontal branch (HB), and asymptotic giant branch (AGB) are labelled.

1.1 Evolution of Low and Intermediate Mass Stars

The colour-magnitude diagram of the globular cluster M3 is illustrated in Figure 1.1, using data from Buonanno et al. (1994). We will use M3 as a representative example of an intermediate metallicity globular cluster for the purposes of this introduction. Whilst most of the stars are located on the main sequence, there are distinct bands to the top right of the diagram and an almost horizontal band above the main sequence. Each of these bands, including the main sequence, represents a distinct phase in the life of low to intermediate mass stars. In the luminous red part of the diagram (top-right corner) there are a few AGB stars. These stars are highly evolved giants and most of this thesis is spent discussing the evolution and nucleosynthesis of these interesting objects. Before we introduce what AGB stars are, we need to review the evolution of low to intermediate mass stars prior to the AGB phase. This is only a brief introduction and we leave the detailed discussion of pre-AGB evolution and nucleosynthesis to Chapter 3. All stars begin their nuclear-burning life on the main sequence, burning hydrogen to CHAPTER 1. INTRODUCTION 3 helium in their cores. The majority of a star’s nuclear-burning life is spent on the main sequence, which is why we find most of the stars observed in M3 in this phase of evolution. The stars in M3 typically have masses between about 0.18 to 0.8M (Marconi et al. 1998), which means that their main-sequence lifetimes are exceedingly long (&10 gigayears or Gyr). In the following discussion we describe the evolution of a low-mass ( 0.8M ) star, and note that all low to intermediate mass stars undergo the same evolution. The∼ evolution of an intermediate mass ( 5M ) star diverges from that of a low-mass counterpart after the beginning of core helium∼ b urning and we also discuss this briefly. Following core hydrogen exhaustion, the star becomes a red giant for the first time. A first giant branch star can be loosely characterized by an inert He core, a hydrogen-burning shell, and a deep convective envelope which extends to the stellar surface. It is during the ascent of the first giant branch (FGB) that a star undergoes the first dredge-up (FDU). The FDU is caused by the deep convective envelope moving inward (in mass) to the region where partial hydrogen burning has taken place. This region was previously occupied by the core during central hydrogen burning, but is external to the H-shell. In Chapter 3 we review in detail the surface abundance changes caused by the first dredge-up, but here we note that the main results are an increase in the 4He, 13C and 14N abundances and a decrease in the 12C abundance. The star is now very big (up 100 times the size it was on the main sequence) but most of the mass in the core is within a small fraction of the total radius. A consequence of this is that the outer layers are only tenuously held onto the star and can be lost through an outflow of gas called a stellar wind. About 30% of a star’s total mass can be lost on the first giant branch, depending upon the length∼of time a star spends in this phase of evolution. The FGB lifetime is terminated when the necessary temperatures for central helium ignition are reached, at about 100 million K. The time it takes to reach this temperature in the core is dependent on the initial mass and composition, with low-mass stars taking longer to reach the critical temperature than higher mass (or lower metallicity) stars. For the predominantly low-mass stars found in M3, the first giant branch is the next longest phase of evolution after the main sequence, and the star will typically spend & 0.5 Gyr there. This is why the first giant branch of M3 is the second most populated branch in Figure 1.1. After helium is ignited in the core, the whole star contracts and moves down and to the left (blue) part of the HR diagram. Even though there is now a central energy source to halt the contraction of the core, it is the hydrogen-burning shell that provides most of the energy radiated by the stellar surface. The star is now in the core He-burning phase of evolution and is found on the horizontal branch (HB), as indicated in Figure 1.1. There are fewer stars here because the core He-burning timescale is about 100 million years for globular cluster stars, which is less than the time spent on the first giant branch and main sequence. Stars with M & 4M can move into the classical Cepheid instability strip during the core helium burning phase. Cepheid variables have pulsation periods strongly coupled with their photometric luminosity and thus are powerful tools for estimating distances. Owing to the age of M3, which is about 12.5 0.1 Gyr (VandenBerg 2000), stars over 2M have evolved beyond the core He-burning phase of evolution and consequently there are no Cepheid variables. RR Lyrae stars, with progenitor masses of about 0.75M (Gautschy & Saio 1996) are present on the HB and provide an accurate chronometer from

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¤¡¤¡¤¡¤¡¤ §¡§¡§¡§¡§¡§¡§¡§¡§¡§ Hydrogen burning¨¡¨¡¨¡¨¡¨¡¨¡¨¡¨¡¨¡¨ shell

Figure 1.2: Schematic structure of an AGB star showing the degenerate CO core surrounded by a He-burning shell above the core and a H-burning shell below the deep convective envelope. The burning shells are separated by an intershell region rich in helium ( 75%) and carbon ( 22%) with some oxygen and 22Ne. ∼ ∼ the luminosity-metallicity relation (Sandage, Katem & Sandage 1981; Carney, Storm & Jones 1992) Following core helium exhaustion, the core contracts and there is a structural re-adjustment to shell helium burning. The re-adjustment drives a strong expansion of the outer layers and the star becomes a red giant for the second time. Stars in this phase are said to be on the asymptotic giant branch, and as mentioned earlier, are located in the most luminous, red part of Figure 1.1. The strong expansion extinguishes the hydrogen-burning shell in the more massive stars (roughly 4M to 8M ), and the inner edge of the convective envelope moves inward in mass a second time. The second dredge-up mixes the products of complete hydrogen burning (mostly 4He and 14N) to the surface, and is described in detail in Chapter 3. For low-mass stars, the convective envelope also moves inward in mass during the early AGB, but the hydrogen-burning shell is not extinguished so there is no second dredge-up. The AGB phase is the final nuclear burning phase for stars with masses between about 0.8 to 8M . The helium-burning shell is thermally unstable, and flashes or pulses every 104 years or so. After the occurrence of the first thermal pulse, the star is said to be on the thermally-pulsing AGB (TP–AGB). A TP–AGB star is characterized by a CO core, two nuclear burning shells (one burning helium to carbon episodically, the second burning hydrogen to helium), an intershell region between the burning shells, and a deep convective envelope. In Figure 1.2 we schematically illustrate the structure of an AGB star. We note that this diagram is not to scale, and that the ratio of the radial thickness of the H-exhausted 5 core to the envelope is 10− : 1! In other words, AGB stars are very large and distended, CHAPTER 1. INTRODUCTION 5 often occupying a radius that is many hundreds of times the solar radius. Toward the end of the AGB phase, rapid episodic mass loss expels the envelope into the interstellar medium (ISM). The AGB phase is terminated by the final (non-explosive) ejection of the envelope, when the star becomes a planetary nebula and finally a white dwarf. Figure 1.1 shows that there are fewer AGB stars in M3 than there are on the FGB or horizontal branch. The reason for this is again the relative timescales. The AGB phase is short ( 106 years), comprising much less than 1% of a star’s main-sequence lifetime. ∼ After the occurrence of a thermal pulse, mixing episodes may occur which bring the products of nuclear burning from deep inside the star to the stellar surface. These mixing events, or so-called third dredge-up events, bring the products of partial He-burning, which is mostly 4He and 12C, to the stellar surface. Thus stars on the TP–AGB can become carbon-rich where the carbon to oxygen ratio (by number) is greater than unity, unlike most FGB stars and the Sun, which have C/O ratios 0.4. Carbon-rich AGB stars, or carbon stars, were known long before stellar evolution theory∼ could explain their existence. The first carbon star was found by Secchi in 1868, and Annie Cannon (1918) devised a classification scheme for carbon stars, based on the strength of a number of absorption lines, including the C2 line. The cool, luminous carbon stars were classified as N-type giants, where the C/O > 1 (Keenan & Boeshaar 1980; Ohnaka & Tsuji 1996). There are also S and SC-type stars which show varying degrees of carbon enhancement. The S stars show definite enhancements of 12C (Scalo 1976; Smith & Lambert 1990) and the SC stars have C/O ratios 1 (Ohnaka & Tsuji 1996). The carbon content in the envelope is expected to increase along≈ the spectral sequence M S SC CS C (N-type) (for example Abia et al. 2003). Besides carbon overabundances,→ → the→N-type→carbon stars and some S-type and SC-type stars show overabundances of heavy elements (Smith & Lambert 1986, 1990) created by the slow-neutron capture process (s-process) which is discussed further in Chapter 2. The S-type stars can be classified by the presence of ZrO lines (Keenan 1954) and some S stars also show lines of LaO and VO (Smith & Lambert 1990). The abundances of many s-process elements including Sr, Y, Zr, Ba, La, Nd, Sm and Ce are observed in N-type carbon stars (Abia et al. 2002) as well as the S-type (Utsumi 1985) and SC-type (Abia & Wallerstein 1998) stars. The first direct evidence that mixing occurs in AGB stars was found by Merrill in 1952, who detected Tc in several stars primarily of type S. As there is no stable isotope of Tc, and the longest lived isotope is 99Tc with a half-life of 2.11 105 years, this detection was direct confirmation that heavy-element nucleosynthesis was× occurring in these stars, and that a recent mixing episode had brought material from the nucleosynthesis site to the surface. The S, SC and N-type giants are now believed to be TP–AGB stars, and that the third dredge-up is responsible for their carbon and s-process overabundances. Other classes of carbon-rich stars include the R-type (Dominy 1985; Knapp, Pourbaix & Jorissen 2001), the J-type and the CH-type stars. The R-type stars are believed to be either at the tip of the FGB or on the horizontal branch, and there is still some debate as to the cause of their carbon overabundances (Fujimoto 1977; McClure 1997). The J-type stars are possibly AGB or early-AGB stars, but since they do not show the same spectroscopic signature as the N-type stars (Abia & Isern 2000) it is unclear what is the origin of their carbon overabundances. CHAPTER 1. INTRODUCTION 6

There are other types of chemically-peculiar stars with s-process enhancements, including the Population I barium stars (Zacs 1994; Barbuy et al. 1992; Liang et al. 2003), the extrinsic or binary S-type stars (Jorissen et al. 1998; Van Eck & Jorissen 2002), the Pop- ulation II CH stars (McClure 1984) and the dwarf carbon stars (Dahn et al. 1977; Green & Margon 1994). The dwarf carbon stars are on the main sequence, and the barium and CH-type stars are (mostly) ﬁrst giant branch stars. For this reason neither of these stars could have obtained their s-process enhancements from internal nucleosynthesis. The barium stars (McClure & Woodsworth 1990), and many of the CH stars (McClure 1984) and extrinsic S-type stars (Jorissen et al. 1998) are found to be in binary systems with relatively long orbital periods. It was thus assumed that these chemically-peculiar objects obtained their carbon and s-process enhancements through mass transfer from a now extinct AGB companion (Boﬃn & Jorissen 1988).

1.2 Chapter Synopsis

We now give a synopsis of each chapter, starting with Chapter 2, where we present results of numerical simulations carried out in order to study the formation of the barium stars.

1.2.1 The Barium Stars The barium stars were first identified by Bidelman & Keenan (1951) as a class of chemically peculiar Population I red giants with an observed frequency of about 1 per cent of the total population of G and K giants. They exhibit unusually strong absorption lines of barium and strontium, as well as enhanced CN and CH bands. Detailed abundance analyses reveal that the barium stars are typically enhanced in the s-process elements by factors of 2 to 30 with respect to normal giants (Tomkin & Lambert 1979; Kovacs´ 1985; Liang et al. 2003). Despite carbon being found to be overabundant by a factor of 3 in the most extreme cases (Lambert 1988; Barbuy et al. 1992), the barium stars are not carbon stars i.e. C/O < 1 at the surface. All observed barium stars were found to belong to spectroscopic binary systems (McClure, Fletcher & Nemec 1980; McClure 1984; Jorissen & Mayor 1988; McClure & Woodsworth 1990) with eccentricities significantly lower than a sample of spectroscopically normal G and K giants (McClure & Woodsworth 1990). This was taken as evidence that the large s-process overabundances might be the result of mass transfer from a former AGB companion (Webbink 1986; Boffin & Jorissen 1988), rather than the result of internal nucleosynthesis. According to the mass-transfer hypothesis, the former AGB star would now be an optically invisible white dwarf (McClure 1984; Brown et al. 1990; Luck & Bond 1991; McClure 1997). Several investigations with the IUE satellite looked for the characteristic UV excesses associated with white dwarfs. Whilst some white dwarfs were found, there were also some negative results. McClure & Woodsworth (1990) explain these by suggesting that the s-processed material was transferred a very long time ago so that the white dwarfs have cooled down below detectable levels. Further evidence for the mass-transfer hypothesis came from the discovery of main-sequence barium stars, first by Bidelman in 1985 and later by by North & Duquennoy (1991), North, Berthet & CHAPTER 1. INTRODUCTION 7

Lanz (1994) and Bergeat & Knapik (1997). Bergeat & Knapik (1997) comment that while most barium stars are first giant branch stars, consistent with their original classification, they are found over five different luminosity classes (or evolutionary phases). There are two distinct mass-transfer processes which could result in the formation of the barium stars. The first is Roche-lobe overflow (RLOF; Webbink 1986; Iben & Tutukov 1985) and the second is stellar-wind accretion. We discuss the two mass transfer processes in more detail in 2.1 but we mention here that one of the characteristics of RLOF is to circularize orbits.§ Because the eccentricities of the barium stars are on average, non zero, Webbink (1986) ruled out such a process particularly with either common-envelope (CE) or contact-binary evolution (McClure & Woodsworth, 1990). To circumvent the problem of rapid orbit circularization, Boffin & Jorissen (1988) suggested that the mass was transferred by a stellar wind. There have been a few studies of barium star formation via the stellar wind accretion hypothesis, notably by Boffin & Jorissen (1988), Han et al. (1995), Theuns et al. (1996) and more recently by Liu et al. (2000) and Liang et al. (2003). Han et al. (1995) also investigated stable RLOF and CE evolution as channels in the production of barium and CH stars even though both these scenarios tend to completely circularize the orbits. The simulations of Han et al. (1995) are evidence that the dominant formation mechanism is stellar wind accretion, because most of their barium stars which formed via RLOF had orbital periods well below 80 days, which is the observed lower- limit for barium star orbital periods. In this thesis we test the stellar wind accretion hypothesis further, by including tidal evolution in a rapid binary evolution algorithm developed by Tout et al. (1997), modified to include stellar wind accretion and nucleosynthesis. We carry out the population synthesis in a similar way to that described by Han et al. (1995), but with an improved treatment of the barium enhancement in the TP–AGB phase of the mass-losing star, and dilution into the atmosphere of the pre-barium stars, and add a tidal evolution model. The details of the model we used, and the results obtained are presented in Chapter 2 and published in Karakas, Tout & Lattanzio (2000).

1.2.2 Parameterizing the Third Dredge-Up The nucleosynthesis model used in the above mentioned synthetic binary algorithm included a parameterization of the third dredge-up for the AGB star. The parameterization of the third dredge-up required a fit for the core mass at the first thermal pulse, the core mass at the first dredge-up episode, and the efficiency of the third dredge-up. Unfortu- nately, the papers in the literature only covered a small section of the mass range that we required for the binary evolution algorithm (which required data from about 0.8 to 8M ). In Chapter 2 we used a fit for the low-mass stars obtained from the models of Straniero et al. (1997) and we used the models of Frost (1997) to fit the efficiency of the third dredge- up for M > 4M . There was simply a lack of stellar models published in the literature for masses between about 2 and 3.5M . For instance, Straniero et al. (1997) only include data for a 1.5 and 3M model. Boothro yd & Sackmann (1988b,c) also examined the operation of the third dredge-up in low-mass models but again, only for a narrow mass range. Also, the calculations of Boothroyd & Sackmann (1988b,c) are much older than either the Frost CHAPTER 1. INTRODUCTION 8

(1997) or Straniero et al. (1997) models which means that different input physics, notably different opacity tables, were used. Lattanzio (1986) covered the 1 to the 3M mass range, but not enough thermal pulses were calculated, and again older input physics was used. As an improvement, our next project focused on the occurrence of the third dredge- up in detailed AGB models. We produced a homogeneous set of models covering a large range in mass and composition. We calculated models between 1M and 6M with Z = 0.004, 0.008, 0.02. We evolved the models from the pre-main sequence to near the end of the AGB, through all intermediate stages including the core helium flash for the low-mass models. Owing to large uncertainties in the treatment of mass loss on the AGB, we calculated models both with and without mass loss. The models without mass loss provided the limiting behaviour of the third dredge-up, in other words we obtain the deepest dredge-up in these models. The mass-loss models were initially calculated to provide information on the cessation of dredge-up. This was to be done by studying how the efficiency of the third dredge-up changed as the envelope mass approached zero, but owing to convergence difficulties the evolution of the massive AGB models was halted before the envelope was reduced below about 1M , and the third dredge-up efficiency was not effected. Instead, we used the mass-loss models to calculate stellar yields, discussed below. The results of this extensive set of AGB models was fitting formulae for the core mass at both the first thermal pulse and the first third dredge-up episode, as well as to constrain the third dredge-up efficiency parameter, λ. This is the first time such a parameterization has been available. The results from this project are presented in Chapter 6 and published in Karakas, Lattanzio & Pols (2002).

1.2.3 Stellar Yields Even though the timescale of AGB evolution is short compared to earlier phases, AGB stars are numerous. Over 10,000 carbon stars have been observed in the Large Magellanic Cloud and about 3,000 in the Small Magellanic Cloud (Blanco, Blanco & McCarthy 1980; Catchpole & Feast 1985). Owing to the shape of the initial mass function, which favours stellar formation at low mass for stars with near to solar metallicities (for a recent example see Kroupa, Tout & Gilmore 1993), and the short lifetimes of very massive stars, AGB stars are more numerous than massive stars that explode as supernovae or Wolf-Rayet stars that lose their mass through a very strong stellar wind. AGB stars are so numerous that they contribute more mass to the interstellar medium per year than supernovae (Os- terbrock 1989). If AGB stars contribute so much mass to the ISM, then the combination of chemical processing and mixing events means that AGB stars are likely to be important contributors to the chemical evolution of say, carbon, nitrogen and the s-process elements. The contribution made by AGB stars can be estimated both from observations of planetary nebulae and from theoretical models. Whilst the mass-loss models did not help constrain the third dredge-up efficiency parameter, they were ideally suited to the study of the contribution made by AGB stars to the ISM. Thus we performed detailed nucleosynthesis calculations on the mass-loss models, using a post-processing nucleosynthesis code with 74 species and 506 reactions. The numerical details of the stellar structure and the post-processing nucleosynthesis codes are CHAPTER 1. INTRODUCTION 9 discussed in Chapter 3, as are the results of the evolution and nucleosynthesis for the pre- AGB phase. We present the evolutionary tracks and the surface abundance variations of selected species for all of the mass-loss models in Appendix A and C, respectively. The results of the evolution during the AGB phase are given in Chapter 4, and the results of the nucleosynthesis calculations during the AGB are presented in Chapter 5. In these chapters we compare our results with recent calculations and the observations where possible. In Chapter 5 we focus the discussion to the production of the 22Ne, 25Mg and 26Mg isotopes in low to intermediate mass AGB stars. We compare our results for the neutron-rich Mg isotopes to the observations of Kahane et al. (2000), Gay & Lambert (2000) and Yong et al. (2003). The last part of Chapter 5 documents the comparison between the neon abundances predicted by the Z = 0.02 and Z = 0.008 models to the observations of elemental neon in galactic and LMC planetary nebulae. As previously noted, convergence difficulties set in for models with M > 2.5M before the very end of the AGB was reached. In practical terms, this meant that some of these models still had relatively large envelope masses and could undergo more thermal pulses and dredge-up episodes. To account for the enrichment from these few remaining thermal pulses, we used a simple analytic model based on the principles of synthetic AGB algorithms (see Groenewegen & de Jong 1993; Marigo, Bressan & Chiosi 1996). From the combination of detailed AGB models for most of the AGB and synthetic models for the remaining thermal pulses, we calculated a set of stellar yields suitable for galactic chemical evolution. The stellar yields are presented in Chapter 7, along with a comparison between our results and the yields of other authors, notably van den Hoek & Groenewegen (1997), Forestini & Charbonnel (1997), Marigo (2001), Ventura et al. (2002) and Izzard et al. (2003). Most of these other calculations, with the exception of Ventura et al. (2002), use semi-analytic (van den Hoek & Groenewegen 1997; Marigo 2001) or fully analytic models (Izzard et al. 2003) and only follow the CNO isotopes. Hence this new set of stellar yields is a valuable contribution to the area because we calculate yields from detailed stellar models for the CNO nuclei, fluorine, sodium and the neon, magnesium, aluminium and silicon isotopes. Chapter 2

The Formation of the Barium Stars

This chapter presents the stellar wind accretion model used to study the formation of the barium stars, and the results obtained for that model. The chapter is based on a self- contained body of work and is published in the Monthly Notices of the Royal Astronomi- cal Society, 2000, volume 316, pages 689 to 698 by Karakas, Tout & Lattanzio. The work presented in this chapter provided the inspiration for parameterizing the third dredge-up (which we present in Chapter 6). The initial aim was to use the new parameterizations to improve the barium-star model, and to also extend the model to the Population–II CH stars. However, the large and homogeneous set of asymptotic giant branch (AGB) models were instead studied in more detail, as explained in later chapters. In 2.1 we begin the chapter with a discussion of the two types of mass transfer. In 2.2 we§ present the observed properties of the barium stars. The details of the model, including§ the mass-loss law used, the nucleosynthesis model and the Monte-Carlo simulation parameters are given in 2.3, and the tidal evolution model in 2.4. The results and discussion are presented in 2.5§ and 2.6 concludes the chapter. § § §

2.1 Introduction

In Chapter 1 we introduced the mass-transfer formation mechanism that has been proposed to explain the chemical peculiarities of the barium stars. The hypothesis states that mass is transferred from a donor AGB star, enriched in heavy elements, onto a main-sequence companion. The basic hypothesis begins with a pair of stars in a detached binary system, which evolve as single stars until the more massive star, which we will call the primary, reaches the thermally-pulsing AGB phase (TP–AGB). During the TP–AGB phase, the envelope of the primary becomes enriched in 12C and heavy elements created by slow- neutron capture. The AGB star is also losing mass through a strong stellar wind, some of which is accreted onto the surface of the main-sequence companion, which we will call the secondary. As an aside, there are two regimes under which heavy elements are created, the rapid and slow neutron capture processes. Which regime occurs in a star depends on the number of free neutrons available. The rapid or r-process, requires neutron densities of the

10 CHAPTER 2. THE FORMATION OF THE BARIUM STARS 11

Figure 2.1: Roche equipotential in the orbital plane z = 0. The centres of the stars are for the primary (star 1) at O and for the secondary (star 2) at S . The centre of mass of the system is at G. The plane shown is in the orbital plane of the binary system. The inner Lagrangian point L1 and the outer Lagrangian point L2 are also marked (from Pringle 1985).

20 3 order of nn & 10 cm− , and which means that the relative lifetime of β-decay to neutron capture favours neutron capture. The astrophysical site of the r-process is still uncertain, but the required neutron densities and timescales ( 1second) imply a violent event such as a core-collapse supernova or neutron star–neutron–star∼ merger (Cowan, Thielemann & Truran 1991). The most neutron-rich isotopes are created under this regime, including the radioactive elements uranium and thorium. The slow or s-process occurs when the neutron 8 3 densities are of the order of nn . 10 cm− , and which means that the relative lifetime of β-decay to neutron capture favours β-decay. Thus the s-process elements are generally found along the so-called “valley of β–stability”, and include yttrium, zirconium, barium, technetium, strontium, lanthanum, samarium and lead (Busso et al. 2001). The s-process is believed to occur in TP–AGB stars either in the intershell during the interpulse phase (Straniero et al. 1995) or in the convective pocket during the thermal pulse. The region where the s-process occurs depends on the initial mass of the star, which in turn determines which reaction is responsible for providing the neutrons. In low-mass stars the neutrons are believed to come from the 13C(α,n)16O reaction whilst the 22Ne(α,n)25Mg reaction is believed to be dominant in more massive (M > 4M ) AGB stars (Gallino et al. 1998; Busso et al. 2001). We do not discuss the creation of the s-process elements further but refer the reader to Gallino et al. (1998), Busso, Gallino & Wasserburg (1999) and Busso et al. (2001) for further details. If we assume the basic hypothesis that the barium stars were created by mass transfer from a TP–AGB star onto a main-sequence companion, it is natural to ask the question: what type of mass transfer process is the most likely, given the indirect observational ev- CHAPTER 2. THE FORMATION OF THE BARIUM STARS 12 idence? Two hypotheses have been put forward to explain the barium stars. The first is tidal disruption (Webbink 1986), which occurs when the primary star becomes a giant and fills or nearly fills it’s Roche Lobe. In this scenario, the AGB star has grown larger than the equipotential surface that joins the two stars (see Figure 2.1), forcing the binary system into a semi-detached state and initiating mass-transfer. The change to the orbital parameters from mass transfer via Roche lobe overflow (RLOF) depends on the masses and evolutionary state of the two stars involved. In the case of barium star formation, an AGB giant is dumping mass onto a main-sequence companion, which causes the orbital period to shrink and the eccentricity to decrease. RLOF in this case is dynamically unstable, and leads to common envelope ejection (CE). Han et al. (1995) argue that stable RLOF might occur if the primary has lost a substantial fraction of it’s mass via a tidally- enhanced stellar wind. Han et al. (1995) go on to state that stable RLOF might account for the few barium stars with orbital periods 100 days and zero eccentricities. In 2.2 we show that most barium stars∼do not have circular orbits, but have, on average, eccentric§ orbits. Boffin & Jorissen (1988) proposed stellar wind accretion as an alternative to RLOF, to circumvent the problem of rapid circularization by allowing the two stars to always remain detached. The effect of stellar wind accretion on the orbital parameters of the barium star system is to reduce the eccentricity of the system (but not completely circularize) and to drive the two stars apart due to changes in the angular momentum (Han et al. 1995). The main theoretical uncertainty associated with the stellar wind accretion scenario is the amount of mass lost in the stellar wind from the primary, the velocity of the stellar wind and the accretion process onto the secondary. In this chapter we aim to test a theoretical model proposed to explain the existence of barium stars, based on the wind accretion hypothesis with tidal evolution. The simplest way to test such a model is to run population synthesis simulations and compare theoretical distributions of observable properties to actual observations. We do not investigate the formation of barium stars via RLOF and hence we stop the calculation when the stellar radius, R, is larger than the Roche lobe radius RL. Very few barium stars have zero eccentricities so we do not expect many to result from RLOF. This is borne out by the simulations of Han et al. (1995), with significantly enhanced mass loss (which we shall show is necessary), where the fraction of barium stars formed from RLOF and CE is less than 1% of all the barium stars. We now discuss the observational properties of the barium stars.

2.2 Observations of the Barium Stars

A decade worth of radial velocity observations led to the conclusion that all barium or Ba-II stars are in binary systems with orbital periods typically longer than 100 days (d) but less than 10 years. (McClure, Fletcher & Nemec 1980; McClure 1984; McClure & Woodsworth 1990). Jorissen et al. (1998) reported barium stars with periods outside this range. Figure 2.2 shows the observed eccentricity–period (e, P) diagrams and the eccentricity distributions of the barium stars (taken from Jorissen et al. 1998) and a similar set of normal giants from Boffin, Cerf & Paulus (1993). In the Jorissen et al. (1998) sample CHAPTER 2. THE FORMATION OF THE BARIUM STARS 13 of barium stars, there are 62 with known orbital periods and eccentricities (large black dots) and 8 with lower-limit estimates for the orbital period but with unknown eccentricity (open stars). The barium stars with unknown eccentricity have been arbitrarily assigned a high eccentricity of about 0.9 to distinguish them from the other stars. The large-filled triangle represents the star BD +38◦ 118 (a+b) , which has been excluded from the sample of barium stars used later to test the models on the grounds of being in a triple system. For comparison we plot a sample of 213 spectroscopically normal giant binaries from Boffin et al. (1993) which have G and K spectral types as do the barium stars (small grey dots). We also include a shaded region between 104 and 105 days to represent the few barium stars with no evidence for binary motion. There are a few barium stars with known orbital parameters in this period range; and it is also quite probable that the Ba-II stars with no evidence for binary motion sit in this region of the (e, P) diagram if they are binaries. The sample of barium stars includes both “mild” and “strong” according to the classification system devised by Warner (1965). Warner’s classification scheme for the barium stars uses an estimate of the Ba II λ4554 line strength to indicate the extent of abundance anomalies on a scale of 1 to 5, with higher numbers indicating higher overabundances. Strong barium stars have a Ba index of 3,4 or 5 and the mild or weak barium stars have an index of either < 1 or 2. The sample of spectroscopically normal giant binaries have MK spectral classifications indicating giant luminosity and G or K spectral type as the barium stars. The average eccentricities are 0.17 for the barium stars and 0.23 for the normal giants. A Kolmogorov-Smirnov statistical test determines the probability that two data sets have been drawn from different population distribution functions. If a Kolmogorov- Smirnov test results in a low probability then the two data sets are different whilst a high probability means the two data sets are similar. A Kolmogorov-Smirnov test reveals that the probability of the barium star eccentricities being chosen from the same distribution as 3 those of the normal giants is about 10− . Because lower eccentricities are much easier to measure than higher eccentricities, selection effects will alter the shape of the observed eccentricity distributions and this must be accounted for when comparing the observed distributions with the simulated distributions produced by our models. This means that any statistical tests conducted as a comparison with the observed eccentricity distributions will favour models which produce more lower eccentricity barium stars than higher ones. This is not the case for periods, however. All but two of the 37 known strong barium stars have known orbital periods and all but 3 of the sample of mild barium stars taken by Jorissen et al. (1998) have known orbital periods. Of the sample of 70 mild and strong barium stars investigated by Jorissen et al. (1998) only 8 do not have well determined eccentricities. We conclude that selection effects are minor for periods < 104 d. The cumulative frequency of mass functions f (M) observed for the barium stars yields information on the mass of the barium stars and their companions, such that

M3 = 2 3 3 f (M) 2 sin i Q sin i, (2.1) (M1 + M2) ≡ where M1 and M2 are the masses of the barium star and its companion respectively. Web- CHAPTER 2. THE FORMATION OF THE BARIUM STARS 14

Figure 2.2: The observed (e, P) diagram (top) and eccentricity distribution (bottom) for the barium stars, compared to a sample of spectroscopically normal G and K giants. Note that the eccentricity histogram has the number of normal giants scaled to the number of barium stars. CHAPTER 2. THE FORMATION OF THE BARIUM STARS 15 bink (1986) and McClure & Woodsworth (1990) show that the f (M) associated with the barium star binary systems is very different from that of normal G and K giants. Mc- Clure & Woodsworth (1990) find a narrow range of Q values, with a mean value of Q = 0.041 0.010M for their sample of 16 barium stars. The narrow range of Q values indicate that the companions of the barium stars span a narrow range in mass, which is expected if the companions are white dwarfs (Jorissen et al. 1998). Jorissen et al. (1998) derive an average Q value for their sample of barium stars and find Q = 0.043. If we assume the companion mass is M2 = 0.6M then, on average, the masses of the barium stars are 1.65M , whilst if we take M2 = 0.67M the barium star masses are on average

M1 = 2M (Jorissen et al. 1998).

2.3 Description of the Model

Formulae for the computation of stellar evolution as a function of initial mass, age and current mass (reduced by mass loss) are to be found in detail in the paper by Tout et al. (1997). The formulae are good empirical ﬁts to the detailed stellar models described in that paper. They give the radius, luminosity, stellar type and core mass, for giants throughout the evolution. In the following sections we describe the additional components that we must add to this model speciﬁcally for this work.

2.3.1 Mass Loss Throughout this work we use the mass-loss law of Reimers (1975),

1 13 (R/R )(L/L ) M˙ R/M yr− = 4 10− ηR , (2.2) × (M/M ) where R, L and M are the radius, luminosity and mass of the mass-losing star respectively (in solar units) and ηR is the Reimers parameter which we set to unity here. Observations indicate that formulae that concentrate mass loss toward the tip of the AGB may be more appropriate and we comment further on such prescriptions when we discuss our results. Tout & Eggleton (1988) added a term to the Reimers mass-loss law to allow for tidal eﬀects of a companion star and to account for the formation of certain semi-detached Algol-like binaries, such as

6 1 R 1 M˙ TE/M yr− = M˙ R 1 + B min , , (2.3)  ×  R ! 26    L        where RL is the radius of the Roche lobe (which is assumed to bespherical) around the mass losing star and the constant B is a parameter determining the size of the increase. When B = 0, equation (2.3) reduces to that of Reimers (1975). Tout & Eggleton (1988) required B = 104 to obtain mass-loss rates high enough to fit the observed parameters of the RS CVn system Z Her but we find a milder enhancement is sufficient for the Ba-II stars. CHAPTER 2. THE FORMATION OF THE BARIUM STARS 16

2.3.2 Stellar Wind Accretion A fraction of the mass lost in a stellar wind from the primary can be accreted by the secondary star. The accretion rate M˙ 2 for the secondary is estimated by Boﬃn & Jorissen (1988) and we use

2 1 1 GM2 αacc M˙ 2/M yr− = min 1, M˙ 1, (2.4)  2 2 2 2 3/2  −  √1 e Vwind ! 2a 1 + v   −      1 where v = Vorb/Vwind and Vwind is the wind velocity from the primary star (in kms− ). The mass-loss rate from the primary is M˙ 1 < 0, M1 and M2 are the primary and secondary mass respectively, e is the orbital eccentricity, a the semi-major axis and Vorb the orbital velocity given by 1 Vorb/kms− = G(M1 + M2)/a, (2.5) where G is the gravitational constant. The effipciency of accretion is given by the parameter αacc, which we vary between 0.05 to 1.5 in our models to examine which value leads to the best fit to the observed period distribution. The appropriate value of αacc for Bondi-Hoyle accretion is 1.5 but Theuns, Boffin & Jorissen (1996) suggest a value as low as 0.05 may be more appropriate for stellar wind accretion in the case of the formation of the barium stars following detailed hydrodynamic simulations. In the case of barium-star formation the binary motion strongly disturbs the shape of the accretion column as Vorb Vwind when r a, where r is the radius of the accretion column and a the semi-major axis≈of the binary system.≈ We take the wind velocity Vwind to be the escape velocity from the primary

1 GM1 Vwind/kms− = Vesc = , (2.6) r R1 where R1 is its radius. This is the most uncertain quantity in equation (2.4), because it is hard to determine accurately through observations. Decreasing αacc is equivalent to increasing the ratio of the kinetic energy of the wind to the escape energy. We evaluate the change in orbital separation and eccentricity owing to wind accretion according to the formalism of Huang (1956) and ﬁnd

2 a˙ M˙1 1 2 (1 + e ) = + M˙ 2 (2.7) a − M − M M ! (1 e2) 2 − and e˙ 1 2 = M˙ 2 + < 0, (2.8) e − M M2 ! where M = M1 + M2, if we assume that the change in momentum of the accreting star is given by M2δv2 = δM2V, where δv2 the change to the instantaneous velocity of the secondary star and V−is the total instantaneous velocity of the system. Both equations (2.7) and (2.8) take account of changes to the separation and eccentricity caused by the CHAPTER 2. THE FORMATION OF THE BARIUM STARS 17 accretion of more matter at periastron than at apastron in the case of an already eccentric orbit. These formulae assume that the momentum and angular momentum lost by the accreting component, owing to the drag it experiences in moving through the primary’s wind, is lost from the system in the escaping wind. The corresponding change, owing to mass transfer, in the instantaneous relative velocity v of the two stars is then m˙ v˙ = 2 v, (2.9) − M2 where m˙ 2 is the instantaneous accretion rate. They differ from the formulae derived by Theuns et al. (1996) and also those derived by Eggleton, Kiseleva & Hut (1998), both of which can lead to growing eccentricity in certain circumstances. The latter implicitly assume that any momentum lost by the accreting secondary star is somehow returned to the primary. This is difficult to envisage when the stars are well separated and m˙ 2 M˙ 1 so we prefer the assumption that we make here. In any case eccentricity changes that | are| due to formula (2.8) are small compared with those due to tides in the case of barium star formation. A more careful examination of the slow wind approximation (Snellgrove & Gair, private communication) leads to e˙ 1 1 = M˙ 2 + , (2.10) e − M 2M2 ! and a˙ M˙ 1 + e2 2 e2 M˙ = 1 + − 2 , (2.11) a − M − ( M M ) 1 e2 1 2 − but using these instead of equation (2.7) and equation (2.8) does not affect the results.

2.3.3 Nucleosynthesis Model We include the eﬀects of TP–AGB evolution with a synthetic evolution model that evalu- ates composition changes at each thermal pulse. The core mass at the ﬁrst pulse (in solar units) is taken from Lattanzio & Malaney (1989) to be

M (1) = 0.53 (1.3 + log Z)(Y 0.2), Z 0.01 (2.12) c − − ≥ where Y is the mass fraction of helium and Z is the metallicity at the ﬁrst thermal pulse. Barium stars are Population–I objects with a solar abundance of Z 0.02 and Y 0.28 so ∼ ∼ that Mc(1) = 0.562 M for all initial masses. We take the interpulse period to be a constant

4 tip = 10 yr, (2.13) for all stars. Although this assumption is crude it is acceptable because the interpulse period is not critical to our calculations. We test this by varying the interpulse period between 105 and 5 103 years and find no appreciable difference in the models. The same number of barium stars× are produced although the Kolmogorov–Smirnov test results and the average eccentricities are slightly but not significantly different. CHAPTER 2. THE FORMATION OF THE BARIUM STARS 18

In the population syntheses we are only interested in the production of the s-process elements, particularly barium, so we do not include nucleosynthesis and dredge-up prior to the thermally pulsing phase. We assume that third dredge-up begins when the core mass min min is higher than a critical value Mc . The exact value of Mc is still a matter of debate for stars with solar metallicity but for the present calculations we set

min 0.61 M , M0 3.0 M Mc = ≤ (2.14) (0.795 M , M0 > 3.0 M .

where M0 is the initial mass of the star. The value for M0 3.0 M is consistent with low-mass stellar models of Lattanzio (1989) and Straniero ≤et al. (1997) while that for M0 > 3.0 M is from Frost (1997). Between pulses on the AGB the H-exhausted core mass will grow during the interpulse period according to tip dMH ∆MH = dt, (2.15) Z0 dt 1 with the rate of core mass growth (M yr− ) to a good approximation given by

dMH 12 LH = 9.555 10− , (2.16) dt × X where LH is the luminosity provided by hydrogen burning (in solar units) and X the hy- min drogen abundance (by mass) in the envelope. If there is dredge-up (MH > Mc ) then a fraction of ∆MH will be dredged up

∆Mdredge = λ ∆MH, (2.17) where λ is the third dredge-up eﬃciency parameter. For M0 < 3.0 M the calculations of

Straniero et al. (1997) indicate a linear relation between λ and M0, while for M0 3.0 M the calculations of Frost (1997) favour a constant λ: ≥

0.2 + 0.0866M0, M0 < 3.0 M , λ = (2.18) ( 0.9, M0 3.0 M . ≥ These values of λ are not continuous. This reﬂects the lack of reliable AGB stellar models for stars around 3 and 4 M and the uncertainties in ﬁnding a consistent numerical treatment for the third dredge-up (see Frost & Lattanzio 1996 for a review). These values of min Mc and λ are crude estimates and we have attempted to improve upon these in Chapter 6. min Now, if MH > Mc , the core mass MH will also decrease in size owing to dredge- up so that over an interpulse phase the core mass increases by ∆MH then decreases by ∆Mdredge while the envelope mass Menv will increase by ∆Mdredge but decrease by ∆MH and by mass-loss (Groenewegen & de Jong 1993). Barium is produced in the intershell zone under radiative conditions (Straniero et al. 1995) where the logarithm of the barium CHAPTER 2. THE FORMATION OF THE BARIUM STARS 19 overabundance with respect to the solar value reaches an approximately constant value of

[Bashell/Fe] log (Bashell/Ba ) 2.6, (2.19) ≈ ≈ (Gallino, private communication). The mass of barium added to the envelope at each third dredge-up episode is then ∆MBa = Bashell∆Mdredge. (2.20) The actual abundance of barium in the primary’s envelope between the ith and the (i + 1)th dredge-up episode is Ba(i 1)M (i 1) + ∆M (i) Ba(i) = − env − Ba , (2.21) M (i 1) + ∆M (i) env − dredge where Ba(i 1) is the barium mass fraction in the envelope just before the ith dredge-up episode and−Ba(0) = Ba . We compute the abundance of barium in the envelope following dredge-up after each thermal pulse and this is the abundance of barium in the wind of the primary, a fraction of which is accreted by the secondary. The mass transferred from th th the primary to the secondary between the i and the (i + 1) dredge-up episode is Macc. Over the life of the star (in dredge-up episodes) this sums to give a total accreted mass of n n Macc = i=1 Macc(i) of which i=1 Macc(i)Ba(i) is barium. The pre–Ba-II star polluted by the TP–APGB star evolves to becomeP a red giant with a deep convective envelope that has a mass of Menv,2. The accreted barium is diluted into the envelope of the secondary at the deepest extent during the ﬁrst dredge-up, which has a mass of

M , (M + M ) M , (2.22) env 2 ≈ 2 acc − c,bgb where Mc,bgb is the core mass of the secondary star at the beginning of the giant branch. Subsequently the surface barium abundance remains constant for the ﬁrst giant branch lifetime (as we assume no further mixing) at

n ®nal i=1 Macc(i)Ba(i) + (Menv,2 Macc)Ba Ba = − , (2.23) P Menv,2 where the entire AGB envelope has been lost in the wind before the (n + 1)th pulse which shows dredge-up.

2.3.4 Monte-Carlo Simulation Parameters For each Monte-Carlo simulation we model 106 pairs of stars with initial separations, masses and eccentricities determined randomly from the following distributions. The initial separation distribution is ﬂat in log a

X amax a = amin , (2.24) amin ! CHAPTER 2. THE FORMATION OF THE BARIUM STARS 20 from Eggleton, Fitchett & Tout (1989), where X is a random number between 0 and 1, 4 amax = 10 a.u. and amin = 0.1 a.u. We are not interested in very close systems that interact before the primary has left the giant branch but note that they will aﬀect the overall fraction of Ba-II stars to a small extent. We give the initial distribution of eccentricities a linear slope to approximate the spread of observed red-giant eccentricities between 0 and 0.99. Such a linear slope results from the generating function

e = 0.99(1 √X), (2.25) − where X is the same random number used for equation (2.24). Most observed normal giants with periods in the same range as the barium stars should have eccentricities very close to their initial eccentricities because separations remain too large for tidal circularization to become eﬀective. The initial mass distribution is generated according to the formula of Eggleton et al. (1989), 0.19X M = , (2.26) (1 X)0.75 + 0.032(1 X)0.25 − − where X is the same random number used for equation (2.24). Note that equation (2.26) leads to a mass function similar to that of Miller and Scalo (1979). The masses of the two stars in each binary system are each chosen at random with equation (2.26). Duquennoy & Mayor (1991) comment that this is acceptable for long period binaries but for tight binaries the hypothesis of random association is not likely to be valid. The binary evolution algorithm always makes the heavier of the two stars star 1, the mass-losing star, thus setting the lower mass binary component to be star 2, the mass-gaining star. The mass range chosen for both the primary and the secondary is 0.08 M1, M2 8 M . Stars below 0.08M are brown dwarfs while those above 8M are rare≤ and will≤not experience thermal pulses. We evolve each pair of stars from the ZAMS to the age of the galaxy, T = 1.2 1010 years. gal ×

2.4 Tidal Evolution Model

Tidal forces tend to drive a binary system to its lowest energy state for a given angular momentum. This, if stable, is one in which the orbit is circular, the spins of the stars are synchronized with the orbital motion and the spin axes are parallel to the orbital axis (Hut 1980). When a star, particularly a giant, fills its Roche lobe it is assumed, and confirmed by observations, that its orbit is already circular and that its spin is synchronized with the orbit. During Roche-lobe overflow tidal forces are strong enough to maintain this circular, synchronous state and it is indeed the fact that the majority of barium stars have eccentric orbits that leads us to reject Roche-lobe overflow as the dominant formation mechanism. After mass transfer, although neither star need remain synchronous, there is no force within the binary that can drive it eccentric. Under some conditions the binary system may develop a circumbinary accretion disk instead of the standard internal accretion disk. According to calculations by Artymowicz et al. (1991) the development of CHAPTER 2. THE FORMATION OF THE BARIUM STARS 21 such an accretion disk could provide the mechanism to drive the eccentricity of the orbit upward. In the standard case however, the resonant-tidal mechanism proposed would be rapidly damped by stellar tides and can be ignored. A more thorough treatment would include the effects of circumbinary accretion disks on orbital properties if we know how many binaries end up with such disks as opposed to standard internal accretion disks. As this fraction is largely unknown we do not include the effects of circumbinary accretion disks in our models. Although the orbits of barium stars are eccentric they still tend to be more circular than ordinary giant binaries (Figure 2.2). This is a fortuitous situation because we can conclude that tidal circularization has begun but not proceeded to its conclusion. If our model for the formation of barium stars is correct then we must be able to include the effects of tides and recover the distribution of barium-star eccentricities alongside their other properties. The timescale on which tidal circularization takes place is given by Verbunt & Phinney (1995) or Rasio et al. (1996) as

1 f M R 8 = 0 env q(1 + q) , (2.27) τcirc τconv M a where Menv is the mass of the convective envelope, q = M1/M2 the mass ratio, M = M1+M2 the total mass of the system, a the semi-major axis and τconv the timescale on which the largest convective cells turn over. From mixing-length theory this can be approximated by 1 2 3 MenvR τconv , (2.28) ≈ 12L ! where L is the stellar luminosity. Rasio et al. split the factor f into

2 Ptid f 0 = f min , 1 , (2.29)  2τconv !      where Ptid is the tidal pumping timescale given by

1 1 Ωspin = , (2.30) Ptid Porb − 2π

with Ωspin the spin angular velocity of the star on which the tides are raised. The second part of the factor takes account of the fact that convective cells that turnover in a time longer than the tidal pumping timescale cannot contribute to the viscous damping of the tides (Goldreich & Keeley 1977). Because convective turnover timescales are about a year while orbital periods of barium stars are typically a few years, this eﬀect is not large but it is important for the closest systems and those with large eccentricity in which the stellar spin pseudosynchronizes with the periastron orbital angular velocity. The remaining factor f we are free to vary in the absence of a complete understanding of tidal damping. It has been found both theoretically (Zahn 1977) and observationally (Verbunt & Phinney 1995) to be about 1. CHAPTER 2. THE FORMATION OF THE BARIUM STARS 22

Because formula (2.27) is only valid for small eccentricity and nearly synchronous spin we use the formalism of Hut (1981), which is valid for all e and Ωspin on the assumption that damping leads to a constant lag time between the tide raised and the tidal force that raises it. Comparison of Hut’s equation (10) with our equation (2.27) for small eccentricity and spin-orbit synchronism reveals that Hut’s k 2 f M = 0 env , (2.31) T 21 τconv M and we use this in his equations (10) and (11) to find de/dt and dΩ/dt as a result of the tides. We calculate the change in semi-major axis a by conserving the total, spin and orbital, angular momentum of the system. We also take into account spin down of an expanding star by conserving the spin angular momentum Jspin of a star as it expands with the approximation 2 2 Jspin = MenvkgR Ωspin, (2.32) 2 with the ratio of the radius of gyration to the stellar radius kg = 0.1. It turns out that a significant fraction of the system’s angular momentum is required by the spin of a giant that is close to filling its Roche lobe so that incorporation of tidal effects not only allows us to follow the eccentricity evolution of barium stars but also leads to a significant, and hitherto ignored, shrinkage of their orbital separation which in turn affects the efficiency of mass transfer via a stellar wind. We calculate tidal effects throughout the giant evolution of all our binary stars so that tides are also operating when the contaminated barium stars ascend the first giant branch themselves. However any further circularization at this stage is negligible because the systems are now much wider owing to their earlier mass loss.

2.5 Results and Discussion

We evolve over 600 population syntheses models, each containing 106 binary systems, varying B, f and αacc to find where the highest Kolmogorov-Smirnov (KS) probabilities occur for both the period and eccentricity distributions. A small selection of the models is given in Table 2.1. Note that this table includes our best models with the exception of the last 5 models listed which are included to allow us to narrow the range of the three parameters. Column 1 gives the model number, column 2 the wind enhancement parameter B, column 3 the tidal-strength factor f defined in 2.4. Note that full strength tides § are given by f = 1. Column 4 gives the value of the accretion efficiency parameter αacc, column 5 the weighted percentage of Ba-II stars ([Ba] 0.2) to normal giants, column 6 the KS probability of association between the distribution≥ of eccentricities from the model and the observed distribution of barium star eccentricities (see below), column 7 the KS probability of association between the distribution of orbital periods from the model and the observed distribution of periods, column 8 the percentage of model barium stars with periods between 104 and 105 d and column 9 the percentage of model barium stars with periods greater or equal to 105 d. We test our models with the KS statistical test to calculate a probability of association CHAPTER 2. THE FORMATION OF THE BARIUM STARS 23 d 5 10 0 with 1.78 2.09 1.35 1.11 17.4 7.51 12.9 8.15 7.61 8.07 13.1 1.20 2.26 1.21 1.26 ≥ 0.135 0.082 0.103 % P xt. te 5 10 the < in en (d) with 20 50 v 57.8 34.5 34.4 P 33.7 29.2 32.3 30.6 29.5 30.9 32.5 32.8 26.3 30.9 29.0 32.8 35.5 37.8 gi % < is 4 10 table d 3 3 3 the 4 − − − of 10 prob 10 10 10 0.1 0.17 0.13 0.12 0.13 ≤ 0.057 0.036 0.057 0.052 0.071 0.047 0.055 0.054 0.021 0.027 0.031 periods 3.1 7.6 1.2 KS P contents 4 d 11 − the 4 − 10 of 10 prob 10 0.17 0.12 0.15 0.15 0.20 0.17 0.24 0.42 0.15 0.11 0.61 0.32 0.44 ≤ 0.028 0.305 0.347 0.0348 KS 7.2 3.89 P eccentricities description % stars 2.68 2.51 2.02 2.19 2.04 2.46 2.22 1.40 1.13 1.85 1.16 1.41 1.39 1.39 1.24 0.852 0.647 0.464 detailed 0.0106 eighted Ba-II A W acc 1 results. 0.5 0.2 0.2 0.2 0.5 0.2 0.1 α 0.05 0.05 0.05 0.01 0.075 0.075 0.075 0.075 0.075 0.075 0.075 of f 0.1 0.1 0.2 0.1 0.1 1.0 0.25 0.05 0.05 0.05 0.05 0.05 0.01 0.075 0.075 0.075 0.075 0.075 0.075 Selection 3 3 2 2 2 2 3 3 3 3 3 3 3 3 3 3 2 2 10 10 10 10 10 10 2.1: 0 B 10 10 10 10 10 10 10 10 10 10 10 10 le 2 2 2 2 2 1 2 1 2 1 5 1.5 1.5 7.5 7.5 7.5 7.5 ab T 3 2 8 4 5 6 7 9 1 10 11 12 15 19 16 13 14 17 18 Model Number CHAPTER 2. THE FORMATION OF THE BARIUM STARS 24 between our model distributions of orbital period and eccentricity and the observed distributions. The observed eccentricity distribution is given by a sample of 62 strong and weak Ba-II stars from Jorissen et al. (1998). To compare model orbital period distributions, we draw a sample of observed orbital periods from the same set of stars used to test the eccentricity distribution but include eight stars which have only lower limits given on orbital periods and unknown eccentricities. In both cases, we only perform the KS tests on model Ba-II stars with periods up to 10,000 d to match the observed range of barium star periods. There are a small number of barium stars observed with no evidence for binary motion and these stars could well be very long period barium stars (with P 104 d). However the data are consistent with an upper limit to barium star orbital periods≥of about 10,000 d. In Jorissen et al.’s sample, about 20% of the mild barium stars show no evidence for binary motion or there are only lower limits for the orbital periods and no eccentricity measurements.

2.5.1 Number of Ba stars to Red Giants Any time a barium star is produced in the 1.2 1010 year timespan we note how long it spends on the red-giant branch and as a core-helium× burning star. We do the same anytime either star in the binary pair becomes a normal red-giant or core-helium burning star. From each run we then weight the final percentage of barium stars to the number of normal red- giants according to the length of time each star has spent on the RGB (plus core-helium burning) normalized to the time a 1M star would spend in the same phase. With a constant star formation rate, the effect of weighting the statistics by the normalized lifetimes means that the final percentage will not be an integrated value but a value representing the fraction of barium stars to red-giants in the galaxy at the present time. Note that we are calculating Number(model Ba-II stars) , (2.33) Number(model RG binaries) and comparing to Number(observed Ba-II stars) . (2.34) Number(observed RG stars) These two fractions are not equal because

Number(RG binaries) , Number(RG stars). (2.35)

However we expect Number(RG binaries) 0.4 < < 1, (2.36) Number(RG stars) (Eggleton et al. 1989) so that our percentages are over-estimates of the observed percentage of barium stars to normal giants. In fact, if we take model 10, which has a simulated percentage of 1.24%, as an example we get the correct fraction of 1% if Number(RG binaries) = 0.81. (2.37) Number(RG stars) CHAPTER 2. THE FORMATION OF THE BARIUM STARS 25

Table 2.1 shows that most models overproduced the fraction of barium stars to red giants compared with the observed fraction of 1%. Models with B 2 103 produce barium stars at a higher frequency, up to 2.68% in the case of model 1 compared≈ × with lower values of B. All are therefore consistent with a reasonable binary fraction amongst all stars.

2.5.2 Dependence of Results on Free Parameters Figure 2.3 shows the (e, P) diagrams for models 1 and 10. Comparing these results with the observed barium star (e, P) diagram in Figure 2.2 we see the observations are consistent with some of the models from Table 2.1 using viscous tidal evolution in convective envelopes and the wind accretion model for Ba-II star production. Our favoured value for B is at least five times smaller than that originally proposed by Tout & Eggleton (1988) and that most models required reductions in f of at least four (preferably, at least ten) times smaller than the expected tidal strength. Figure 2.3 (b) is the best fit to the observed eccentricity and period distributions according to KS probabilities calculated from the observed distributions. While our best model almost entirely covers the observed range in eccentricity and orbital period and gives the highest KS probabilities, the area covered by the simulated distribution in the (e, P)-plane is not the same as the area covered by the observed distribution. By altering B, f and αacc we can actually correct this discrepancy as shown in Figure 2.3 (a) which covers more of the observed distribution but has a higher overproduction of high period systems. The period distributions are highly dependent on αacc and B while mostly independent of the tidal strength f . Reducing αacc from 1 to 0.05 greatly reduces the number of high period barium systems while increasing B from zero to 104 increases the number of low period, zero eccentricity systems. It is for these reasons that B must be greater than zero 4 but less than 10 together with a low value of αacc to match the observed distribution of barium star orbital periods. The eccentricity distributions are seen to be highly sensitive to changes in the tidal strength f and moderately dependent on B and αacc, as Table 2.1 and Figure 2.3 demonstrate. There is not a unique set of the three parameters, B, f and αacc that give a significantly better fit to the observations than other combinations of parameters. This is demonstrated by Table 2.1 where every model is a reasonable fit to the observed distributions (for P < 104 d) according to the KS probabilities with model 10 chosen as the best fit because the KS probability for both the period and eccentricity distributions were the largest for the combination of parameters such that B = 7.5 102, f = 0.075 × and αacc = 0.075. From running a large number of models we can rule out extreme values of the three parameters, f , B and αacc with some confidence in their application to the modelling of barium star formation. Notably, we can rule out strong and very weak tidal strengths, so that 0.01 < f < 1.0. From Table 2.1 we require a tidal-wind enhancement of 102 < B 2 103. For accretion efficiency, we find applicable values close to that ≤ × expected by Theuns et al. (1996), where 0.01 < αacc < 0.2. Models with αacc 0.2 can still produce barium stars which fit the observed eccentricity distribution but produce≥ too many high period barium star systems and so can be ruled out. From observations we expect no more than 20% of (mild) barium stars to have periods CHAPTER 2. THE FORMATION OF THE BARIUM STARS 26

Figure 2.3: Simulated (e, P) diagram for the barium stars produced using (a) f = 0.1, 2 B = 2000 and αacc = 1.0 (model 1) and (b) f = 0.075, B = 7.5 10 and αacc = 0.075 (model 10). We have assumed that an overabundance of [Ba] 0.×2 is necessary for spectroscopic detection. ≥ CHAPTER 2. THE FORMATION OF THE BARIUM STARS 27 between 104 and 105 days and 0% of barium stars to have periods greater than 105 days. Only model 15 adequately satisfies this criteria, although this model’s low KS probabilities rule it out as a good fit to the observed distributions. Another motivation for choosing model 10 as the best model is that the number of high period barium stars (over 105 d) is effectively zero ( 1%) owing to it’s low value of αacc; models with αacc larger than 0.2 produce too man≈y barium stars with periods greater 105 d, up to 17.4% in the case of model 1, where αacc is equal to 1. All models except model 15, produced 26% to 37% of barium stars between 104 and 105 d, greater in all cases than the upper limit of 20% expected from observations. If we use high-KS probabilities and low numbers of high period barium stars as criteria in choosing which models fit the observations then only models 9, 10 and 11 from Table 2.1 can be considered good fits. But all models, even those that were good fits to the observed distributions, produce too many barium stars in the range 104 < P < 105 d. We can not solve this problem by reducing the wind accretion efficiency parameter or by some special combination of the three parameters, f , B and αacc. This may indicate a fundamental problem with the wind accretion model as a comprehensive formation mechanism in the production of barium star binary systems or some aspect of our modelling or parameterizations give an inadequate description of the stellar physics.

2.5.3 Barium Over-abundances Our detailed modelling of the barium enhancement allows us to plot the barium overabundances for each star against orbital period to see if any correlation exists in the models. The upper diagram in Figure 2.4 is a plot of the (P, ∆(38 41)) diagram from Jorissen et al. (1998), which has been shown to give a reasonable measure− of the heavy element overabundances. The lower diagram is a plot of the correlation between the average barium overabundances (large black dots) and the orbital periods for the best model, model 10. In both figures, the actual abundances for each observed and model barium star are also plotted (grey dots). Note that we set a spectroscopic detection threshold of [Ba] 0.2 in our models to distinguish barium stars from normal giants. This detection limit is∼the cause of the apparent hump in average barium abundances near 104 d for model 10. Reducing the detection limit by a factor of ten completely eliminates the hump altogether so the average trend for the model follows the observed. Even though the two distributions are plots of different quantities, they are in essence plotting the same thing – an indication of the level of heavy element overabundances in barium stars. We find that the model distributions illustrated by the lower plot of Figure 2.4 are similar to the observed (P, ∆(38 41)) diagram from Jorissen et al. (1998), particularly in the ten- dency for greater enhancement− among shorter period systems. Models which use tidally enhanced stellar winds are observed to produce more strong-barium low-period stars than models which use Reimers alone. This is because stars which experience a tidally-driven wind will lose more matter before growing large enough to fill their Roche lobes and consequently enable the secondary to accrete more heavily polluted matter and thus end up with higher barium overabundances. CHAPTER 2. THE FORMATION OF THE BARIUM STARS 28

Figure 2.4: (Top) Orbital period versus the average ∆(38 41) colour index from Jorissen et al. (1998) (large black dots). Further explanation is given− in the text. (Lower) The correlation between the average barium overabundance (large black dots) and the orbital period found for the best model, model 10. CHAPTER 2. THE FORMATION OF THE BARIUM STARS 29

2.5.4 The Mass Distributions Lastly we plot the mass distribution of the simulated barium stars and their companions from model 10 in Figure 2.5. We can see that the barium stars produced by the model (all models produce a similar distribution) span a large range, from 0.9 M up to about 6 M with an average mass (for model 10) of about 1.63M for the barium stars, which is consistent with the observations. Decreasing the size of the wind enhancement parameter decreases slightly the mass of the barium stars in the models. The companion stars also span a fairly large range from 0.6 M up to 1.3 M with an average mass of about 0.8 M for all models.

2.6 Summary and Further Work

The eccentricity distribution of barium stars is consistent with the stellar wind accretion and tidal evolution models for the production of barium stars, though we require weaker tides than expected to model the eccentricity distribution adequately. The shape of the eccentricity distributions are strongly dependent on the strength of the tides used in the models, and also to some extent on the mass loss and on the efficiency of wind accretion. To fit the observations, a decrease of at least four times the expected tidal strength is required. A tidally-enhanced wind produces the best fit to the observations for both the eccentricity and period distributions, but we find that this enhancement should not be as strong as originally proposed by Tout & Eggleton (1988). Even so, the stellar wind accretion model is unable to account for the lowest period system and the highest eccentricity system. The period distribution is sensitive to changes in efficiency in wind accretion for long period systems, and changes in the wind tidal enhancement parameter for the short period systems. Changing tidal strength has little effect on the period distributions. Models that fit well the observed eccentricity and period distributions below 10,000 d are prone to produce too many high period barium stars, a problem we can not reconcile by reducing the efficiency of wind accretion, or any other combination also involving tidal wind enhancement and tidal strength. Further work would be to improve the evolution algorithm using the results of the parameterization presented in Chapter 6 and to include the latest results for s-process nucleosynthesis in the synthetic algorithm. For example, the most massive AGB stars (with M & 4M ) are not expected to be as efficient at producing the s-process elements. This is because the mass in the intershell is about an order of magnitude less than in lower mass stars (Gallino et al. 1998). In our model, we assumed that all AGB stars are equally as efficient at producing barium, which is not the case in detailed stellar models. Thus we should really use a barium intershell abundance that is a function of the initial mass and metallicity, and that is consistent with the current s-process models, such as those by Gallino et al. (1998) and Busso et al. (2001). Barium stars also show overabundances of carbon, even though they are not carbon stars. Thus it is equally as important that a detailed model reproduce the carbon overabundances observed in the barium stars. With the CHAPTER 2. THE FORMATION OF THE BARIUM STARS 30

Figure 2.5: Diagram showing the distribution of masses for the barium stars and white- dwarf companions from model 10. The average mass of the barium stars for this model is 1.63 M and the average mass of the companion 0.84 M .

CHAPTER 2. THE FORMATION OF THE BARIUM STARS 31 inclusion of carbon (and nitrogen and oxygen) into the nucleosynthesis model, we should be able to explain the carbon overabundances observed in the dwarf carbon stars (Green & Margon 1994). The evolutionary algorithm should also be extended to study the Popula- tion II CH-type stars. This would involve further improvements to the parameterization of the third dredge-up, because the lowest metallicity considered in Chapter 6 is Z = 0.004, and we really need to extend the stellar models down to about Z = 0.0001. It would also be useful to include a more realistic mass-loss law on the AGB, instead of the modified Reimers law given by Tout & Eggleton (1988). Either using the mass-loss law of Block¨ er (1995) or Vassiliadis & Wood (1993) would provide further insight into the dependence of the model results on the adopted mass-loss rates. It would also be useful to use RLOF to study the formation of the barium stars as Han et al. (1995) did, but using an improved model. Such a study could explain the existence of the shortest-period barium stars with circular orbits. We finish the chapter with the comment that the model is subject to many uncertainties. The most obvious are the modelling of the third dredge-up and the nucleosynthesis model. Both of these were taken from the detailed stellar models available at the time. The wind- accretion model also suffers from many free parameters, which could be constrained by the observations but not uniquely determined (if that is possible) given the uncertainties. Even with these uncertainties, the wind-accretion model presented here is still the most successful at reproducing the qualitative features of the empirical data, but we note that further improvements in the modelling are required to reproduce all the aspects of the barium star observations. Chapter 3

Pre–asymptotic Giant Branch Evolution and Nucleosynthesis

We calculate a large set of stellar models with and without mass loss, covering a wide range in mass (from 1M to 6M ) and composition (Z = 0.02, 0.008 and 0.004). We perform detailed nucleosynthesis calculations on the mass-loss models, with the aim of calculating the contribution made by asymptotic giant branch (AGB) stars to the enrichment of the galaxy. The models without mass loss are used to parameterize the third dredge-up and discussed further in Chapter 6. We begin in 3.1 with the details of the numerical methods and the input physics used to create the stellar§ models and the nucleosynthesis calculations. In 3.3 we present the stellar lifetimes, and in 3.4 and 3.5 we present the results of the§surface abundance changes from the first and§ second dredge-up.§ In 3.6 we discuss the core He-flash and we finish with a summary. §

3.1 Numerical Details

We calculate stellar evolution and nucleosynthesis in two steps. First we calculate the stellar structure, from the pre-main sequence to near the end of the thermally-pulsing AGB (TP–AGB). Second, we perform the nucleosynthesis calculations. Parts of 3.1 are taken from Karakas, Lattanzio & Pols, 2002, Publications of the Astronomical Society§ of Australia, volume 19, pages 515 to 526.

3.1.1 The Stellar Evolution Code We use the Monash version of the Mount Stromlo Stellar Structure code (Wood & Zarro 1981; Lattanzio 1986; Frost & Lattanzio 1996) referred hereafter in the text as the evolution code. The evolution code is a based on the Henyey-matrix method to solve the equations of stellar structure, where we divide our star into N 1 Lagrangian mass shells, where spacing between mesh points is determined by setting− maximum diﬀerences in mass, radius, luminosity, temperature and pressure as well as the chemical abundances

32 CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 33

Table 3.1: Initial compositions (in mass fractions) used for stellar models: Z = 0.02 Z = 0.008 Z = 0.004 solar LMC SMC H 0.6872 0.7369 0.7484 4He 0.2928 0.2551 0.2476 12 3 4 4 C 3.4080 10− 9.6959 10− 4.8229 10− 14 × 4 × 4 × 5 N 1.0542 10− 1.4240 10− 4.4695 10− 16 × 3 × 3 × 3 O 9.6000 10− 2.6395 10− 1.2830 10− × 3 × 3 × 3 Other Z 5.9378 10− 4.2484 10− 2.1899 10− × × × between two adjacent mesh points. Time steps between models are similarly constrained, to ensure that abundances do not vary greatly between each mesh point from one model to the next. Typically we have 600 mesh points on the main sequence and as many as 1800 mesh points during the core-helium (He) ﬂash of a low-mass model or during a thermal pulse (TP). The evolution code includes 6 species involved in the main energy-producing nuclear reactions: H, 3He, 4He, 12C, 14N and 16O. Table 3.1 gives the initial compositions used (in mass fractions) in the stellar models. Note that we set 3He = 0 initially. For the solar composition models (Z = 0.02) we set the initial 4He abundance to be Y = 0.2928, chosen to be consistent with the models of Frost (1997). We take the initial 12C, 14N and 16O abundances from Grevesse, Noels & Sauval (1992). We also calculate stellar models for compositions appropriate for the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC). We set Z = 0.008 and Y = 0.2551 for LMC models and Z = 0.004 and Y = 0.2476 for SMC models. We take initial 12C, 14N and 16O abundances for the LMC and SMC from Russell & Dopita (1992) and assume a scaled-solar mix for the rest to obtain the required total metallicity. The solar oxygen abundance has recently been measured by Allende Prieto, Lambert & Asplund (2001) to be log ε(O) = 8.69 0.05 dex. This value is at the lower end of the distribution of previously published ﬁgures, but is in good agreement with the oxygen abundance measured in the interstellar medium (Allende Prieto et al. 2001). About half of the matter that makes up the global metallicity Z, is in the form of 16O, so a reduction in the solar oxygen abundance means a reduction in the global metallicity of the Sun. Using the solar abundances of Anders & Grevesse (1989), we obtain a global solar metallicity of Z = 0.01899 (which we assume is Z = 0.02), and which is reduced to Z = 0.0149 if we use the oxygen abundance given by Allende Prieto et al. (2001) (and keep all the other abundances constant). The reduction of the Sun’s metallicity implies a reduction in the metallicity of all objects, because the observed [Fe/H]a abundance is used as a metallicity indicator. This has implications for the occurrence of the third dredge-up in AGB models, which we discuss further in Chapters 4 and 6.

awe use the standard spectroscopic notation, where [A/B] = log(A/B) ± log(A/B) , where A and B are atomic number densities ∗ CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 34

We include mass loss on the ﬁrst giant branch using the Reimers (1975) formula

dM 13 (R/R )(L/L ) = 4 10− η , (3.1) dt × R (M/M ) where R, L and M are the radius, luminosity and mass of the star (in solar units) respectively, and ηR is the Reimers parameter which we set to 0.4. On the AGB, we use the mass-loss prescription of Vassiliadis & Wood (1993) (hereafter VW93). VW93 parameterised the mass-loss rate as a function of pulsation period,

dM log = 11.4 + 0.0125P, (3.2) dt ! −

1 where the mass-loss rate is in M yr− and P is the pulsation period, given by

log P/days = 2.07 + 1.94 log R/R 0.9 log M/M , (3.3) − − where R and M are in solar units. For P > 500 days, the mass-loss rate given by equation (3.2) is truncated at dM L = , (3.4) dt c vexp corresponding to a radiation-pressure driven wind. The luminosity L is in solar units, and 1 c is the speed of light (in km s− ). The wind-expansion velocity, vexp is also taken from VW93 and is given by 1 v /km s− = 13.5 + 0.056P, (3.5) exp − 1 where vexp is limited to 15 km s− . Note that we do not include the modification to equation (3.2) for masses greater than 2.5M given in VW93. This modification was made to delay the onset of the super-wind phase for the more massive AGB stars which are observed to be optically visible out to periods of 750 days (Hughes & Wood 1990). We include convection using the∼mixing-length prescription with the assumption of instantaneous mixing, setting the mixing-length to be 1.75 times the pressure scale height. We do not include any overshoot in the usual sense, but we recognize the discontinuity in the ratio of the adiabatic to radiative gradients at the bottom edge of the convective envelope during the dredge-up phase. We search for a neutral border to the convective zone, in a manner described by Frost & Lattanzio (1996). We evaluate the ratio of the radiative to adiabatic temperature gradients, / , at the last two mesh points defined ∇rad ∇ad by the Schwarzschild criteria to be convective (where rad > ad) and linearly extrapolate in mass to find the ratio at the first radiative mesh point.∇If the∇extrapolated value is greater than unity then the radiative mass shell is included in the convective zone even if the mass shell has rad < ad. If the extrapolated value is less than unity then the radiative mesh point remains∇ radiati∇ ve. We perform this test on all convective boundaries after each Henyey iteration when we mix abundances (Lattanzio 1986), including the inner and outer edges of the flash-driven convective zone during the core He-flash and helium-shell flashes. We note that searching for convective neutrality has little effect on the boundaries CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 35 of the convective zones in these regions Sometimes convergence problems arise during a TP when mixing after each iteration. When this situation arises, we only allow abundances to be mixed after the first N iterations, where N is some number between about 10 and 30. If the evolution code has not converged to a model after N iterations, we hold the convective boundaries at the Nth iteration and converge to a structure model. Even though this method only allows one mass shell to be added per iteration, it allows for the growth of the convective core caused by semiconvective mixing and also allows for the occurrence of core-breathing pulses during the core helium-burning (He-burning) phase when the central 4He abundance is small (Y < 0.1) (Lattanzio 1986). There is some debate concerning the occurrence of core-breathing pulses in real stars (see Castellani, Chieffi & Pulone 1989) and the inclusion of this phenomena will affect the structure of the model and the burning lifetimes. The search for a neutral border to the convective zone also increases the efficiency of the third dredge-up compared with models that define convective boundaries purely on the grounds of the Schwarzschild criteria for convection (Frost & Lattanzio 1996; Mowlavi 1999b). The opacity tables used in the evolution code are taken from a variety of sources (Frost & Lattanzio 1996). The updated OPAL radiative opacity tables of Iglesias & Rogers (1996) are used exclusively for temperatures above 7000 K and which extend to 500 106 K. When the temperature falls below 6000 K, we use a combination of the Huebner et× al. (1977) and the Cox & Stewart (1970a,b) tables. The Huebner et al. (1977) tables are valid for T > 104K, and the Cox & Stewart (1970a,b) tables cover temperatures down to 1500 K. For temperatures in the region between 6000 K and 7000 K, we calculate opacity from both the OPAL and the Cox & Stewart tables and linearly interpolate between the two values to ensure a smooth transition from one set to the other. Both the high and low-temperature opacities assume a scaled-solar heavy element distribution. The Cox & Stewart (1970a,b) opacity tables do not include a contribution from bound-bound transi- tions in molecules, which become important for the opacity at temperatures below 4000 K (Alexander, Rypma & Johnson 1983). To remedy this, the evolution code also includes a contribution to the opacity from a number of molecules (Frost 1997). In particular CN, CO, H2O and TiO using the formulations prescribed by Bessell et al. (1989) and corrected by Chiosi, Wood & Capitanio (1993). These are fits to the molecular opacities of Alexan- der (1975) and Alexander et al. (1983). Note that we only include the effects of molecules for temperatures below 8000 K. Conductive opacities are calculated from a program sup- plied by MacDonald (1992, private communication) that uses opacities from a variety of sources including a modified fit to the Hubbard & Lampe (1969) conductive tables for non-relativistic electrons (Iben 1975), the results of Itoh et al. (1983) and Mitake, Ichi- maru & Itoh (1984) for relativistic electrons and the results of Raikh & Yakovlev (1982) for solids. The equation of state (EOS) is the relationship between the temperature, pressure and density. In the evolution code, the independent variables associated with the EOS are the temperature and pressure with the density calculated from the appropriate formula. In the evolution code we use the fitting formulae of Beaudet & Tassoul (1971) and the ideal gas equation with radiation pressure for determining those variables required from the EOS, namely the density, ρ, the internal energy, U, the specific heat at constant pressure, cp, and CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 36 the electron degeneracy, ψ, and their derivatives. Ionization is included for hydrogen and both ionization states of helium. All nuclear reaction rates involved with hydrogen-burning (H-burning) are taken from Caughlan & Fowler (1988) (CF88), with the exception of the reaction rate for 1H(p, e + ν)2H which comes from Harris et al. (1983). All other nuclear burning rates are taken from CF88. We calculate CNO branching ratios by allowing for the temperature dependence of the nuclear reactions. The branching ratio of 17O + p is highly temperature dependent (see Arnould, Goriely & Jorissen 1999), and we use the reaction rates taken from CF88 to calculate the ratio. In each of the given rates there is an uncertainty factor f , which we set to 1. We are aware that these reaction rates are not the latest available but we re-calculate the composition for each mass-loss model using a post-processing nucleosynthesis code with updated reaction rates. Energy losses due to neutrino interactions are taken from Beaudet, Petrosian & Salpeter (1967) for the pair neutrino, photoneutrino and plasma neutrino process rates. Bremsstrah- lung neutrino rates are from Festa & Ruderman (1969). The effect of neutral currents is included using the results of Ramadurai (1976) to provide corrections to the pair annihi- lation, photoneutrino and plasma processes, and Dicus et al. (1976) provide corrections to the bremsstrahlung process rate.

3.1.2 The Nucleosynthesis Code The wealth of stellar abundance observations and high precision pre-solar grain abundance measurements demands the inclusion of more chemical species into stellar evolution models than have been traditionally included. It is quite common for evolution codes to only include the six to ten species that are relevant to the major energy-generating reactions and have the greatest impact on the structure of a model. But to try and explain the abundances patterns observed in many stars, such as the oxygen–sodium and magnesium–aluminium anti-correlations observed in some globular cluster stars (see for example Shetrone 1996; Kraft et al. 1998; Yong et al. 2003), or the excesses of 29Si and 30Si compared to 28Si measured in silicon carbide grains from the Murchison meteorite (Hoppe et al. 1994), requires the inclusion of more species into nuclear networks. The energy generated from these extra nuclear reactions is negligible compared to that produced via the pp–chains, the CNO cycles, or the triple–α process, so they do not have to be included into the nuclear network used by the evolution code. The approach that we use is to perform the evolution calculations ﬁrst and the detailed nucleosynthesis calculations afterward. The post-processing nucleosynthesis code used for our calculations was originally developed to study Thorne–Zytk˙ ow objects by Cannon (1993) before being substantially modiﬁed to receive input from the evolution code. The post-processing code tracks 74 species from H to 62Ni (see Figure 3.1) as a function of mass and time. The nuclear network includes 506 reactions. The species used in the nucleosynthesis code are listed in Table 3.2, with the solar abundance in mole fraction YA given in parentheses for stable isotopes. Note that the mass fraction is simply equal to the total number of nucleons, A, multiplied by the mole fraction i.e. X = A Y . The stable isotopes are shown in bold lettering and the unstable isotopes A × A CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 37

S 32 33 34 35 P 29 30 31 32 33 34 Si 27 28 29 30 31 32 33 Al 25 26 27 28 Mg 23 24 25 26 27 Na 21 22 23 24 Ne 19 20 21 22 F 17 18 19 20 O 14 15 16 17 18 19 N 13 14 15 C 12 13 14 g

Atomic Number, Z B 8 Ni Be 7 58 59 60 61 62 64 Co 59 60 61 Li 7 Fe 59 60 61 He 3 4 56 57 58 H 1 2 n 1 Neutrons Figure 3.1: Nuclear network used in the nucleosynthesis code. The most abundant isotope of an element is shaded dark grey, the stable isotopes are shaded light grey and the unstable isotopes are white. in red. There are 74 nuclear species: from neutrons and protons up to sulphur there are 59 nuclei, with another 14 iron group species to allow neutron capture on iron seed nuclei. 27 26 26 We include two isotopes of aluminum (besides Al), Alg and Alm representing the ground state and meta-stable state of 26Al. There is also an additional “particle” g which has the function of counting the number of neutron captures occurring beyond 60Ni. The reaction network is terminated by a neutron capture on 61Ni followed by an ad hoc decay, producing the particle represented by the symbol g: 61Ni(n,γ)62Ni 61Ni + g. Following the method of Jorissen & Arnould (1989), neutron captures on the→missing nuclides are modelled by neutron sinks, meaning that the 34S(n,γ)35S and the 61Ni(n,γ)62Ni reactions are given some averaged cross section values in order to represent all nuclei from 34S to 54Fe and from 61Ni to 209Bi respectively. The initial solar abundances listed in Table 3.2 are mostly from Anders & Grevesse (1989), who give estimated solar abundances for all stable species. Abundances for C, N, O, Ne and Ar are taken from Grevesse et al. (1992). The abundance of 34S also includes the abundance of species from 34S to 54Fe that are not included in our network. Similarly, all species beyond 64Ni are in the form of our artiﬁcial species g. In the nucleosynthesis code, the initial abundances of H, 4He, 12C, 14N and 16O are taken from the ﬁrst evolution model, and the initial abundances of all the other species are either solar (taken form Table 3.2) or scaled solar. For the non-solar compositions, we simply scale all the other species listed CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 38

Table 3.2: Species included in the nuclear network:

element A (initial abundance YA for Z = 0.02.) n 1 H 1(0.6872),1 2(0.0)2 He 3(0.0),3 4(0.0732)1 9 Li 7(1.53 10− ) Be 7× B 8 4 1 6 C 12(2.84 10− ), 13(3.16 10− ), 14 × 5 1 × 7 N 13, 14(7.53 10− ), 15(2.76 10− ) 4×1 7× 6 O 14, 15, 16(6.00 10− ), 17(2.28 10− ), 18(1.20 10− ) × × 8 × F 17, 18, 19(2.44 10− ), 20 5 × 7 6 Ne 19, 20(9.04 10− ), 21(2.20 10− ), 22(6.60 10− ) × × 6 × Na 21, 22, 23(1.66 10− ), 24 5 × 6 6 Mg 23, 24(2.46 10− ), 25(3.10 10− ), 26(3.24 10− ), 27 × × 6 × Al 25, 26g & 26m, 27(2.46 10− ), 28 5 6 × 7 Si 27, 28(2.67 10− ), 29(1.35 10− ), 30(8.98 10− ), 31, 32, 33 × × 7 × P 29, 30, 31(3.01 10− ), 32, 33, 34 5 × 7 6 4 S 32(1.42 10− ), 33(1.12 10− ), 34(5.49 10− ), 35 × 5 × 7 × 8 Fe 56(2.39 10− ), 57(5.73 10− ), 58(7.30 10− ), 59, 60, 61 × × 8 × Co 59(6.52 10− ), 60, 61 7 × 7 8 8 Ni 58(9.76 10− ), 59, 60(3.74 10− ), 61(1.61 10− ), 62(5.13 10− ) × × 6 5 × × g 64(5.40 10− ) × 1 The initial abundance for this species is taken from the ﬁrst calculated evolutionary model. 2 We approximate the initial deuterium abundance to zero as it is not important in our calculations. 3 He3 is initially set to be zero inside the star in the evolution model so we do the same for the nucleosynthesis model. 4 Represents abundances from 34S to 54Fe. 5 Represents abundances from 64Ni to 209Bi. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 39

M r M tot

M2,top

top of shell M1,top

These points are interpolated

M2,bottom

M1,bottom bottom of shell non Lagrangian points

0 Evolution model 2 Evolution model 1 Time Figure 3.2: How the mesh is set up in the nucleosynthesis code. The positions of the nuclear burning shells are determined from data from the evolution code, but allowed to move so the position of the bottom, middle and top of the shells follow chosen abundances. The code interpolates the positions of the shells using preset H and He abundances for the top, middle and bottom. Once the positions of the shells are set the code adds mesh points above and below the shells by linearly interpolating in the logarithm of the mass. in Table 3.2 to obtain the required metallicity. The nucleosynthesis code needs as input from the evolution code structure variables such as the time, the temperature and density as well as the details of the convection zones: the mixing length and mixing velocity as functions of mass. With this information, the nucleosynthesis code independently calculates abundance changes caused by convective mixing and nuclear reactions, calculating it’s own time step and mass mesh, using a combination of Lagrangian and non-Lagrangian points. The non-Lagrangian points are not Eulerian either, but rather follow a ﬁxed H and He abundance, and are used to track the hydrogen and helium shells. The nucleosynthesis code only calculates the abundance changes in the stellar model and not the structure. The slowest computational parts of a model star’s evolution, which is typically during the interpulse phase, becomes the fastest for the nucleosynthesis code. This is because the maximum time step used in the evolution code is governed so that properties at each mesh point do not vary too much between models. During the interpulse phase, the evolution code typically calculates 5000 or so models, with time steps between each of the order of a few years to resolve the progress of the nuclear-burning shells. The CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 40

Downward moving stream

Upward moving stream

Figure 3.3: Schematic representation of the convective mixing mechanism used by the nucleosynthesis code, taken from Cannon (1993, Figure 2). The composition in each cell is linked to those cells from which material ¯ows into it. So the composition of a cell in the upward moving stream can receive material from the cell directly below or from the cell adjacent to it. Likewise for a cell in the downward moving stream. interpulse phase becomes fast for the nucleosynthesis code because we calculate a much smaller number of nucleosynthesis models, and by using non-Lagrangian points inside the H and He-shell, we can keep track of the shell positions so that they remain at a constant H and He mass fraction. This is similar to the shell-shifting techniques used in early calculations of stellar evolution models (see for example Eggleton 1967, 1973 and Sweigart 1971). The nucleosynthesis code does not select which evolution models are to be read in, but rather the evolution code outputs selected models according to specific criteria. The criteria are determined by changes in the stellar structure; for example, a large increase in the He-burning luminosity during the TP–AGB phase indicates the start of a thermal pulse, so we output a model every time the He luminosity changes by a specified amount. There are similar criteria for the other evolutionary phases, including the core-burning phases, the first giant branch, the core He-flash, and the occurrence of the third dredge-up and hot bottom burning during the TP–AGB. During the interpulse phase we typically output every 10 to 100 evolution models, depending on the whether or not there are changes in the H and He-burning luminosities. The nucleosynthesis code then reads in the positions of the burning shells from two selected evolution models (which may be more than 100 evolution CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 41 models apart), and interpolates the position of the shells whilst the nucleosynthesis code steps forward in time (see Figure 3.2) using it’s own time steps. To mix the convective regions, we do not assume instantaneous mixing but make an approximation to diffusive mixing using a method devised by Cannon (1993). Figure 3.3 schematically demonstrates the mixing algorithm used in the nucleosynthesis code. We have an upward and downward moving stream, with velocities taken from the evolution code (calculated from the mixing-length theory) to calculate the mass-flow rate at a given mass shell. It is assumed that the material in an upward moving cell is influenced by upward moving material from the cell below and likewise the material in a downward moving cell is influenced by downward moving material in the cell above. We also allow for some horizontal mixing between adjacent cells. If we remove the motion of material between adjacent cells we get a conveyor-belt mixing scheme otherwise the algorithm described in Figure 3.3 approximates the diffusion equation when adjacent mixing is very efficient, and one takes the continuum limit. Reaction rates are taken from Lugaro (1998). Briefly, the bulk of the 506 reaction rates are taken from the Reaclib Data Tables of nuclear reaction rates based on the updated compilation of Thielemann, Arnould & Truran (1991). New theoretical or experimental reaction rates are included when possible, and fitted to the same formula used in the Rea- clib Data Tables in the temperature range of interest for low to intermediate mass AGB stars (Lugaro 2001).

3.2 Stellar Models

For each composition (Z = 0.02, 0.008 and 0.004) we calculate evolutionary sequences for masses: 1, 1.25, 1.5, 1.75, 1.9, 2.25, 2.5, 3, 3.5, 4, 5 and 6M . We also have evolutionary sequences for a 2M , Z = 0.02 model and a 2.1M , Z = 0.008 model. The evolutionary tracks for the Z = 0 .02 models are shown in Figure A.1, the Z = 0.008 models in Fig- ure A.2, and the Z = 0.004 models in Figure A.3. We deﬁne low-mass models to be those with M < 2.25M and intermediate mass models to be between 2.25 6 M(M ) 6 6.5. The dividing mass which separates low to intermediate mass models is not arbitrarily chosen but corresponds to the transition from degenerate to non-degenerate core He ignition. We perform detailed nucleosynthesis calculations on each evolutionary sequence modelled. The remainder of the chapter includes a discussion of the stellar lifetimes and the surface abundance changes as a result of the ﬁrst and second dredge-up. We include a comparison to other calculations and observations where possible.

3.3 Stellar Lifetimes

In this section we present the stellar lifetimes for each nuclear-burning phase, and provide a ﬁt to the total (nuclear) stellar lifetime as a function of the mass and metallicity. We begin with a discussion of the uncertainties. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 42

3.3.1 Uncertainties Affecting the Stellar Lifetimes The largest uncertainty affecting stellar lifetimes from detailed evolution calculations is the lack of understanding about convective mixing in stellar interiors. There has been much discussion about the need to include some form of convective overshoot during core hydrogen and helium-burning in intermediate to massive stars. Contributions to this discussion have been made by Maeder & Meynet (1987, 1988, 1989), Renzini (1987), Chiosi et al. (1989) Stothers & Chin (1990), Alongi et al. (1991), Schaller et al. (1992), El Eid (1995), Freytag, Ludwig & Steffan (1996), Deupree (1998), Canuto (1997, 2000), Woo & Demarque (2001) and Barmina, Girardi & Chiosi (2002). Latest attempts to include core overshoot in stellar models indicate that, if present, the overshooting should be moderate (Schaller et al. 1992; Stothers & Chin 1992; Bressan et al. 1993; Demarque, Sarajedini & Guo 1994; Mermilliod, Huestmendia & del Rio 1994; Shroder¨ , Pols & Eggleton 1997; Bono et al. 2001), possibly with a composition dependence (Cordier et al. 2002; Guo & Li, 2002) on the amount of core overshooting required. The exact amount of overshoot needed is usually found by comparing the stellar models with some selected observable quantities (Bressan et al. 1993). Owing to the dependence on the input physics used, notably the opacity and also possibly the initial composition, there is some spread in the amount of overshoot needed to match the observations. For simplicity, we do not include core overshooting in our models. The effect of convective overshoot is to lengthen the main sequence and core He-burning lifetimes by increasing the amount of fuel available and decreasing the time spent on the giant branch. Toward the end of core He-burning, the outer layers of the convective core can become semiconvective. Semiconvection is the slow mixing that occurs in a region where the chemical composition is changing rapidly with radius. This usually occurs in convective cores of massive stars during the main sequence, but also affects low-mass stars during the central He-burning stage. Semiconvective mixing occurs in a temperature layer where the Ledoux criterion for convective stability is fulfilled, but the molecular gradient, µ, is so large that the Schwarzschild criterion is not: ∇

ϕ d ln u < < + , (3.6) ∇ad ∇rad ∇ad δ d ln P! where ϕ = (∂ ln ρ/∂ ln T) and δ = (∂ ln ρ/∂ ln µ) (Kippenhahn & Weigert 1990). Allow- ing for semicon− vective mixing and core-breathing pulses during core He-burning increases the time spent in this phase because more fuel is mixed into the convective core. The core He-burning lifetime is also dependent on the rate used for the 12C(α, γ)16O reaction (Chin & Stothers 1991; Cassisi et al. 1998; Bono et al. 2000; Imbriani et al. 2001). In the evolution code we use the CF88 rate for the 12C(α, γ)16O reaction but in the nucleosynthesis code we use 2.7 the CF88 rate, consistent with recent measurements. We note that the stellar lifetimes ×are only effected by the rate used in the evolution code. Recent theoretical studies of massive stars (Woosley & Weaver 1995; Hoffman et al. 1999) suggest that the rate should be at least to 1.7 times faster than suggested by CF88. Increas- ing the rate for the 12C(α, γ)16O reaction increases the core He-burning lifetime. Bono et al. (2000) vary the rate of the 12C(α, γ)16O reaction to estimate the effect on the core He- CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 43

Table 3.3: Stellar lifetimes for Z = 0.02 models. All lifetimes are in units of 106 years. See the text for a detailed description.

M0 τms τfgb τ¯ash τheb τeagb τagb τest τtotal 1.0 9097.5 2863 1.331 108.28 17.63 18.63 0.0000 12088.7 1.25 3735.0 1483 1.320 111.50 14.94 15.72 0.0000 5346.24 1.5 2025.8 700.2 1.294 111.98 13.73 14.64 0.0000 2853.88 1.75 1276.2 315.8 1.108 113.21 13.17 14.36 0.0000 1720.63 1.9 1005.2 194.3 0.319 123.87 12.70 14.02 0.0000 1337.67 2.0 817.07 197.6 0.150 132.02 15.09 16.69 0.0107 1163.51 2.25 623.95 74.09 – 192.73 21.12 23.36 0.0113 914.147 2.5 463.40 36.06 – 207.34 15.30 17.70 0.0953 724.598 3.0 281.27 16.68 – 115.39 9.964 11.71 0.0936 425.137 3.5 185.95 9.189 – 66.510 4.948 5.829 0.0627 267.540 4.0 133.51 5.667 – 39.098 3.019 3.334 0.0638 181.669 4.5 99.328 3.794 – 28.181 1.957 2.147 0.0515 133.501 5.0 77.906 2.597 – 19.885 1.379 1.559 0.0539 102.000 6.0 51.439 1.259 – 11.847 0.689 0.851 0.0376 65.4339 6.5 43.016 0.921 – 9.5599 0.522 0.646 0.0325 54.1752

burning lifetimes, the mass of the carbon–oxygen core and the C/O ratio inside the star at the end of core He-exhaustion. Bono et al. (2000) use the CF88 rate, 1.7 CF88 and the Caughlan et al. (1985) rate (which is approximately 2.35 faster than the×CF88 rate). These authors ﬁnd that 5 to 10M , Z = 0.02 models which use× the Caughlan et al. (1985) rate, have core He-burning lifetimes that are about 10% longer than lifetimes calculated with the CF88 rate.

3.3.2 Model Results In Tables 3.3, 3.4 and 3.5 we present the stellar lifetimes for the Z = 0.02, 0.008 and 0.004 models respectively. The initial stellar mass (in solar units) is given in column one, in column two the main-sequence lifetime τms, which is from the zero-aged main sequence to the end of core H-burning. In column three we present the FGB lifetime τfgb, which is from the end of core H-burning to the tip of the first giant branch. For low-mass stars, the first-giant branch is terminated by the core He-flash. There is usually a short time between the peak of the first (and most extreme) core He-flash and the beginning of quiescent core He-burning, which is typically about 1 million years. We define the start of core He- burning when the central 4He mass fraction drops below 0.9b. The time between the core flash and the start of core He-burning is noted by the “flash” lifetime, τ¯ash , in column four. In column five we present the core He-burning lifetime τheb, which is from the beginning

bwhilst some 4He is burnt during the preceding core He-¯ash, the central 4He will still be 0.95 following the ¯ash ∼ CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 44

Table 3.4: Same as for Table 3.3 but for the Z = 0.008 models.

M0 τms τfgb τ¯ash τheb τeagb τagb τest τtotal 1.0 7782.3 2089 1.144 103.38 15.19 16.77 0.0000 9992.8 1.25 3132.8 1218 1.082 101.86 14.18 15.91 0.0000 4470.4 1.5 1765.4 580.0 1.008 100.50 13.59 15.47 0.0000 2462.4 1.75 1101.6 284.8 0.906 103.08 12.84 14.77 0.0000 1505.2 1.9 879.64 184.0 0.357 110.55 12.48 14.58 0.0000 1189.1 2.1 668.49 104.7 – 130.56 15.51 18.12 0.0310 921.90 2.25 557.43 61.31 – 230.43 17.93 21.50 0.0251 870.69 2.5 424.20 32.52 – 192.18 14.81 17.92 0.0197 666.85 3.0 265.28 15.43 – 97.427 6.991 8.848 0.0288 387.00 3.5 180.42 8.847 – 58.520 3.398 4.079 0.0677 251.94 4.0 131.63 5.524 – 37.480 2.386 2.738 0.0454 177.45 4.5 99.936 3.733 – 26.788 1.642 > 132.2 5.0 79.194 2.589 – 20.235 1.012 1.628 0.0497 103.70 6.0 53.798 1.368 – 12.400 0.580 0.897 0.0365 68.496 6.5 45.496 1.007 – 10.119 0.431 > 57.1

Table 3.5: Same as for Table 3.3 but for the Z = 0.004 models.

M0 τms τfgb τ¯ash τheb τeagb τagb τest τtotal 1.0 6730.6 1597 1.066 99.358 14.05 16.44 0.0000 8444.9 1.25 2710.4 1038 0.926 98.610 12.77 15.06 0.0000 3863.0 1.5 1787.0 689.3 0.964 98.816 14.27 16.69 0.0000 2592.8 1.75 1134.3 341.4 0.925 99.169 13.78 16.07 0.0000 1591.9 1.9 908.98 221.2 0.687 104.70 12.73 15.10 0.0000 1250.7 2.25 503.70 53.32 – 210.78 13.30 16.69 0.0153 784.49 2.5 384.29 30.01 – 154.87 10.22 13.01 0.0373 582.22 3.0 244.53 14.93 – 81.926 4.639 5.932 0.0207 347.34 3.5 169.29 8.432 – 52.839 2.657 6.334 0.0146 236.90 4.0 124.59 5.370 – 35.052 1.766 2.197 0.0427 167.25 4.5 95.561 3.636 – 25.606 1.160 > 126.2 5.0 76.350 2.473 – 18.766 0.880 1.473 0.0335 99.096 6.0 52.452 1.315 – 11.580 0.477 0.828 0.0284 66.203 6.5 47.500 1.086 – 10.295 0.443 > 59.4 CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 45

Table 3.6: The constants a, b, c and d for equation (3.7). Z a b c d 0.02 6.78946 0.76170 0.09513 0.0071405 0.008 6.46931 0.71161 −0.08755 0.0065228 0.004 5.87525 0.53732 −0.03846 0.0017090 − of quiescent core He-burning to core He exhaustion. In column six we give the early AGB lifetime τeagb, which is from core He exhaustion to the first thermal pulse. In column seven we give the total AGB lifetime τagb, which is from the end of core He exhaustion to the time the model moves off the AGB track or the calculation ceases owing to convergence difficulties. The definition we use to determine when a model moves off the AGB track is when the effective temperature, Teff, changes by ∆ log Teff = 0.3 (Vassiliadis 1992). In column eight we estimate the extra time, τest, spent on the TP–AGB phase owing to the remaining envelope mass, discussed below. In column nine, we give the total stellar lifetime, τtotal, which we define as the total time spent in all nuclear-burning phases. Thus we ignore the time spent as a white dwarf, which could certainly exceed the main-sequence lifetime. We also ignore the time spent contracting during the pre-main sequence phase, as this phase is relatively short ( 107 years) for low and intermediate mass stars. ∼ The only model to leave the AGB, i.e. to move more than ∆ log Teff 0.3 away from the AGB track, is the 1M , Z = 0.02 case. Mass loss removes the entire∼outer envelope and the calculation ends on the white-dwarf cooling track, with a final log Teff = 4.118. The low-mass models (M < 2M ) almost lose their entire outer envelopes, even if they do not leave the AGB track completely . For most of these models the remaining envelope mass is small, and much less than 0.1M . For these models, the total AGB lifetime given in the table is an excellent indicator of the “true” AGB lifetime for that model. For all models with M > 2M , the AGB lifetimes given in Tables 3.3, 3.4 and 3.5 are lower limits to the actual AGB lifetimes. The total AGB lifetimes for these models can be easily estimated if we know the time between the last two thermal pulses, and the rate at which mass is being lost per interpulse period, then (assuming these values are constant) we can determine how long it would take to remove the remaining envelope. Column 7 is the total AGB lifetime for each model, and column 8 is an estimate of the time, τest, spent on the AGB owing to the remaining envelope mass. See 7.1 for the details of this calculation. We evolved a number of extra models§ to the TP–AGB and through the first few thermal pulses to obtain lower limits to their total lifetimes. These models are the 4.5M for all metallicitiesc and the 6.5M models for Z = 0.008 and Z = 0.004. Because these models only evolved through the first few thermal pulses, we leave the total AGB lifetime blank in the tables and put a ’>’ symbol in front of the total lifetime, to indicate these are lower- limits. For each metallicity, we fit the total stellar lifetime using a rational polynomial of the

cthe solar case terminated near the end of the TP±AGB phase, so we obtained a good estimate of the AGB lifetime CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 46

Figure 3.4: The ®t to total stellar lifetime for the Z = 0.02 Z = 0.008 and Z = 0.004 models. The model results are shown by the red points and the ®t by the solid line. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 47

Table 3.7: The main-sequence lifetime (in Myr) for a small selection of Z = 0.02 models with a lower initial 4He abundance of Y = 0.28. Y 1.2M 1.5M 2M 2.5M 3M 4M 5M 6M

0.2928 4311 2025 817 463 281 133 77.9 51.4 0.28 4750 2202 934 497 300 141 82.7 54.5 ∆τms 439 177 117 34 19 8.0 4.8 3.1

form a 6 = log(τtotal/10 ) 2 3 , (3.7) 1 + bM0 + cM0 + dM0 where M0 is the initial mass and a, b, c and d are constants that depend on the metallicity and are given in Table 3.6. In Figure 3.4 we show the fit to the total stellar lifetime (solid line) and the total lifetimes from each calculation (red points). In the upper panel we show the fit for the Z = 0.02 models, in the middle panel the fit for the Z = 0.008 models and in the lower panel the fit for the Z = 0.004 models. We plot the fit between 1 and 8M even though the fit is only valid between 1 and 6.5M .

3.3.3 Comparison With Other Work We first compare our stellar lifetimes to those given in VW93, who use a very similar stellar evolution code but with the older opacities of Huebner et al. (1977). VW93 do not include core overshoot but include semiconvective mixing during core He-burning using the method of Castellani et al. (1985). VW93 suppress core-breathing pulses during the core He-burning phase, on the grounds that models which suppress breathing pulses are best able to match the observed ratio of AGB to horizontal branch stars. All evolutionary sequences have an initial 4He mass fraction of 0.25, regardless of metallicity. The differences between our main-sequence lifetimes and those given in VW93 are as large as 30%. The differences are the largest for the solar-like metallicity models, where VW93 use Z = 0.016 and Y = 0.25 compared to our Z = 0.02 and Y = 0.2928. To estimate the change to the main-sequence lifetime caused by a decrease in the initial helium content, we ran a small selection of Z = 0.02 models with Y = 0.28 to the end of core H-burning. The main-sequence lifetime depends on the amount of hydrogen present. In our Z = 0.02 models with Y = 0.28, the mass fraction of hydrogen initially present is 0.7, whereas there is 0.704 in the VW93 calculations. In Table 3.7 we present the main- sequence lifetimes of the models with Y = 0.28 and Y = 0.2928 (for comparison) and the 4 difference, ∆τms, between the two calculations. If we decrease the initial He in our models by 3 times ∆Y = 0.0128, we have an initial He content of Y = 0.2544, similar to that used by VW93. We therefore estimate the main-sequence lifetime for a given mass at this lower He abundance. At 2.5M , the main-sequence lifetime would increase to 559Myr (c.f. 619.2Myr in VW93) and at 5M the lifetime would increase to 92.5Myr∼(c.f. 95.6Myr in VW93). In both cases the diff erences between the core H-burning∼ lifetimes are . 10%. We therefore conclude that the largest difference between the model lifetimes is the result CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 48 of a different initial 4He content (or different initial hydrogen content). This is easily explained. At a given mass, the model with the largest amount of hydrogen will have the longest main-sequence lifetime. We also compare our results to the more recent computations by Bono et al. (2000) (hereafter B2000) and Dominguez et al. (1999) (hereafter D99). Neither B2000 nor D99 include convective overshoot but both use the OPAL opacities. D99 include a treatment of semiconvective mixing and core-breathing pulses during the central He-burning stage. B2000 include semiconvective mixing but do not include a treatment for core-breathing pulses. There is very good agreement between our H-burning lifetimes and those presented in D99, who set Y = 0.28 in their Z = 0.02 models. The differences are less than 10% when comparing our models with Y = 0.2928 and less than 3% when comparing our models with Y = 0.28 presented in Table 3.7. We observe differences of up to 10%, when comparing our main-sequence lifetimes to those presented in B2000. We conclude that our main-sequence lifetimes are in good agreement with other recent stellar evolution calculations (without convective overshoot). Core He-burning lifetimes are subject to larger uncertainties than those found for core H-burning lifetimes owing to the treatment (or lack) of semi-convective mixing, breathing pulses or convective overshoot. The VW93’s core He-burning lifetimes are consistently larger than the core-He burning lifetimes found in our models, for all compositions considered. For example, the core He-burning lifetimes of the 1M , solar composition models differ by 33.3 Myr, where the VW93 lifetime is 141.6 Myr compared to our 108.3 Myr. The core He-burning lifetimes of the 5M , solar composition models only differ by 3 Myr, where the VW93 lifetime is 23 Myr compared to our 20 Myr. The cause of the difference between our models and VW93 is most likely due to a different treatment of semiconvective mixing, the suppression of core-breathing pulses in the VW93 models and different opacities (we use the larger OPAL opacities). There is very good agreement (with differences less than 15%) between our Z = 0.02 core He-burning lifetimes and those presented in D99. There are very large differences at all metallicities (up to 50%) between our core He-burning lifetimes and those presented in B2000. This is most likely because we include core-breathing pulses and B2000 do not. Since the breathing pulses only occur when the central helium abundance, Y is less than 0.1, it should be possible to test this e ≈ explanation by comparing the helium-burning lifetimes at Ye 0.1 from the present study with the corresponding lifetimes from B2000. Unfortunately≈B2000 do not provide these lifetimes. The total AGB lifetime is mostly determined by the onset of the superwind phase (and hence is dependent on the mass-loss law) and it changes in a complicated way with both mass and metallicity. Both VW93 and our calculations use the same mass-loss formula on the AGB so the total AGB lifetimes should, in principal, be similar. We do not find this to be the case, and for all models we could compare, the total AGB lifetimes from our models are longer. The difference is the most significant for the low-mass models while the total AGB lifetimes are similar for the intermediate mass models. For example, the difference between the total AGB lifetime is about 6 million years at 1M but less than 1 million years at 5M . The early AGB lifetimes from our calculations (again, only comparing the solar- composition models) are longer at low mass and similar at an intermediate mass compared CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 49 to those presented in VW93. For example, the difference was about 5 million years in the case of the 1M model but only 200,000 years for the 5M model. The TP–AGB lifetimes from our models are similar to those presented by VW93, especially for the 1.5, 2 and 2.5M , Z = 0.02 models. The TP–AGB lifetimes for our 1 and 3.5M , Z = 0.02 models are about twice as long compared to the same mass models in VW93. The situation is reversed for the 5M model, where our TP–AGB lifetime is about 44% shorter than the VW93 lifetime for this mass (0.18 Myr for our model compared to 0.26 Myr for VW93). The TP–AGB lifetimes are quite different because the interpulse periods and the number of thermal pulses are also quite different for many of the solar composition models. For example, we calculate 9 thermal pulses for the 1M model, whereas VW93 only find 6 for their 1M model. In the case of the 1M model we suspect that the total mass at the beginning of the AGB will influence the total number of thermal pulses and hence is dependent on the previous evolution. For the 3.5M model, we calculate 21 thermal pulses with interpulse periods that are about 104 years longer than the typical interpulse periods calculated by VW93. Since our 3.5M model experiences more thermal pulses (c.f. our 21 compared to 16) with longer interpulse periods, it is not surprising that our TP–AGB lifetime is longer. The 5M , solar composition model is the reverse of the 3.5M case. The VW93 model went through 37 thermal pulses (they don’t give an exact figure, this is an estimate from their Figure 18) compared to 24 for our 5M model. The duration of the interpulse is similar. However, the entire envelope of the VW93 model is removed by mass loss, whereas our model has about 1.5M of envelope mass left when convergence difficulties end the calculation. If we include the estimated time it would take to remove the remaining envelope (see Table 3.3) then the TP–AGB lifetime becomes 0.23 Myr, which is now similar to the VW93 value of 0.26 Myr (the difference of 0.03 Myr can be most likely attributed to the difference in the number of thermal pulses).

3.4 First and Second Dredge-Up: the CNO Isotopes

Prior to the TP–AGB, the surface composition of a star will be altered by the first and possibly second dredge-up events. Following core hydrogen exhaustion, a hydrogen shell is established around the contracting helium core. Simultaneously, the outer layers expand and as a consequence become convective, due to an increase in the opacity. During the star’s ascent of the giant branch, the convective envelope moves inward, mixing the outer layers with internal matter that has experienced partial H-burning. This mixing event is known as the first dredge-up (FDU) and effects the surface composition of low to intermediate mass stars. The main surface abundance changes are an increase in the 4He abundance, a decrease in the 12C abundance, and an increase in the 14N and 13C abundances. The oxygen isotopes experience only small changes, with an increase in the 17O and depletions in 16O and 18O depending on the initial mass (El Eid 1994; Charbonnel 1994; Boothroyd & Sackmann 1999, hereafter BS99). The FDU leaves behind a sharp composition discontinuity exterior to the position of the H-burning shell. In the lowest mass-stars the relatively long lifetime on the first giant branch allows the H-shell to reach the composition discontinuity and erase it. This does not occur in intermediate mass stars, CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 50 which leave the first giant branch before the H-shell can erase the discontinuity. The second dredge-up (SDU) occurs after core He-exhaustion as the star ascends the giant branch for the second time. Unlike the FDU, only stars with masses greater than about 4M have their surface abundances effected by the SDU. For these stars, the SDU reaches deeper than the FDU, mixing the products of complete H-burning to the surface. The main result is a large increase in the 4He abundance at the expense of H, large enhancements to the surface 14N abundance and a large increase in the 14N/15N ratio. There are also small changes to the 12C/13C ratio and the oxygen isotopic ratios (El Eid 1994; BS99). The SDU also reduces the mass of the H-exhausted core by mixing H-rich material inward to regions previously depleted in hydrogen. In this section we first discuss the conflict between theoretical abundance predictions and observations for stars on the first giant branch. We next discuss the maximum inward extent of the first and second dredge-up in the models, followed by the surface abundance changes caused by these mixing events.

3.4.1 Conflict Between Theory and Observation Theoretical predictions of the 12C/13C ratio after the FDU for intermediate mass stars are in good agreement with the observations, to within 25% (El Eid 1994; Charbonnel 1994; Wasserburg, Boothroyd & Sackmann 1995; BS99).∼The situation is quite the opposite for low-mass stars where the predicted trend, which has the 12C/13C ratio increasing with decreasing mass, has been found to be clearly incorrect when compared with the observations (Gilroy 1989; Charbonnel, Brown & Wallerstein 1998). The same is true for Population II giants in globular clusters, where the deviation between theory and observation is more extreme (Smith & Tout 1992; Shetrone 1996; Pilachowski et al. 1996; Kraft et al. 1998). The same is true for the 12C/14N ratio, which is also observed to be lower in red-giant stars than predicted by standard stellar evolution theory (Charbonnel 1994). The situation is different for the oxygen isotopic ratios, where theoretical predictions and observations are not in conflict (Schaller et al. 1992; Bressan et al. 1993; El Eid 1994; BS99). The observations of low 12C/13C and 12C/14N ratios have been interpreted as evidence for extra mixing taking place between the base of the convective envelope and the H- shell. Observations of low 12C/13C and 12C/14N ratios indicate that the conflict between theory and observation does not arise until after the deepest first dredge-up (Charbonnel 1994; Charbonnel et al. 1998; BS99), hence this mixing most likely takes place after the H-shell has erased the composition discontinuity left by the FDU event. One theory proposes to explain the deep mixing in terms of meridional circulation caused by rotation on the giant branch (Sweigart & Mengel 1979; Zahn 1992). Charbonnel et al. (1998) find there is some observational evidence to support this theory. Whilst the exact nature of the cause of the deep mixing below the base of the envelope is still uncertain, most algorithms used in deep mixing models are still without a physical basis (Smith & Tout 1992; Wasserburg et al. 1995). BS99 use a “cool-bottom processing” method which takes material from the base of the convective envelope into regions just hot enough for some nuclear processing (at the top of the H-shell) and then transports the material back to the envelope. Charbonnel (1995) and Denissenkov & Tout (2000) use a mechanism developed CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 51

Figure 3.5: Innermost mass layer reached by the convective envelope during the ®rst (solid lines) and second dredge-up (dashed lines) as a function of the stellar mass. by Zahn (1992) to simulate mixing induced by rotation. Denissenkov & Weiss (1996) use diﬀusive mixing models to explain the abundance anomalies observed in globular clusters with some success.

3.4.2 Depth of the First and Second Dredge-Up In Figure 3.5 we show the innermost mass layer reached by the convective envelope during the first (solid lines) and second (dashed lines) dredge-up as a function of the stellar mass. The black lines represent the Z = 0.02 models, the red lines the Z = 0.008 models and the blue lines the Z = 0.004 models. The deepest first dredge (for a given mass) occurs in the Z = 0.02 models, but there is little difference between the maximum depth for models with M 6 3M . We can compare Figure 3.5 to Figure 2 in BS99. Our models show the same trend with metallicity and mass, and also very similar values for the deepest extent of the convective envelope during the first dredge-up. The FDU does not extend very far inward in the most massive models, but as we see in Figure 3.5, the convective envelope extends further inward during the SDU. The depth reached by the SDU is approximately the same for all the 5 and 6M models, regardless of the initial metallicity. We observe the same trend with mass and metallicity for the SDU as reported by BS99, and find the same metallicity dependence on the depth of the SDU in the mass range 2.25M to 4M . The lowest masses in this range do not experience a reduction in the mass of the H-exhausted core, and only the 3 to 4M models show any change to their surface composition as a result of the convective envelope moving inward after the end of core He-exhaustion. Hereafter, we define the SDU to only occur in those CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 52

Figure 3.6: (Top) the mass at the bottom edge of the convective envelope (solid line) and the mass at the top (dashed) and bottom (dot±dashed) of the H-shell as a function of time for the 3.5M , Z = 0.008 model. (Bottom) same but for the 3M , Z = 0.004 model. The de®nition of the top and middle of the H-shell is given in the text. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 53 models that show a reduction in the mass of the H-exhausted core. There are borderline cases for the SDU, where the convective envelope moves into the H-shell but not below. In these cases, the mass of the H-exhausted core is not altered but some partially burnt material is mixed to the stellar surface. In Figure 3.6 we show the mass at the lower edge of the convective envelope (solid line) and the mass at the top (dashed) and bottom (dot–dashed) of the H-shell during the beginning of the AGB phase, for two models that experience mild surface abundance changes caused by the SDU: the 3.5M , Z = 0.008 model and the 3M , Z = 0.004 model. We deﬁne that top of the H-shell to be where the H mass fraction is equal to 0.689, the middle of the H-shell to be where the H mass fraction is equal to 0.345 (this also deﬁnes the mass of the H-exhausted core), and the bottom of the H-shell to be where the H mass fraction equals 0.05. For both models considered in Figure 3.6 the bottom of the convective envelope reaches into the top of the H-shell but neither model experiences a reduction in the size of the H-exhausted core at this stage.

3.4.3 Results from the FDU and SDU: Carbon In Figure 3.7, 3.8 and 3.9 we show the changes made to the 12C/13C ratio, the 14N/15N ratio and the oxygen isotopic ratios for the Z = 0.02, Z = 0.008 and Z = 0.004 models after the FDU and SDU events. All isotopic ratios are number ratios and can be found in tabular form in Appendix B. Appendix B also contains the ratios (by mass) of C/Z, N/Z and O/Z for each model. For solar metallicity models, the 12C/13C ratio shows a noticeable decline after the FDU from the initial value of 90. In Figure 3.7 we observe ratios between 20 and 30, which are similar to those reported by other authors, notably Charbonnel 1994, El Eid 1994 and Boothroyd & Sackmann 1999. Note that we can not directly compare our results for Z = 0.008 and Z = 0.004 because we use an initial CNO abundance spread that is not scaled from solar. For the solar composition models, we see the same trend with mass as other standard calculations (Charbonnel 1994) where the 12C/13C ratio increases as the mass of the model decreases. In all models, the convective envelope moves inward to a region interior to the 13C pocket. To understand what we mean by this, we show the composition profiles for the stable CNO species and 19F for the 1, 3 and 6M , Z = 0.02, Z = 0.008 and Z = 0.004 models in Figures 3.10, 3.11 and 3.12, respectiv ely. In each figure, we plot the composition as a function of the stellar mass (in M ) in the region reached by the convective envelope during the FDU; and we note the position of the maximum inward penetration of the convective envelope. In each figure, we see that hydrogen burning leaves behind a 13C pocket, and in each model, regardless of mass or composition, the convective envelope fully engulfs this pocket resulting in an increase in the surface abundance of 13C. We also find a reduction in the surface abundance of 12C. Figures 3.11 and 3.12 demonstrate that even in the 6M models which experience very shallow first dredge-up, the convective envelope moves into the region depleted in 12C from CNO cycling during core H-burning. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 54

Figure 3.7: The surface abundance ratios of 12C/13C, 14N/15N, 16O/17O and 16O/18O as a function of the stellar mass for the Z = 0.02 models. The solid line (with points) show the ratios after the FDU and the dashed red line (with stars) show the ratios after the SDU. The initial ratio is indicated on each panel.

3.4.4 Nitrogen From Figures 3.7, 3.8 and 3.9, we observe that the 14N/15N ratio increases dramatically in all models after the FDU and SDU events, regardless of the initial mass or metallicity, and that our results are very similar to those found by other authors, notably Charbonnel (1994), El Eid (1994) and BS99. The composition proﬁles demonstrate the mechanism for the large increase in the 14N/15N ratio. The increase can be attributed to the production of 14N at the expense of 12C but also a reduction in the abundance of 15N from CNO cycling. The equilibrium value of the 14N/15N ratio is 2.5 104 (Clayton 1983), much higher than the 14N/15N ratios seen in our models after either× the FDU or SDU. In all models, the maximum extent of the convective envelope reaches the ﬁrst 14N peak but only in models with the deepest FDU, or for those models that experience the SDU, does the convective envelope reach the second 14N plateau caused by ON cycling. Figure 3.10, 3.11 and 3.12 show that this is indeed the case for the 3M models. For the most massive LMC and SMC models, the FDU does not penetrate deep enough to reach the second 14N plateau and the 14N/15N ratios for these models are similar to that found for the for the lowest mass models. However, for the more massive LMC and SMC metallicity models, the action of the SDU produces a nearly linear increase in the 14N/15N ratio with mass. This can be explained by noting that CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 55

Figure 3.8: Same as for Figure 3.7 but for the Z = 0.008 models

‘

Figure 3.9: Same as for Figure 3.7 but for the Z = 0.004 models. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 56

Figure 3.10: The composition pro®le for the CNO isotopes as a function of the interior mass for the 1M , 3M and 6M solar composition models. The units on the y-axis is the logarithm of the mole fraction not mass fraction. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 57

Figure 3.11: Same as for Figure 3.10 but for the Z = 0.008 models. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 58

Figure 3.12: Same as for Figure 3.10 but for the Z = 0.004 models. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 59 during the SDU, the convective envelope reaches the ashes of complete H-burning, which is mostly 4He and 14N. For this reason, there is little change to the 12C/13C ratio but a large change to the 14N/15N ratio. The most extreme case is the 6.5M , Z = 0.02 model, where the 14N/15N ratio after the SDU is approximately 2500.

3.4.5 Oxygen The behaviour of the 16O/17O ratio after the FDU depends crucially on the depth reached by the convective envelope and hence on the initial mass and metallicity of the model. In most cases the 16O/17O ratio is decreased substantially from the initial value. There are exceptions in the lowest mass models and for the most massive models that do not experience deep FDU events. For example, in the case of the 1M model (see the top panel of Figure 3.10), the convective envelope reaches in far enough to mix the entire 13C pocket into the envelope but leaves the oxygen isotopes unchanged. Figure 3.12 demonstrates that for the 1M , Z = 0.004 model, the convective envelope moves into the region depleted in 18O and slightly enhanced in 17O but leaves the 16O abundance unchanged. In most massive Z = 0.004 models (and to a lesser extent the Z = 0.008 models) the FDU does not reach the 17O pocket or into the region depleted in 16O. This is clearly seen in Figure 3.9 for the 6M , Z = 0.004 model where the 16O/17O ratio is essentially unchanged after the FDU. Our results for the 16O/17O ratio follow the same trend with mass as reported in El Eid (1994) and BS99, except that our results are systematically higher. This is probably caused by a higher initial 16O/17O ratio in the Z = 0.02 models plus different rates for the 17O + p reactions. The 16O/18O ratio increases by up to 30% in the solar composition models following the FDU. The models of El Eid (1994)∼and BS99 also find increases of up to 30%, in excellent agreement with our results. El Eid (1994) begins with a lower initial value of the 16O/18O ratio to ensure that the values after the FDU are in agreement with the observed value of about 500 50 (El Eid 1994). Our post-FDU values (and those reported in BS99) are higher than theobserved values. The change to the 16O/18O ratio for the LMC and SMC models depends on the depth of the FDU. Hence there is little change to the least and most massive models, with an increase of up to 40% for the intermediate mass models. For the lower metallicity models, we see the same∼ trend with mass for both the 16O/17O and 16O/18O ratios as reported by BS99. Following the SDU, the oxygen isotope ratios are not altered for the Z = 0.02 models but there are changes for the Z = 0.008 and Z = 0.004 models. The 16O/17O ratio decreases and there is a small increase to the 16O/18O ratio, following the same trend with mass as seen in the Z = 0.02 models. Further evidence for extra mixing in low-mass stars comes from the range of oxygen isotopic ratios observed in asymptotic giant branch stars measured in pre-solar grains. Boothroyd & Sackmann (1999) and previously, Wasserburg et al. (1995) comment that the oxygen isotopic values observed in AGB stars are in a region inaccessible by either the first dredge-up or hot bottom burning (which occurs during the TP–AGB phase). Some of the pre-solar oxide grains also have values in this region (Nittler et al. 1994). Wasserburg et al. (1995) point out that some form of extra mixing, as discussed earlier to solve the CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 60 carbon isotopic problem, could also alter the oxygen isotopic abundances such that they are moved into this inaccessible region.

3.5 First and Second Dredge-Up: non-CNO Elements

All stars within a given globular cluster are observed to have the same iron abundance to within a narrow range although the metallicity varies from cluster to cluster. Most well- studied globular clusters have star–to–star abundance variations for the light elements, including carbon, nitrogen, oxygen, sodium, magnesium and aluminium (C, N, O, Na, Mg and Al). The amplitude of the variation differs from cluster to cluster but the trends are the same: low C and high N, low O and high Na and low Mg and high Al (Yong et al. 2003). These globular cluster abundance anomalies have been studied in detail in many clusters, including M3 (Smith 2002), M5 (Cohen, Briley & Stetson 2002), M13 (Shetrone 1996; Kraft et al. 1998; Briley, Cohen & Stetson 2002) and NGC 6752 (Gratton et al. 2001; Yong et al. 2003) Two theories have been proposed to explain the abundance anomalies. The first scenario assumes that the abundance anomalies are created inside the star itself. In this case, some matter from the convective envelope is transported to the H-burning region, where nuclear processing may occur. Models such as ours do not predict this extra mixing, but the inclusion of rotation may provide the necessary mechanism, as discussed earlier in the context of the 12C/13C ratio in 3.4.1. The depth of the extra mixing determines the elements that are altered: if shallow§, then only C and N are effected, if deeper, the O and Na and possibly Mg and Al abundances might be altered. There is the danger that if the mixing is too deep, the evolution of the star will be dramatically changed as fresh H is constantly being added to the H-shell. The second scenario assumes that the star obtains the abundance anomalies from pollution from a previous generation of stars, or is born from a cloud of gas polluted from earlier stars. It is assumed that in this scenario, a previous generation of AGB stars are responsible for the abundance anomalies and not supernovae. The reason for this is that supernovae would change the iron abundance, thus we would also expect star–to–star variations in the iron abundance and this is not observed. Pollution from an AGB star would not effect the iron abundance, but possibly alter the O, Na, Mg and Al abundances of a star. There are problems with this scenario, in that pollution from an AGB star would possibly alter the 4He abundance and that most of the matter from the polluting AGB star would need to be accreted by lower mass objects. We discuss the Mg isotopic ratios observed in stars in NGC 6752 by Yong et al. (2003) further in Chapter 5. The first and second dredge-up effect the surface abundances of 7Li, 19F, 23Na and to a lesser extent the Ne, Mg and Al isotopes. We present the fractional surface abundance changes (as a fraction of the initial abundance) for these species in Appendix B in tabular form. Tables B.4, B.5 and B.6 contain the fractional change to the 7Li, 19F, 23Na and the Ne isotopes for the Z = 0.02, Z = 0.008 and Z = 0.004 models, respectively. Tables B.7, B.8 and B.9 contain the fractional changes to the Mg, Al and Si isotopes for the Z = 0.02, Z = 0.008 and Z = 0.004 models, respectively. The silicon isotopes are not effected by CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 61

Figure 3.13: The fractional changes to the surface abundance of 7Li (left) and 19F (right) after the FDU (solid line) and SDU (dashed line). the FDU or SDU and we do not discuss these species further. We discuss the eﬀects of the FDU and SDU on the surface abundances of each of these other elements in turn, starting with lithium.

3.5.1 Lithium and Fluorine In Figure 3.13 we show the changes to the surface abundance of 7Li (left panel) and 19F (right panel) after the first and second dredge-up. The Z = 0.02 model results are denoted by black solid points, the Z = 0.008 model results by red stars and the Z = 0.004 results by blue triangles. The solid lines refer to abundance changes caused by the FDU and the dashed lines refer to abundance changes caused by the SDU. In Figure 3.13 we see that the extent to which lithium is depleted at the surface after the FDU does not depend on the metallicity of the model. This makes sense since 7Li is easily destroyed in stellar atmospheres at relatively low temperatures and is depleted by burning during the pre-main sequence phase. Therefore depletions in lithium occur much further out in mass compared to the other elements. For this reason the convective envelope will extend into regions heavily depleted in lithium regardless of the maximum depth reached by the FDU in our models. The effect of the SDU on the surface 7Li abundance is to further reduce the lithium abundance in the envelope. The 6M models have the most extreme 7Li depletions: where 7 7 4 Li/ Li0 10− at the surface after the SDU. From ∼Figure 3.13 we see that there are slight enhancements in the surface abundance of 19F after the FDU for the low-mass models (below 2M ) and in some of the LMC and SMC models where the first dredge-up is not very efficient. To understand this, we need to examine the composition profiles of those models at the end of core H-burning. We will focus on the Z = 0.004 models for this discussion. If we examine the composition profile for the 1M , Z = 0.004 model in Figure 3.12, we note that the convective envelope reaches a small 19F pocket produced from proton capture on 18O via the CNO cycle. Below this 19F pocket, fluorine is completely destroyed by the 19F(p,α)16O reaction. For the 3M , Z = 0.004 model, the situation is similar except the 19F pocket is smaller and the con vective envelope moves further inward, into the region depleted in 19F. This agrees CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 62

Figure 3.14: The fractional changes to the surface 22Ne abundances (Left) and the 23Na abundances (Right) after the FDU (solid lines) and SDU (dashed lines).

19 19 with Figure 3.13 which shows the F/ F0 abundance to be less than 1.0 for this model. For the 6M Z = 0.004 model, the 19F pocket is very small but the FDU only reaches down to 2.6M , just deep enough to reach this very small 19F pocket. Therefore we ﬁnd a (very small) enhancement in ﬂuorine after the FDU, as shown in Figure 3.13. The SDU results in surface depletions in the 19F abundance, with the largest changes (up to 20%) observed in the 6M models.

3.5.2 Neon and Sodium The surface abundance of 20Ne is not eﬀected by the FDU and we observe only very small depletions after the SDU for the most massive models. There is also little change to the abundance of 21Ne after the FDU but modest enhancements after the SDU, with the largest enhancements found for the 5 and 6M , Z = 0.008 models, where the increase compared to the initial abundance is 70%. In Figure 3.14 (left panel) we show the changes to the surface abundance of 22N∼e after the FDU and SDU (using the same convention as in Figure 3.13). The FDU only eﬀects the 22Ne isotope in the mass range 2 to 3M , where depletions of about 15% are observed to occur. In the most massive models, the SDU causes the largest changes∼ to the surface abundance of 22Ne, with depletions of about 26% for the 6M , Z = 0.004 model. Sodium is produced at the expense of 22Ne via the Ne–Na cycle. In Figure 3.14 (right panel) we see that the surface abundance of sodium is enhanced in all models after the FDU and/or SDU. The extent of the enhancement depends on the mass and metallicity of the model and certain trends are clear. At a given mass, the most metal-poor model has the largest 23Na enhancement; and at a given metallicity, the models in the mass range 2 to 3M experience the largest enhancements after the FDU, and the most massive models the largest increase after the SDU. The largest change to the surface abundance of 23Na occurs in the 6M , Z = 0.004 model, where the increase from the initial abundance is greater than 200%. For comparison to Figure 2 in El Eid & Champagne (1995), we show the [Na/H] versus [14N/H] abundances from our Z = 0.02 models in Figure 3.15. There is a positive linear CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 63

Figure 3.15: [14N/H] as a function of [Na/H] for the Z = 0.02 models. correlation between the [14N/H] and [Na/H] abundances for M > 2M as found by El Eid & Champagne (1995). The slope of the line drawn in Figure 3.14 is 1.16, similar to, but lower than the value of 4/3 found by El Eid & Champagne (1995).∼ For M < 2M , the positive correlation between the surface 14N and 23Na abundances is not linear, because a large nitrogen abundance is coupled with a relatively small sodium abundance. The reason for this is that whilst 14N is enhanced at the surface of the low-mass models, the sodium abundance is not, since the 22Ne(p,γ)23Na reaction is not eﬃcient until the temperature reaches about 50 million K (Arnould et al. 1999).

3.5.3 Magnesium and Aluminium The magnesium isotope, 24Mg, shows no change after the FDU and SDU for all masses and metallicities modelled. This is not a surprising result, since the rate for the 24Mg + p reaction is slow unless the temperature reaches at least 70 million degrees K. Therefore even if the H-shell of a model reached this temperature, the ﬁrst dredge-up has to be very deep to reach the layer where 24Mg has been altered by H-burning. In most models, the temperature in the H-shell during the ﬁrst giant branch does not reach 70 million K. The abundance of the heavy magnesium isotopes, 25Mg and 26Mg, remains unaltered by the FDU. The SDU produces depletions of up to 14% in the surface abundance of 25Mg and enhancements of up to 10% in 26Mg for the most massive models. There is no change to the surface abundance of the Mg nuclei in any of the low-mass models. We also include the change to the elemental Mg abundance in Appendix B, which is negligible for all masses and metallicities after both the FDU and SDU. We do not include the radioactive aluminium isotopes, 26Al, with a half–life of approxi- CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 64

Figure 3.16: (Left) radiated (dashed line) and He-burning (solid line) luminosity during the core helium ¯ash for the 1M , Z = 0.004 model. (Right) mass of the temperature maximum as a function of time. At the ¯ash peak, the maximum temperature is 0.2M from the centre of the model. mately 717,000 years. This is because any enhancements from the FDU will have decayed to 26Mg by the time the model reaches the tip of the ﬁrst giant branch or the end of core He- burning. During the SDU, there can be quite large enhancements of 26Al, which may be observable in the more luminous early AGB stars. The abundance of the stable aluminium isotope, 27Al, shows negligible enhancements after the FDU and SDU for all models and metallicities.

3.6 The Core Helium Flash

During the ascent of the first giant branch, the core contracts until the necessary temperature for He-burning is reached, at about 100 million K. Low-mass models need to contract substantially before reaching this temperature and hence the central regions become highly electron-degenerate, and helium is ignited under explosive conditions. During a core He- flash, the surface luminosity does not change very much but the helium-luminosity increases by many orders of magnitude, reaching up to 109L in some models (see Fig- ure 3.16). From Figure 3.16 we see that the helium-burning luminosity oscillates a few times after the main He-flash, before settling down to quiescent core He-burning. We also show the mass at the temperature maximum as a function of time in Figure 3.16, noting that for this model the peak temperature occurs 0.2M from the centre. The maximum initial mass for the core He-flash to occur, M¯ash , is about 2.25M with the evolution code, depending on metallicity. Models with convective overshoot in the core find that the upper limit occurs at lower mass, where M¯ash . 1.6M (Bertelli et al. 1986). There has been debate for some time as to whether the core helium flash is a violent hydrodynamic event or not (Deupree 1996). Early results from two-dimensional hydrodynamic simulations (Deupree & Wallace 1987) suggest that the flash could be a relatively quiescent or violent hydrodynamic event, depending on the degeneracy of the stellar model. Deupree & Wallace find that for the most violent flashes, there is enough mixing CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 65 from the core to the H-shell or envelope to alter the surface abundances of C, N, Ne, Mg, Al, Ar, S and maybe Si. It has been ruled out that the core-helium flash is responsible for the abundance anomalies observed in globular cluster stars, as the flash does not produce sodium to the extent observed. The anti–correlation between Mg and Al is not reproduced, because enhancements in Mg are also accompanied by enhancements in Al. Also, most of the observed globular cluster stars are generally below or near the tip of the first giant branch and thus have not experienced the core-He flash. There are some exceptions, including the core He-burning stars in the globular cluster M67, observed first by Gilroy & Brown (1991) and later by Tautvaisien˘ e˙ et al. (2000) to have CN and possibly Na surface abundance anomalies. Perhaps the ultra-metal poor stars recently observed by Norris, Ryan & Beers (2001) and Christlieb et al. (2002) are better objects to study for signs of a violent core-He flash. The location of the He ignition point during the first (and strongest) He-flash has important consequences for the resulting evolution (Paczynski´ & Tremaine 1977). If the ignition point is close enough to the H–He discontinuity, then there can be some mixing between the flash-driven convection zone and the hydrogen-rich layers above. Fujimoto, Iben & Hollowell (1990) and Schlattl et al. (2001) find this mixing for their low-mass Z = 0 models. The low-mass Z = 0 models of Siess, Livio & Lattanzio (2002) find no such mixing, indicating that mixing between the convective pocket and the overlying H-rich layers strongly depends on the details of the input physics used. Schlattl et al. (2002) investigate if this mixing can explain the large carbon and nitrogen overabundances (with respect to iron) in ultra-metal poor halo stars ([Fe/H] . 2.5) (see Norris, Ryan & Beers 1997; Norris et al. 2001; Aoki et al. 2002). Schlattl et−al. (2002) conclude that there are two main problems explaining the carbon and nitrogen overabundances in the ultra-metal poor stars: first, too much carbon and nitrogen is transported to the surface and second, mixing during the core He-flash can not account for the stars present day metallicity. Schlattl et al. (2002) suggest some solutions to this problem, including efficient pollution of heavy elements to raise the [Fe/H] abundance. We do not want to diverge too much, but note that other authors explain the abundances in ultra-metal stars with some success using hyper- novae (Norris et al. 2002) or combinations of supernovae models of different mass (Chieffi & Limongi 2002). The most recent results by Deupree (1996) for low-mass solar composition models, using improved but still uncertain input physics, now suggest that the core flash is not a violent hydrodynamic event, and that there is no mixing between the flash-driven H- exhausted core and the envelope. Deupree also states that hydrostatic evolutionary calculations do not treat the core He-flash correctly, because they do not allow energy to be transported into the neutrino-cooled inner core or treat the flash-driven convective region as super-adiabatic. For this reason, the true nature of the core helium flash can only be roughly approximated by the evolution code, but it is better to evolve the models through, for reasons of self consistency in the structure and abundances. Some researchers do not model the core He-flash (see for an example Charbonnel et al. 1996 and Castellani, Chieffi & Straniero 1992), but instead create zero-aged horizontal branch models. This is done by assuming that the model just after the He flash should have the same distribution of chemical species as the model before the flash, but with 2% of the He-core mass processed into CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 66

Figure 3.17: The composition pro®le for the CNO nuclei against interior mass for the 1.5M , Z = 0.004 model.

12C (Castellani et al. 1992).

3.6.1 Results From the Core-helium Flash Convergence difficulties during the core helium flash usually halt the evolution calculation temporarily, as the luminosity in the degenerate core can be at least L 1010L , 4 ≈ with time steps of the order of 10− years. To evolve through the core-helium flash, we typically relax convergence criteria and only let the evolution code mix abundances after a model has converged. Results from the stellar models that experience the core helium flash are presented in Tables 3.8, 3.9 and 3.10. We include the initial mass, the mass at the tip of the RGB, Mtip, the mass interior to the helium ignition point, Mcore, the maxi- max min mum (log(LHe /L )) and minimum He-burning luminosity (log(LHe /L )), the maximum 6 temperature reached in the core (in units of 10 K), Tmax, and the resulting helium (Ycore) 12 and carbon ( Ccore) core mass fractions following the main flash. We do not include models that experience “mild” core He-flashes, where the He-burning luminosity increases sharply (perhaps reaching 106L briefly), but causes no convergence problems during the evolution. In these mild cases, such as the 2.1M , Z = 0.008 or the 2.25M for all metallicities, the 4He burnt during the flash is negligible and the time from the flash to quiescent He-burning is short. The severity of the core flash increases as the mass or metallicity of the model decreases (owing to a higher degeneracy), resulting in higher core temperatures and helium burning luminosities. An increase in the severity of the core flash also results in a larger amount of 12C produced. This can be seen in Table 3.8, 3.9 and 3.10, where the largest carbon CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 67

Table 3.8: Helium-core ¯ash and resulting core carbon production. Z = 0.02 models Initial mass (M )

Parameters 1.0 1.25 1.5 1.75 1.9

Mtip 0.8145 1.1237 1.4047 1.6779 1.8460 Mcore 0.4659 0.4644 0.4639 0.4609 0.4499 Mtempmax/Mtotal 0.2046 0.1966 0.1942 0.1796 0.1320 max log(LHe /L ) 9.435 9.418 9.414 9.414 9.368 min log(LHe /L ) 0.953 0.909 0.057 0.983 0.06 T ( 106 K) −195.3 −194.7 −194.1 −191.8 −186.9 max × Resulting Ycore 0.9527 0.9535 0.9539 0.9556 0.9610 12 Resulting Ccore 0.0280 0.0272 0.0268 0.0255 0.0196

Table 3.9: Helium-core ¯ash and resulting core carbon production. Z = 0.008 models Initial mass (M )

Parameters 1.0 1.25 1.5 1.75 1.9

Mtip 0.8471 1.1438 1.4195 1.6877 1.8510 Mcore 0.4759 0.4732 0.4723 0.4695 0.4610 Mtempmax/Mtotal 0.2476 0.1762 0.1416 0.1110 0.0826 max log(LHe /L ) 9.820 9.795 9.789 9.783 9.741 min log(LHe /L ) 0.716 0.595 0.591 0.679 0.223 T ( 106 K) −201.9 −200.1 −198.9 −197.0 −193.0 max × Resulting Ycore 0.9624 0.9633 0.9660 0.9654 0.9656 12 Resulting Ccore 0.0299 0.0290 0.0262 0.0264 0.0240

mass fractions are seen in the lowest mass or the lowest metallicity models. Boothroyd & Sackmann (1988c) also find this trend, even though core flash ignition takes place centrally in their models, rather than off centre as found in hydrodynamical calculations and also in our calculations (and all other calculations with neutrino-energy losses). In Figure 3.17 we show the composition profile for the 1.5M , Z = 0.004 model during the core-helium flash. The shaded regions corresponds to regions that are convective. In the flash-driven convective region we observe a large increase in the 12C abundance (at the expense of 4He), and a large decrease in 13C. Just before the flash peak the closest the flash-driven convection zone gets to the bottom of the H-shell (defined earlier in the chapter) is 0.0084M , and the closest to the convective envelope is 0.013M . The situation is similar for the other low-mass models and the core He-flash has no effect on the surface abundances for this reason. The 12C mass fraction after the core flash is about 2%, consistent with the calculations of Castellani et al. (1992). For example, for the 1.5M , Z = 0.004 model in Figure 3.17, the resulting 12C mass fraction is 2.68%. CHAPTER 3. PRE–AGB EVOLUTION AND NUCLEOSYNTHESIS 68

Table 3.10: Helium-core ¯ash and resulting core carbon production. Z = 0.004 models Initial mass (M )

Parameters 1.0 1.25 1.5 1.75 1.9

Mtip 0.8683 1.1588 1.4254 1.6907 1.8499 Mcore 0.4787 0.4750 0.4764 0.4747 0.4707 Mtempmax/Mtotal 0.2453 0.1678 0.1683 0.1126 0.0968 max log(LHe /L ) 9.956 9.926 9.931 9.937 9.920 min log(LHe /L ) 0.560 0.437 0.470 0.499 0.549 T ( 106 K) −203.3 −201.5 −201.3 −200.3 −197.6 max × Resulting Ycore 0.9630 0.9628 0.9674 0.9690 0.9708 12 Resulting Ccore 0.030 0.0294 0.0290 0.0270 0.0250

3.7 Summary

The phases prior to the AGB are important because they determine the structure and composition of a model at the beginning of the thermally-pulsing AGB phase. The first dredge- up is also important when studying stars on the first giant branch. Here we present the stellar lifetimes from our evolution models and the surface abundance results after both the first and second dredge-up. We present surface abundance predictions for not just the CNO elements as commonly found in the literature, but also for elements linked to other H-burning cycles, including fluorine, sodium, neon, magnesium and aluminium. Where possible, we have compared our results to other theoretical calculations and observations. For reasons of self consistency we evolve the low-mass models through the core He-flash, and include a brief summary of results from our models and the literature. There is a strong mass dependence on the abundance changes for all but the light element 7Li, which is easily destroyed in stellar atmospheres. The 12C/13C ratio is observed to decrease in all models, in a manner consistent with other calculations without any form of extra mixing. The 14N/15N ratio increases, with the largest increases found in the most massive models after the SDU. There is an increase in the 16O/18O ratio (the largest increase 40%) along with a decrease in the 16O/17O ratio. After the FDU, the surface abundances∼ of most of the non-CNO nuclei remain essentially unchanged in low-mass stars (with M . 2.25M ), but we do observe small enhancements in the 19F and 23Na abundances. The more massi ve models show depletions of 19F and 22Ne and show strong enhancements of 23Na, especially after the SDU. The abundance of the magnesium, aluminium and silicon isotopes are essentially unchanged after the FDU, with slight changes for some species (for example 25Mg) after the SDU. Chapter 4

Asymptotic Giant Branch Evolution

In this chapter we continue our discussion of the evolution of the mass-loss models introduced in Chapter 3. We now focus on the thermally-pulsing asymptotic giant branch (AGB) phase of evolution, which alters the surface abundances of the models in two distinct and important ways. The ﬁrst is through the operation of the third dredge-up, which can occur periodically after each thermal pulse (TP) and is the mechanism for turning (single) stars into N-type carbon stars. The second mechanism is hot bottom burning (HBB).

4.1 Evolution on the AGB

There are many reviews of the AGB phase of stellar evolution, including the articles by Block¨ er (2001), Busso & Gallino and Wasserburg (1999), Wood (1997), Frost & Lattanzio (1995) and Iben (1991) for a theoretical overview. Here we briefly review the main features of AGB evolution. Following core He-exhaustion, the central regions of the star, which are now composed primarily of carbon and oxygen, contract. At the same time, the outer layers expand and the star ascends the giant branch for the second time. Whilst the core contracts, He-burning ignites in a thin layer around the degenerate CO core but instabilities quickly set in, due to the thinness of the burning shell. At this stage the star is said to have entered the thermally-pulsing AGB (TP–AGB) phase of evolution. This stage of evolution is characterized by relatively long periods of quiescent H-shell burning, known as the interpulse phase, interrupted by instabilities of the He-shell. The ashes of the H-shell increase the thickness of the hydrogen depleted region, until eventually a thermonuclear runaway or He-shell flash, is ignited. The He-shell burns fiercely, producing up to 108L for a few hundred years. This enormous quantity of energy does not reach the∼ surface but goes into expanding the outer layers of the star, causing the H-shell to become extinguished. The energy produced by the flash also powers a convective pocket in the He-shell, which has the effect of mixing the abundances in the region. When the flash dies down, the outer convective envelope moves inward in mass to the region mixed by the intershell convective pocket. This phase is known as the third dredge-up (TDU), and can occur after each thermal pulse. The result of the third dredge-up is to mix the products of partial He-burning (mostly 4He and 12C) to the surface. This is the process which

69 CHAPTER 4. AGB EVOLUTION 70 turns (single) stars into N-type carbon stars. Following dredge-up, the star contracts and the H-burning shell is re-ignited and the star enters a new interpulse phase. The cycle, interpulse–thermal pulse–dredge-up will occur many times on the AGB, depending on the initial mass and composition of the model, as well as the mass-loss rate. In the most massive AGB models, the convective envelope can dip into the top of the H-shell, resulting in nuclear burning at the base of the convective envelope. This phenomena is known as hot bottom burning (HBB), and is important when the temperature at the base of the convective envelope exceeds about 50 million K. It has been known for some time that intermediate mass stars over about 5M develop deep convective envelopes with very high temperatures at the base, allowing for nuclear burning and some energy generation (Sugimoto 1971; Sackmann, Smith & Despain 1974; Iben 1975; Scalo, Despain & Ulrich 1975). Whilst there is observational evidence that HBB is occurring in massive AGB stars (Wood, Bessell & Fox 1983), the first detailed calculations to show envelope burning were by Block¨ er & Schonberner¨ (1991) and Lattanzio (1992). Block¨ er & Schonberner¨ (1991) found that the linear core–mass–luminosity relation that was first proposed by Paczynski´ (1970), does not apply to stellar models with HBB. HBB also prevents carbon star formation, by burning the 12C dredged to the surface into 14N (Block¨ er & Schonberner¨ 1991; Boothroyd, Sackmann & Ahern 1993). The luminosity range of the models with HBB are also consistent with the oxygen rich, luminous LMC AGB stars observed by Smith & Lambert (1989, 1990) and Plez, Smith & Lambert (1993) to have large lithium abundances. In 4.1.1 we first review the third dredge-up occurring in the mass-loss models. In 4.2 we examine§ some structural quantities that influence the nucleosynthesis (and hence§the stellar yields) during the AGB, and in 4.3 we review the evolution during the interpulse phase. §

4.1.1 The Third Dredge-Up The third dredge-up occurs when the convective envelope moves inward (in mass), following a thermal pulse. The depth, or efficiency of the third dredge-up is measured by the parameter λ, which is defined in Chapter 6. Briefly, λ is the ratio defined by the mass dredged up into the envelope divided by the amount by which the hydrogen-exhausted core massa grew during the preceding interpulse phase. From this definition it should be clear that when λ = 1, the core mass does not grow but remains constant. AGB nucleosynthesis not only depends on the efficiency of the TDU but also on the minimum core mass at which the thermal pulses first begin, which we will call Mc(1), and the minimum min core mass at which the third dredge-up begins, which we call Mc . These parameters: λ, min Mc(1) and Mc vary as a function of the total mass, envelope mass and metallicity in a complex way. In Chapter 6 we provide a fit to these three parameters as a function of the total mass and metallicity, obtained from the models without mass loss. We do not repeat that analysis here (which also includes a discussion of the uncertainties) but instead give a summary of the third dredge-up in the mass-loss models.

ahereafter the core mass refers to the H-exhausted core mass; all masses are in solar units, unless speci®ed CHAPTER 4. AGB EVOLUTION 71

Table 4.1: A selection of results for the Z = 0.02 models. See the text for a detailed description. min max tdu M0 No. Mc(1) Mc TP λmax C/Of THe log Tbce HBB HBB TPs TDU begin end 1.0 10 0.544 – – 0.000 0.4021 230 – – – 1.25 10 0.553 – – 0.000 0.3654 220 – – – 1.5 12 0.558 – – 0.000 0.3377 240 – – – 1.75 15 0.561 – – 0.000 0.3194 246 – – – 1.9 16 0.561 – – 0.000 0.3186 254 – – – 2.0 21 0.553 – – 0.000 0.3168 258 – – – 2.25 27 0.546 0.620 20 0.241 0.4176 269 6.727 – – 2.5 26 0.533 0.608 16 0.564 0.8146 282 6.768 – – 3.0 26 0.576 0.614 9 0.777 1.4230 302 6.803 – – 3.5 21 0.667 0.678 4 0.879 1.5748 319 6.928 – – 4.0 18 0.773 0.777 4 0.975 1.0111 332 7.312 – – 5.0 24 0.861 0.865 4 0.961 0.7014 352 7.505 9 21 6.0 38 0.915 0.918 4 0.934 0.3496 369 7.653 3 35 6.5 40 0.951 0.952 4 0.910 0.3695 372 7.708 1 36

In Tables 4.1 and 4.2 we present some results for the Z = 0.02, the Z = 0.008 and the min Z = 0.004 models, respectively. In this section we discuss Mc(1), Mc and λmax, and the other quantities will be discussed in later sections. Column 1 is the initial mass, column 2 the total number of thermal pulses calculated during the TP–AGB, column 3 the core mass min at the first thermal pulse, Mc(1), column 4 the core mass at the first TDU episode, Mc , and in column 5 the pulse number at which the TDU begins. Column 6 is the maximum λ for each model, λmax. In models with efficient TDU, λ approaches an asymptotic value, which we give as λmax, whereas in other models we present the maximum value, calculated in most cases after the last thermal pulse. Column 7 is the last calculated 12C/16O ratio, column 8 the maximum temperature in the He-shell during a thermal pulse (in millions of K) and column 9 the logarithm of the temperature at the base of the convective envelope when dredge-up first occurs (in units of log K). Finally, in column 10 and column 11 we present the pulse number after which HBB begins and ends. We first compare our results for the core mass at the first thermal pulse, Mc(1). From Table 4.1 we see that for the 1, 3, 4 and 5M , Z = 0.02 models, the core mass at the first thermal pulse is Mc(1) = 0.544, 0.576, 0.773 and 0.861M respectively. In comparison,

Block¨ er (1995) ﬁnds smaller values of Mc(1) at 1, 3 and 4M , Z = 0.02. His values at these masses are 0.53, 0.54 and 0.65M , which are very similar for the 1 and 3M models, but considerably smaller for the 4M case. However, Block¨ er ﬁnds Mc(1) = 0.91M for the

5M , Z = 0.02 model, considerably larger than our value of Mc(1) = 0.86M . Lattanzio

(1986) ﬁnds (for his models with Y = 0.2) Mc(1) = 0.528, 0.536, 0.715 and 0.867M for the 1, 3, 4 and 5M , Z = 0.02 models. Thus we also ﬁnd larger values of Mc(1) for the 1 and 3M , Z = 0.02 models compared to Lattanzio (1986), but approximately the same value at 4 and 5M , Z = 0.02. For the 3M , Z = 0.02 model, there is excellent agreement

CHAPTER 4. AGB EVOLUTION 72

Table 4.2: Same as Table 4.1 but for the Z = 0.008 and Z = 0.004 models respectively. See the text on page 71 for a description of the contents. Z=0.008 min max tdu M0 No. Mc(1) Mc TP λmax C/Of THe log Tbce HBB HBB TPs TDU begin end 1.0 11 0.532 – – 0.000 0.4028 250 – – – 1.25 12 0.541 – – 0.000 0.3629 254 – – – 1.5 15 0.544 0.610 13 0.084 0.5036 265 6.568 – – 1.75 15 0.552 0.596 10 0.325 1.9054 272 6.672 – – 1.9 22 0.549 0.594 11 0.500 2.5472 278 6.709 – – 2.1 20 0.538 0.585 10 0.638 4.4514 286 6.748 – – 2.25 26 0.522 0.585 14 0.727 5.3257 293 6.745 – – 2.5 27 0.541 0.588 10 0.805 6.6943 301 6.795 – – 3.0 28 0.629 0.647 4 0.893 10.134 319 6.854 – – 3.5 20 0.750 0.756 3 0.982 5.8842 353 7.342 – – 4.0 22 0.829 0.833 3 0.987 3.8922 349 7.474 14 20 5.0 59 0.870 0.872 2 0.978 1.0653 367 7.560 9 56 6.0 69 0.930 0.933 3 0.954 1.7767 375 7.694 2 66 Z=0.004 1.0 14 0.532 – – 0.000 0.4077 265 – – – 1.25 15 0.540 – – 0.000 0.6580 272 – – – 1.5 16 0.540 0.587 9 0.285 2.4498 274 6.613 – – 1.75 15 0.543 0.584 8 0.431 5.7715 275 6.681 – – 1.9 16 0.548 0.585 7 0.563 8.1795 282 6.709 – – 2.25 26 0.538 0.577 8 0.767 15.436 300 6.744 – – 2.5 29 0.575 0.603 5 0.833 21.371 308 6.806 – – 3.0 26 0.695 0.702 2 0.949 19.268 337 7.121 – – 3.5 25 0.806 0.809 2 0.998 10.738 351 7.444 15 20 4.0 32 0.842 0.845 2 0.978 3.8236 366 7.556 12 29 5.0 83 0.888 0.890 2 0.959 4.4819 379 7.599 7 80 6.0 105 0.959 0.960 4 0.935 5.6994 386 7.785 1 101 CHAPTER 4. AGB EVOLUTION 73

between our value for Mc(1) and the value given by Straniero et al. (1997). These authors find 0.572M compared to our 0.576M . In our calculations, the TDU only occurs in models with M > 2M at Z = 0.02, and M > 1.5M at Z = 0.008 and Z = 0.004. There is substantial evidence that the dredge-up of carbon occurs in galactic disk stars with initial masses as low as 1.2M (Wallerstein & Knapp 1998), which our models fail to reproduce. The stellar models∼ do a little better at lower metallicity where the minimum mass for carbon-star production is 1.5M at an SMC composition. Even so, the observations indicate that in the LMC and SMC, carbon stars exist with masses as low as 1M (Bessell, Wood & Evens 1983). If we were to use the most recent estimate for the solar oxygen abundance from Allende Prieto, Lambert & Asplund (2001) in our stellar models, we would reduce the global solar metallicity from Z = 0.02 (or 0.01899 from Anders & Grevesse 1989) to Z 0.01. This has important consequences for the TDU, which is easier to obtain in lower∼metallicity models. Thus it might be possible to obtain the TDU in a solar composition 1.5M model. By reducing the solar metallicity, we would also reduce the metallicity of the LMC and SMC. Ignoring α– element enhancements, the metallicity, Z, is related to the [Fe/H] abundance via 10[Fe/H] 0.3 × Z . Thus the metallicity of the LMC (assuming Z = 0.01) would become 10− 0.01 = 0. 005b, closer to the current assumed metallicity of the SMC (Z = 0.004). This× means it would be easier to obtain the TDU in low-mass LMC and SMC models. We note that many authors do not find the TDU, even in intermediate mass models, without including some form of convective overshoot. For example, both Mowlavi (1999b) and Herwig (2000) need to include convective overshoot to find dredge-up in a 3M , Z = 0.02 model whereas for the same mass and composition (see Figure 4.1) we obtain substantial dredge- up because we search for a neutral border to the convective boundary (Frost & Lattanzio 1996). These issues are discussed further in Chapter 6. min We compare our λ and Mc values to the mass-loss models published in Straniero et al. (1997; hereafter S97). In Figure 4.1 we show the variation of λ with core mass for the 3M , min Z = 0.02 models. At this mass and composition, S97 find Mc = 0.611 and λmax = 0.389, min min whereas we find Mc = 0.614 and λmax = 0.777. Thus there is good agreement for Mc , but we find deeper third dredge-up. The decrease of λ with core mass observed in the S97 model is not seen in our model. The different behaviour of λ with core mass will be related to the numerical differences between the codes, notably the differences in the choice of mass-loss law used on the AGB and the treatment of convective boundaries. We can examine how the mass-loss law might influence the results. The Vassiliadis & Wood (1993; hereafter VW93) mass-loss formula results in smaller mass-loss rates compared to the Reimers (1975) formula (used by Straniero et al. 1997) for most of the AGB. The consequence of using the VW93 mass-loss law is that the envelope remains quite large for most of the TP–AGB, ensuring the maximum amount of dredge-up. The VW93 mass-loss law terminates the AGB with a superwind phase, which means that most of the envelope is lost in a few thermal pulses, whereas the Reimers mass-loss law, which does not have a superwind phase, leads to more thermal pulses as can be seen in Figure 4.1. The final core and envelope mass of the S97 model are 0.727M and 0.758M , compared to our

busing the [Fe/H] value for the LMC from Russell & Dopita (1992) CHAPTER 4. AGB EVOLUTION 74

Figure 4.1: (Top) λ as a function of the core mass for the 3M , Z = 0.02 model. (Bottom) Same, but from Straniero et al. (1997). CHAPTER 4. AGB EVOLUTION 75

0.680M and 0.675M . The ﬁnal CO core mass of our model is 0.668M , consistent with the initial–ﬁnal core mass relation of Weidemann (2000), which gives Mf = 0.68M for

M0 = 3M . S97 do not give the CO core mass of their model, but we suspect it would be about 0.7M , which is also consistent with the initial–ﬁnal mass relation. If we examine the production of carbon stars at Z = 0.02, we see from Table 4.1 that only the models in the mass range from 3M to 4M become carbon stars. Models with masses less than 3M either have no dredge-up, or the TDU is not eﬃcient enough to produce a carbon star . In models with masses greater than 4M , HBB prevents carbon star formation. This result is also somewhat dependent on the choice of mass-loss law. If we were to use the Reimers (1975) mass-loss law there would be more thermal pulses since there is no superwind to terminate the evolution, and the most massive models might become carbon stars after the cessation of HBB. All the LMC and SMC composition models with λmax > 0.1 become carbon stars, including the 1.5M and 1.75M models at Z = 0.004. This also includes all the models that experience HBB, which become carbon stars near the end of the AGB. This occurs because the third dredge-up continues even after HBB has ceased when the envelope mass is reduced below about 1.5M (see also Frost et al. 1998). ∼

4.2 Evolution During a Thermal Pulse

In this section we examine the structural features of a thermal pulse, including the mass and duration of the convective pocket and the range of temperatures found in the He- burning shell. We also discuss the occurrence of a secondary convective pocket which forms in low-mass models as a result of the luminosity oscillations after the main He- instability.

4.2.1 Mass and Duration of the Convective Pocket The enormous quantity of energy produced by a thermal pulse results in the formation of a convective pocket in the He-rich intershell region, which lasts for a few hundred years, depending on the core mass. The maximum mass of the convective pocket, ∆Mcsh, has been parameterized by Iben (1977) as a function of the core mass

2 log(∆Mcsh/M ) = 1.835 + 1.73MH 2.67MH. (4.1) − − Iben (1977) also provides a relationship between the size of the H-exhausted core and the duration of the convective pocket, ∆τcsh,

log(∆τ / seconds) = 9.66 + 0.9M 2.063M2 . (4.2) csh H − H Iben (1977) derived these relationships from solar composition models with H-exhausted core masses over 0.9M or initial mass & 6M . In the upper panels of Figures 4.2, 4.3 and 4.4 we plot the maximum mass (in M ) of the

CHAPTER 4. AGB EVOLUTION 76

0.04 M=6.5 Z=0.02 0.035 M=6 Z=0.02 M=5 Z=0.02 0.03 M=4 Z=0.02 M=3.5 Z=0.02 M=3 Z=0.02 0.025 M=2.5 Z=0.02 M=2 Z=0.02 csh 0.02 M=1.5 Z=0.02 M M=1 Z=0.02 0.015

0.01

0.005

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 H-exhausted core mass

600 M=6.5 Z=0.02 M=6 Z=0.02 500 M=5 Z=0.02 M=4 Z=0.02 M=3.5 Z=0.02 400 M=3 Z=0.02 M=2.5 Z=0.02 M=2 Z=0.02 300 M=1.5 Z=0.02 M=1 Z=0.02 200

Duration of pocket (years) 100

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 H-exhausted core mass Figure 4.2: (Upper panel) maximum mass of the convective pocket (in M ) as a function of the core mass for the Z = 0.02 models. (Lower panel) duration of con vection in the intershell (in years) as a function of the core mass for the Z = 0.02 models. CHAPTER 4. AGB EVOLUTION 77

0.04 M=6 Z=0.008 0.035 M=5 Z=0.008 M=4 Z=0.008 0.03 M=3.5 Z=0.008 M=3 Z=0.008 M=2.5 Z=0.008 0.025 M=2.1 Z=0.008 M=1.5 Z=0.008 csh 0.02 M=1 Z=0.008 M 0.015

0.01

0.005

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 H-exhausted core mass

600 M=6 Z=0.008 M=5 Z=0.008 500 M=4 Z=0.008 M=3.5 Z=0.008 M=3 Z=0.008 400 M=2.5 Z=0.008 M=2.1 Z=0.008 M=1.5 Z=0.008 300 M=1 Z=0.008

200

Duration of pocket (years) 100

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 H-exhausted core mass Figure 4.3: Same as for Figure 4.2 but for Z = 0.008. CHAPTER 4. AGB EVOLUTION 78

0.04 M=6 Z=0.004 0.035 M=5 Z=0.004 M=4 Z=0.004 0.03 M=3.5 Z=0.004 M=3 Z=0.004 M=2.5 Z=0.004 0.025 M=2.25 Z=0.004 M=1.5 Z=0.004 csh 0.02 M=1 Z=0.004 M 0.015

0.01

0.005

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 H-exhausted core mass

600 M=6 Z=0.004 M=5 Z=0.004 500 M=4 Z=0.004 M=3.5 Z=0.004 M=3 Z=0.004 400 M=2.5 Z=0.004 M=2.25 Z=0.004 M=1.5 Z=0.004 300 M=1 Z=0.004

200

Duration of pocket (years) 100

0 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 H-exhausted core mass Figure 4.4: Same as for Figure 4.2 but for Z = 0.004. CHAPTER 4. AGB EVOLUTION 79 convective pocket as a function of the core massc for a number of Z = 0.02, Z = 0.008 and Z = 0.004 models, respectively. The mass of the convective pocket is a strong function of the core mass (or the initial mass), with only a slight dependence on the initial composition. The mass of the convective pocket decreases with core mass, so the maximum mass of the convective pocket is smaller at MH 0.95M than at 0.6M by almost an order of ∼ magnitude. For example, a typical value of ∆Mcsh for the 1.5M , Z = 0.02 model is about

0.025M , whilst ∆Mcsh 0.004M for the 5M , Z = 0.02 model. ∼ For individual models, the behaviour of ∆Mcsh with core mass is qualitatively diﬀerent for the low and intermediate mass cases. In the low-mass models, the maximum mass of the convective pocket decreases with increasing core mass (or time). Conversely, for the more massive models, the maximum mass of the convective pocket increases with increasing core mass. We compare our massive solar composition AGB models to equation (4.1). For the 6.5M , Z = 0.02 model, thermal pulses begin when the core mass is equal to Mc(1) = 0.9509M and end when MH = 0.9628. For these core masses, the maximum mass of the convective pocket is ∆Mcsh = 0.00143M and 0.00263M respectively. From

Iben’s relationship, ∆Mcsh = 0.00248 at MH = 0.9509 and ∆Mcsh = 0.00226 at MH = 0.9628. Whilst these values are very similar to our model results, equation (4.1) does not predict the increase of ∆Mcsh with time that is seen in the most massive models. In the lower panels of Figures 4.2, 4.3 and 4.4 we show ∆τcsh (in years) as a function of the core mass for a number of the Z = 0.02, Z = 0.008 and Z = 0.004 models, respectively. Whilst there is some scatter in the data, there is a clear relationship between the core mass and the duration of convection in the intershell during a pulse. Similar to ∆Mcsh, ∆τcsh is strongly dependent on the core mass with only a small dependence on the initial composition. The duration of the convective pocket, ∆τcsh, decreases with increasing mass, from about 500 years for the early pulses of the low-mass models to about 10 years for the most massive models. At a given core mass, ∆τcsh increases slightly with decreasing Z. For example, at a core mass of 0.6M , ∆τcsh 180 for the Z = 0.02 models, and ∆τcsh 250 ∼ ∼ for the Z = 0.004 models. For most of the stellar models, ∆τcsh decreases with increasing core mass (or time). If we compare equation (4.2) to our Z = 0.02 models, we ﬁnd 105 years for 0.55M (c.f. our 350 years) and 14.26 years for 0.95M (c.f. our values range from 20 years to 12 years). There is very good agreement for the most massive models and poor agreement for the lowest mass cases. This result is not surprising considering Iben’s ﬁt was derived from AGB models with large core masses.

4.2.2 Temperature in the Helium-burning Shell The temperature in the He-shell is also a function of the mass and composition. We demonstrate this in Tables 4.1 and 4.2, where we give the maximum temperature in the middle of d max the He-shell for each model, THe . The location (in mass) of the middle of the He-shell approximately corresponds to the location of the base of the convective pocket. At a constant mass, the lowest metallicity models experience the highest temperatures in the He-shell.

cwhere the core mass increases with time along the AGB, unless the TDU eﬃciency parameter, λ > 1. Note that λ < 1 in all our models dde®ned as the point at which the4He mass fraction equals 0.45. CHAPTER 4. AGB EVOLUTION 80

Figure 4.5: Temperature in the middle of the He-shell plotted as a function of time for the (upper panel) 3M Z = 0.02 model and the (lower panel) 3M Z = 0.004 model.

CHAPTER 4. AGB EVOLUTION 81

For example, in Figure 4.5 we compare the temperature in the middle of the He-shell for two models of the same initial mass (3M ) but diﬀerent composition. Many thermal pulses of the 3M , Z = 0.004 model have temperatures in excess of 300 million K whereas this temperature is only reached in the very last thermal pulse of the 3M , Z = 0.02 model. The temperature in the He-shell has important consequences for nucleosynthesis, and in particular the 22Ne(α,n)25Mg reaction is only active for T & 300 million K. For many of the models, the temperature in the He-burning shell increases with evolution along the AGB, where the maximum temperature occurs at the last thermal pulse (see the top panel of Figure 4.5), except in the most massive or lowest metallicity models where the temperature is seen to reach an asymptotic value (for example in the 3M , Z = 0.004 model in Figure 4.5). Iben (1977) provides a relationship between the temperature at the base of the He-shell and the core mass for solar composition massive AGB stars

T /T = 310 + 285(M 0.96), (4.3) He 6 H − 6 where T6 = 1 10 K. Malaney & Boothroyd (1987) derive a similar relationship for solar composition lo×w-mass AGB stars

250 + 305(MH 0.53) if MH < 0.65, THe/T6 = − (4.4) ( 290 + 67(MH 0.53) if 0.65 < MH < 1.0. − Boothroyd & Sackmann (1988c) find that neither relation gave a good fit to their models. These authors find instead that THe increases with evolution along the AGB, and that the relationship provided by Iben (1977) could not be extrapolated down to smaller core masses. A direct comparison between our Z = 0.02 models and the fits given by Boothroyd & Sackmann (1988c) and Iben (1977) is not exactly valid, as the opacities used in the stellar evolution calculations have changed significantly in the last few years. In particular, we use the OPAL tables provided by Iglesias & Rogers (1996), which provide opacity over a finer grid of temperature and density than was previously available from the Los Alamos opacity tables (which were used by Boothroyd & Sackmann 1988c). Iben (1977) used analytic fits to the opacities of Cox & Stewart (1970a,b). Many details of the calculations are dependent on the opacities, and there will be an indirect effect on the maximum temperature reached in the He-shell during a thermal pulse, and therefore we would not expect our models to exactly match earlier calculations. Boothroyd & Sackmann (1988c) also stress that the temperature THe is sensitive to the size of the mass grid used in the intershell. We are interested in the relationship between THe and the core mass, so a comparison is still useful. Although we do not expect either relation to predict the increase of THe with time, we do expect both relationships to predict the temperatures in the right range. The 3M , Z = 0.02 model evolves from Mc(1) = 0.576M at the beginning of the TP–AGB to a final core mass of 0.680M near the end. In the same core mass range, equation (4.4) gives values between 264 and 300 million K, consistent with our model results. We also compare our massive AGB models to equation (4.3). The 6.5M , Z = 0.02 model

CHAPTER 4. AGB EVOLUTION 82

Figure 4.6: He-burning luminosity for the second and third thermal pulse of the 1.9M , Z = 0.008 model.

evolves from Mc(1) = 0.9509 to a ﬁnal core mass of 0.9628M . In the same core mass range, equation (4.3) produces values between 307 and 311 millions of K. The evolution of the temperature in the He-shell is not shown for this model, but we note that the range of temperatures are between about 250 million K at the beginning of the TP–AGB to just over 350 million K near the end. Thus our Z = 0.02 models produce temperatures in excess of those predicted by equation (4.3). If we use equation (4.4) instead over the range in core mass between 0.9509 to 0.9628 we get temperatures between 318 and 319 million K, again inconsistent with our model results. Both of the relationships probably fail because they are based on models that do not experience deep TDU, whereas many of our models experience deep which prevents the core mass from substantially growing in mass.

4.2.3 Secondary Convective Pockets At the beginning of the TP–AGB phase, all low-mass models (M < 2.5M ) experience oscillations of the He-burning luminosity following the main thermal pulse. In Figure 4.6 we show the He-burning luminosity for the second and third thermal pulse of the 1.9M , Z = 0.008 model. In this ﬁgure we observe that up to two oscillations follow the main ther - mal instability, with each being progressively weaker than the previous oscillation. With time evolution along the TP–AGB phase, the secondary oscillations become weaker and occur closer together to the main pulse before disappearing entirely. This phenomena has CHAPTER 4. AGB EVOLUTION 83

Figure 4.7: Diagram showing convective regions for the 1.9M , Z = 0.008 model. In this diagram we plot the intershell region for the ®rst ®ve thermal pulses, clearly showing a second convective pocket develop after the ®rst and strongest He-¯ash. The convective regions are shaded in green and the radiative in magenta. Note we are not plotting evolutionary model number on the x-axis but the nucleosynthesis time-step number. been described before by Schwarzschild & Harm¨ (1967) and Iben (1982). Both of these papers report that the main pulse causes the formation of an intershell convective pocket but the secondary instability does not. Here we find that the secondary oscillations do produce enough energy to drive a secondary intershell convective pocket. This pocket initially lasts almost as long as the main convective pocket, but is smaller in mass i.e. extends over a smaller region in the intershell than the main pocket. Over time, the duration and mass of the secondary pocket decreases before disappearing entirely. We demonstrate this for the 1.9M , Z = 0.008 model in Figure 4.7, where we show the convective regions for the first five thermal pulses. Following the first He-burning instability, a convective pocket develops that is about 0.03M in size and the secondary instability that follows generates a convective pocket that is about 0.01M in size. In this diagram, we can clearly see that dredge-up has not started, and does not be gin until the core mass reaches 0.594M . In Table 4.3 we present data relating to the first five thermal pulses. In each ro w, the results for the main or primary pulse are given by ’(p)’ and the results for the secondary pulse by ’(s)’. In the first column we give the pulse number, in the second column the time (in 106 years) from the ZAMS, in the third column the maximum He-burning luminosity CHAPTER 4. AGB EVOLUTION 84

Table 4.3: The ®rst ®ve thermal pulses for the 1.9M , Z = 0.008 model. In each case, data relating to the ªprimaryº pulse is denoted by (p) and for the ªsecondaryº pulse by (s). max max Pulse No. Time (Myr) log LHe Mcsh ∆τcsh ∆t(Pocket a ± Pocket b) (L ) (M ) years years

1.0 (p) 1187.1520 4.745 0.030432 392.29 – ...... (s) 1187.1566 3.940 0.013757 328.05 4600 2.0 (p) 1187.2435 5.100 0.030755 362.25 – ...... (s) 1187.2464 4.052 0.012900 263.11 2900 3.0 (p) 1187.3644 5.687 0.032172 238.33 – ...... (s) 1187.3661 4.153 0.010481 201.23 1700 4.0 (p) 1187.5059 6.014 0.031833 240.28 – ...... (s) 1187.5072 4.158 0.007427 158.85 1300 5.0 (p) 1187.6590 6.341 0.031369 259.84 – ...... (s) 1187.6600 4.149 0.004613 105.30 1000

(in log L ), in the fourth column the maximum mass of the convective pocket (in M ), in the fifth column the duration of the convective pocket (in years) and in the sixth column the difference in time between the main and secondary convective pockets. The time in the sixth column is measured from the maximum He-burning luminosity. The data in Table 4.3 clearly shows the decrease of the mass and duration of the secondary pulse with time, and also demonstrates that the time difference between the main and secondary oscillations decreases with evolution along the TP–AGB. By the sixth thermal pulse there is no secondary convective pocket. What influence does the secondary convective pocket have on the nucleosynthesis? Most likely none. If we examine Figure 4.7 we can see that the region in mass mixed by each secondary convective pocket is engulfed by the next primary convective pocket. Therefore any nucleosynthesis that results from the secondary convective pocket is swept up by the next thermal pulse. Also, as dredge-up does not begin until much later in the evolution, the He-burning products of these early pulses are not mixed to the surface.

4.3 Evolution During the Interpulse Phase

In this section we examine some of the main features of evolution during the interpulse phase. We look at the interpulse period and how that changes as a function of the core mass and the temperature at the base of the convective envelope, and how that eﬀects hot bottom burning. We ﬁnish with an examination on how the pre-pulse surface luminosity maximum varies as a function of the core mass. CHAPTER 4. AGB EVOLUTION 85

4.3.1 The Interpulse Period

The interpulse period, τip, is an important variable for synthetic AGB evolution algorithms, and has been parameterized by Boothroyd & Sackmann (1988c) as a function of the core mass for two metallicities:

4.50(1.689 MH) Z = 0.02, log(τip/years) = − (4.5) ( 4.95(1.644 MH) Z = 0.001. −

Wagenhuber & Groenewegen (1998) give an improved ﬁt to τip, which is dependent on the core mass, metallicity and mixing-length parameter α:

log(τ /years) = ( 3.628 + 0.1337ζ)(M 1.9454) (4.6) ip − H − 2.080 0.353ζ+0.2 (Mm+α 1.5) 10− − − − 0.626 70.30 (Mc(1) ζ) ∆MH 10− − − , − where α is the mixing-length parameter (set to 1.75 in our models), ζ = log(Z/0.02), Mc(1) is the core mass at the first thermal pulse (taken from Table 4.1) and Mm is the mass contained in the convective mantlee, which is very well approximated by the envelope mass i.e. M M . The quantity ∆M is equal to the difference between the core m ' env H mass at the current and first thermal pulse i.e. MH(τ) Mc(1). Note that ∆MH grows with evolution along the TP–AGB. − In Figure 4.8 we show log τip from the Z = 0.02 models. In both panels, the black- dashed line corresponds to equation (4.5). In the upper panel, the black dots show the fit given by equation (4.6) over the same range in core mass as our models. In the lower panel, the black dots show the fit given by Izzard et al. (2003) (described below) plotted over the same range in core mass as our models. In the low-mass Z = 0.02 models, we observe that the interpulse period initially increases before decreasing as a linear function of the core mass. From Figure 4.8 we note that the fit given by Boothroyd & Sackmann (1988c) is a fit to the general trend of the data but little more. In particular, it does not reproduce the increase of log τip at the beginning of the TP–AGB phase, and under-estimates log τip toward the end of the TP–AGB. For models with M > 3M (or MH & 0.7M ) the interpulse period increases with increasing core mass (or time). From the upper panel of Figure 4.8 we see that the Wagenhuber & Groenewegen (1998) fit well approximates the increase of log τip at the beginning of the TP–AGB, and is a satisfactory fit to all models. However, equation (4.6) does not account for the largest increase in the interpulse period observed in the intermediate mass models. In the more massive models, the increase of τip with increasing core mass can be attributed to deep dredge-up. Izzard et al. (2003) use the detailed models presented here to modify the fit given by Wagenhuber & Groenewegen (1998) to include a dependency on the dredge-up parameter λ:

log(τ /years) = a (M b ) 10cs 10ds + 0.15 λ2, (4.7) ip s H − s − − ede®ned in Wagenhuber & Groenewegen (1998) as the region between the photosphere and the core CHAPTER 4. AGB EVOLUTION 86

Figure 4.8: The interpulse period as a function of the core mass. In each diagram, the coloured symbols show the interpulse period, log τip, as a function of the core mass for the Z = 0.02 models. The dashed-line corresponds to the core±mass±interpulse relation of Boothroyd & Sackmann (1988c) and the black dots the ®t given by Wagenhuber & Groe- newegen (1998) (upper panel) and Izzard et al. (2003) (lower panel). CHAPTER 4. AGB EVOLUTION 87

Table 4.4: The constants as and bs for equation (4.7). Constant 0.02 0.008 0.004 0.0001 as -3.821 -4.189 -4.255 -4.5 bs 1.8926 1.8187 1.8141 1.79

where as and bs are constants that depend on the metallicity, given in Table 4.4. The expressions cs and ds are taken directly from Wagenhuber & Groenewegen (1998):

c = 2.080 0.353ζ + 0.2 (M + α 1.5), (4.8) s − − env − and d = 0.626 70.30 (M (1) ζ) ∆M , (4.9) s − − c − H where α is the mixing-length parameter and ζ = log(Z/0.02). In the lower panel of Fig- ure 4.8 we see that equation (4.7) is an excellent fit for all models, even those with deep TDU and HBB. This time, there is a noticeable increase in the interpulse period for the even the most massive models. In Figure 4.9 we show the interpulse periods for the Z = 0.008 (upper panel) and Z = 0.004 (lower panel) models. In each diagram we include the Izzard et al. (2003) fit for the respective metallicity and the Z = 0.001 fit by Boothroyd & Sackmann (1988c). The Z = 0.001 fit given by Boothroyd & Sackmann (1988c) is remarkably good for the lower mass Z = 0.008 and Z = 0.004 models, but quite poor for the more massive models. Similar to the Z = 0.02 models, the Izzard et al. (2003) fit is excellent for all models, including the most massive models with very deep TDU and HBB. Nucleosynthesis from HBB depends on the interpulse period. Older synthetic AGB calculations, such as those by Groenewegen & de Jong (1993) and van den Hoek & Groe- newegen (1997), use the core-mass–interpulse-period relation of Boothroyd & Sackmann (1988c). More recent calculations, such as those by Marigo (2001), use the fit provided by Wagenhuber & Groenewegen (1998). The fit by Wagenhuber & Groenewegen (1998) is quite successful at reproducing the interpulse periods of the lower mass models, and provides a satisfactory fit for the intermediate mass models, but does not account for the largest increase in the interpulse periods. The synthetic calculations by Izzard et al. (2003) use a modified fit to the Wagenhuber & Groenewegen (1998) formula (equation (4.7)), which includes a dependency on the TDU efficiency, λ. This fit is the most successful at reproducing our models results, and it is not surprising then that the synthetic calculations of Izzard et al. (2003) produce surface abundance results that are consistent with our detailed calculations (they also use our prescription for the TDU given in Chapter 6, but modified so dredge-up begins at a smaller core mass).

4.3.2 Temperature at the Base of the Convective Envelope

The temporal variation of the temperature at the base of the convective envelope, Tb, is shown for two models in Figure 4.10. The details of this diagram are discussed below, CHAPTER 4. AGB EVOLUTION 88

Figure 4.9: The interpulse period, log τip, as a function of the core mass for the Z = 0.008 models (upper panel) and the Z = 0.004 models (lower panel). The ®t by Izzard et al. (2003) is shown as the black dots over the model results and the dashed line corresponds to the Z = 0.001 ®t by Boothroyd & Sackmann (1988c). CHAPTER 4. AGB EVOLUTION 89

but here we review the qualitative features. At the beginning of the TP–AGB, Tb does not change very much over the course of an interpulse phase, but steadily increases over many successive thermal pulses. From Figure 4.10 we see that Tb drops sharply during a thermal pulse but increases again very shortly afterward to reach a maximum value. The maximum during a thermal pulse is higher than the temperature reached during the interpulse for models without HBB (see the lower panel of Figure 4.10). For models with HBB, the temperature maximum during a thermal pulse is less than the maximum during the preceding interpulse phase (see the top panel of Figure 4.10). The temperature, Tb, is also dependent on the mass of the convective envelope, and drops sharply when the envelope is reduced in size by mass loss. Hot bottom burning becomes significant when the temperature at the base of the convective envelope exceeds about 50 million K. At this temperature, there can be significant 7Li production, along with an increase in the 14N abundance at the expense of 12C. The mass at which this occurs depends on the composition, and is 5M at Z = 0.02 and 4M at Z = 0.008 and Z = 0.004. Mild HBB can occur in models when∼ the temperature reaches about 30 million K at the base of the convective envelope. In this case the temperature is hot enough to produce some 7Li but not hot enough to effect the abundances of the CNO isotopes. In Figure 4.10 we show the temperature at the base of the convective envelope for the 5M , Z = 0.02 model (upper panel) and the 3.5M , Z = 0.004 model (lower panel). In each diagram the grey box indicates the region where HBB becomes significant. For the 5M , Z = 0.02 model, HBB begins after the 9th thermal pulse, when the temperature at the base of the envelope exceeds about 56 million K, and ends after the 20th thermal pulse, when the temperature falls below about 46 million K. The 3.5M , Z = 0.004 model experiences mild HBB after the 15th thermal pulse, when the temperature reaches about 33 million K, and HBB ceases when Tb falls after the 20th thermal pulse. The temperature at the base of the convective envelope is also important for the occurrence of the TDU. Marigo, Bressan & Chiosi (1998) include an extra parameter in their synthetic AGB algorithm to determine the onset of the TDU. If the temperature at the base of the convective envelope exceeds a specified value, dredge-up is assumed to occur regardless of the core mass. This idea was originally proposed by Wood (1981). He de- fines a variable, TN, to be the temperature at a hydrogen mass fraction of X = 0.1, at the time of the surface luminosity maximum following a thermal pulse. If the temperature at the base of the convective envelope does not exceed TN, dredge-up does not occur, while Tb = TN indicates the onset of dredge-up. Wood finds that TN approaches a limiting value of log TN = 6.7 0.1 for all core masses studied. In Tables 4.1 and 4.2 we give the temperature at the base of the convective envelope at the surface luminosity maximum during tdu the first TDU episode, Tbce. The Z = 0.02 models only experience the TDU when the temperature at the base of the envelope exceeds log T = 6.7, in agreement with Wood’s early results, whereas the Z = 0.008 and Z = 0.004 models experience the TDU when Tb exceeds log Tb 6.5. Marigo (1998b) requires log TN = 6.4 in her calculations to match the carbon-star ∼luminosity functions of the Magellanic Clouds, and in 6.2.4 we discuss how this effects the minimum core mass at the first TDU episode and the§ stellar yields. CHAPTER 4. AGB EVOLUTION 90

Figure 4.10: Temperature at the base of the convective envelope, Tb, as a function of time (in years) for the 5M , Z = 0.02 model (upper panel) and the 3.5M , Z = 0.004 model (lower panel). The shaded region corresponds to the temperatures required for signi®cant HBB. CHAPTER 4. AGB EVOLUTION 91

4.3.3 The Core–mass–luminosity Relationship Paczynski´ (1970) was the ﬁrst to derive a linear relationship between the maximum surface luminosity during the quiescent interpulse phase and the H-exhausted core mass

L/L = 59250 (MH 0.522). (4.10) − Paczynski’´ s calculations estimate that there is a maximum luminosity an AGB model can have, determined by the maximum possible core mass: the Chandrasekhar limit of 1.4M . At Mch = 1.4M , equation (4.10) results in a predicted maximum luminosity of 52021L∼ . Subsequent calculations (Iben 1977; Wood & Zarro 1981) verified Paczynski’´ s result and derive similar linear fits, the only difference being the numerical factors in equation (4.10). However, recent computations of intermediate mass AGB stars reveal that hot bottom burning violates the conditions of the core-mass luminosity relationship (Tuch- man, Glasner & Barkat 1983; Block¨ er & Shonberner¨ 1991; Lattanzio 1992), and that an AGB model can have luminosities far larger than predicted by Paczynski.´ This phenomena has been well studied by many authors including Block¨ er & Shonberner¨ (1991), Lattanzio (1992), Boothroyd & Sackmann (1992), Block¨ er (1995), Wagenhuber & Groenewegen (1998) and Herwig, Shonberner¨ & Block¨ er (1998). There is also evidence that deep third dredge-up effects the core–mass–luminosity relationship (Frost 1997; Marigo et al. 1999), even for low-mass AGB stars that do not undergo HBB (and are therefore expected to follow the linear core–mass–luminosity relationship). Frost (1997) finds large deviations from the linear MH L relationship in models that do not experience HBB, but have deep TDU. She concludes− that the luminosity on the AGB depends on more than just the core mass, MH, and more work needs to be done to establish just what these extra factors are. Herwig et al. (1998) find the same effect as Frost (1997), and conclude that the deep TDU in their models is causing a breakdown of the MH L relationship. Marigo et al. (1999) studied Herwig et al.’s models in more detail, and conclude− that the M L relationship H − is not violated by deep TDU. Instead, these authors argue that the change to the MH L relationship is caused by an increase in the mean molecular weight, µ, in the convecti−ve envelope, because material from the H-exhausted core is added to the envelope at each TDU event. The core–mass–luminosity relationship on the AGB is also a key ingredient in a synthetic AGB model because it determines many fundamental features of AGB evolution including the growth of the H-exhausted core with time. Up until recently, many synthetic models used MH L relationships that ignored any deviation (see for example, Groenewe- gen & de Jong; Marigo,− Bressan & Chiosi 1996) caused by HBB (and possibly the TDU). However the situation has changed with very accurate fits to the luminosity by Wagenhu- ber & Groenewegen (1998). These authors fit the luminosity maximum during the quiescent interpulse phase, which is the standard linear relationship for smaller core masses plus a correction for HBB at larger core masses. Marigo (1998a) uses the formula provided by Wagenhuber & Groenewegen (1998), but also performs detailed envelope integrations in her synthetic AGB models to solve for the radiated luminosity. In this way, she quite accurately accounts for the extra-luminosity provided by HBB. Marigo et al. (1999) argue CHAPTER 4. AGB EVOLUTION 92

Figure 4.11: Maximum pre-pulse luminosity as a function of the core mass for a selection of the Z = 0.02 (upper panel) and Z = 0.004 models (lower panel). The ®t used by Marigo et al. (1996) is shown as the solid black dots over the model results. CHAPTER 4. AGB EVOLUTION 93

Figure 4.12: Maximum pre-pulse luminosity as a function of the core mass for a selection of the Z = 0.02. The ®t of BlÈocker (1993) for low-mass models is shown as the solid black dots over the model results. that the synthetic TP–AGB calculations of Groenewegen & de Jong (1993) and Marigo et al. (1996) use non-linear MH L relationships that include substantial deviations owing to the first few thermal pulses, and− changes to the surface composition caused by dredge-up (by including a dependence on the mean molecular weight). Our models experience both deep dredge-up and HBB, so we can make a comparison between our model results and some of the linear and non-linear MH L relationships in the literature. We make a comparison to three different type of fits. The− first type are the M L relationships used in older synthetic calculations, the second type is a non-linear H − MH L relationship with corrections for HBB but derived from models without the TDU, and −the third type is also a non-linear fit but derived from our detailed models. We compare to:

1. the MH L relationships used by Marigo et al. (1996), which is derived from the results of−Boothroyd & Sackmann (1988b) and Groenewegen & de Jong (1993); and the linear core–mass–luminosity relationship of Block¨ er (1993) given in Herwig et al. (1998) for low-mass models.

2. The non-linear, metallicity dependent ﬁt by Wagenhuber & Groenewegen (1998) which provides a correction for HBB.

3. The non-linear M L relationship provided by Izzard et al. (2003). H − CHAPTER 4. AGB EVOLUTION 94

Figure 4.13: Same as Figure 4.11 but using the ®t given by Wagenhuber & Groenewegen (1998). CHAPTER 4. AGB EVOLUTION 95

We begin with a discussion and comparison of the formulae used by Marigo et al. (1996). These authors use the core–mass–luminosity relationship given by Boothroyd & Sackmann (1988b) for core masses below 0.66M

3 0.04 2 L/L = 238000 µ Zcno (MH 0.0305MH 0.1802), (4.11) − − where µ is the mean molecular weight in the envelope (which is assumed to be ionized) given by µ = 4/(5X + 3 Z) and X is the mass fraction of hydrogen. Equation (4.11) − also includes a dependence on the total CNO abundance in the envelope, given by Zcno, which is the mass fraction of the carbon, nitrogen and oxygen isotopes. For core masses above 0.95M , Marigo et al. (1996) use the relationship between core mass and luminosity derived by Iben & Truran (1978) and modiﬁed by Groenewegen & de Jong (1993)

2 0.19 L/L = 122585 µ (MH 0.46) Mtot , (4.12) − where Mtot is the total mass. For core masses between 0.66 6 MH 6 0.95, a linear interpolation is adopted. In Figure 4.11 we show the maximum luminosity from each interpulse period for a selection of our Z = 0.02 (upper panel) and Z = 0.004 models (lower panel). We also plot equations (4.11) and (4.12) over the same core mass range as the models (solid black dots). Whilst this ﬁt does not account for the luminosity increase during the early pulses, it is nevertheless satisfactory for smaller core masses, where MH . 0.7M . Block¨ er (1993) derived a linear M L relationship from low-mass Z = 0.02 models H −

L/L = 62200 (MH 0.487), (4.13) − valid in the core mass range 0.55 . MH/M . 0.80. For core masses > 0.95M we use equation (4.12), and in the range 0.80 6 MH/M 6 0.95 we linearly interpolate between the two relationships. In Figure 4.12 we sho w the maximum luminosity from each interpulse period for a selection of our Z = 0.02 against the fit (solid black dots) by Block¨ er (1993) for low-mass models and equation (4.12) for massive models. Similar to the Boothroyd & Sackmann (1988b) fit, the formula given by Block¨ er (1993) is very good at reproducing the MH L relationship for the low-mass solar composition models. Both the Wagenhuber− & Groenewegen (1998) and Izzard et al. (2003) formulae make an allowance for HBB and the luminosity during the first few thermal pulses, which do not follow the linear MH L relationship owing to a substantial contraction of the core. Wagenhuber & Groenewe−gen (1998) fit the maximum pre-pulse luminosity using a linear core–mass–luminosity relation plus a correction for HBB

L/L = (18160 + 3980 Z) (MH 0.4468) (4.14) − +10(2.705+1.649 MH) 2 2 (0.0237 (α 1.447)Mc(1) Menv[1 exp ( ∆MH/0.01)]) 10 − − − × (3.529 (Mc(1) 0.4468) ∆MH/0.01) 10 − − , − CHAPTER 4. AGB EVOLUTION 96

Figure 4.14: Same as Figure 4.11 but using the ®t given by Wagenhuber & Groenewegen (1998) with α = 2.5 for the Z = 0.02 models (upper panel) and α = 2.75 for the Z = 0.004 models (lower panel). CHAPTER 4. AGB EVOLUTION 97

Figure 4.15: Same as Figure 4.11 but using the ®t given by Izzard et al. (2003). CHAPTER 4. AGB EVOLUTION 98

where α is the mixing-length parameter, Mc(1) is the core mass at the first thermal pulse and ∆MH is the difference between the core mass at the current and first thermal pulse. In Figure 4.13 we compare equation (4.14) to our Z = 0.02 and Z = 0.004 models. The fit is good for small core masses (MH . 0.7) but poor for the more massive models, and does not account for the luminosity increase in the models with HBB. Equation (4.14) also fails to predict the range of luminosities observed in intermediate mass models with deep TDU but no HBB. The luminosity increase observed in the models with HBB strongly depends on the mixing-length parameter. Wagenhuber & Groenewegen (1998) comment that they obtain a “remarkably good fit” to the intermediate mass models of VW93 if they increase α to 2.25 (VW93 used 1.6). If we increase α from 1.75 to 2.5, equation (4.14) results in an excellent fit for all the Z = 0.02 models (see the upper panel of Figure 4.14), and if we increase α to 2.75, equation (4.14) results in an excellent fit for many of the Z = 0.004 models (see the lower panel of Figure 4.14). Hence increasing α not only allows for the luminosity increase caused by HBB, but also accounts for the luminosity of models with deep TDU. Izzard et al. (2003) fit the maximum pre-pulse surface luminosity as the sum of the core–mass–luminosity relationship plus a term for HBB:

L/L = fd( ftLCMLR + Lenv), (4.15)

where Lenv is the luminosity from envelope burning

4 2 2 Lenv/L = max 1.5 10 Menv(MH + 0.5∆MH,nodup 0.75) , 0 , (4.16) × − where ∆MH,nodup is the amount by which the core would grow from the start of the TP– AGB to the present pulse if there was no dredge-up. When we plot equation (4.15), we set ∆M equal to (M (τ) + λ∆M ) M (1). The core-mass-luminosity relationship, H,nodup H H − c LCMLR is given by

LCMLR = max a ( max[(MH b)(c MH) , 1.2 (MH 0.48) ] ) , 10 , (4.17) h − − − i where a = 3.7311 104, b = 0.52659, c = 2.7812, and L is in solar units. The × CMLR expressions fd and ft are given by

f = 1 0.21807 min 1.0, exp [ 11.613(M 0.56189)] , (4.18) d − − H − and 0.2 ∆MH,nodup f = min , 1 . (4.19) t  0.04 !    Izzard et al. (2003) include ft to account for the early thermal pulses, before the TP–AGB phase has reached the full-amplitude stage. Equation (4.15) is a reasonably good fit for all of the Z = 0.02 models (see the upper panel of Figure 4.15) and a reasonable fit for most of the Z = 0.004 models (see the lower panel of Figure 4.15). The luminosity at the very first thermal pulse is greatly under-estimated by the equation (4.15). Whilst some of the luminosity increase caused by envelope burning is accounted for, the fit does not reproduce CHAPTER 4. AGB EVOLUTION 99 the luminosities of the most massive, lowest Z models. We suggest that equation (4.15) needs an explicit metallicity dependence to better fit these models. In summary, the non-linear MH L relationships in the literature are quite successful at reproducing our model results. The− fit given by Izzard et al. (2003) is the most accurate (assuming no modification by increasing α) at reproducing the Z = 0.02 models, but requires further modification at lower metallicity. The Wagenhuber & Groenewegen (1998) is the best fit if we substantially increase the mixing-length parameter from the value used in the stellar models (1.75) to α 2.5. ≈

4.4 Summary

The results of the AGB calculations presented here are consistent with other stellar models published in the literature, including those by Straniero et al. (1997), Block¨ er (1995) and the older calculations of Iben (1977), Lattanzio (1986) and Boothroyd & Sackmann (1988b,c,d). The intermediate mass models experience deeper TDU than the models of Straniero et al. (1997); but there is no TDU in our 1.5M , Z = 0.02 model whereas Straniero et al. (1997) find enough dredge-up to turn their 1.5M , Z = 0.02 with mass loss into a carbon star. Both the core mass at the first TDU episode and the efficiency of the TDU are metallicity dependent. At a given mass, the deepest TDU is obtained in the lowest metallicity model. There is very little difference between the efficiency of the TDU for the M & 3M models of different compositions. There is efficient hot bottom burning in all models with M > 5M , and mild HBB in the 3.5M , Z = 0.004 model. At a given mass, a reduction in the initial metallicity results in a decrease in the minimum mass for HBB. HBB ceases after the mass of the convective envelope is reduced below about 1.5M , and all the LMC and SMC models with HBB become carbon stars. During a thermal pulse, the duration and maximum mass of the convective pocket is a strong function of the core mass, and only mildly dependent on the initial composition. The maximum temperature in the He-shell increases with evolution along the AGB, a result that is not predicted by the Z = 0.02 fits for the He-shell temperature by Iben (1977) and Malaney & Boothroyd (1987). These relationships predict the range of He- shell temperatures observed in the 3M , Z = 0.02 model but not in the 6.5M , Z = 0.02 model. The reason for the failure is probably deep TDU, where λ 1, so the core mass increases only slightly over the duration of the TP–AGB phase. Compared∼ to the results of Boothroyd & Sackmann (1988c), the He-shell temperatures are higher in our models. There are many possible reasons for this including our use of the OPAL opacities, and a different treatment of convective mixing. Secondary convective pockets are observed to develop as a result of the secondary He- shell flash following the first and strongest He-shell flash. These secondary convective pockets only develop with the first few thermal pulses of the low-mass models (M < 2.5M ) and occur at all the initial compositions considered. The secondary luminosity oscillations have been reported previously in the literature (notably by Iben 1982), but the secondary convective pockets have not. Whilst these secondary pockets are interesting CHAPTER 4. AGB EVOLUTION 100 because they are new, we believe they have no effect on the surface abundances during the TP–AGB phase. We compare the core–mass–interpulse period and core–mass–luminosity relationships in the literature to our AGB models. These quantities are important for synthetic AGB calculations, because they effect the evolution along the AGB and hence the stellar yields. The core–mass–interpulse period relations of Boothroyd & Sackmann (1988c), Wagenhu- ber & Groenewegen (1998) and Izzard et al. (2003) are compared to our model results, with very good agreement for all models for the latter two fits. The fit by Wagenhuber & Groenewegen (1998) does not account for the largest increase in the interpulse periods observed in the intermediate mass models, possibly because these masses experience very deep TDU. Similar trends exist for the maximum pre-pulse surface luminosity. There are many fits for the MH L relation in the literature, including the relationships derived by Pacynski´ (1970) and −later by Boothroyd & Sackmann (1988b) and Block¨ er (1993), and the non-linear relationships derived by Wagenhuber & Groenewegen (1998) and Izzard et al. (2003). The linear MH L relationships are very good at reproducing the luminosities in the lower mass models,−but they do not account for the luminosity increase caused by HBB. The non-linear fit of Wagenhuber & Groenewegen (1998) is an excellent fit for all models if we increase the mixing-length parameter, but less so otherwise. The formula of Izzard et al. (2003) is the best fit to the stellar models if no modifications are made. The fit for the Z = 0.02 models is satisfactory, but the luminosities observed in the Z = 0.004 HBB models are not reproduced. For this reason, we believe that the formula of Izzard et al. (2003) would benefit from the addition of an explicit dependence on the initial composition. Chapter 5

Asymptotic Giant Branch Nucleosynthesis

In this chapter, we continue our discussion of the mass-loss models presented earlier in Chapters 3 and 4. In this chapter we review the nucleosynthesis that occurs during the TP–AGB phase, with an emphasis on the production and destruction mechanisms for the Ne, Na, Mg and Al isotopes. We do not discuss the nucleosynthesis that occurs in Z = 0 models but refer the reader to Chieﬃ et al. (2001) and Siess, Livio & Lattanzio (2002). Parts of this chapter are taken from the paper “Production of aluminium and the heavy magnesium isotopes in asymptotic giant branch stars”, by Karakas & Lattanzio, accepted for publication in the Publications of the Astronomical Society of Australia.

5.1 Introduction

In recent years our attempts to understand many aspects of nucleosynthesis and stellar evolution have come to rely on our understanding of the production of the magnesium and aluminium isotopes. For example, abundance anomalies in globular cluster stars have been a problem for many years, and the role of Mg and Al is central and far from understood (Shetrone 1996; Kraft et al. 1998; Yong et al. 2003). At the heart of this problem is the quest for the origin of the Mg and Al anomalies: are they produced in the star itself, and mixed to the surface by some form of deep mixing (Denissenkov & Weiss 1996) or are they the result of pollution from an earlier generation of stars? The latter would seem to implicate asymptotic giant branch (AGB) stars (Denissenkov et al. 1998; Ventura et al. 2001), where Mg and Al can be produced by thermal pulses (and mixed into the envelope by the subsequent third dredge-up, hereafter TDU) and hot bottom burning (HBB). Although all isotopes of magnesium are produced by massive stars, low-metallicity supernovae models mostly produce 24Mg, with very little 25Mg and 26Mg synthesized. Only when the initial metallicity of the model reaches about [Fe/H] 1 is there substantial 25Mg and 26Mg production (Fenner et al. 2003). Observations of∼the−Mg isotopic ratios in metal-poor stars (Gay & Lambert 2000) show that there is more of the neutron-rich Mg isotopes in these stars than expected from detailed chemical evolution models using su-

101 CHAPTER 5. AGB NUCLEOSYNTHESIS 102 pernovae yields alone (Timmes, Woosley & Weaver 1995). Other possible sources of the neutron-rich magnesium isotopes include the winds from Wolf-Rayet (WR; Maeder 1983; Woosley, Langer & Weaver 1995) and AGB stars (Forestini & Charbonnel 1997; Siess et al. 2002). There are currently no quantitative studies of the production of the neutron-rich Mg isotopes in low-metallicity WR stars. There are quantitative studies of magnesium production in low-metallicity AGB stars (Forestini & Charbonnel 1997; Siess et al. 2002) but these studies do not cover a sufficiently large range of mass or composition to produce yields suitable for galactic chemical evolution models. Further, the yields of Forestini & Charbonnel (1997) involved extrapolations from parameterized “synthetic” models, rather than being the result of detailed stellar evolutionary calculations. For this reason, a quantitative estimate of the production of the neutron-rich Mg isotopes from detailed AGB models of different mass and metallicity is the main aim of this chapter. We also examine the production of 22Ne in AGB stars, and compare the results from our models with observations of elemental neon in planetary nebulae. Magnesium is processed together with aluminium through the Mg–Al chain, and hence a discussion of one requires a discussion of the other. The main product of the Mg–Al 26 26 chain is the unstable Al, which has a half-life of τ1/2 = 710, 000 years. The decay of Al into 26Mg results in the emission of 1.809 MeV photons, which can be used to probe the spatial distribution of the stars responsible for the 26Al production (Chen, Gehrels & Diehls 1995). The COMPTEL satellite mapped the distribution of 26Al in the Galaxy, and the future INTEGRAL satellite will reveal details of that map (Prantzos 1998). Whilst most of the 26Al observed in the Galaxy today probably originated in young massive WR stars (Prantzos 1993) contributions from other sources such as classical novae (Jose´ & Hernanz 1998) and low and intermediate mass AGB stars might be important (Meynet 1994). The production and destruction of 26Al in AGB stars has been discussed in detail by Mowlavi & Meynet (1999). These authors find that HBB in massive AGB stars could be an important source of 26Al. Nollett, Busso & Wasserburg (2003) recently studied parameterized extra-mixing processes in low-massa AGB models. They find that, depending on the mixing parameters used, 26Al can be produced in sufficient amounts to explain the 26Al/27Al ratio inferred to have been present in some circumstellar oxide grains at the time of their formation. We begin with a discussion of the nucleosynthesis that results from the third dredge-up.

5.2 Nucleosynthesis Resulting From the Third Dredge- Up

There are three sites of nucleosynthesis in AGB stars: the hydrogen-burning shell (H- shell), the helium-burning shell (He-shell) and the base of the convective envelope (HBB). The third dredge-up changes the surface composition of models by mixing the products of the H and He-burning shells to the surface. The TDU is the mechanism by which the

ahereafter low-mass refers to models with M 6 2.25M and intermediate refers to models with M > 2.25M

CHAPTER 5. AGB NUCLEOSYNTHESIS 103

(p,γ ) (p,γ ) (p,γ ) 21Ne 22Na 25Mg 26Al 27Si

(β+) (β+) (β+) (β+)

+ 21Na 22Ne 25Al 26Mg (β )

(p,γ ) (p,γ ) (p,γ ) (p,γ )

20 23 24 27 28 19F Ne Na Mg Al Si (p,γ ) (p,α) (p,γ ) (p,α) (p,γ )

Ne−Na Mg−Al

Figure 5.1: Reactions of the Ne±Na and Mg±Al chains. Unstable isotopes are denoted by dashed circles.

surface of single stars becomes enriched in carbon and s-process elements (Iben & Renzini 1983), and is discussed in Chapters 4 and 6. In this section we review the nucleosynthesis that occurs in the H and He-burning shells.

5.2.1 The Hydrogen-burning Shell Hydrogen is burnt to 4He via the CNO cycles, but the Ne–Na and Mg–Al chains also operate. The main result from the CNO cycle, besides the conversion of H to He, is an increase in the abundance 13C and 14N from the destruction of other CNO species. The isotopes 12C and 15N are first destroyed by the CN cycle, which comes into equilibrium quickly. Later, the oxygen isotopes 16O and 18O are also destroyed to produce 14N. The abundance of 17O can be enhanced by the CNO-bi cycle, depending on the uncertain rate of the 17O + p branching reactions. The H-shell ashes are important because they affect the nucleosynthesis that occurs in the He-shell. This point is discussed further in 5.2.2. In the left part of Figure 5.1 we show the reactions of the Ne–Na chain (Arnould,§Goriely & Jorissen 1999; Rolfs & Rodney 1988), where unstable isotopes are denoted by dashed circles. The main result of the Ne–Na chain is the production of 23Na at the expense of the neon isotopes, primarily 22Ne, which begins to be destroyed at about 20 million K. The production of sodium by the Ne–Na chain has been examined in detail by Mowlavi (1999a), who predicted that AGB stars could play an integral role in the chemical evolution of sodium in the Galaxy. The rare neon isotope, 21Ne, can be substantially enhanced by an unnoticeable destruction of 20Ne at temperatures below 40 million K, but at temperatures above this 21Ne is completely destroyed. 20Ne is slightly destroyed at temperatures above 50 million K, but the destruction of 23Na at temperatures over about 80 million K can lead to a slight enhancement in the 20Ne abundance (Arnould et al. 1999). The rate by CHAPTER 5. AGB NUCLEOSYNTHESIS 104 which 23Na is destroyed is important for determining sodium yields. Current rates from the NACRE compilation (Angulo et al. 1999), suggest 23Na can be destroyed when T & 60 million K. Whether there is leakage out of the Ne–Na chain into the Mg–Al chain depends on the relative rates of the 23Na(p,α)20Ne and 23Na(p,γ)24Mg reactions. The NACRE rates favour the latter reaction at temperatures below about 80 million K, but at hotter temperatures the reaction, 23Na(p,α)20Ne, is faster. Magnesium and aluminium are altered in the H-burning shell via the activation of the Mg–Al chain, which begins operation at temperatures of about 30 million K (Arnould et 26 26 al. 1999). This involves the radioactive nuclide Al which has a ground state Alg that has to be considered a separate species from the short-lived (τ1/2 = 6.35 s) isomeric state 26 Alm, since they are out of thermal equilibrium at the relevant temperatures (Arnould et 26 26 al. 1999). Hereafter, when we refer to Al we are referring to the ground-state, Alg. In the right part of Figure 5.1 we show the reactions involved in the Mg–Al chain (Arnould et al. 1999; Rolfs & Rodney 1988). The first isotope in the Mg–Al chain to be affected is 25Mg, which is burnt to 26Al. The lifetime of β–decay relative to proton capture generally favours proton capture within the H-burning shell. This produces the unstable 27Si which β–decays (with a lifetime on the order of a few seconds) to 27Al. The rate of 26Mg + p is slow until the temperature reaches about 60 million K, which means that there are only small reductions in 26Mg in the H-shells of most AGB models. The abundance of 26Mg is enhanced by the β–decay of 26Al in the H-shell ashes. Proton capture on 24Mg requires higher temperatures than those required for the other reactions in the Mg–Al chain. For that reason most of our models show little change in the abundance of this isotope due to the slow rate of proton capture at temperatures below about 70 million K. The lowest temperatures in the H-burning shells of our AGB models are just over 40 million K for the 1M , Z = 0.02 model, and the highest temperatures are about 98 million K for the 6M , Z = 0 .004 model. Thus even in the lowest mass models the Ne–Na and Mg– Al chains are active, depleting 22Ne and 25Mg, and producing 23Na and 26Al. However, the operation of the TDU is required to mix the products of the H-shell to the surface. From Chapter 4 we know that the TDU only occurs in models with M > 2M at Z = 0.02 and M > 1.25M at Z = 0.008 and Z = 0.004. Thus there are no surface ab undance changes during the TP–A GB phase for less massive models. In the low-mass models, the change to the surface abundance of the Na, Mg and Al isotopes with efficient TDU is as follows. There is a very small depletion in the abundance of 25Mg and a slight increase in 26Mg, 26Al and 27Al. The 24Mg abundance remains unchanged. The 23Na abundance increases after each dredge-up episode, but the increase over the entire TP–AGB phase is small compared to the increase observed in models with HBB. We demonstrate the effect of H-burning nucleosynthesis in Figure 5.2. In the top panel of Figure 5.2 we show the composition profile of the 1.5M , Z = 0.004 model just before the 14th thermal pulse, showing the ashes of the H-burning shell. The shaded region denotes the convective envelope. The maximum extent of the convective pocket during the 14th thermal pulse is indicated on the diagram. In the bottom panel of Figure 5.2 we show the composition profile at the maximum extent of the TDU, after the pulse. The composition of 26Al in the intershell has been homogenized by the convective pocket, but is not destroyed by neutron capture because the 22Ne neutron source has not been activated CHAPTER 5. AGB NUCLEOSYNTHESIS 105

Figure 5.2: Composition pro®le for the 1.5M , Z = 0.004 model just before the 14th thermal pulse (top panel) and at the maximum e xtent of the following dredge up episode (lower panel). CHAPTER 5. AGB NUCLEOSYNTHESIS 106

Figure 5.3: The intershell abundances for the 4M , Z = 0.008 model. We show the abundance of 22Ne (black plus signs), 25Mg (red open circles) and 26Mg (blue closed circles) as a function of pulse number. In this diagram we show the intershell abundances for the 15th to the 20th pulse, but only during the time when the convective shell is present; the x±axis is the (scaled) duration of the convective pocket. at this low mass. After the next dredge-up episode, the surface abundance of 26Al has 26 27 3 increased by about 30%, and the Al/ Al ratio is 4 10− . In conclusion, the operation of the H-shell in lo∼w-mass× models is quantitatively unim- portant to the production of the Mg isotopes. Some 23Na and 26Al is produced in low-mass, low-metallicity AGB models, but this conclusion suﬀers from many uncertainties. In the next two sections we show that the operation of the He-burning shell and HBB are much more important in intermediate mass AGB models than the H-burning shell.

5.2.2 The Helium-burning Shell The He-burning shell in AGB stars is a rich source of nucleosynthesis. The main result is the production of 12C, which when mixed to the surface may produce N-type carbon stars (Becker & Iben 1980; Lattanzio 1986; Straniero et al. 1997). There is also a wealth of other He-burning products such as 22Ne, 25Mg and 26Mg (Forestini & Charbonnel 1997), plus species produced through the combined operation of the He and H-burning shells such as 19F (Forestini et al. 1992), and 23Na (Mowlavi 1999a). S-process elements are also created in the intershell region, where free neutrons are released by the reactions 13C(α,n)16O and 22Ne(α,n)25Mg. Substantial 22Ne is created during a thermal pulse by α–capture onto the 14N left by the H-burning shell during the preceding interpulse period. If the temperature exceeds about 300 million K, 25Mg and 26Mg are produced in substantial quantities by α–capture onto 22Ne via the reactions 22Ne(α,n)25Mg and 22Ne(α, γ)26Mg. In Figure 5.3 we show the time CHAPTER 5. AGB NUCLEOSYNTHESIS 107 variation of the intershell abundances of 22Ne, 25Mg and 26Mg for the 4M , Z = 0.008 model for five thermal pulses. This model experiences very deep dredge up, which begins following the third thermal pulse. The abundance for each species initially decreases due to the growth of the convective shell into the region previously processed by the H-shell. At the end of the preceding interpulse phase this region has been depleted in 22Ne, 25Mg and 26Mg by H burning at temperatures near 80 million K. The thermal pulse increases the temperature in the region, allowing successive α–captures onto 14N to first produce an increase in the 22Ne abundance, followed by an increase in 25Mg and 26Mg when the temperature reaches 300 million K. For the 4M , Z = ∼0.008 model, only about 20% of the 22Ne created during the thermal pulse is converted into 25Mg and 26Mg. In the most massive models, up to 75% of the 22Ne created during the thermal pulse is converted to the heavy Mg isotopes. For example, in the last few pulses of the 6M , Z = 0.004 model, as much as 60 to 75% of the 22Ne produced by the pulse is converted to Mg. In this model, the 22Ne mixed into the envelope is further destroyed by HBB. Although the massive AGB stars can produce the heavy Mg isotopes, the lower mass stars should produce 22Ne in substantial quantities. Neon abundances can be reliably measured in planetary nebulae (PNe) so observations of these objects can be used as a probe of our knowledge of AGB nucleosynthesis. We discuss this further in 5.5. If we examine the abundance of matter in the intershell from standard models (with-§ out convective overshooting) we observe (see Figure 5.3) that the mass fraction of 22Ne is . 2%, making it the third most abundant isotope, after 4He and 12C. The abundance of 16O is usually much lower, with the maximum value of the order of 1%. We can compare these values to the surface abundance observations of PG 1159–type objects, which are thought to be in transition from central star of PNe to white dwarf (Napiwotzki 1998). These objects are quite rare, with only about two dozen known, and their atmospheres are mostly composed of helium, carbon and oxygen (Werner & Rauch 1994). Spectroscopic observations of the PG 1159 central stars reveal oxygen mass fractions as high as 20% (Werner & Rauch 1994), clearly at odds with standard stellar models. Spectroscopic observations of neon reveal mass fractions of about 2 to 5% (Werner & Wolff 1999), consistent with the models. To reproduce the large oxygen abundances observed, extra-mixing processes are required to bring 16O from the CO core into the intershell region. The diffusive convective overshooting models of Herwig (2000) have intershell abundances that are consistent with the abundance patterns observed in PG 1159 central stars. The degenerate thermal pulses found by Frost, Lattanzio & Wood (1998) could also have a similar effect. In this case, the third dredge-up following the degenerate pulse can mix material from the CO core into the envelope, thus enhancing the envelope in 16O. Whilst this idea is certainly promising, a quantitative study is required. The relative amount of 25Mg to 26Mg produced in the He-shell is also dependent on the other nuclear burning sites: in the hottest H-shells, 25Mg can be substantially depleted compared to 26Mg. Since the ashes of the H-burning shell are engulfed by the next thermal pulse, the pre-pulse abundance of the two heavy Mg isotopes will influence the nucleosynthesis during the subsequent thermal pulse. For example, in the 6M , Z = 0.004 model, the abundance ratio of 25Mg/26Mg can be as low as 0.2 at the beginning of a thermal pulse (c.f. the initial ratio 25Mg/26Mg 0.9). For this model, even though the temperature in the ∼ CHAPTER 5. AGB NUCLEOSYNTHESIS 108

Figure 5.4: Composition pro®le for the 3M , Z = 0.004 model just before the 23rd thermal pulse (top panel) and during the 23rd thermal pulse (lower panel). The maximum temperature in the He-shell for this pulse was just over 330 million K. CHAPTER 5. AGB NUCLEOSYNTHESIS 109

He-shell favours the production of 25Mg over 26Mg, we still find that 25Mg/26Mg 0.65 just prior to the TDU. ∼ Temperatures exceed 300 million K in the He-shell of models with M & 3M , depending on the composition. From Figure 4.5 on page 80 we see that the temperature in the He-shell of the 3M , Z = 0.02 model does not reach 300 million K until the very last calculated pulse, whereas this temperature is reached in the He-shell of the 3M , Z = 0.004 after the eighth thermal pulse. In Figure 5.4 we show the composition profile for the 3M , Z = 0.004 model just prior to (top panel) and during (bottom panel) the 23rd thermal pulse. This model began dredge up after the second thermal pulse and by the third thermal pulse λ 0.5. This diagram demonstrates the large increase in 25Mg and 26Mg in the intershell when∼ the temperature exceeds 300 million K. During a thermal pulse, the reaction 22Ne(α,n)25Mg will produce a short burst of neu- 10 3 trons. The neutron density from this reaction is higher (nn 10 cm− ) than the neutron density produced from the 13C(α,n)16O reaction (Busso, Gallino∼ & Wasserburg 1999), which operates during the interpulse at much lower temperatures (T 1.5 108 K). The theoretical and observational evidence strongly suggests that the main∼neutron× producing reaction in low-mass AGB stars is the 13C(α,n)16O reaction, and that the 22Ne neutron source is only marginally activated (Gallino et al. 1998). This is consistent with no Mg production in low-mass stars, and efficient Mg production in the most massive AGB models, consistent with our results. For the 13C neutron source to operate efficiently, models require more 13C in the intershell than is left behind by the H-burning shell. However, a large abundance of 13C could be achieved locally if some protons are mixed from the envelope into the 12C-rich intershell region, allowing for the reaction 12C(p,γ)13N(β+)13C to occur. The 13C-rich region, or 13C pocket, will only occupy a small part of the intershell region, estimated to be 1/20th of the mass involved in a thermal pulse (Gallino et al. 1998). It is important that there are not too many protons, otherwise further proton capture onto 13C produces 14N, which will capture any free neutrons and therefore inhibit the formation of s-process elements. The exact nature of how the 13C pocket forms is still unknown, but it is believed that protons are somehow mixed into the intershell during the deepest extent of the convective envelope during the third dredge-up, but as Gallino et al. (1998) point out, the details will require hydrodynamical multi-dimensional modelling. Travaglio et al. (2001a,b) show that the more massive AGB stars only play a minor role in the production of the s-process elements from Ba to Pb. The reason given by Travaglio et al. is that the interpulse periods (and hence the intershell masses) are at least an order of magnitude smaller in these models than in the lower mass objects. Consequently the amount of s-processed material mixed into the envelope by the TDU is also reduced by the same factor. If intermediate mass AGB stars do not efficiently produce s-process elements, then observations of massive AGB stars should reveal non-solar Mg isotopic ratios with either no s-process elements or only weak absorption lines. If there are s-process elements present in the spectra, these elements should be neutron rich, and we should expect more rubidium to strontium (Lugaro, private communication). This test is difficult because the distance, and hence the luminosities and masses, are still very uncertain for many galactic AGB stars. The distances to the Magellanic Clouds are quite well known, but even the most luminous and massive AGB stars are too faint to separate the Mg isotopes in all but CHAPTER 5. AGB NUCLEOSYNTHESIS 110 the largest telescopes available today. In conclusion, the He-shell is the most important production site of 22Ne and the heavy magnesium isotopes, but the amount produced depends in a complicated way on the temperatures in the He-shell and the abundance of matter left by the H-shell at the end of the interpulse phase. Sodium and the aluminium isotopes are not produced in the He- shell. The isotope 26Al is depleted by neutron capture by the reactions 26Al(n,p)26Mg and 26Al(n,α)23Na, where the neutrons come from the 22Ne(α,n)25Mg reaction. The abundances of 23Na and 27Al are not altered at the temperatures found in the He-shells of AGB stars.

5.3 Nucleosynthesis Resulting from Hot Bottom Burn- ing

For AGB models with M & 4M , depending on the composition, the bottom of the convective envelope reaches into the top of the H-burning shell. H-burning occurs primarily via the CNO cycles, but also via the Ne–Na and Mg–Al chains if the temperature is high enough. This site then becomes important for the production of many elements, including nitrogen (Frost et al. 1998; Chieﬃ et al. 2001), lithium (Travaglio et al. 2001c) and sodium (Mowlavi 1999a) as well as magnesium and aluminium (Mowlavi & Meynet 2000). The region in the envelope that is hot enough for H-burning is quite thin, but owing to eﬃcient mixing, where the convective turnover time scale is of the order of one year, the matter in the entire envelope passes through the hot region at least 1000 times during every thermal pulse cycle. This means that the CN cycle operates in equilibrium after a few interpulse periods, reducing the 12C/13C ratio from the pre-AGB value near 20 to the equilibrium value of about four. Hot bottom burning also prevents carbon star formation (Boothroyd, Sackmann & Ahern 1993), because the 12C mixed into the envelope by the TDU is burnt mostly to 14N.

5.3.1 The Creation of 7Li via HBB The observations of Wood, Bessell & Fox (1983) were the ﬁrst to suggest that the oxygen- rich luminous AGB stars in the Magellanic Clouds are undergoing CNO cycling at the base of the convective envelope, converting the dredged up carbon to nitrogen. The discovery of the existence of Li-rich giants in the Magellanic Clouds by Smith & Lambert (1989, 1990) and Plez et al. (1993) gave further credibility to the idea that HBB was actually occurring in massive AGB stars. The production of 7Li is thought to occur via the Cameron–Fowler mechanism (Cameron & Fowler 1971), which we summarize. Some 3He, created earlier in the evolution, captures an α–particle to create 7Be. The 7Be can either 1) capture a proton to complete the PPIII chain or 2) capture an electron to produce 7Li. Whether the 7Be follows path 1) or path 2) depends critically of the temperature of the region. Owing to eﬃcient mixing in the convective envelope, some of the 7Be is mixed into a cooler region which prevents proton capture. The 7Be can still undergo electron–capture in this cooler region, producing 7Li. The 7Li is also subject to proton capture. From the CHAPTER 5. AGB NUCLEOSYNTHESIS 111

Figure 5.5: The time variation of the envelope 7Li abundance for three different Z = 0.004 models. (Top) the 3.5M model, (middle) the 4M model and (lower) the 6M model. In each panel, the grey shaded region indicates the Li-rich region, which approximately corresponds to log ε(Li) & 3. CHAPTER 5. AGB NUCLEOSYNTHESIS 112 above description it should be clear that time-dependent mixing is required to produce 7Li in a HBB calculation. This is because the nuclear timescale for the reactions involved in the Cameron–Fowler mechanism are similar to the convective turnover timescale (see Figure 2 in Sackmann & Boothroyd 1992). Indeed, many calculations which include time- dependent convective mixing find large 7Li enhancements during the AGB phase (Block¨ er & Schonberner¨ 1991; Sackmann & Boothroyd 1992; Frost et al. 1998; Ventura et al. 2002). The super-Li rich phase is destined to be relatively short because eventually the star runs out of 3He, and the 7Li in the envelope is mixed into a region hot enough for proton capture. How long the 7Li enhanced phase lasts will depend on the temperature at the base of the envelope. In Chapter 4 we comment that models with “mild” HBB (such as the 3.5M , Z = 0.004 model) have convective envelopes that are just hot enough to produce some 7Li. The temperature required for 7Li production is reached after 15 thermal pulses. The consequence of is that the mild HBB is shut off before all the 3He is burnt, leaving an envelope enriched in 7Li. In the more massive models, the situation is quite different. The temperature at the base of the convective envelope is much hotter, so the 3He is burnt at a faster rate. This means that the envelope becomes enriched in 7Li near the beginning of the HBB phase, but the supply of 3He is exhausted before HBB ceases. By the time HBB ends, all the 3He and 7Li created earlier in the evolution has been destroyed. The 7Li rich phase only lasts for a few 105 years. In Figure 5.5 we show the time variation of 7Li at the surface for three Z = 0×.004 models. In each panel, the grey shaded region indicates the Li-rich region, which approximately corresponds to log ε(7Li) & 3b, appropriate for Li-rich stars in the Small Magellanic Clouds (Plez et al. 1993). For the 3.5M , Z = 0.004, the surface abundance of lithium is less than log ε(7Li) = 3 for the majority of time spent on the AGB, except for a brief period near the beginning and end. The 4M , Z = 0.004 model is lithium rich for about 230,000 years, whereas the 6M , Z = 0.004 case is only lithium rich for 50,000 years at the beginning of the TP–AGB phase. Plez et al. (1993) suggest that the lithium-rich phenomena is easier to achieve in higher metallicity environments. Our model results support this idea. The largest 7Li enrichment is seen in the most metal-rich models. For example, both the 5 and 6M , Z = 0.02 models are super-Li rich for a short time, where the maximum enrichment is log ε(7Li) & 4. The lifetimes of the super-Li rich phases for the 5 and 6M , Z = 0.02 models are about 60,000 and 80,000 years respectively. It is still an open question whether or not AGB stars contribute to the production of 7Li in the Galaxy (D’Antona & Matteucci 1991; Romano et al. 2001; Travaglio et al. 2001c). There are many uncertainties involved in the production of 7Li in AGB models, including the mass-loss rates used and the treatment of convective mixing. Standard mass- loss rates for AGB stars, such as the formulae given by Vassiliadis & Wood (1993) and Block¨ er (1995), have a superwind phase which occurs during the final few thermal pulses. The superwind phase results in a period of rapid mass loss, and most of the convective envelope is lost during this time. Thus the composition of the envelope at the start of the superwind phase critically determines the contribution of AGB stars to the interstellar medium. Because most of the 7Li has been destroyed by the time the superwind phase

bwhere log ε(X) = log(X/H) + 12, where (X/H) is the abundance by number of the element X CHAPTER 5. AGB NUCLEOSYNTHESIS 113 starts, the final yields are small. For this reason, Romano et al. (2001) conclude that AGB stars cannot be considered as important 7Li producers because the 7Li yields from current AGB models are too low. Romano et al. calculate detailed AGB models over a wide range of initial metallicity, and include mass loss using the Reimers (1975) formula. They find that the lithium production is very efficient, but that there is a rapid decline in the envelope lithium abundance near the end of the TP–AGB phase, consistent with our model results. In contrast to the results of Romano et al. (2001), Travaglio et al. (2001c) conclude that the superwind phase from AGB stars can have a significant positive effect on the 7Li enrichment of the Galaxy. Travaglio et al. (2001c) compute AGB models with the same range of initial composition that we calculate, but vary the amount of mass loss by modifying the Vassiliadis & Wood (1993) mass-loss formula. If the high mass-loss rates start earlier on the AGB than currently predicted from the Vassiliadis & Wood (1993) formula, and if stars of 3 to 4M at low metallicities contribute to the lithium production, then Travaglio et al. conclude that AGB stars would be one of the major producers of lithium in the Galaxy. We note that contributions from other sources, such as novae or Type II supernovae, do not make enough 7Li to account for present day abundances (Travaglio et al. 2001c). The results from our model calculations support the findings of Romano et al. (2001). Whilst we observe efficient 7Li production during the TP–AGB phase, most of the 7Li is destroyed before the superwind phase starts so the final yields are small. Only a few models show a positive net production of lithium. Particular cases are the 3.5M , Z = 0.004 and the 4M , Z = 0.008 models. Whilst our results, using the standard VW93 mass- loss rates, support the view given by Romano et al. (2001), the uncertainties should be considered and we view this conclusion as tentative.

5.3.2 Nucleosynthesis from HBB: CNO Isotopes and Fluorine Hot bottom burning first alters the envelope abundance of the CNO isotopes by the CN cycle and later, by the ON cycles. There is also a fourth cycle involving the destruction of 19F to produce 16O. The CN cycle burns 12C first into 13C and later into 14N, which (as previously noted) reduces the 12C/13C ratio from the pre-AGB value near 20 to the equilibrium value of about 4. Since the 14N abundance increases from CN cycling and the rare nitrogen isotope, 15N, is destroyed by proton capture, the 14N/15N ratio dramatically increases from the post-SDU value of 103 to values between 104 and 105 in the convective envelope. ∼ There are two burning cycles that involve the oxygen isotopes. Both operate simultaneously but what effect each has on the envelope abundance depends on the branching ratio between the 17O(p,γ)18F and 17O(p,α)14N reaction rates. We observe a rapid reduction in the 18O abundance, which results in a large increase in the 16O/18O ratio, followed by a slow increase in the 18O abundance when the ON cycles come into equilibrium. Overall there is a net decrease in the 16O, 17O and 18O abundances in the envelope though the decrease in the surface 16O abundance is only significant in the most massive AGB models. For this reason, the elemental oxygen abundances observed in AGB stars should be indicative of their initial abundances, except in the most massive objects. CHAPTER 5. AGB NUCLEOSYNTHESIS 114

HBB very quickly destroys the 19F in the envelope via the reaction 19F(p,α)16O. Thus a star experiencing HBB would be observed to have low 12C/13C and 19F/16O (if detectable) ratios, a large 16O/18O ratio, and would be enriched in nitrogen and possibly lithium. There are some observational contradictions to this standard picture, an example of which is the SC–type AGB star WZ Cas. WZ Cas is observed to have an atmosphere enriched in both lithium and 13C, and as this star is also quite bright it was initially assumed that HBB was the cause. The envelope of WZ Cas is also observed to be enriched in 19F (Mowlavi, Jorissen & Arnould 1996), so we are possibly witnessing a modification to the standard nucleosynthesis picture. WZ Cas is an SC star (Aoki, Tsuji & Ohnaka 1998), where the C/O ratio is very near unity, which makes observations of 19F very uncertain (Jorissen, Smith & Lambert 1992). If it is the case that WZ Cas is enriched in both lithium and 19F, then we are possibly observing the effects of a slow extra-mixing process. In Figure 5.6 we show the time variation of the surface abundances of the CNO isotopes and 19F for the 6M , Z = 0.004 model. This figure demonstrates the most extreme behaviour in the HBB models, with temperatures exceeding 94 million K at the base of the convective envelope. At these temperatures the CN cycle comes into equilibrium quickly, with the ON cycle also acting to reduce the 16O abundance. The large production of 14N is clearly demonstrated in Figure 5.6, where the abundance steadily climbs over the TP– AGB phase except at the end when HBB is switched off. In this model, the 12C/13C ratio 4 for most of the TP–AGB, except during the last eight thermal pulses when it increases to∼ about 50 because of dredge-up. When the mass of the envelope has been reduced below about 2M , HBB ceases but the continuation of dredge up turns the model into an obscured carbon star, where the final C/O ratio 5.7 (see Frost et al. 1998). Note that the surface abundance of 16O has been depleted by∼ HBB, so less 12C is needed to produce a carbon star. The time variation of the 18O abundance in Figure 5.6 is interesting. This species is initially depleted but the destruction of 17O and 19F by proton capture produces enhancements of 18O. Overall the 18O at the end of the TP–AGB is still lower than the initial abundance.

5.3.3 The Production of Primary 14N Before discussing the production of nitrogen in AGB models, we need to deﬁne what we mean by the terms primary and secondary elements. Primary elements are those that are synthesized directly from the hydrogen and helium present in the star. An example is oxygen production in massive stars, which is quite insensitive to the initial metallicity. Secondary elements depend on the initial abundance of their progenitors. Examples are 14N and 22Ne production in massive stars, which depend on the number of CNO nuclei initially present. The 14N produced by HBB has a primary and secondary component, depending on whether the carbon and oxygen used in the CNO reactions is produced by the He-burning shell or was initially present when the star ﬁrst formed. Meynet & Maeder (2002) point out that primary 14N is likely to be formed with a He-burning core and a CNO burning shell, provided there is some transport mechanism to convey the primary 12C from the He-core to the H-burning shell. Primary 14N can also be produced when the TDU mixes primary CHAPTER 5. AGB NUCLEOSYNTHESIS 115

Figure 5.6: Surface abundance evolution during the AGB for the 6M , Z = 0.004 model for selected species: (top) for 12C, 13C, 14N and 16O, (bottom) for 15N, 17 O, 18O and 19F. CHAPTER 5. AGB NUCLEOSYNTHESIS 116

Figure 5.7: Abundances of N and O in damped Lyman-α systems (triangles) and extra- galactic HII regions (small dots). The solar abundances is given by the large blue dot, where we use the solar oxygen abundance from Allende Prieto et al. (2001). The HII region measurements are from Kobulnicky & Skillman (1996), Thurston et al. (1996), van Zee et al. (1998), Ferguson, Gallagher & Wyse (1998), Izotov & Thuan (1999) and van Zee (2000). The DLA system measurements are represented by the red triangles (Pettini et al. 2002). The dashed lines are the approximate levels of primary and and secondary nitrogen production.

12C from the He-shell into the envelope of an AGB star with HBB (Wood et al. 1983; Frost et al. 1998; Chieffi et al. 2001). Standard models of Type II supernovae cannot account for the primary nitrogen component in the Galaxy (see Timmes et al. 1995), which make low-metallicity intermediate mass AGB stars the most likely candidates. If the principal source of primary nitrogen is intermediate mass stars, then there should be a lag between the release of nitrogen into the interstellar medium and the release of oxygen, which is produced by Type II supernovae (Pettini et al. 2002), which explode soon after a period of star formation. Observations of metal-poor galaxies undergoing star formation, and in particular, observations of Damped Lyman–α systems (DLA’s) should show values of the N/O ratio that are below the primary plateau of about log (N/O) 1.5 (Pettini et al. 2002). In Figure 5.7 we show the nitrogen and oxygen abundances in ∼HII− regions and DLA’s, taken from a variety of sources. The sample of DLA abundances is from Pettini et al. (2002), and are represented by the red triangles. We show the position of the primary nitrogen plateau at log (N/O) 1.5, and give an approximate trend between the secondary nitrogen component and oxygen≈ − at high metallicity, extrapolated down to the lower metallicity region occupied by the DLA’s. Note that this figure is almost identical to Figure 6 in Pettini et al. (2002). It is interesting that many of the measurements for the DLA’s sit below the primary plateau. The measurements of the DLA’s support the idea that primary nitrogen is produced in intermediate mass stars, and that there is a delay be- CHAPTER 5. AGB NUCLEOSYNTHESIS 117 tween the oxygen and nitrogen enrichment in the interstellar medium. That up to 40% of the sample of DLA’s (Pettini et al. 2002) are nitrogen deficient means that either the lag is larger than that estimated by Henry, Edmunds & Koppen¨ (2000) (about 250 million years), or that we are observing the effects of small number statistics. There are a number of other explanations to account for the nitrogen deficiency of the DLA’s, including systematic differences between abundances measured via absorption lines in cold gas and those deduced from nebular emission from hot HII regions, measurement error or that an unrecognized ionization correction is responsible for the low N/O values (Pettini et al. 2002). If we are observing a larger time–lag between the oxygen and nitrogen enrichment of the interstellar medium than predicted, then perhaps we are observing the effect of low- metallicity stellar evolution on the DLA’s. It is now quite well established that in solar metallicity models, HBB only occurs in masses greater than about 5M (Boothroyd & Sackmann 1992; Frost et al. 1998; Block¨ er, Herwig & Driebe 2000), and that the minimum mass for HBB decreases with a decrease in the initial metallicity. HBB is observed to occur in a 3M model at Z = 0.0001 (Lattanzio; private communication) and Siess et al. (2002) estimate that HBB could occur in masses as low as 2M at Z = 0. At Z = 0.0001, the lifetime of a 2M model is about 800 million years, and the lifetime of a 3M model is about 300 million years (Lattanzio, private communication). Depending on the metallicity of the DLA’s, the lag could be as much as 800 million years and would explain why many of the measurements are below the primary nitrogen plateau. The mean redshift of the DLA’s is z 2.5, which corresponds to an age of about 11.7 Gyrc. Pettini et al. (2002) note that there∼ is some evidence that star formation began at about z 6 (Shapley et al. 2001), which corresponds to an age of about 13.5 Gyr. The difference∼ between the mean age of the DLA’s and the first epoch of star formation is about seven times the time delay suggested by Henry et al. (2000). Naively, this would imply that about one in seven of the DLA’s would be nitrogen poor, but instead 40% of the sample is nitrogen poor. If we take the lag to be 800 million years instead, then we would expect about 44% to be nitrogen poor, consistent with the observed statistics. Given the low number of DLA abundance observations, we cannot make any firm conclusions, but the circumstantial evidence seems to suggest a substantial primary nitrogen component from low-metallicity AGB stars of about 2M .

5.3.4 The Neon, Magnesium and Aluminium Isotopes When the temperature is high enough for the Ne–Na and Mg–Al chains to operated, both 23Na and 26Al are produced at the expense of 22Ne and 25Mg. HBB is responsible for the largest increase in the surface abundance of 23Na, 25Mg and 26Al, and is the only nucleosynthesis site in intermediate mass stars capable of altering the 24Mg abundance (according to standard models). The 26Al production can be substantial, and 26Al/27Al 1 ratios of the order of a few 10− are observed in the most massive models. The Ne–Na and Mg–Al chains× follow the same sequence as seen in the H-shell, except

c using the cosmology parameters Ωm = 0.3, ΩΛ = 0.7, h = 0.65, the age of the universe is 14.5 Gyr. dhigher for HBB than in the H-shell, because the density is lower at the base of the convective envelope CHAPTER 5. AGB NUCLEOSYNTHESIS 118

Figure 5.8: Surface abundance evolution during the AGB of the neon, sodium and magnesium isotopes for the 6M , Z = 0.004 model. HBB ceases after the 101st thermal pulse, which corresponds to t 3 .5 in the diagram. In this ®gure we also show the enrichment from six synthetic thermal≈ pulses (see 7.1 for more details). § that temperatures of at least 90 million K are required before 24Mg is substantially depleted. In Figure 5.8 we sho∼w the time variation of various surface abundances for the 6M , Z = 0.004 model. Large depletions in 16O (see the upper panel of Figure 5.6), 24Mg and 22Ne are followed by signiﬁcant enhancements in 25Mg, 26Mg and 26Al. We also observe moderate enhancements in 23Na and 27Al. The 23Na enrichment from HBB is less in this model than in the 5M , Z = 0.004 model (see Figure C.38). This is because in the 6M model, more 22Ne is destro yed by α–capture in the He-burning shell so there is less mix ed into the envelope by each TDU episode, and hence there is less available to produce sodium. The ﬁnal 26Al/27Al ratio for this model is about 0.6.

5.4 Comparison of Model Predictions to Observations

Owing to the many uncertainties in the stellar modelling process, such as the lack of a reliable theory governing convection and mass loss, and uncertainties in the nuclear reaction rates and the opacity tables, it is unclear whether current models present a realistic picture of nucleosynthesis in AGB stars. There is evidence that we do make some accurate predictions. For example, the cool (N–type) carbon stars are nicely explained by the prediction CHAPTER 5. AGB NUCLEOSYNTHESIS 119 that the third dredge-up mixes 12C from the intershell into the envelope (Iben 1991). There is also a correlation observed between the s-process elements and carbon overabundances in AGB stars, which can also be qualitatively explained by the models (Smith & Lambert 1986). The details of the s-process nucleosynthesis came much later (Busso et al. 2001); and current models still suffer from considerable uncertainties, such as the details of the 13C–pocket. Nonetheless, it seems that we now have a good understanding of s-process nucleosynthesis in AGB stars. The lack of luminous carbon stars in the LMC and SMC is explained by theoretical models with HBB (Wood et al. 1983; Block¨ er 1991; Boothroyd, Sackmann, Ahern 1992) as is the existence of luminous super-Li rich AGB stars (Plez et al. 1993). Whilst observations and models are qualitatively correct, there are still many unsolved problems. There are many examples, such as the J-type carbon stars, with low 12C/13C ratios, no s-process enhancements and Li enhancements (Abia & Isern 2000). There is also the puzzle of WZ Cas, which seems to show the effects of HBB but is highly enriched in 19F (Mowlavi et al. 1996). Both the J-type carbon stars and WZ Cas are possibly explained by including some form of deep mixing into the standard models, and we do not discuss these problems further. We will instead discuss two particular problems related to nucleosynthesis in the He-burning shell. The first problem is that non-solar Mg ratios have not been observede in AGB stars even though this is what the models predict for intermediate mass stars (Smith & Lambert 1986; Guelin´ et al. 1995). Lastly, we examine the observations of planetary nebulae, which show constant Ne/O ratios despite the models predicting enhancement in the 22Ne abundance.

5.4.1 Observations of the Mg Isotopes in IRC+10216 Up until recently, there has been little evidence of non-solar Mg isotopic ratios in stars. Recent observations by Gay & Lambert (2000) and Yong et al. (2003) have shown that non-solar Mg isotopes do indeed exist in some stars. There are few observations of the Mg isotopes in AGB stars. Observations by Smith & Lambert (1986) show that galactic MS and S–type AGB stars with s-process enhancements do not show enhancements in the neutron-rich Mg isotopes. The bolometric luminosities of the stars observed by Smith & Lambert (1986) are between 1 6 Mbol 6 5.5, with an average of about Mbol 2.9, indicating a low-mass population.− We do not−give these observations further consideration∼ − because it is now widely accepted that low-mass ( 1.5M ) AGB stars are responsible for the production of the main component of the s-process∼ elements (Busso et al. 2001), and we do not predict Mg production to occur in low-mass AGB stars. The millimetre observations of the dusty carbon star IRC+10216 by Guelin´ et al. (1995), and updated by Kahane et al. (2000) observe the split in the Mg isotopes and we discuss these further. The mass estimates of the central star of IRC+10216, CW Leo, lie between 2 and 5M , depending on the distance used (Winters et al. 1994; Weigelt et al. 1998). The observations of IRC+10216 reveal solar values for the Mg isotopes. Using detailed models of 1.5, 3 and 5M , Z = 0.02 stars, Kahane et al. (2000) conclude that the mass of IRC+10216

epossibly because of the great diﬃculty in separating the Mg isotopes in the stellar spectra CHAPTER 5. AGB NUCLEOSYNTHESIS 120

Figure 5.9: Computed isotopic ratios of 26Mg/24Mg vs 25Mg/24Mg in the envelope of a number different models. See the text for more details. The red circle is the abundance of IRC+10216. is low, with M . 2M . This implies that the 22Ne neutron source is only marginally activated, leaving the Mg isotopic ratios unchanged. Supporting evidence comes from the Cl isotopic ratio, 35Cl/37Cl, which is altered by neutron capture. The observed value is also inconsistent with the mass of CW Leo exceeding about 3M . The carbon and oxygen isotopic ratios hint at some extra-mixing taking place in CW Leo. In Figure 5.9 we show the computed isotopic ratios of 26Mg/24Mg vs 25Mg/24Mg (divided by the solar value) in the envelope of a 3, 4 and 5M model of Z = 0.02 (black symbols) and a 2.5 and 5M model of Z = 0.008 (blue symbols). We also include a low- mass 1.9M , Z = 0.008 model (because it has substantial TDU), denoted by the green symbol. The abundance of IRC+10216 is shown by the solid red circle. Note that we do not include all the thermal pulses for the 5M , Z = 0.008 model; the final ratios are (25Mg/24Mg)/solar = 3.65 and (26Mg/24Mg)/solar = 3.25, respectively. If we assume that the error in the observed Mg ratios of IRC+10216 are about 0.1, then all but the later pulses of the 4 and 5M , Z = 0.02 and 5M , Z = 0.008 models are a good fit to the observed abundances. If we assume the error is smaller, then the agreement is not so good and even the 1.9M model no longer matches the observed data, which is surprising if CW Leo is of low mass. It is interesting that the 5M models intersect the position of the observed abundance. The reason that the 5M models sit at a position with more 26Mg compared to 25Mg is that the second dredge up has mixed material into the envelope that is slightly depleted in 25Mg and enhanced in 26Mg, so the envelope composition at the start of the AGB is shifted relative to the lower mass models. IRC+10216 looks to have undergone a similar type of mixing episode. If CW Leo is of low mass, then only the first dredge up has occurred CHAPTER 5. AGB NUCLEOSYNTHESIS 121 prior to the AGB. In our standard models with no extra mixing, the Mg isotopic ratios of low-mass models are unaltered by the first dredge up. Perhaps the measured Mg isotopes are simply reflecting the initial composition, which is enriched in 26Mg compared to the solar value? There is evidence that extra mixing is occurring in CW Leo, supported by the low 12C/13C ratio 45 with C/O 1.4 (compared to our models, where C/O 1.4 implies a 12C/13C ratio > ∼100) and the oxygen∼ isotopic ratios (see Nollett et al. 2003).∼ However, to alter the Mg isotopes, the extra mixing has to occur much deeper than is required to alter the CNO nuclei, and this might change the evolution of the star. The Mg isotopic ratios of IRC+10216 are solar (within the errors), suggesting that very deep mixing is not occurring. Nollett et al. (2003) give an estimate of the effect that deep mixing would have on the evolution of a TP–AGB star and conclude the effect would be minimal. However, this result needs to be tested in a self-consistent manner, where the energy generated by the slow-extra mixing is fed back into the evolution of the star.

5.4.2 Observations of the Mg Isotopes in Other Stars Gay & Lambert (2000) observed the Mg isotopic ratios in a sample of 20 stars. The stars are mostly field dwarfs, with one giant in the sample. Many of the stars have Mg isotopic ratios that deviate from the solar value, and the authors suggest that a few of the stars are possibly the result of mass transfer from an AGB star. Gay & Lambert (2000) compare their observations to the galactic chemical evolution model of Timmes et al. (1995), which utilizes the supernovae yields from Woosley & Weaver (1995). The chemical evolution model does not predict the amount of 25Mg and 26Mg observed in the field stars, especially at low metallicities. Unless one invokes non-standard mixing scenarios in the supernovae models, another production site for the neutron-rich Mg isotopes has to be found. Other possible sources of the neutron-rich Mg isotopes include low-metallicity Wolf-Rayet (WR) (Maeder 1983; Woosley, Langer & Weaver 1994) and AGB stars (Siess et al. 2002; Forestini & Charbonnel 1997). Yong et al. (2003) observed the Mg isotopic ratios in 20 stars near the tip of the giant branch in the globular cluster NGC 6752. These authors find that many of the stars have Mg isotopic ratios that deviate highly from the solar value, with an excess of 26Mg compared to 25Mg in most of the stars. Yong et al. (2003) find no correlation between the effective temperature (used as an indicator of evolutionary state) and the abundance anomalies and thus conclude that the giants were polluted by an earlier generation of stars. The authors suggest the following scenario: the most abundant isotope of magnesium, 24Mg, is produced in supernovae. However a population of Z = 0 AGB stars should produce a substantial amount of 25Mg and 26Mg. The population of Z = 0 AGB stars pol- lute the cluster gas, so the next generation of stars form with 24Mg:25Mg:26Mg ratios that are near solar. When the current generation of stars form, some of them are further pol- 4 luted by intermediate mass AGB stars of Z 10− , which increases the relative amounts of 25Mg and 26Mg. Many of our models, whilst∼ not Z = 0, produce as much 25Mg and/or 26Mg as 24Mg. This is shown in Figures 5.10 and 5.11, where we show the time variation of the relative proportions of the three Mg isotopes during the TP–AGB for the 3 to 6M models. We observe in Figures 5.10 and 5.11 that the largest deviations from solar occur CHAPTER 5. AGB NUCLEOSYNTHESIS 122

Figure 5.10: The abundance of 24Mg, 25Mg and 26Mg scaled to the total Mg abundance during the TP±AGB phase for the 3M and 4M models.

CHAPTER 5. AGB NUCLEOSYNTHESIS 123 in the lowest metallicity (Z = 0.004 in this case) models. The variation of the Mg isotopic ratios with metallicity [Fe/H] has been calculated by Fenner et al. (2003), who incorporate Mg yields from our AGB models and supernovae yields from Chieffi & Limongi (2001) (see Figure 5.12) into a chemical evolution model for the solar neighbourhood. Two models are shown. The solid line corresponds to the predicted trend for the solar neighbourhood with the contribution from AGB models and the dotted line is the predicted trend without the AGB contribution. Both models arrive at similar present-day values of the 25Mg/24Mg and 26Mg/24Mg ratios, however only the model including AGB yields matches the empirical data at low metallicities. The Mg isotopic ratios predicted by our models are consistent with many of the stars in the Gay & Lambert sample, but not with the stars in the Yong et al. sample. Most of the field stars in the Gay & Lambert sample have Mg isotopic ratios with more 25Mg relative to 26Mg, similar to our models. Quite a few of the stars have 25Mg/24Mg and 26Mg/24Mg ratios that are less than solar, and only a few have ratios much larger than solar. Conversely, the globular cluster stars observed by Yong et al. have more 26Mg relative to 25Mg, and many of the stars have Mg isotopic ratios that are much greater than the solar ratios. The observations of Yong et al. (2003) are very puzzling and possibly present a problem for the stellar models. Perhaps the nuclear reactions rates are wrong, and we should really be producing more 26Mg compared to 25Mg? Arnould et al. (1999) comment that there are still large uncertainties in the 22Ne α capture (and the Mg–Al) reactions at the relevant temperatures for AGB stars. An exploration− of the effect that the nuclear-reaction rate uncertainties have on the production of the Mg isotopes would help clarify the situation. If we assume for now that the reaction rates are correct, then what other solutions are there? We know that the rates for the Mg–Al chain, and the 22Ne α capture reactions are extremely temperature dependent. We know also that the temperature− in a given burning region will be higher in a lower Z model (at a given mass), than in a solar or SMC composition AGB model. Thus more 25Mg will be burnt by the H-shell, and even if the He-shell temperature favours the production of 25Mg relative to 26Mg, more 26Mg will eventually be mixed to the surface. This occurs in a few of our models, but perhaps at very low metallicity this would occur more frequently? We also assume that the initial Mg isotopic ratios in our models are either solar or scaled solar. This will certainly not be the case for the second generation of AGB stars, formed at [Fe/H] 1.62 (Yong et al. 2003). The initial Mg will be mostly 24Mg, with a small fraction of∼ 25−Mg and 26Mg from the Z = 0 population of AGB stars. A quantitative study of intermediate mass AGB stars with [Fe/H] 1.62, and that have initial Mg ratios that are 24Mg-rich compared to a scaled-solar composition∼ − is required before we can rule out AGB stars as the source of the heavy Mg isotopes. If we are observing the effect of a reduced metallicity on the production of the Mg isotopes, then why did we not see this effect in the stars observed by Gay & Lambert (2000)? The reason is that the stars observed by Gay & Lambert have a range of metallicities between 1.8 < [Fe/H] < 0.0, and only four stars out of the sample of 20 have a metallicity lower −than [Fe/H]= 1.2. In summary−, most of our intermediate mass models produce more 25Mg relative to 26Mg. These models are consistent with the observations of Gay & Lambert (2000), since many of the stars in their sample have a larger proportion of 25Mg relative to 26Mg. The situation CHAPTER 5. AGB NUCLEOSYNTHESIS 124

Figure 5.11: The abundance of 24Mg, 25Mg and 26Mg scaled to the total Mg abundance during the TP±AGB phase for the 5M and 6M models.

CHAPTER 5. AGB NUCLEOSYNTHESIS 125

Figure 5.12: Variation of the magnesium isotopic ratios with metallicity [Fe/H]. 25Mg/24Mg and 26Mg/24Mg are shown in the upper and lower panels. Blue circles correspond to the stellar abundances observed by Gay & Lambert (2000), while orange circles represent a sample of halo and thick disk stars observed by Yong (2003, private communication). Solar values appear as red squares. The predicted trend for the solar neighbourhood model of Fenner et al. (2003) incorporating Mg yields from AGB stars (solid line) is shown against a model without the AGB contribution. Courtesy of Yeshe Fenner. CHAPTER 5. AGB NUCLEOSYNTHESIS 126 is reversed for the stars observed by Yong et al. (2003). Whilst our models fail to reproduce these observations, we may simply be observing the eﬀects of a reduced metallicity plus initial non-solar Mg abundances on the production of the Mg isotopes. A qualitative study of low metallicity AGB stars will help resolve these problems.

5.5 Neon in Planetary Nebulae

After the AGB phase is terminated, low to intermediate mass stars evolve through the planetary nebulae (PNe) phase before finally ending their lives as white dwarfs. The gaseous nebula that makes up a PNe is the remnant of the deep convective envelope that once surrounded the core, which is now exposed as the central star of the PNe. Thus the abundances of the nebula should reveal information about the chemical processing that took place during the AGB; and more precisely, information about the final thermal pulses. The spectra of PNe are usually a composite, a mixture from the nebula and the illumi- nating central star. Many PNe are sufficiently large or bright that the nebular spectrum can be isolated (Kaler 1997). In the following discussion, we do not examine the “very late thermal pulses” that seem to be required to convert a H-rich central star to a He-rich central star; instead we refer the reader to Block¨ er (2001). We can accurately observe helium and neon abundances in PNe, unlike in the solar photosphere (Kaler 1978; Henry 1989; Dopita et al. 1997; Stanghellini et al. 2000). This is because a strong ultraviolet flux from the central star of the PNe has ionized the surround- ing gas. The ionized gas recombines at excited levels, with the decay producing observable emission lines such as the Balmer series. Some of the strongest emission lines observed in PNe are the so-called forbidden lines, such as those produced by doubly ionized oxygen, O2+, [O III] (Kaler 1997). Other forbidden emission lines are [Ne III] and [O II], observed in the ultraviolet, and [N II] and [S II], which are observed at longer wavelengths. The forbidden emission lines form in PNe (and diffuse nebulae) because of the low densities, where collisional de-excitation is so slow that a photon is emitted after a time of the order of seconds (Pagel 1997). The C, N and O abundances can also be accurately observed, as well as a host of rare elements including S, Cl and Ar (Dopita et al. 1997). The observations of neon in PNe can be used as a powerful tool to constrain models of AGB stars. What do the observations tell us? A correlation is observed to exist between the Ne/H and O/H abundance in PNe in the Galaxy, LMC, SMC and M31, within a small spread (Henry 1989). Dopita et al. (1997) find the same result for the Magellanic Clouds, and conclude that there is no sign of the dredge-up of 22Ne. Stasinska, Richer & McCall (1998) extend the work of Henry (1989) to a larger range of PNe, including a small sample in M32, and also observe a tight correlation between the neon and oxygen abundance. It seems that all PNe, regardless of location, seem to have the same Ne/O ratio (by mass) of 0.20; and it is assumed therefore that the Ne, Ar, and S abundances are unaltered by stellar∼ evolution (Henry 1989; Stasinska et al. 1998), and that these elements are produced in the same relative proportions in massive stars (Stasinska et al. 1998). Earlier in this chapter we comment that the surface abundances of AGB models can become enriched in 22Ne. Do these observations present a problem for standard models? Before we answer CHAPTER 5. AGB NUCLEOSYNTHESIS 127 that question, let us summarize the chemical signature expected from the operation of the TDU in AGB models. There would be surface enhancements in 4He, 12C and 22Ne in the models with M . 3M ; and also surface enhancements in the heavy Mg isotopes in the most massive models, but in these cases HBB would alter the CNO nuclei and destroy 22Ne. How does this affect the Ne/O ratio? The Ne measured in PNe is the elemental neon abundance, and hence a sum of all the isotopes. The 20Ne isotope is the most abundant, and it is not altered significantly by H or He-burning during the AGB. Therefore only when the 22Ne abundance exceeds or is equal to the 20Ne abundance should we expect an enhancement in the Ne/O ratio. In 5.5.1 we examine the observations more closely, followed by a detailed study of the model§ results. We conclude by answering the question: Do the Ne abundances observed in PNe present a problem for the stellar models?

5.5.1 The Observations The linear relationship between the Ne and O abundance in PNe from many different locations is illustrated in the upper panel of Figure 5.13. In this diagram we include the observed abundances for over 300 PNe in the Galaxy, LMC, SMC and M31, taken from Stasinska et al. (1998) (hereafter S98) and Henry (1989) (hereafter H89). The galactic data from S98 is taken from the bulge of the Milky Way, and Exter et al. (2001) note there is no difference between the average abundances and the abundance relationships for PNe in the bulge and disk. Even though there is some dispersion in the data, the correlation between the Ne/H and O/H abundance is clearly demonstrated. The errors in this diagram are probably on the order of 0.2 dex for the Ne/H and O/H abundances (Stanghellini et al. 2000). Stanghellini et al. (2000) examined the systematic differences between PNe abundances from different sources and finds that the differences are within the errors for all elements but nitrogen. The elemental abundances are derived by using an ionization correction factor, which can be large for low ionization PNe (Alexander & Balick 1997), giving artificially high neon and sulphur abundances. Stanghellini et al. (2000) comment that there is no correlation between high neon abundances and low ionization systems, and most of the systems analyzed have high ionization. We will restrict the following discussion to PNe in the Galaxy and LMC, as most of the data are for these two systems. In the lower panel of Figure 5.13 we show the Ne/O ratio (by mass) as a function of the oxygen abundance for PNe in the Galaxy and LMC using the same data set. In this diagram, we include the observed abundances for just over 200 PNe, most of which have a Ne/O ratio of about 0.2. Kaler (1978) finds no variation to the Ne/O ratio for galactic PNe, and Henry (1989) extends that observation to other systems. Even so, the spread in the data from S98 is rather large, even allowing for large errors bars of 0.2 to 0.3 dex. Many of the LMC and bulge PNe observed by S98 have Ne/O ratios fargreater than 0.2, where the largest ratio in the sample is about 1.2. The Ne/O ratio for the bulge PNe also seems to have a noticeable positive gradient, where PNe with the lowest log (O/H)+12 values are observed to have slightly lower Ne/O ratios. This effect is probably connected to the initial chemical composition of the PNe, and related to galactic chemical evolution effects. CHAPTER 5. AGB NUCLEOSYNTHESIS 128

9.5

8.5

7.5

7 H89 MW

log (Ne/H) + 12 S98 MW H89 SMC 6.5 S98 SMC H89 M31 6 S98 M31 S98 LMC H89 LMC 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 log (O/H) + 12

1.2 H89 MW S98 MW 1 S98 LMC H89 LMC

0.8

0.6

Ne/O (by mass) 0.4

0.2

0 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 log(O/H) + 12

Figure 5.13: (Upper panel) Observed Ne/H versus O/H for PNe in the Galaxy LMC, SMC and M31, taken from Stasinska et al. (1998) and Henry (1989). In the lower panel, the Ne/O ratio (by mass) for the Galaxy and LMC from the same data set. CHAPTER 5. AGB NUCLEOSYNTHESIS 129

We conclude from Figure 5.13 that some of the dispersion is real, and that the PNe with the largest Ne/O ratios are either enriched in neon or depleted in oxygen. If it is oxygen depletion, then this is unlikely to have taken place during the AGB, and the most plausi- ble scenario is that the star was born with a low oxygen abundance. There is, however, evidence that PNe can become enriched in neon. Comparing the properties of elliptical and bipolar PNe, Corradi & Schwarz (1995) conclude that bipolar PNe are produced from more massive progenitors than the other PNe classes, and that bipolar PNe have overabundances of helium, nitrogen and neon. Stanghellini et al. (2000) provide evidence that the morphology of PNe are related to the abundances, and ﬁnd that all asymmetric PNe, which supposedly originate from a more massive progenitor (Corradi & Schwarz 1995), have higher neon abundances on average than the symmetric PNe. It is unclear from the discussion in Stanghellini et al. (2000) if this eﬀect is caused by chemical evolution, i.e. intermediate mass stars formed from a more metal rich gas than the older, less massive stars responsible for the symmetric PNe, or of an actual Ne enhancement from internal nucleosynthesis.

5.5.2 Model Results In the nucleosynthesis calculations we take the initial C, N and O abundances for the Z = 0.008 models from Russell & Dopita (1992) and simply scale the other species, including neon, to obtain the required metallicity. In doing this, we obtain initial neon abundances that are larger than those observed in PNe and HII regions (Vermeij & van der Hulst 2002) i.e. initial Ne/O ratios that are larger than 0.2. Thus we calculated a second set of Z = 0.008 nucleosynthesis models for each mass,∼ using the same evolution models as input. The initial CNO abundance remains unchanged. This second set has the initial Ne, Na and Mg abundances taken from Russell & Dopita (1992), and we ensure that the Ne/O ratio (by mass) is equal to 0.21 (see Table 5.1). The Ne and Mg isotopic ratios are solar. From Table 5.1 we see that in the second set of models the total Ne abundance is reduced by about 60%, sodium is about 9 times more abundant and the Mg abundances remain unchanged. In the following discussion, the nucleosynthesis results for the Z = 0.008 models are from this second set of calculations. The neon isotope 22Ne is efficiently produced in the He-burning shell of AGB stars. In Figure 5.14 we show the 22Ne mass fraction in the intershell of all the stellar models we compute. The 22Ne abundances are taken from the intershell at the end of the last calculated thermal pulse, and therefore reflect the abundances that are mixed into the envelope by the subsequent TDU episode. The M 3M models have the largest 22Ne intershell mass fractions. This conclusion does not∼depend strongly on the metallicity, but there is a shift toward lower mass as the initial metallicity is decreased. The reason that the peak 22Ne intershell abundance does not depend on the metallicity is because of the third dredge-up, which increases the 12C abundance in the envelope. Hence, there is more 12C available in the H-shell to convert to 14N. In Figure 5.4 on page 108 we see that there is an almost one-to-one correspondence between the 14N abundance in the H-shell ashes and the 22Ne produced by the thermal pulse. Since the 3M models also have efficient TDU, where λ & 0.7, we should expect these models to produce the most 22Ne. In the CHAPTER 5. AGB NUCLEOSYNTHESIS 130

Table 5.1: Initial abundances (in mass fraction) of Na and the Ne and Mg isotopes in the ®rst and second set of Z = 0.008 models. We also give the Ne/O ratio (by mass). The last column is the percentage difference between the second and ®rst set of abundances. species first second % diff 20 3 4 Ne 1.269 10− 5.215 10− 58.9 21 × 6 × 6 − Ne 3.243 10− 1.333 10− “ 22 × 4 × 5 Ne 1.019 10− 4.188 10− “ 23 × 5 × 4 Na 2.680 10− 2.396 10− 794 24 × 4 × 4 3 Mg 4.144 10− 4.129 10− 3.62 10− 25 × 5 × 5 − × Mg 5.440 10− 5.420 10− “ 26 × 5 × 5 Mg 6.242 10− 6.219 10− “ Ne/O 0.517× 0.212× 58.9 − more massive models, 22Ne is efficiently destroyed by α capture and at lower masses the temperature is not hot enough for efficient 22Ne production.− We know from 5.2.2 that spectroscopic observations of neon in PG 1159 objects reveal mass fractions of§about 2 to 5% (Werner & Wolff 1999), consistent with the abundances in Figure 5.14. Thus the models are predicting the right amount of neon in the intershell; it is the amount of dredge-up that determines the neon enrichment in the gaseous nebulae. In Figure 5.15 we show the surface abundance of 20Ne, 22Ne and Ne/H as a function of time for the 1.5, 3 and 6M models. The left-hand column presents results for the Z = 0.02 models and the right-hand column presents results for the Z = 0.008 models. The Z = 0.02 models are assumed to be representative of galactic PNe and the Z = 0.008 models are assumed to be representative of LMC PNe. From Figure 5.15 we see that the lower mass models (M < 2.5M ) experience very little or no dredge-up, so the change to the elemental Ne/H abundance is negligible. In the most massive models, α capture destroys 22Ne in the intershell, plus 22Ne is destroyed by the Ne–Na chain to produce− 23Na by HBB. However in these models, the abundance of 20Ne is slightly enhanced from the destruction of 23Na. Overall, the change to the Ne/H ratio is small. In the 6M , Z = 0.02 model the change to the Ne/H ratio is much less than 0.1 dex and in the 6M , Z = 0.008 model, the Ne/H abundance increases by about 0.1 dex. These results imply that there will be little change to the neon abundances in PNe originating from low-mass progenitors, and a only small increase in the neon abundance in PNe from the most massive progenitors. An increase of 0.1 dex, as predicted by the models, is probably not observable. Because the M 3M models produce the most 22Ne during a thermal pulse (see Fig- ure 5.14), we would∼ expect considerable 22Ne enrichment in these models. In Figure 5.15 we see this to be the case, with the largest 22Ne enhancements occurring in the 3M models, regardless of the metallicity. This large increase actually occurs over a small mass range, from 2.5M to 3.5M , for both the solar and LMC metallicity models. In the case of the 3M Z = 0 .02 model, the change to the Ne/H ratio during the TP–AGB is about 0.2 dex. Since the errors are also about 0.2 dex (Stanghellini et al. 2000), this is probably not a significant increase. The change to the Ne/H ratio for the 3M , Z = 0.008 model is about 0.6 dex, and should be observable. We note for comparison that the change to CHAPTER 5. AGB NUCLEOSYNTHESIS 131

0.035 Z = 0.02 22 0.03 Z = 0.008 Z = 0.004 0.025 0.02 0.015 0.01 0.005 intershell abundance: Ne 0 1 2 3 4 5 6 7

Initial mass (Msun) Figure 5.14: The 22Ne mass-fraction in the intershell as a function of the initial mass. The 22Ne abundances are taken from the last calculated thermal pulse, just after the convective pocket has died down. Note that the 22Ne production in the He-shell is dependent on the 12C abundance in the envelope, which is the same in the ®rst and second set of Z = 0.008 models. CHAPTER 5. AGB NUCLEOSYNTHESIS 132 the Ne/H ratio for the 2.5 and 3.5M , Z = 0.008 models are about 0.4 dex and 0.3 dex respectively. This analysis also highlights another important aspect of the models, that the most metal-poor models have the largest increase in the surface Ne/H ratio. Most PNe are observed to have Ne/O ratios of about 0.2, with a linear relationship between the Ne/H and O/H abundances. Can these observations help constrain the stellar models? In other words, is the TDU too eﬃcient in the low-metallicity models with M 3M ? To try and answer this question, we now compare the stellar models directly to the∼observ ations.

5.5.3 Discussion Before we go on, we should point out that a direct comparison is not exactly valid, because there are metallicity gradients observed in both the Galaxy and the LMC, and we are using models based on a constant composition. But we can say something about the Ne enhancement expected at a given metallicity, or log (O/H) value. The stellar models presented in Figure 5.15 do not lose their entire envelopes. For the lower mass models the remaining envelope is small, usually less than 0.2M but for the intermediate mass models there can be up 1.5M of envelope mass remaining. For the intermediate masses, thermal pulses and the∼TDU may in principal continue, so we use an algorithm to simulate the evolution and enrichment of these few remaining thermal pulses. We do not discuss the algorithm here but refer the reader to Chapter 7 for details. Here we simply mention that the efficiency of the third dredge-up becomes a free parameter. We do not know how dredge-up is affected by small envelope masses but there is some evidence that the efficiency is decreased (Straniero et al. 1997; see also Chapter 6). Thus we calculate the remaining thermal pulses using three values of λ: 1) λ = λlast, 2) λ = 0.3 and 3) λ = 0.0, where λlast is the efficiency of the third dredge-up evaluated from the last two thermal pulses, and is presented in Table 7.1 on page 165. In the following discussion, we only discuss cases 1) and 3), which are the two extremes. The gaseous nebulae that comprise the PNe are only a few 0.1M , hence we are seeing the nucleosynthetic results of the last thermal pulse. For the purposes× of comparison to the observations in Figure 5.13 we will focus on the surface abundance after the last dredge-up episode. From the upper panel of Figure 5.16 we see that the Z = 0.02 models with λ = 0 (blue stars) fit well within the range of the galactic PNe observed by H89, and the Z = 0.02 models with λ = λlast sit well within the range of Ne/O occupied by the bulge PNe. From the lower panel of Figure 5.16 we see that most of the Z = 0.008 points fit within in the Ne/O range occupied by the observations, with the exception of two points. Both of these points are for the 3M , Z = 0.008 model. The blue star at Ne/O 0.9 represents the 3M model with λ = 0 for the remaining dredge-up episodes and the∼pink box at Ne/O 1.2 ∼ represents the case where λ = λlast = 0.86. The point at Ne/O 0.9 fits within the errors, if we assume the errors are at least of the order of 0.2, but the∼ point at Ne/O 1.2 does not. S98 reported PNe in the bulge with Ne/O ratios of at least 1.0 at the log (O∼/H) + 12 ratios consistent with the LMC. If the models are correctly predicting that a narrow mass range of stars produce enough 22Ne to increase the Ne/O ratios, then perhaps we have not observed enough LMC PNe to find the few with large Ne/O ratios? Since there are over 150 LMC PNe shown in Figure 5.16, the question then becomes how many PNe is enough CHAPTER 5. AGB NUCLEOSYNTHESIS 133

Figure 5.15: The abundance of Ne/H, 20Ne, 22Ne and 23Na during the TP±AGB phase for the 1.5, 3M and 6M , Z = 0.02 (left) and Z = 0.008 (right) models.

CHAPTER 5. AGB NUCLEOSYNTHESIS 134 to adequately cover this narrow mass range? Perhaps instead the M 3M models are simply producing too much 22Ne. This could certainly be the case for the∼ 3M , Z = 0.008 with λ = λlast, but the 3M , Z = 0.008 case with λ = 0 sits within the errors of the observed points. Maybe these observed neon abundances can be used as a constraint on the cessation of the third dredge-up in the AGB models? The third dredge-up is governed somewhat by the mass-loss law used, so perhaps we require stronger mass loss earlier on the AGB than predicted by the Vassiliadis & Wood (1993) mass-loss law? We mention for comparison that the models of Herwig (2000), with diffusive convective overshoot, also experience very efficient TDU (λ & 1) but because these models also produce more 16O in the intershell than our models, it is unclear how the Ne/O ratio will change during the TP–AGB phase. In summary, we conclude that the Z = 0.02 models fit well within the range of neon abundances observed in galactic PNe. Most of the Z = 0.008 model points also fall within the range of neon abundances observed in LMC PNe, with the exception of the 3M , Z = 0.008 model. That we only find disagreement for a very narrow range of mass is suspicious, and more observations of neon in LMC PNe are needed to understand this problem. It cannot be ruled out that the TDU is too efficient in the 3M , Z = 0.008 model, at least at the end of the TP–AGB phase. Now we go back to our original question: Are the neon abundances observed in PNe a problem for standard stellar models? The answer to that question would have to be no, for the galactic objects but possibly yes for the LMC objects, where the models predict too much neon in a narrow mass range. In this discussion, we have concentrated on the observations of neon in PNe but there are also abundances for He, C, N and O that should also be considered. Dopita et al. (1997) state that there is clear evidence for the dredge-up of carbon, consistent with the efficient operation of the TDU. There are also PNe with large He/H and high N/O ratios, perhaps consistent with the operation of HBB (Dopita et al. 1997). But Dopita et al. (1997) also mention that agreement between the observations and theory would be better reached if HBB occurred at M & 2M , perhaps indicating that some form of deep mixing is yet again required.

5.6 Summary and Further Work

This chapter is an extensive review of the nucleosynthesis that occurs during the TP– AGB phase of evolution. We examine the production of 26Al in the hydrogen-burning 26 27 3 shell of low-mass AGB stars, and find that the Al/ Al ratio can be as high as 10− . 26Al is not destroyed by neutron capture in the intershell because of the lack of an efficient neutron source in the low-mass models; the inclusion of a 13C pocket will effect this result. Hence further work would be to include a 13C pocket into our nucleosynthesis calculations, and examine the effect on the 26Al intershell abundance. We study the production of 22Ne, 25Mg and 26Mg in the He-burning shell of AGB stars. There is a peak in the 22Ne production at about 3M , and only the most massive AGB stars produce enough 25Mg and 26Mg to significantly alter the Mg isotopic ratio. The models produce more 25Mg relative to 26Mg, a trend which is consistent with many of the stars observed by Gay & Lambert CHAPTER 5. AGB NUCLEOSYNTHESIS 135

1.2 H89 MW S98 MW Z=0.02 1 λ λ Z=0.02 = max

0.8

0.6

Ne/O (by mass) 0.4

0.2

0 6.5 7 7.5 8 8.5 9 9.5 log(O/H) + 12

1.4 H89 LMC S98 LMC 1.2 Z=0.008 λ λ Z=0.008 = max 1

0.8

0.6 Ne/O (by mass) 0.4

0.2

0 6.5 7 7.5 8 8.5 9 9.5 log(O/H) + 12

Figure 5.16: (Upper panel) The Ne/O ratio for the galactic PNe, with the results for the Z = 0.02 models included. (Lower panel) Same, but for the LMC and over-plotted with the results for the Z = 0.008 models. CHAPTER 5. AGB NUCLEOSYNTHESIS 136

(2000) but not with the recent observations of Yong et al. (2003). The likely reason for the discrepancy is the effects of nucleosynthesis in low-metallicity stars, plus different initial conditions to those that we use. The initial Mg ratios are solar in our models but perhaps for low-metallicity stars it would be more appropriate to use non-solar Mg isotopic ratios, where the proportion of 24Mg to 25Mg and 26Mg is greater than that observed in the solar system (where about 80% of the Mg in the solar system is 24Mg). Whilst our models predict that intermediate mass AGB stars are enriched in 25Mg and 26Mg relative to the solar abundance, this has not been observed. The study by Smith & Lambert (1986) only observed MS and S-type stars with s-process enhancements, and if the recent calculations of s-process nucleosynthesis (by Gallino et al. (1998) and Travaglio et al. 2001a,b) are any guide, then we should not expect a correlation between large s- process enhancements and an enrichment in the heavy Mg isotopes. High resolution spectroscopic observations of luminous AGB stars are needed to resolve this issue. If non-solar Mg isotopic ratios fail to be observed in luminous AGB stars then a major rethink of the nucleosynthesis processes in AGB stars is required! The neon abundances observed in planetary nebulae are compared to our models, with very good agreement between the Z = 0.02 models and the galactic PNe. There is also good agreement between most of the Z = 0.008 models and the LMC PNe, but these models produce too much 22Ne at about 3M . This result might help constrain the amount of dredge-up required near the end of the TP–AGB, when the convective envelope is small. We discuss the implications of the observations and note that further work needs to be done to determine if and when deep mixing takes place on the AGB. There is substantial evidence that some form of extra-mixing beyond the formal boundary of the convective envelope is required to match the range of observed abundances in real AGB stars. The physical mechanism for this mixing is unknown, but it might be similar to that required on the first giant branch. Several investigators have included simple extra-mixing schemes (see Nollett et al. 2003) with some success to model AGB nucleosynthesis. Further investigation into this area is required, but success will be limited until we have a better understanding of convection and turbulent mixing in stars. Also the uncertainties in the nuclear reaction rates need to be addressed in more detail. In particular, a study of how the uncertainties in the reactions involved in the Mg–Al chain, and the 22Ne α capture reactions effects the surface abundances of AGB models is needed. The impact of−the different mass-loss laws on AGB nucleosynthesis is still poorly understood, but as Travaglio et al. (2001c) show, the amount of mass loss is important for species effected by HBB such as 7Li. Owing to the complex nature of mass loss in AGB stars, we most likely have some time to wait before we will have a complete theory. At present, we know that mass loss seems to be a dual process, involving both radial pulsations and the formation of dust grains, which are acted upon by radiation pressure and pushed away from the star (Vassiliadis & Wood 1993). There is also the matter of pre-solar grains, many of which were probably produced inside AGB stars, such as the mainstream silicon carbide grains. These grains provide an unparalleled wealth of information that the models must be compared to. Ultimately, it is the observations, either from AGB stars, PNe or pre-solar grains that the models must face. In conclusion, the models presented here provide a good qualitative fit to the observations, but they do not CHAPTER 5. AGB NUCLEOSYNTHESIS 137 yet provide a definitive understanding of all nucleosynthetic processes that occur during the TP–AGB phase. Chapter 6

Parameterizing the Third Dredge-Up

In this chapter we use the models without mass loss to find an approximate fit for the core min mass at the first thermal pulse, Mc(1), the core mass at the first dredge-up episode, Mc , and the third dredge-up efficiency parameter λ, as a function of the total mass, metallicity and pulse number. The bulk of this chapter is published in the paper “Parameterizing the third dredge-up”, by Karakas, Lattanzio & Pols, 2002, Publications of the Astronomical Society of Australia, volume 19, pages 515 to 526.

6.1 Introduction

The efficiency of the third dredge-up (TDU) is quantified by the parameter λ (see Fig- ure 6.1), which is the ratio of mass dredged up by the convective envelope, ∆Mdredge, to the amount by which the core mass increases owing to H-burning during the preceding interpulse period, ∆MH, ∆Mdredge λ = . (6.1) ∆MH The value of λ depends on physical parameters such as the core mass, metallicity (and hence opacity) as well as time and the total mass of the star. Exactly how λ depends on these quantities is still unknown. The two main reasons for this are the difficulty in locating the inner edge of the convective envelope during the dredge-up phase (Frost & Lattanzio 1996; Mowlavi 1999b) and the huge computer resources required to explore an appropriate range of mass and composition over such a computationally demanding evolutionary phase. Without a systematic investigation of the dredge-up law, only certain trends have been identified by extant models, such as the increase of λ with decreasing Z and increasing mass (Boothroyd & Sackmann 1988d), and the fact that below some critical envelope mass, the third dredge-up ceases altogether (Wood 1981; Straniero et al. 1997). The convenient fact that the stellar luminosity on the AGB is a nearly linear function of the H-exhausted core mass has stimulated the development of “synthetic” AGB evolution models, as a quick way of simulating stellar populations on the AGB. The main observational constraint which models must face is the carbon-star luminosity function (CSLF) for the Magellanic Clouds. In some synthetic AGB evolution calculations, for example, as

138 CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 139

Interpulse period

∆ Mdredge ∆MH

∆M λ = ∆ dredge Mass of H−exhausted core MH

Time Figure 6.1: The de®nition of λ, shown schematically, where the x±axis represents time and the y±axis represents the mass of the H-exhausted core. performed by Groenewegen & de Jong (1993) and Marigo, Bressan & Chiosi (1996), λ is treated as a constant free parameter, calibrated by comparison with the CSLF. Synthetic codes enable us to investigate a diverse range of problems, such as binary population synthesis (Hurley, Tout, & Pols 2002), AGB population studies (Groenewegen & de Jong 1993), and the calculation of stellar yields from AGB stars (van den Hoek & Groe- newegen 1997; Marigo 2001; Izzard et al. 2003). Most parameterisations used in synthetic evolution studies are found either empirically from observations (such as the mass-loss law) or from results from full stellar calculations, such as the core–mass–interpulse–period relation. Currently there are no parameterisations in the literature based on detailed evolutionary models that describe the behaviour of λ with total mass, metallicity, age and/or core mass. However with current computing power the problem becomes time consuming rather than impossible. Hence we have embarked on just such an exploration of relevant parameter space using full detailed evolutionary models. Our aim is to determine the dependence of evolutionary behaviour (such as the dredge-up law) on the various stellar parameters, and to provide these in a form suitable for use in synthetic population studies. The stellar models are presented in 6.2, including a discussion of the model uncertainties. In 6.3 we present our parameterization§ of the third dredge-up, including an § approximate fit to the core mass at the first thermal pulse, Mc(1), and the core mass at the first dredge-up episode. The fit to the third dredge-up efficiency parameter, λ is presented in 6.4, and in 6.5 we give a discussion of the results. We finish with a summary in 6.6. § § § CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 140

6.2 Stellar Models Without Mass Loss

6.2.1 Model Details We calculate evolutionary sequences without mass loss using the evolution code described previously in 3.1 on page 32. We evolve models with masses between 1M and 6M , for the three compositions§ discussed in previous chapters: solar (Z = 0.02), Lar ge Magellanic Cloud (LMC) (Z = 0.008) and Small Magellanic Cloud (SMC) (Z = 0.004). The initial compositions for H, 4He, 12C, 14N and 16O are presented in Table 3.1 on page 33.

6.2.2 Convection and the Third Dredge-Up The amount of third dredge-up in evolutionary calculations crucially depends on the numerical treatment of convective boundaries: many codes do not find any dredge-up for low-mass stars without some form of overshoot (Herwig et al. 1997; Mowlavi 1999b). Herwig (2000) finds very efficient dredge-up, with λ > 1, in a 3M Z = 0.02 model with diffusive convective overshoot on all convective boundaries, but no TDU for the same mass without overshoot. Pols & Tout (2001) find very efficient dredge-up, with λ 1, in a 5M Z = 0.02 model using a completely implicit and simultaneous solution for∼the stellar structure, nuclear burning and convective mixing. Frost & Lattanzio (1996) find the treatment of entropy to affect the efficiency of the TDU, and Straniero et al. (1997) find the space and time resolution to be important. In view of this strong dependence on numerical details, it is important to specify carefully how we treat convection. We use the standard mixing-length theory for convective regions, with a mixing-length parameter α = l/HP = 1.75, and determine the border by applying the Schwarzschild criterion. Hence we do not include convective overshoot in the usual sense. We do, however, recognize the discontinuity in the ratio r of the radiative to adiabatic temperature gradients at the bottom edge of the convective envelope during the dredge-up phase. We search for a neutral border to the convective zone, in the manner described in Frost & Lattanzio (1996) and in 3.1.1. We do not repeat that discussion here, but note that this algorithm does seem to increase§ the efficiency of the third dredge-up in the stellar models. The initial 4He abundance and the mixing-length parameter α are set by matching the observed solar radius and luminosity. Hence the value of α depends not only on the opacities used but also on the initial solar composition and the numerical treatment of convection. There is some spread in the values of α found by different authors, reflecting the uncertainties in the modelling process. Using an earlier version of the evolution code but with the older opacity tables from the Los Alamos Opacity Library (LAOL; Huebner et al. 1977), Lattanzio (1986) sets α = 1.0 and does not find the TDU in any of his calculations. Lattanzio argues that increasing α to approximately 1.5 is required to match the observed effective temperature and luminosity of the carbon star TW Hor. Herwig (2000), using the OPAL tables of Iglesias & Rogers (1996), requires α = 1.7 and Y = 0.28 to obtain a solar model. Using the OPAL opacities, Straniero et al. (1997) require α = 2.2 to match the solar radius, but using the LAOL tables Chieffi & Straniero (1989) require α = 1.6. Girardi CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 141 et al. (2000) require Y = 0.273 and α = 1.68 to match the solar luminosity and radius, whereas the older calculations of Bressan et al. (1993) require Y = 0.282 and α = 1.63. We note that these calculations use an older version of the OPAL tables (Iglesias, Rogers & Wilson 1992). Boothroyd & Sackmann (1999) find α = 1.67 using the OPAL tables but require α = 1.0 using the LAOL tables (Boothroyd & Sackmann 1988c,d). These authors comment that the appropriate value of α is uncertain, and values up to 2 or more are not unreasonable. These authors find, as suggested by Wood (1981) and Lattanzio (1986) afterward, that increasing α to 1.5 (from 1.0) is necessary to obtain a carbon star, and an increase in α leads to conditions more favourable for dredge-up. We test this in one model (to be described in more detail later) by increasing α to 2.2 in a 1.5M , Z = 0.02 model (because this is the value used by Straniero et al. 1997). Finally, although we believe our treatment of convection is physically motivated and phenomenologically sound, our results cannot be regarded as the definitive solution to the difficult problem of the third dredge-up. However, the important point is that all our models are computed using the same algorithm. Together they constitute an internally consistent set of models covering a wide range in mass and metallicity.

6.2.3 Mass Loss Mass loss is a crucial part of AGB evolution, and seriously affects the TDU in two ways. Firstly, for the more massive stars, dredge-up can be terminated when the envelope mass decreases below some critical value. Secondly, for lower masses, mass loss may terminate the AGB evolution before the H-exhausted core reaches the minimum value for the TDU to occur. However, the mass-loss rate in AGB stars is very uncertain, and for this reason we calculate each stellar sequence both with and without mass loss. By neglecting mass loss, we find the limiting behaviour of dredge-up for each model we calculate. In 6.3 we parameterize this dredge-up behaviour in the absence of mass loss. § When this parameterisation is used in synthetic evolutionary calculations, the chosen mass-loss law will determine if the models reach the limiting behaviour we provide. The subsequent AGB evolution and dredge-up will then be modified by the choice of mass-loss law. For example, we will determine a minimum core mass for the TDU at a given mass and composition in the case without mass loss, and whether the model reaches this core mass or not will depend on the chosen mass-loss rate. Alternatively, a particular mass-loss law may or may not prevent a model from reaching the asymptotic value for λ, which can only be determined from full stellar models without the inclusion of mass loss.

6.2.4 Model Results The behaviour of models that experience the second dredge-up (SDU) (generally masses M0 & 4 depending on Z, or core masses greater than 0.8M ) differs qualitatively from of lower-mass models. To more accurately find the minimum mass for the SDU, Msdu, for the three different compositions we considered, we ran a few extra model with different masses to the start of the thermally-pulsing AGB only. The SDU occurs in models with M > 4.05M at Z = 0.02, M > 3.8M at Z = 0.008, and M > 3.5M at Z = 0.004.

CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 142

Figure 6.2: (Top) Mass of the H-exhausted core against time (in years) for the 4M , Z = 0.008 model without mass loss. (Bottom) The dredge-up eﬃciency parameter, λ as a function of the core mass. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 143

The 4M , Z = 0.008 model (see Figure 6.2) is a good example of the type of behaviour observed in the massive models that experience the SDU. This sequence shows 30 thermal pulses; and the dredge-up parameter, λ is seen to increase quickly, reaching a value near 0.9 in about six pulses and maintains that value until the end of the calculation. We see that λ oscillates a little as unity is approached; this may indicate the imprecision of the dredge-up algorithm. Mass loss has little effect on the depth of the TDU for models with M & 3M (see 4.1.1). For models that experience the SDU, λ reaches an asymptotic value of about 0.§9 or higher, regardless of the composition and the inclusion of mass loss. The most efficient TDU (where λ 1) occurs in the 4M models, regardless of the composition. For the solar and LMC composition∼ models, the deepest dredge-up occurs at this mass, where λ 0.97. For the SMC composition models, the deepest TDU occurs max ≈ in the mass range 3 to 5M , where λmax is between 0.96 and 0.99. For the most massive models (M > 5M ) we observe that λmax decreases with increasing initial mass. The 4 and 5M models generally experience deeper TDU compared to the 6M models, regardless of metallicity. For example, the 4 and 5M models have λmax 0.97 whilst the 6M models ∼ generally have λmax 0.91. Not only is the TDU deeper for the 4 and 5M cases but λmax ∼ is reached more quickly, in under five thermal pulses whilst the 6M model reaches λmax slowly, taking up to 10 thermal pulses for all metallicities modelled. We ran two calculations to test the effectiveness of extrapolating the fitting formulae out to larger masses. Both models are of solar composition, with masses of 6.5M and min 6.75M respectively. For the 6.5M case, Mc = 0.957 and λmax = 0.910, and this model min is observed to reach λmax after about 20 thermal pulses. For the 6.75M case, Mc = 0.978 and λmax = 0.870, where λmax is reached after 15 thermal pulses. Both of these models have shallower dredge-up than the 6M , Z = 0.02 model, highlighting the point mentioned above, that over a certain mass, the depth of the TDU decreases with increasing mass. We now turn our attention to the lower mass models. From Chapter 4 we know that mass loss can have a significant effect on the efficiency of the TDU in the low mass models, which show either shallower dredge or none at all compared to the models without mass loss. Because many of the mass-loss models do not reach the minimum core mass for the min TDU, we use the models without mass loss to parameterize λmax and Mc . An example of a low-mass model with efficient TDU is the 1.5M , Z = 0.004 model without mass loss, the results of which are shown in Figure 6.3. The 1.5M , Z = 0.004 with mass loss also experiences some dredge-up, where λmax 0.2 (compared to 0.375 for the model without mass loss). ∼ Straniero et al. (1997) find substantial dredge-up in a 1.5M Z = 0.02 model without mass loss. These authors observe λ to increase to 0.34 before declining to almost zero; min and the minimum core mass for the TDU is Mc = 0.625 (Straniero et al. 1997). For min the same mass and composition, we obtain λmax 0.05 and Mc = 0.658; less efficient dredge-up which begins at a large core mass. The∼ difference might be due to a different mixing-length parameter α. For the purposes of comparison, we increase α to 2.2 in a min 1.5M , Z = 0.02 model, and find Mc = 0.634 and λmax = 0.44, with λ declining to 0.30 after 10 TDU episodes. The final core mass is 0.70M when convergence difficulties ceased the calculation, compared to 0.724M for Straniero et al. (1997). In summary, the 1.5M , Z = 0.02 model calculated with α = 2.2 shows remarkably similar behaviour to

CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 144

Figure 6.3: (Top) Mass of the H-exhausted core against time (in years) for the 1.5M , Z = 0.004 model without mass loss. (Bottom) The dredge-up eﬃciency parameter, λ as a function of the core mass. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 145

min Table 6.1: Mc(1), λmax, and Mc for the Z = 0.02 models without mass loss. See the text for a detailed description of the contents. Z=0.02 min M0 Mc(1) Mc λmax TP(λmax) No. TPs 1.25 0.556 – 0.0 – 24 1.5 0.561 0.658 0.0486 (L) – 24 1.75 0.561 0.634 0.224 9 28 2.0 0.554 0.632 0.457 (L) – 27 2.25 0.540 0.624 0.709 14 37 2.5 0.549 0.625 0.746 12 36 3.0 0.595 0.635 0.790 13 25 3.5 0.662 0.676 0.850 12 26 4.0 0.793 0.799 0.977 5 22 5.0 0.862 0.866 0.955 9 28 6.0 0.915 0.918 0.922 12 65

that found by Straniero et al. (1997). In Table 6.1 and 6.2 we present the results for the Z = 0.02, Z = 0.008 and Z = 0.004 models without mass loss, respectively. In each table we include the initial mass, the core min mass at the first thermal pulse, Mc(1), the core mass at the first TDU episode, Mc and the maximum TDU efficiency parameter for each model, λmax. In models with efficient TDU, λ approaches an asymptotic value, which we give as λmax, whereas in other models we present the maximum value, calculated in most cases after the last thermal pulse. For these cases, we put the symbol “L” next to the value of λmax, noting it is not the asymptotic value for that particular model but the largest value of λ calculated. Column 5 is the number of thermal pulses taken to reach the asymptotic value and in column 6 we give the total number of thermal pulses calculated for each model without mass loss. Note that the calculation is terminated when λ reaches an asymptotic value so there may be less thermal pulses for some models compared to the cases with mass loss. In Figure 6.4 we min show Mc(1), Mc and λmax as a function of the initial mass for the Z = 0.02, Z = 0.008 and Z = 0.004 models without mass loss. In each panel, the blue solid line refers to the Z = 0.02 models, the black dashed line to the Z = 0.008 models and the red dash-dotted line to the Z = 0.004 models. A comparison with current synthetic calculations is useful. Most calculations assume min a constant value of Mc (Groenewegen & de Jong 1993; van den Hoek & Groenewegen 1997), but Marigo (1998b) attempted to improve on this. She assumes that the TDU occurs if, following a thermal pulse, the temperature at the base of the convective envelope dred dred reaches a specified value, Tb . Both the TDU efficiency parameter, λ, and Tb are free parameters, which Marigo sets by fitting the carbon-star luminosity functions in the dred Magellanic Clouds. Marigo (1998b) varies Tb between 6.4 and 6.7, and in her Figure 3 we see the results for the Z = 0.008 models. The best fit to the LMC CSLF is obtained dred with log Tb = 6.4, and λ = 0.5 (note that λ = 0.65 is required for the SMC). For CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 146

min Figure 6.4: The Mc(1), Mc and λmax against initial mass for the Z = 0.02, Z = 0.008 and Z = 0.004 models without mass loss. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 147

min Table 6.2: Mc(1), λmax, and Mc for the Z = 0.008 and Z = 0.004 models without mass loss. See the text on page 145 for a detailed description. min M0 Mc(1) Mc λmax TP(λmax) No. TPs Z=0.008 1.0 0.535 0.657 0.0016(L) – 22 1.5 0.550 0.624 0.306 6 21 1.75 0.555 0.609 0.523(L) – 21 1.9 0.551 0.581 0.605(L) – 21 2.1 0.540 0.596 0.656 11 22 2.5 0.540 0.591 0.792(L) – 27 3.0 0.629 0.639 0.882 11 20 3.5 0.744 0.748 0.957 9 22 4.0 0.830 0.833 0.990 6 17 5.0 0.870 0.871 0.974 7 27 6.0 0.926 0.929 0.932 9 26 Z=0.004 1.0 0.541 – 0.0 – 22 1.5 0.551 0.588 0.375(L) – 15 1.75 0.558 0.589 0.611(L) – 16 1.9 0.558 0.589 0.669 10 18 2.1 0.550 0.578 0.717(L) – 16 2.25 0.537 0.577 0.770 15 26 2.5 0.578 0.607 0.783 7 15 3.0 0.699 0.706 0.963 4 16 3.5 0.804 0.808 0.982 5 20 4.0 0.842 0.845 0.990 4 20 5.0 0.889 0.891 0.970 6 24 6.0 0.962 0.963 0.933 11 30

dred the three diﬀerent values of log Tb shown in Figure 3, Marigo (1998b) also provides min the values of Mc as a function of the initial mass for her LMC models. For M 6 2.5, min dred our values for Mc agree well with her values calculated with log Tb = 6.7. At a dred value of log Tb = 6.4, Marigo’s models begin the TDU at a much smaller core mass ( 0.53M ) than we ﬁnd in our LMC models. For example, in a 1.5M , Z = 0.008 model ∼ min min she obtains Mc 0.53 compared to our value of Mc = 0.624 and van den Hoek & ≈ min Groenewegen (1997)’s value of Mc = 0.58. This result has serious implications for the surface abundance changes caused by the TDU during the TP–AGB phase, and for the stellar yields. We will show in Chapter 7 that Marigo’s models produce more 4He, 12C and 16O than any other calculation. The reason for this should now be clear: her assumption that the TDU occurs at such as small core mass compared to all the other calculations. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 148

min 6.3 The Fit for Mc(1) and Mc We discuss the fit for the core mass at the first thermal pulse and the fit for the core mass at the first TDU episode separately.

6.3.1 The Core Mass at the First Thermal Pulse Wagenhuber & Groenewegen (1998) provide a fitting formula for the core mass at the first thermal pulse, Mc(1) as a function of mass and metallicity (their equation 13). We compare their Population I fit to our results for Z = 0.02, and find qualitative agreement in the shape of the formula, but there are significant quantitative differences. The same is true for lower metallicities, when we linearly interpolate the coefficients given in Wagenhuber & Groenewegen (1998) for Z = 0.008 and Z = 0.004 and compare the resulting relation to our models. Here we provide modified coefficients for the fitting formula given by Wagenhuber & min Groenewegen (1998), instead of providing a completely new fit to Mc(1) as we do for Mc and λmax (see Sections 6.3.2 and 6.4). We choose to do this for two reasons. Firstly, the shape of the function provided by Wagenhuber & Groenewegen (1998) for Mc(1) (equations 13a–c) is a very good approximation to the shape of the Mc(1)–initial-mass relation that we find from our models. Secondly, researchers who already use the Wagenhuber & Groenewegen (1998) Mc(1) fit for Population I and II stars in their synthetic evolution codes can easily convert to our fit for Population I, LMC, and SMC models. The equations used by Wagenhuber & Groenewegen (1998) to fit Mc(1) are

2 Mc(1) = ( p1(M0 p2) + p3) f + (p4 M0 + p5)(1 f ), (6.2) − M p− 1 − 0− 6 − f = 1 + e p7 , (6.3) where p1 to p7 are constants that depend on the metallicity and M0 is the initial mass (in solar units). These equations are almost constant for stars with M0 6 2.5M and almost linear for stars that experience the second dredge-up (for masses greater than about 4M ).

The constant coeﬃcients, p1 to p7, that best ﬁt the model results are given in Table 6.3.

Table 6.3: Coeﬃcients for equations (6.2) and (6.3): p1, p2, p3, p4, p5, p6, and p7

Z p1 p2 p3 p4 p5 p6 p7 0.02 0.038515 1.41379 0.555145 0.039781 0.675144 3.18432 0.368777 0.008 0.057689 1.42199 0.548143 0.045534 0.652767 2.90693 0.287441 0.004 0.040538 1.54656 0.550076 0.054539 0.625886 2.78478 0.227620

In Figure 6.5 we show the modified fit for Mc(1) (solid line) as a function of the initial mass for the Z = 0.02 (top panel), Z = 0.008 (middle panel) and Z = 0.004 (bottom panel) models. In each panel we also include the results from the models as solid points. We fit Mc(1) for each composition between 1M and 6M , and the lines between 6M and 8M

CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 149

Figure 6.5: The ®t for Mc(1) using the Wagenhuber & Groenewegen (1998) equation with modi®ed coefficients (solid line). In each panel we include the model results for comparison. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 150 are extrapolations from the fitting functions and may not reflect real model behaviour. For the 6.5M and 6.75M models, we obtain Mc(1) = 0.954 and 0.975 respectively. The extrapolated values from equations (6.2) and (6.3) are Mc(1) = 0.934 for the 6.5M and 0.943 for the 6.75M models. Thus these equations under-estimate the minimum mass for the first thermal pulse for models with masses larger than 6M . The amount by which the fit under-estimates Mc(1) is small and thus should be no concern up to about 7M . However, we do not recommend using the fit for more massive models.

6.3.2 The Core Mass at the First TDU Episode The value of the core mass at the ﬁrst TDU episode depends critically on the occurrence min of the second dredge-up. For models with masses below Msdu, the behaviour of Mc is well approximated by a third-order polynomial of the form

min 2 3 Mc = a1 + a2 M0 + a3 M0 + a4 M0, (6.4) where a1, a2, a3, and a4 are constants that depend on Z and are given in Table 6.4, and M0 is the initial mass of the star (in solar units). min For masses greater than Msdu, we observe that Mc rises nearly linearly with total mass. min From the results presented in 6.2 it is clear that if Mc(1) > 0.7M , then Mc has a § value very close to Mc(1) (diﬀering by less than 0.005M ). Hence it is justiﬁed to take min Mc = Mc(1) in this case. Since equation (6.4) diverges for large masses, in practice we min recommend calculating Mc by the following procedure:

min min Mc = max Mc(1), min (0.7M , Mc ∗) , (6.5) h i min min where Mc ∗ is given by equation (6.4). This ensures that Mc > Mc(1) as required, while min Mc = Mc(1) if Mc(1) > 0.7M .

Table 6.4: a1, a2, a3, and a4 for equation (6.4) min Z Mc a1 a2 a3 a4 0.02 0.732759 -0.0202898 -0.0385818 0.0115593 0.008 0.672660 0.0657372 -0.1080931 0.0274832 0.004 0.516045 0.2411016 -0.1938891 0.0446382

min In Figure 6.6 we show the fit for Mc (solid line) as a function of the initial mass for the Z = 0.02 (top panel), Z = 0.008 (middle panel) and Z = 0.004 (bottom panel) models. We also include the results from the models without mass loss for comparison. We only min fit Mc between 1.5M and 6M for the solar composition models, and between 1M and 6M for the LMC and SMC composition models. We note that the fits between 6M and 8M are extrapolations and may not reflect model behaviour. For the 6.5M and 6.75M min min models, we obtain Mc = 0.958 and 0.978 respectively. We set Mc = Mc(1) when CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 151 the core mass is larger than 0.7M thus the minimum core mass for the TDU to begin is simply the value of Mc(1), given previously for the 6.5M and 6.75M models. For both min of these models, the fit for Mc under-estimates the observed model behaviour. However, the deviation from the model value is small, so we conclude that the fit is adequate to about 7M . For more massive models, a new fit is likely to be required.

6.4 The Fit for λmax

The behaviour of λmax is nearly linear at low M, rising steeply with mass until M 3M before turning over and flattening out to be almost constant at high mass. This beha∼viour is seen in the lower panel of Figure 6.4. We fit λmax using a rational polynomial of the form b + b M + b M3 = 1 2 0 3 0 λmax 3 , (6.6) 1 + b4 M0 where b1, b2, b3, and b4 are constants given in Table 6.5, and M0 is the initial mass (in M ). We provide a separate fit for each composition, but interpolation between the coefficients of the polynomials is possible for arbitrary Z in the range 0.02 to 0.004. In Figure 6.7 we show the fit for λmax (solid line) as a function of the initial mass for the Z = 0.02 (top panel), Z = 0.008 (middle panel) and Z = 0.004 (bottom panel) models. The fit for the Z = 0.02 models is valid between 1.5 to 6M , whereas the fits for the LMC and SMC models are valid between 1 to 6M . The line between 6M and 8M is an extrapolation from the fitting functions and may not reflect real model beha viour. The 6.5M model has λmax = 0.894 and the 6.75M model has λmax = 0.871. These values are in excellent agreement with the fit for Z = 0.02 extrapolated out to larger masses, where we obtain λmax = 0.891 and 0.879 for the 6.5 and 6.75M models respectively. This result suggests that we could possibly use the fit for λmax out to 7 or even 8M , noting that we do not guarantee the fit for these masses.

Table 6.5: b1, b2, b3, & b4 for equation (6.6) for λmax

Z λmax

b1 b2 b3 b4 0.02 -1.17696 0.76262 0.026028 0.041019 0.008 -0.609465 0.55430 0.056878 0.069227 0.004 -0.764199 0.70859 0.058833 0.075921

min For Z = 0.02, we only fit λmax and Mc down to 1.5M and as a consequence the fit to λmax goes negative for masses below this. Therefore, if equation (6.6) yields a negative value, λmax should be set to zero. Interpolating between the coefficients of equation (6.6) in the range 0.02 < Z < 0.008 will result in negative values of λmax between 1 6 M0(M ) 6

1.5. Again we suggest setting λmax = 0 when this happens. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 152

min Figure 6.6: The ®t for Mc (solid line) against the initial mass for the Z = 0.02, Z = 0.008 and Z = 0.004 models without mass loss. In each panel we include the model results for comparison. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 153

Figure 6.7: The ®t for λmax (solid line) against the initial mass for the Z = 0.02, Z = 0.008 and Z = 0.004 models without mass loss. In each panel we include the model results for comparison. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 154

6.4.1 The Dredge-Up Parameter, λ as a Function of Time To model the behaviour of λ with pulse number, we propose a simple method. When min Mc > Mc , λ starts increasing with pulse number, N, until λ asymptotically reaches λmax for large enough N. Since our models give little information on the decrease of λ with decreasing envelope mass we suggest λ = 0 when Menv 6 Menv,crit, where Menv,crit is some critical value below which the TDU does not occur. Low-mass models with dredge-up suggest that Menv,crit . 0.2M . The behaviour of λ with pulse number can be modelled with the simple function:

(N/Nr) λ(N) = λ (1 exp− ), (6.7) max − where N is pulse number, measured from the first pulse where the core mass is greater min than Mc . Nr is a constant, determining how fast λ reaches λmax. Due to the nature of the exponential function given by equation (6.7), when N & 8Nr, equation (6.7) gives a value indistinguishable from λmax. In finding an appropriate value of Nr for each model, we experiment with different values for each mass. The increase in λ observed in some models can be fitted by a range of Nr, especially for models that exhibit a lot of scatter in λ. In the top panel of Figure 6.8 we show the increase of λ for the 5M , Z = 0.02 model without mass loss. The solid points are the data points from the model, the solid line is the fit using equation (6.7) with Nr = 4 and the dashed line is the fit with Nr = 6. Hence this model is one such case where a range of Nr values results in a reasonable fit to the behaviour of λ with increasing pulse number. In the lower panel of Figure 6.8 we show the increase of λ for the 5M , Z = 0.004 model without mass loss. Similar to before, the solid points are the model results, the solid line the fit with Nr = 2 and the dashed line the fit with Nr = 3. While the fit with Nr = 2 approximates the model behaviour best, the fit with Nr = 3 is a good fit after 5 or more thermal pulses. Since the fitting formulae for the third dredge-up will be used in synthetic evolutionary algorithms with mass loss, we also need to examine the behaviour of the mass-loss models. In Chapter 4 we show that the 5M , Z = 0.004 model experiences 83 thermal pulses, so the composition changes from the first 10 or so pulses will be less important to the final composition of the star than the first 10 pulses of a low-mass model that experiences < 30 thermal pulses. In Table 4.2, we show that the low-mass (M < 2.5M ) LMC and SMC models only experience about 20 thermal pulses, and the 1.5M mass models only experience about 15 thermal pulses. Hence, for a given composition, we should set Nr to be consistent with the low-mass models. For the low- mass models with very small values of λmax . 0.1, equation (6.7) does not result in a good min fit regardless of the Nr value used. We suggest setting λ = λmax when Mc > Mc for these low-mass models. From Table 6.6 we see that there is no systematic trend for Nr with mass. The variation cannot therefore be readily modelled with a simple function. We have argued above that the time dependence of λ for low-mass stars is quite important as they have few thermal pulses, whilst more massive stars have many thermal pulses so the first pulses are not so CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 155

Figure 6.8: Modelling the increase of λ with pulse number. The increase of λ for the 5M , Z = 0.02 model (top panel) and the 5M , Z = 0.004 model (lower panel) are shown. Further description is given in the text. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 156

inﬂuential. Therefore we suggest using a constant Nr value independent of M for a given Z, consistent with the low-mass models, e.g. Nr = 4 for Z = 0.02 and Nr = 3 for Z = 0.008 and Z = 0.004.

Table 6.6: Table of best-®tting Nr values for Z = 0.02, Z = 0.008, and Z = 0.004

M0 Z = 0.02 Z = 0.008 Z = 0.004 1.5 1 1 2 1.75 3 3 3 1.9 3 2 3 2.25 4 3 3 2.5 4 4 2 3 3.5 4 1 3.5 3 2 1 4 2 2 1 5 5 3 2 6 4 3 3

6.5 Discussion

6.5.1 The Core Mass at the First Pulse The value of the core mass at the ﬁrst thermal pulse is perhaps not crucial to synthetic models, because it is the surface composition changes caused by dredge-up that provide min constraints on the models. Hence Mc is more important. Nevertheless, comparisons min with the CSLF in the Magellanic Clouds indicate that detailed models overestimate Mc min and it is useful to know Mc(1) which is in principle the theoretical lower limit for Mc . However, Mc(1) may also be overestimated. There are few parameterizations of this quantity in the literature. Lattanzio & Malaney (1989) give a simple ﬁt for masses between 1 and 3M :

0.53 (1.3 + log Z)(Y 0.2), Z 0.01 − − ≥ Mc(1) =  0.524 + 0.58(Y 0.20) + [0.025 20Z(Y 0.20)M], 0.003 6 Z < 0.01 (6.8)  − − −  (0.394 + 0.3Y) exp[(0.10 + 0.3Y)M], Z < 0.003  where Y is the initial 4He mass fraction and M is the total mass (in M ). Equation (6.8) under-estimates Mc(1) compared to the results from the Z = 0.02 models. For example, the 3M , Z = 0.02 model has Mc(1) = 0.595M , whilst we obtain 0.567M from equation (6.8). At Z = 0.004, equation (6.8) performs much worse. For example, in the 3M ,

Z = 0.004 model, Mc(1) = 0.706 but equation (6.8) yields Mc(1) = 0.565M . We should point out that equation (6.8) was derived from older input physics (notably older opacities) than used in the present calculations. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 157

Both Renzini & Voli (1981) and later, Groenewegen & de Jong (1993) use the fit originally proposed by Becker & Iben (1979, 1980) for massive AGB stars. For the low-mass stars, Groenewegen & de Jong (1993) use equation (6.8). A more detailed fit is given by Wagenhuber & Groenewegen (1998), which is used by Marigo (2001). Since this latter fit reproduces the shape very well, we have simply modified the coefficients as described in Section 6.3.1 to provide a far superior fit to the current results.

min 6.5.2 The Third Dredge-Up: Mc and λmax min Most synthetic calculations assume constant Mc and constant λ. Groenewegen & de Jong min (1993) use the constant values given by Lattanzio & Malaney (1989) for Mc , and then adjust λ to try to fit the CSLF of the Magellanic Clouds. Groenewegen & de Jong (1993) min min find that Mc must be decreased from the theoretical value, they settle on Mc = 0.58 and λ = 0.75 to fit the observations. A similar procedure was followed by Marigo et al. min (1996) and these authors obtain Mc = 0.58 and λ = 0.65. Marigo (2001) now uses a more sophisticated algorithm for determining the onset of dredge-up, as discussed in 6.2.4. The parameterizations we have given here should be a significant improvement to§ the constant values used for most synthetic studies. In the discussion below we will compare our results with other detailed evolutionary calculations. At a given mass, λmax is observed to increase with decreasing Z, so the low-mass models (M0 < 2.25M , Z = 0.008, 0.004 with mass loss) become carbon stars with λmax as high as 0.6 for the 1.75M , Z = 0.004 model. This effect is not so noticeable for the higher mass stars (M & 3M ), where dredge-up quickly deepens with pulse number, and λmax 0.9 for all compositions. In Chapter 4 we discuss that we only obtain the TDU in models with≈ M > 2.25M at Z = 0.02 and M > 1.5M at an SMC and LMC composition. In comparison to our results, Vassiliadis (1992), who uses a different version of the Mount Stromlo stellar evolution code (Wood & Faulkner 1986, 1987) and the older LAOL opacities (Huebner et al. 1977), obtains the TDU for M0 > 2.5M at a LMC composition, and for M0 > 2M at a SMC composition. Clearly, the larger OPAL opacities we use (Frost 1997) and the improved modelling of the TDU by Frost & Lattanzio (1996) make a considerable difference. Straniero et al. (1997), using the OPAL opacities, find λ 0.3 for a solar composition max ≈ 1.5M model without mass loss, whereas we find λmax 0.05 for the same mass and composition. This is probably due to the difference in mixing-length≈ parameter: we use 1.75 and Straniero et al. (1997) use the higher value of 2.2. The test calculation demonstrates that we find considerable deeper TDU when the mixing-length parameter is increased. These discrepancies must also be related to the numerical differences between the codes (Frost & Lattanzio 1996; Lugaro 2001) and the uncertain treatment of convection. Pols & Tout (2001) obtain very similar values of λ for a 5M , Z = 0.02 model compared to our results. These authors use a fully implicit method to solv e the equations of stellar structure and convective mixing, and they observe λ to increase to 1.0 in only six TPs while our models reach λ 0.95 much more slowly (see Figure 6.8).≈ Herwig (2000) includes≈diffusive convective overshoot during all evolutionary stages and on all convective boundaries on two solar composition models of intermediate mass. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 158

Without overshoot, no dredge-up is found for the 3M model. With overshoot, efficient dredge-up is found for both the 3 and 4M models, where λ 1 for the 3M model and λ > 1 for the 4M model, which has the eff ect of decreasing the∼ mass of the H-e xhausted core over time. Clearly , the inclusion of convective overshoot can substantially increase the amount of material dredged up from the intershell to the surface. Langer et al. (1999) use a hydrodynamic stellar evolution code to model the effects of rotation on the structure and mixing of intermediate mass stars, and find some dredge-up in a 3M model of roughly solar composition.

6.5.3 The Carbon Star Luminosity Function The most common observation used to test the models is the reproduction of observed CSLFs. Mass loss has the largest effect on the Z = 0.02 models and as mentioned before, we do not find any TDU in the low-mass models. It thus seems likely that our Z = 0.02 models with mass loss cannot reproduce the low mass end of the Galactic carbon star distribution with progenitor masses in the range 1M to 3M (Wallerstein & Knapp 1998). The lowest mass solar composition model to become a carbon star is the 3M , Z = 0.02 model, which has C/O > 1 after 22 thermal pulses. We do note, however, that the Galactic CSLF is very uncertain. However, for LMC and SMC compositions, the CSLF is very well known (see discussion in Groenewegen & de Jong 1993). It is a long standing problem that detailed evolutionary models fail to match the observed CSLFs in the LMC and SMC (Iben 1981). Although many of our models with LMC and SMC compositions show enough TDU to turn them into carbon stars, we expect that they will not fit the low-luminosity tail of the CSLF. This is because we obtain small values of λ for M 6 1.5M , which is less than the value required by the synthetic calculations of λ 0.5 to fit the CSLF. Also, the values min ∼ min of Mc are larger for our LMC and SMC low-mass models than the Mc values used in synthetic algorithms (Groenewegen & de Jong 1993; Marigo et al. 1996; Marigo 1998b). Within the context of synthetic models, one usually modifies the dredge-up law to en- min sure that agreement is reached with the CSLFs. This usually means decreasing Mc and increasing λ, although this has previously been done crudely by altering constant values for all masses (possibly with a composition dependence). An improvement on this method dred min has been attempted by Marigo (1998b), who modifies the quantity log Tb instead of Mc to match the CSLFs, but λ is still a constant value for all masses and compositions. The models presented here show the variation with mass and composition of all dredge-up parameters. This has not been available previously. Although modifications may be required, perhaps caused by our neglect of overshoot (Herwig et al. 1997; Herwig 2000) or rotation (Langer et al. 1999), we expect the dependence on mass and composition to be maintained. CHAPTER 6. PARAMETERIZING THE THIRD DREDGE-UP 159

6.6 Summary and Further Work

We present extensive evolutionary calculations without mass loss that cover a wide range of masses and compositions. We evolve the models from the pre-main sequence to near the end of the AGB. We have provided fitting formulae for all the dredge-up parameters, in a form suitable for synthetic AGB calculations. As they stand, we expect that these will not fit the observed CSLFs in the LMC and SMC, a long-standing problem. Some adjustments to the fit for synthetic AGB models may be necessary, but must be consistent with the dependence on mass and composition presented here. This may constrain the adjustments and lead to a better understanding of where the detailed models can be improved. The current data set used to fit the TDU parameters only includes models with masses between 1 to 6M for three different initial compositions, Z = 0.02, 0.008, 0.004. Many 4 synthetic AGB calculations have studied Population II stars, where Z 10− (Wagenhu- ber & Groenewegen 1998; Izzard et al. 2003). An extension of the work∼ presented in this chapter would be to extend the fitting formulae to a much wider range of initial mass and composition. Models covering a larger range of initial mass (0.8 to 8M ) and initial composition (Z = 0.0001 to Z = 0.03) would significantly improve the TDU parameterizations presented here. Many synthetic AGB studies (for example Marigo 2001) include the mixing-length parameter, α, in their calculations, and hence the stellar yields become dependent on this quantity. A useful extension to the work presented in this chapter would be to determine how the TDU is modified in our detailed models by varying α. We have done this for one model, by increasing α from 1.75 to 2.2, and find that the TDU efficiency is significantly increased. To properly determine the effect of α on the TDU in our models, we would need to run sequences of stellar models, varying α between say, 1.6 to 2.5. Even doing this over a coarse grid of initial mass and composition would possibly enable us to include min a dependence on α on the TDU parameters. We would expect that the fits to Mc(1), Mc and λmax as functions of the total mass, Z and α would be more useful to synthetic AGB models than our current fits. Chapter 7

Stellar Yields

We present the method used to calculate the stellar yields in 7.1 followed by a discussion of the uncertainties in 7.2. The diﬀerence between the standard§ and the lower-limit set of stellar yields is discussed§ in 7.3, and in 7.4 we examine the physical assumptions of the other calculations we compare§ to. In 7.5§ we present our yields in graphical form and discuss the results. We conclude in 7.6. § Parts of this chapter are taken from§ the paper “Production of aluminium and the heavy magnesium isotopes in asymptotic giant branch stars”, by Karakas & Lattanzio, accepted for publication in the Publications of the Astronomical Society of Australia.

7.1 Stellar Yield Calculation

The deﬁnition of a stellar yield that we use is the same as that used by Marigo (2001)

τ dM Mk = [X(k) X0(k)] dt, (7.1) Z0 − dt where Mk is the yield of species k (in solar masses), dM/dt is the current mass-loss rate, X(k) and X0(k) refer to the current and initial mass fraction of species k, and τ is the total lifetime of the stellar model. The yield can be negative, in the case where the element is destroyed, and positive if it is produced. Equation (7.1) can be re-written as

τ dM τ dM Mk = X(k) dt X0(k) dt. (7.2) Z0 dt − Z0 dt The ﬁrst integral gives the total amount of species k expelled into the interstellar medium (ISM): τ tot dM Mk = X(k) dt, (7.3) Z0 dt tot where Mk is in solar masses, and is always positive. In practice, our models do not lose their entire envelope during the TP–AGB evolution

160 CHAPTER 7. STELLAR YIELDS 161 owing to convergence difficulties near the end of the AGB phase. The convergence difficulties mostly occur during the start of the third dredge-up phase, after the flash peak. The reasons for the convergence problems are unclear, but they are not associated with very small values of β (gas pressure to total pressure) since previous dredge-up episodes occur without problem for similar values of β. We note that convergence problems are sometimes associated with a rapid drop of the temperature at the base of the convective envelope and an increase in the stellar radius. Higher metallicity models, at a given initial mass, experience convergence difficulties earlier (i.e. at a larger envelope mass) than models of lower metallicity. This may indicate that the input physics, notably the opacities, are related to the problem. A thorough investigation into the cause of the convergence difficulties is required. For the lower masses considered, the remaining envelope mass is very small, and is less than what is lost during the last interpulse period. For example, the 1.5M , Z = 0.008 2 model has a final envelope mass of 7.9 10− M , whereas 0.585M is lost during the last interpulse period owing to the very high×mass-loss rates that develop during the superwind regime (Vassiliadis & Wood 1993). Note that much less mass is lost per interpulse period during the first few pulses of the TP–AGB. For these lower mass cases, we calculate the yield by simply removing the small remaining envelope (with its current composition). For the more massive models considered, there may be enough envelope mass remaining for a few pulses to occur. At this stage the envelope mass is relatively small, and the effects of the third dredge-up (hereafter TDU) can be large, as the dilution is smaller. Hot bottom burning (HBB) has ceased by the time the model calculation ends, so we do not need to make allowance for this rather complicated process (see Izzard et al. 2003 for a recent discussion). This means that the species most affected are those which are present in the intershell convective zones. Although the surface abundances may change significantly, in most cases the yields are not dramatically altered during these last few pulses, because the amount of mass involved is small. To calculate the stellar yields in these cases we will use the principles of synthetic AGB evolution to model the few remaining pulses.

7.1.1 Estimating the Surface Enrichment from the Last Thermal Pulses For each of our model sequences, we estimate the number of pulses remaining after the evolution calculation ceases. We do this by calculating the amount of matter lost from the envelope per interpulse period and the change to the mass of the H-exhausted core owing to H-shell burning and the TDU. If we assume the mass-loss rate is constant over the interpulse period (which is approximately correct for later pulses, see Figure 7.1) then the amount of matter lost per interpulse is

∆M = M˙ (r, l, M) τip, (7.4) where τip is the interpulse period (in years) and M˙ (r, l, M) is the mass-loss rate (in units 1 of M yr− ) as a function of the radius, luminosity and total stellar mass taken from Vas- siliadis & Wood (1993; hereafter VW93). The mass-loss rate for the 6.5M , Z = 0.02

CHAPTER 7. STELLAR YIELDS 162

Figure 7.1: The mass-loss rate as a function of time (in years) for the 6.5M , Z = 0.02 model. model is shown in Figure 7.1 as a function of time (in years). When the superwind phase begins (at about 5.407 107 years), the mass-loss rate is approximately constant over an interpulse period. × The total mass at the ith thermal pulse, Mi, is then

i i 1 M = M − ∆M. (7.5) −

If we assume that the H-exhausted core mass grows by a constant ∆MH per interpulse period and decreases in mass after each thermal pulse owing to the TDU, then the change in the mass of the H-exhausted core from the (i 1)th to the ith pulse is given by − i i 1 M = M − + ∆M λ∆M , (7.6) H H H − H where λ is the TDU efficiency parameter, defined in Chapter 6. We use the detailed evolutionary sequences to provide all the necessary input, i.e. from the last calculated model we take the stellar mass, the H-exhausted core mass, the radius and the luminosity. The quantities ∆MH, τip and λ are evaluated between the last two thermal pulses. For the remaining evolution we assume that the radius, luminosity, ∆MH, λ and the interpulse period τip are constant. In Table 7.1 we present, for the last calculated model, the stellar mass, the H-exhausted core mass, the envelope mass (with masses in solar units), the interpulse period, λ, the number of pulses calculated in detail, the number of thermal pulses calculated synthetically and the pulse number at which the first TDU episode occurs (for models that had final envelope masses larger than 0.1M ). From Ta-

CHAPTER 7. STELLAR YIELDS 163

Figure 7.2: Surface abundance evolution during the AGB of the neon, sodium and magnesium isotopes for the 6.5M , Z = 0.02 model. The ®nal 7 dredge-up episodes are calculated from the semi-analytic formulae described in the text. ble 7.1 it is evident that only the more massive AGB models, in particular the 4, 5 and 6M models, are estimated to undergo more than one additional thermal pulse. Since the yields are also the largest from these more massive models, we need to estimate the contribution from these extra thermal pulses to the stellar yield calculation. To evaluate the enrichment of the envelope from the remaining thermal pulses, we need only two additional pieces of information which we obtain from the detailed nucleosynthesis calculations: 1) the composition of the intershell and 2) the composition of the envelope at the last calculated model. If we assume that the intershell abundances are constant for each remaining thermal pulse, we can estimate the mass of species X mixed into the envelope at a given dredge-up episode i.e. ∆MX = Xshell λ ∆MH, where Xshell is the intershell abundance (in mass fractions). This method is an improvement over most other synthetic algorithms because our intershell abundances, while constant in time, are dependent on both the mass and metallicity of the model. The intershell abundances are not really constant in time, as shown in Figure 5.3 on page 106. But as Figure 5.3 also shows, the pre-TDU composition does not change very much over ﬁve thermal pulses, justifying our assumption. To improve the assumption of constant intershell abundances, we could ﬁt the behaviour (as a function of time or pulse number, mass and metallicity) of each species in the intershell from the detailed nucleosynthesis calculations, from the start of the TP–AGB phase to the last calculated model. These results would prove extremely CHAPTER 7. STELLAR YIELDS 164 useful for current synthetic AGB models, which assume constant intershell abundances for all masses and metallicities. The mass fraction of species X at the surface for the ith interpulse phase will then be

(Mi λ ∆M ) Xi 1 + ∆M i = env H − X , X − i (7.7) Menv

(i 1) th where X − is the abundance of species X at the surface for the (i 1) interpulse phase i − and Menv is the current envelope mass. We also note the evolutionary position of the last calculated model: if at the beginning of a thermal pulse then we adjust abundances first before removing mass from the envelope. This was the most common situation for model calculations. However, if the last calculated model was at the beginning of an interpulse phase or some fraction from the start of the current interpulse phase, then we remove the corresponding amount of matter from the envelope before adjusting abundances. For example, for models with the last calculated model half way through the current interpulse phase, then the first calculation removes 0.5∆M from the envelope and increases the core mass by 0.5∆MH before adjusting abundances. Subsequent interpulse periods remove ∆M per interpulse period. Within the intershell, we include species effected by H and/or He-shell burning: 4He, 12C, 16O, 19F, 20Ne, 21Ne, 22Ne, 23Na, 24Mg, 25Mg, 26Mg, 26Al, 27Al, 28Si, 29Si and 30Si. The largest surface abundance changes to the non-CNO elements is for 22Ne or the heavy magnesium isotopes, depending on the mass of the model. In Figure 7.2 we show the surface abundances during the TP–AGB for the 6.5M , Z = 0.02 model, including the enrichment from the 7 remaining thermal pulses using the method described above. For models that have at least one remaining thermal pulse, we present two sets of stellar yields. The first set is calculated with the value of λ given in Table 7.1, which is 0.9 in most cases. There is some evidence that the efficiency of the TDU decreases when∼ the mass of the envelope is reduced below some critical value by mass loss (Straniero et al. 1997; Chapter 6 and Karakas, Lattanzio & Pols 2002). It is unknown to what extent and at what envelope mass this effect becomes important. To estimate the effect of a reduced λ on the stellar yields, we calculate a second set of yields with λ = 0.3 for all remaining TDU episodes. Before presenting the results of the stellar yield calculations, we discuss the main sources of uncertainty that affect the results.

7.2 Stellar Yield Uncertainties

The main source of uncertainty to aﬀect the yield calculation is the choice of mass-loss law to be used during the TP–AGB phase. It is still not well understood how mass loss works on the AGB, or what is the connection between the mass-loss rates and the nucleosynthesis. Uncertainty in the input physics and the treatment of convection will also aﬀect the stellar yields, as we discuss below. CHAPTER 7. STELLAR YIELDS 165

Table 7.1: Data used to estimate the number of thermal pulses missed for each model. See page 162 for a description of the contents.

Z=0.02 M0 Mf MH, f Menv,f τip λ # TPs # Syn. TP(DU) 2.0 0.77934 0.64557 0.13377 6.4448E+04 0.000 21 0 – 2.25 0.79203 0.66611 0.12592 5.5163E+05 0.234 27 0 20 2.5 1.14842 0.66260 0.48582 6.3421E+04 0.564 26 1 16 3.0 1.35827 0.68264 0.67563 5.4718E+04 0.765 26 1 9 3.5 2.04719 0.70990 1.33729 4.3107E+04 0.859 21 1 4 4.0 1.74950 0.79360 0.95590 2.3947E+04 0.975 18 2 4 5.0 2.37355 0.87447 1.49908 1.0522E+04 0.911 24 5 4 6.0 2.46296 0.92902 1.53394 6.6692E+03 0.934 38 5 4 6.5 2.47032 0.96284 1.50748 4.4233E+03 0.910 40 7 4 Z=0.008 1.9 0.86633 0.64457 0.22176 8.7680E+04 0.466 22 0 11 2.1 1.07345 0.64615 0.42730 8.9786E+04 0.638 20 0 10 2.25 1.01461 0.65275 0.36186 8.2240E+04 0.720 26 0 14 2.5 1.01177 0.66836 0.34341 8.0220E+04 0.805 27 1 10 3.0 1.24959 0.69554 0.55405 4.9568E+04 0.861 28 1 4 3.5 1.43926 0.76877 0.67049 4.0320E+04 0.969 20 1 3 4.0 1.75654 0.84163 0.91491 2.0848E+04 0.970 22 2 3 5.0 2.27375 0.88630 1.38745 1.0432E+04 0.960 59 4 2 6.0 2.14493 0.94749 1.19744 6.2960E+03 0.950 69 5 3 Z=0.004 2.25 0.96286 0.66194 0.30092 7.2128E+04 0.767 26 0 8 2.5 1.35739 0.67277 0.68462 5.8560E+04 0.817 29 0 5 3.0 1.23056 0.72634 0.50422 4.3872E+04 0.949 26 0 2 3.5 1.39107 0.81692 0.57415 2.6560E+04 0.950 25 1 2 4.0 1.63163 0.85554 0.77608 1.8192E+04 0.947 32 2 2 5.0 1.85052 0.90610 0.94442 9.3840E+03 0.952 83 3 2 6.0 1.97357 0.97854 0.99503 4.6720E+03 0.913 105 6 4 CHAPTER 7. STELLAR YIELDS 166

7.2.1 Uncertainties From Mass Loss The choice of mass-loss law on the AGB crucially effects the stellar yields in a number of ways. The mass-loss law determines the TP–AGB lifetime, which is terminated when 2 the envelope mass is reduced below about 10− M (Block¨ er 1995). Mass loss also determines the number of thermal pulses a model goes through, which therefore represents the maximum possible number of TDU episodes. The AGB mass-loss rate also determines the structure at the tip of the AGB, which has implications for later evolution in the proto- planetary nebulae stage. We do not discuss the evolution during the post-AGB phase but refer the reader to Vassiliadis & Wood (1994). Not only does mass loss determine the number of thermal pulses, but it can determine if and when the TDU begins and ends. We have shown in previous chapters (both in Chapter 4 and 6) that the inclusion of mass loss can have a dramatic effect on the operation of the TDU in the low-mass models. The main result is more efficient TDU in the models min without mass loss, but we also observe Mc to be reduced in these cases. This effect is the largest for the low-mass Z = 0.02 models, where the inclusion of mass loss results in no dredge-up whatsoever for models with M < 2.25M . Thus mass loss has a direct influence on the enrichment from thermal pulses by affecting the number of pulses and the efficiency of the TDU. Mass loss does not significantly effect the occurrence of the TDU in the intermediate min mass models. In Chapter 6 we saw that the values of Mc and λ are essentially the same for models with and without mass loss. In these cases mass loss is important for determining when the TDU ends. We mentioned earlier that there is some evidence that below some critical envelope mass, the TDU should cease (Straniero et al. 1997). Owing to the large uncertainty in mass loss during the AGB, this critical envelope mass can only be guessed at. In Chapter 6 we put this figure at about 0.2M , but this was determined from the few low-mass models with mass loss that experienced the TDU. It is still highly uncertain when (in terms of envelope mass) the TDU ceases in intermediate mass stars. The occurrence of HBB is also dependent on the mass loss. Block¨ er (1995) studied the effect of mass loss on models with HBB, and found that efficient HBB is not prevented by high mass-loss rates. However, in models where HBB is not so efficient, mass loss not only determines the maximum temperature reached at the base of the convective envelope, but also when that maximum temperature is reached, and for how long HBB will occur. Thus the time variation of many quantities that determine the nucleosynthesis from HBB will depend crucially on the mass loss. The effect on the stellar yields can be quite large. For example, the 7Li yields from AGB models with HBB are strongly dependent on the adopted mass-loss law (Travaglio et al. 2001c). Because the 7Li yields are very sensitive to the mass loss, Travaglio et al. (2001c) ran a sequence of models with different mass-loss rates. By increasing the VW93 mass-loss rate by a factor of 50, the 7Li yields went from negative to positive in all but two cases (for the 4 6M models studied, which cover the same range in composition as considered here). Hence− this example demonstrates just how sensitive the yields can be to the adopted mass-loss law. We use the semi-empirical mass-loss law of Vassiliadis & Wood (1993), which is specif- ically designed for long-period variable AGB stars and relates the mass-loss rate to the CHAPTER 7. STELLAR YIELDS 167

Figure 7.3: Initial±®nal mass relation from Weidemann (2000). Included are the H- exhausted and He-exhausted (CO) core masses for the Z = 0.02 models. pulsation period. This formula is characterized by small mass-loss rates for most of the AGB, followed by a phase with very strong mass loss: the so-called superwind phase. The superwind phase is characterized by a rapid increase in the mass-loss rate, from a few 7 1 4 1 10− M year− to 10− M year− . The superwind lasts for only a short time, because more× than a few tenths∼ of a solar mass is lost per interpulse period. The VW93 mass-loss law is also sensitive to the initial metallicity. This is because the formula depends on the stellar mass, radius and luminosity, all of which are metallicity dependent. From Table 7.1 we can see that at a given mass, the Z = 0.004 models have more thermal pulses than do the Z = 0.02 models. There are a number of other mass-loss laws speciﬁcally designed for either the AGB phase (Bowen 1988; Block¨ er 1995) or for carbon-rich AGB stars (Fleischer, Gauger & Sedlmayr 1992; Arndt, Fleischer & Sedlmayr 1997; Wachter et al. 2002) which are also characterized by an increasing rate of mass loss leading toward a superwind phase. All of these, including the VW93 mass-loss law, are far superior to the Reimers (1975) law at reproducing the range of mass-loss rates observed in LMC and SMC AGB stars (Vassiliadis & Wood 1993; Arndt et al. 1997). This is not surprisingly, since the Reimer’s law was not derived for stars that are very metal poor, are carbon rich, or are Mira variables (Iben & Renzini 1983).

7.2.2 The Initial–final Mass Relation The remnant masses that we use in the yield calculations are given in Appendix D. The final remnant mass is chosen to be the mass of the H-exhausted core at the end of the CHAPTER 7. STELLAR YIELDS 168 evolution calculation, with the synthetic evolution included for the final pulses (where needed). There are two remnant masses for models that have the remaining thermal pulses calculated synthetically. The first remnant mass is from the calculation with λ taken from Table 7.1, and the second from the calculation with λ = 0.3. The second remnant mass is larger than the first, because a larger value of λ results in deeper TDU, which in turn results in a smaller core mass after a thermal pulse than the case with less efficient TDU. From here on, we refer to the first entry for each model as the final mass. The final H-exhausted core mass depends upon a number of factors, including the mass- loss law chosen for the AGB phase, the occurrence and efficiency of the TDU, as well as on the details of the previous central He-burning phase which determines the mass of the He core at the beginning of the TP–AGB. In Figure 7.3 we compare the final masses for the Z = 0.02 models with the initial–final mass relation given in Weidemann (2000). The agreement is quite good for models with M & 3M but less so for models with M . 3M . We note that this is the mass range where observations demand carbon stars to be formed which the models have trouble producing. This is probably more evidence for our need to improve our understanding of dredge-up and its commencement. Deeper dredge-up will produce a smaller final mass than a similar case with shallow dredge-up (Weidemann 2000). Hence if deeper dredge-up is required to match the carbon-star luminosity function in this mass range, then a lower final mass will result, as required by Figure 7.3. It is our educated guess that what is required is for the evolution along the AGB to occur at slightly lower core masses, and thus terminate at lower core masses, with only a small effect on the yields. However, a quantitative estimate is not possible at present.

7.2.3 Uncertainties in the Nuclear Reaction Rates Many nuclear reaction rates are highly uncertain at the temperatures found in low to intermediate mass stellar models. This is because it is very difficult to measure the cross sections of many charged-particle reactions, as the cross sections are so small at the energies of astrophysical interest. To fill the void, theoreticians are required to extrapolate from the lowest measurable energies to the region required by the astrophysicist. The main problem here is that important resonances might not be included, resulting in large uncertainties in some reaction rates. Beginning with the CNO cycles, there are important uncertainties that involve 17O. Even though uncertainties in the reactions 17O(p,α)14N and 17O(p, γ)18F have been greatly reduced over the past few years (Arnould, Goriely & Jorissen 1999), the oxygen isotopic ratios cannot be reliably predicted at a given temperature. We discuss in 3.4.5 that the change to the 17O abundance from the first and second dredge-up is dependent§ on the choice of the 17O(p, γ)18F reaction rate (see also El Eid 1994). The uncertainties in this reaction will also affect the final 17O yield from the HBB models. The abundance of 19F is greatly affected by uncertainties in both the H and He-burning reactions. During H-burning, 19F is destroyed by the reaction 19F(p,α)16O, the rate of which is highly uncertain at temperatures below about 20 million K (Arnould et al. 1999). Arnould et al. (1999) discuss the possibility of producing some 19F from H-burning if the upper limit of the 18O(p, γ)19F NACRE rate was used, which is comparable to the rate CHAPTER 7. STELLAR YIELDS 169 of 19F destruction. This sequence might have an impact on the 19F abundance after the first and possibly second dredge-up, where the H-burning temperatures are lower than on the AGB. Fluorine is produced in AGB models via a complicated series of steps involving α–capture reactions in the He-shell. The most efficient 19F production involves the inclusion of a 13C pocket in the intershell, following the deepest penetration of the convective envelope during the TDU. The 13C has to be included in standard models (without extra mixing) manually, and it is an obvious uncertainty that we do not know how large (in mass) to make the 13C pocket. There are also large uncertainties in the relevant α– capture reactions, as discussed in detail by Forestini et al. (1992) and Mowlavi, Jorissen & Arnould (1996). From models without HBB, our 19F yields should be considered as under-estimates, because we do not include a 13C pocket in the nucleosynthesis calculation. It would be interesting to find out how different the 19F yields would be if we were to re-run these models with a 13C pocket. The Ne–Na chain has important uncertainties effecting the rate of 23Na production and destruction (Arnould et al. 1999; Mowlavi 1999a). Sodium is produced from the reaction 22Ne(p,γ)23Na, and destroyed by the two reactions: 23Na(p,γ)24Mg and 23Na(p,α)20Ne. The rate of the 23Na(p,γ)24Mg reaction is uncertain by a factor of 104 at temperatures over 50 million K (Arnould et al. 1999). The Ne–Na chain produces abundant 23Na in all but the most massive HBB models where we observe significant 22Ne and 23Na destruction (see Figure 5.8 on page 118). Given that there are large uncertainties in the relevant reactions, we consider the sodium yields we present to be tentative. The Mg–Al chain also has important reaction rate uncertainties concerning the destruction of the isotope 26Al (Mowlavi & Meynet 2000; Lugaro et al. 2001) The uncertainty in the 26Al(p,γ)27Si reaction rate is as large as 103 at temperatures over 50 million K (Arnould et al. 1999). The large production of 26Al in our HBB models could be reduced by using the upper bound to this reaction rate. Whilst there are important uncertainties which effect many helium-burning reactions, such as the highly uncertain rate of the 12C(α,γ)16O reaction (Imbriani et al. 2001), we only discuss the 22Ne α capture reactions here. Both the reactions 22Ne(α,n)25Mg and 22Ne(α,γ)26Mg are only −efficient at destroying 22Ne at temperatures over 300 million K. There are still large uncertainties in the 22Ne(α,n)25Mg reaction in the range of temperatures found in the He-burning shells of AGB models (Arnould et al. 1999). In Figure 7.4 we compare the NACRE reaction rates (Angulo et al. 1999) for the two 22Ne α capture reactions at the relevant temperatures for He-shell burning. The reaction producing− 25Mg is predicted to be faster at these temperatures (over 300 million K) than the reaction involving 26Mg. Indeed, at temperatures nearing 400 million K, the difference between the two rates is expected to be almost an order of magnitude. If we were to use the NACRE lower limit for the 22Ne(α,n)25Mg reaction and the upper limit for the 22Ne(α,γ)26Mg, we might produce more 26Mg than 25Mg in the He shell. Thus the 26Mg yields might be larger than the yields of 25Mg, but this result is dependent upon the rate of 25Mg destruction (and production) from the Mg–Al chain during HBB. CHAPTER 7. STELLAR YIELDS 170

-8 -9

/mol/s) -10 3

cm -11 -12 -13 -14 Ne22(a,g)Mg26 Ne22(a,n)Mg25 reaction rate ( -15 -16 0.2 0.25 0.3 0.35 0.4 Temperature (109 K) Figure 7.4: The 22Ne(α,n)25Mg and 22Ne(α, γ)26Mg reaction rates as a function of the temperature.

7.3 Stellar Yields

We present the stellar yields (in solar masses) for 24 isotopes from hydrogen through to 30Si in Appendix D. We also include yields for the unstable aluminium isotope, 26Al, but note that the total mass of 26Al should be added to mass of 26Mg when these yields are used in a chemical evolution model. For each composition, there are three separate tables and in each table we give the initial and final mass of the model. In Tables D.1 to D.3 we present the stellar yields for the Z = 0.02 models, in Tables D.4 to D.6 we present the yields for the Z = 0.008 models and in Tables D.7 to D.9 the yields for the Z = 0.004 models. Earlier we mentioned that there are two sets of yields for each intermediate mass model where the remaining thermal pulses were calculated synthetically. The first set of yields will be referred to as our standard case, and we use these yields for comparison to other authors. This second set of yields will be referred to as the lower-limit seta. The difference between the standard and lower-limit set is small in most cases. This is especially true for the intermediate-mass models which experience many TDU episodes and HBB during the TP–AGB phase, so the change to the surface composition from the few remaining pulses is small. The difference is more significant for the lower mass models (M . 3M ) even though these models only experience one remaining thermal pulse in all cases. There are

aThe true lower limit corresponds to no contribution from the remaining thermal pulses. We will not discuss this case. CHAPTER 7. STELLAR YIELDS 171 two reasons for this. Firstly, the lower mass models do not experiences as many TDU episodes compared to the higher mass models (as shown in Table 7.1) so the final pulse (and TDU) will significantly alter the composition of the envelope. Secondly, the long interpulse periods (> 50, 000 years) means that a significant amount of matter is lost from the time of the last thermal pulse to the time the model leaves the TP–AGB, which in turn results in noticeable change to the stellar yields. An example is the 2.5M , Z = 0.02 12 3 3 model, where the C yield from lower-limit set is 4.3109 10− compared to 4.8532 10− from the standard set, a difference of about 13%. × × Equation (7.1) calculates stellar yields relative to the initial mass of each species in the expelled material. We can convert these yields to the total amount of matter expelled into the interstellar medium, if we know the initial abundances and the remnant mass. The total mass of species k expelled into the ISM by a given model is

Mtot = M + (M M ) X (k), (7.8) k k i − f 0 where Mk is the stellar yield calculated from equation (7.1), X0(k) is the initial abundance of species k (in mass fractions) and Mi is the initial mass and Mf is the ﬁnal mass (in solar units) of the stellar model. Note that the yields calculated using equation (7.8) will be identical to those found from equation (7.3). We therefore provide the initial abundances for the three diﬀerent compositions in Table D.10. We do not include the initial abundance of every species included in the nucleosynthesis code: just the 24 species for which we present yields.

7.3.1 Dependence Upon the Initial Abundances The stellar yields presented in Appendix D depend upon the initial abundances in a number of ways. Firstly, the change in the envelope composition over time depends on the initial abundances of certain species, such as hydrogen, carbon and oxygen. This point is particularly relevant for the Z = 0.008 (LMC) and Z = 0.004 (SMC) models, where we set the initial CNO abundances to be less than scaled-solar, as suggested by Russell & Dopita (1992). For this reason, applying our yields to more general problems, such as the chemical evolution of globular clusters where the α–elements (such as 16O and 20Ne) are enhanced, would not be justiﬁed. This strong dependence upon the initial abundances means that stellar yields are only relevant to the particular problem at hand. The initial composition also determines the stellar lifetime, and as discussed in Chap- ter 3, the lowest metallicity models have shorter main sequence and core He-burning lifetimes than solar metallicity models. The initial composition has a dramatic eﬀect on the stellar structure (mostly via the opacity), and thus upon the evolution along the AGB.

7.4 Comparison to Other Authors

Renzini & Voli (1981) produced the first set of stellar yields from low to intermediate mass stars. These CNO yields were calculated with a fully synthetic evolutionary algorithm, CHAPTER 7. STELLAR YIELDS 172 which included HBB, the TDU and mass loss via the Reimers (1975) formula. Further contributions using synthetic AGB models have been made by Marigo, Bressan & Chiosi (1996), van den Hoek & Groenewegen (1997), Marigo (2001) and Izzard et al. (2003). The biggest difference between the recent calculations listed above and those of Renzini & Voli (1981) is in the improved parameterizations of the AGB phase of evolution, based on detailed models with improved input physics. The latest synthetic models also use parameterizations that depend on the initial metallicity, which is something that Renzini & Voli’s calculation did not do. The increasing speed of modern computers means that the problem of running large grids of stellar models becomes time consuming, rather than impossible. Thus we are now starting to see stellar yields calculated from detailed AGB models, such as those presented by Ventura, D’Antona & Mazzitelli (2002), as well as the yields presented in this chapter. Other studies utilizing detailed models include Forestini & Charbonnel (1997), who use a combination of detailed and synthetic models to estimate the surface abundance changes to the light elements during the TP–AGB; Mowlavi (1999a), who uses a combination of synthetic and detailed evolutionary models to predict the 23Na production in AGB models of different mass; and Boothroyd & Sackmann (1999), who estimate the contribution of the first and second dredge-up to the ISM from a population of stars with M > 1M using detailed models with extra mixing for low-mass models during the ascent of the giant branch. In 7.5 we present the yields in graphical form, and for comparison we include the yields§of van den Hoek & Groenewegen (1997), Forestini & Charbonnel (1997), Marigo (2001), Ventura et al. (2002) and Izzard et al. (2003) where possible. We first discuss the model assumptions that have gone into each calculation.

7.4.1 Physical Assumptions Izzard et al. (2003; hereafter I03) model the evolution from the zero-aged main sequence to the end of the TP–AGB phase using a fully synthetic evolution model. On the AGB, HBB is modelled by including accurate parameterizations of the density and temperature at the base of the convective envelope, derived from the detailed models presented in Chapter 4. If we examine the behaviour of detailed models we observe that HBB occurs over a very 5 thin region in mass, which is typically ∆MHBB 4 10− M (Izzard et al. 2003), much less than the mass of the convective envelope in≈a 4×to 6M model. When the thin HBB shell is burned and mixed into the envelope, the envelope ab undances remain essentially unchanged. Only after a large number of mixing episodes will the envelope composition significantly change as a result of the thin-burning shell. For this reason, I03 approximate HBB by burning a fraction of the envelope, fHBB, for a fraction of the interpulse period, fburn, and fit both fHBB and fburn to match the detailed models. To calibrate the synthetic model, I03 adjust the above parameters until the 13C abundance profile matches the 13C surface abundances in our detailed models. The synthetic model of I03 also uses our min parameterizations for the TDU, but find they need to reduce Mc to fit the carbon-star luminosity function (CSLF) of Magellanic Clouds. I03 use intershell abundances that are consistent with our detailed models i.e. XHe4 = 0.75, XC12 = 0.22, XO16 = 0.01 and XNe22 = CHAPTER 7. STELLAR YIELDS 173

0.02 and assume that these abundances are constant for all masses and metallicities. Marigo (2001; hereafter M01) use a semi-analytical model to calculate stellar yields, which utilizes the detailed stellar models from the Padua group (Girardi et al. 2000) for the pre-AGB phase, and a synthetic model for the AGB phase. Whilst the evolution of 13 species is followed in the detailed and synthetic calculations, M01 only provide yields for H, 3He, 4He and the CNO isotopes. Mass loss is included using the formula from VW93. The synthetic AGB model uses the formulae of Wagenhuber & Groenewegen (1998) for the luminosity, interpulse period, and core mass at the first thermal pulse. The TDU is treated by assuming that λ is constant for the entire TP–AGB phase, where λ = 0.55 for the solar and LMC models, and λ = 0.65 for the SMC models. The onset of the TDU is determined by the temperature at the base of the convective envelope following a thermal dred pulse. If this temperature exceeds a specified value, TB (set at log T = 6.4), then the TDU begins, regardless of the core mass. Complete envelope integrations are performed to include HBB, with a dependence on the mixing-length parameter α. M01 calculates stellar yields for three values of α: α = 1.68, α = 2.0 and α = 2.5. The nucleosynthesis model used by M01, and outlined in Marigo et al. (1996) assumes that the abundances in the intershell (taken from Boothroyd & Sackmann 1988c) are not dependent on the core mass, metallicity or time. M01 does not calibrate her HBB models against detailed model calculations, but instead carefully compares her results to the 7M calculation by Block¨ er (1995) (see Marigo 1998a). van den Hoek & Groenewegen (1997; hereafter HG97) also used a semi-analytical model to calculate stellar yields. Detailed models from the Geneva group (Schaller et al. 1992) are used for the pre-AGB phase of evolution, and a synthetic model for the AGB phase. The AGB model is based on the model presented by Groenewegen & de Jong min (1993), who assume that λ and Mc are constant free parameters to be fit by matching the CSLF in the Magellanic Clouds. HG97 use the same values in their yield calculations, min which are set at λ = 0.75 and Mc = 0.58M . Mass loss is included using the Reimers (1975) mass-loss law, with the parameter η varied between 3 and 5. HBB is only included in a parameterized manner (see Groenewegen & de Jong 1993 for details) and is based on the early models of Renzini & Voli (1981). HG97’s prescription of HBB includes the free parameter, mhbb, which is the minimum core mass for HBB to occur. They find mhbb 0.8 is suitable to match observations of N/O in planetary nebulae. Only six species are∼ included in their synthetic AGB model with initial solar abundances taken from Anders & Grevesse (1989) and LMC and SMC abundances scaled down from solar. Intershell abundances are also from Boothroyd & Sackmann (1988c) and do not depend on the time, the stellar mass or the metallicity. Neither HG97, M01 or I03 include the Ne–Na or Mg–Al chains in their nucleosynthesis models. Marigo et al. (1996) calculates yields for 22Ne, but these will be incorrect for the most massive models because 22Ne destruction rates were not included. Izzard (private communication, 2003) also calculates 22Ne yields, but again, these yields will also be incorrect for the most massive models. We will nonetheless compare Izzard’s 22Ne yields to ours, and expect they will be reasonably consistent for all but the most massive models. We make a comparison with the yields given in Forestini & Charbonnel (1997; hereafter FC97). FC97 only present yields for a small range in mass and metallicity. At Z = 0.02, CHAPTER 7. STELLAR YIELDS 174 the range of mass is 3 to 6M , and at Z = 0.005 (similar to our Z = 0.004 models), the range of mass is 3 to 5M . FC97 used detailed calculations up to the first 10 or 20 thermal pulses after which they e xtend the evolution using a synthetic algorithm. In their synthetic calculation they set λ = 0.6 in all cases, which is probably a cautious choice considering that we find λ 0.9 in the mass range they studied. FC97 perform envelope integrations to solve for the∼ abundance changes caused by HBB. They extrapolate the density and temperature from the early AGB evolution for the whole TP–AGB phase, noting this is most likely incorrect. Reimers (1975) mass loss is used on the AGB, with different values of the parameter η = 2.5, 3, 3.5, 4 for the 3, 4, 5 and 6M models respectively. In the following discussion, we compare their Z = 0.005 yields to the Z = 0.004 yields. We make a comparison with the yields of Ventura et al. (2002; hereafter V02) for 4He, 7Li and the CNO isotopes. These authors calculate detailed stellar models from the pre- main sequence to the TP–AGB phase. The mass-loss rate during the AGB phase is computed from the formulation of Block¨ er (1995). The range of mass calculated is similar to FC97, and is between 3 and 6M , whilst the range of metallicity is much larger, 4 0.01 6 Z 6 2 10− . V02 set Z = 0.01 for their solar metallicity models, which is appropriate since×the solar O abundance was recently revised Allende Prieto, Lambert & 3 Asplund (2001). The TDU only occurs in models with Z 6 10− , and the TDU parameter, λ, is smaller than in our models, where the maximum value is λ 0.5. V02 do not use the mixing-length theory to model convection, but instead use the “full∼ spectrum of turbu- lence” convection model of Canuto & Mazzitelli (1991). Convective overshoot is included during the evolutionary phases prior to the AGB, and V02 find this lowers the minimum mass for HBB by about 0.5M . In the following discussion, we compare their Z = 0.01 yields to the Z = 0.02 yields. We compare our yields to each authors standard set of yields. In the case of I03 we will compare to their set of models which use the VW93 mass-loss prescription and α = 1.75. The standard models of Marigo (2001) have α = 1.68, λ = 0.5 for the solar and LMC models and λ = 0.65 for the SMC models, and to determine the onset of the TDU, the critical temperature at the base of the convective envelope is set to log T = 6.4. HG97’s min standard set of yields are calculated with Mc = 0.58, λ = 0.75, mhbb = 0.8 and η = 4. FC97 and V02 calculate one set of stellar yields.

7.5 Yield Results

We weight the yield, Mk, of species k, by the initial-mass function (IMF), ξ(M0), such that dY y = = ξ(M ) M , (7.9) k dM 0 k where yk is the weighted yield and Mk is given by equation (7.1). We use the three- component IMF of Kroupa, Tout & Gilmore (1993), derived from the stellar distribution toward the Galactic poles as well as the distribution of stars within 5.2pc of the Sun, and CHAPTER 7. STELLAR YIELDS 175 given by 0 if m < m0, a m 1.3 if m < m 6 0.5, ξ(m) =  1 − 0 (7.10)  a m 2.2 if 0.5 < m 6 1.0,  2 −  2.7  a2 m− if 1.0 < m < ,  ∞ where ξ(m) dm is the probabilitythat a star has a mass (in solar units) between m and m + dm. The constants are given by a1 = 0.29056, a2 = 0.15571 and m0 = 0.1M (Hurley, Tout & Pols 2002). Whilst equation (7.10) is applicable to low-mass stars in the solar neighbourhood we will also use it for the Z = 0.008 and Z = 0.004 models. Whilst there are IMFs in the literature for the LMC and SMC these are generally restricted to massive stars (Humphreys & McElroy 1984) and the IMF for low-mass stars in these galaxies is still quite uncertain (Sirianni et al. 2002). In each ﬁgure (unless otherwise stated) we show the weighted yields from our calculations as the black solid points, van den Hoek & Groenewegen (1997) as the open magenta squares, Forestini & Charbonnel (1997) as the solid green squares, Marigo (2001) as the open red circles, Ventura et al. (2002) as the solid aqua triangles and Izzard et al. (2003) as the blue crosses. Note that Forestini & Charbonnel (1997) and Ventura et al. (2002) do not provide yields for Z = 0.008.

7.5.1 Hydrogen For low to intermediate mass stars, most of the mass lost in the stellar wind is the pri- mordial hydrogen the star was born with. Hydrogen yields are negative because each successive mixing event brings material to the surface from either the H or He-burning shells. In each case the material mixed into the envelope is depleted in H and enriched in 4He as well as many other elements. In Figure 7.5 we show the hydrogen yields from each author as a function of the initial mass. FC97 and V02 do not include yields for hydrogen. The yields of I03 match our results very well, especially for higher masses at all metallicities. The yields of Marigo consume more hydrogen at all but the lowest masses, and especially in the range between about 1.25M and 4M (depending on Z). This is related to her treatment of the TDU, which produces more 4He and 12C than other calculations, and hence consumes more hydrogen. We discuss this further below. The H yields of HG97 have a minimum at about 1M at all Z, but this is more noticeable for the Z = 0.02 models. The minimum is caused by the first dredge-up, since HG97 find that models less than 1.2M do not experience the TDU. For masses above about 2M , and notably in the range of masses that experience HBB, our models consume more hydrogen than HG97. This is probably because the parameterized HBB model used by HG97 is not as efficient as the HBB occurring in our detailed models. CHAPTER 7. STELLAR YIELDS 176

Figure 7.5: Weighted yields of hydrogen as a function of the initial mass for the Z = 0.02 models (top), the Z = 0.008 yields (middle) and the Z = 0.004 yields (bottom). See the text on Page 174 for a description of the symbols. CHAPTER 7. STELLAR YIELDS 177

Figure 7.6: Same as Figure 7.5 but for 4He. CHAPTER 7. STELLAR YIELDS 178

7.5.2 Helium: 4He The next most abundant element in the stellar wind is 4He. The 4He yields are positive because each mixing event prior to the AGB brings helium to the surface from either partial or complete H-burning. During the TP–AGB phase, partial He-burning results in only about 25% (by mass) of the 4He being converted into 12C, and thus each TDU episode increases the helium abundance of the envelope. Likewise for HBB, which burns H to 4He via the CNO cycle. Indeed, the most helium-rich models are those with HBB. In Figure 7.6 we see that the trends for the H yields are mirrored for 4He. The yields of I03 agree well with our yields at all metallicities. The 4He yields of FC97 and V02 agree well with the other calculations, and are very similar to our results at Z = 0.02. M01 produce more 4He than any other calculation. To explain this, we need to remind the reader of the TDU prescription used by M01: if the temperature at the base of the convective envelope exceeds a specified value (log T = 6.4), then the TDU begins regardless of the core mass. We note that the temperature of the convective envelope is found by performing envelope integrations but assuming that the envelope is static i.e. the entropy term is min neglected (see Marigo 1998a). In Chapter 6 we discuss Marigo’s (1998b) results for Mc , min and note that for the low-mass LMC models, Mc 0.53. This value is much smaller ≈ min than in our LMC models and smaller than the value of Mc = 0.58M used by HG97. In min Figure 7.6 it is clear that a reduced value of Mc has a large impact on the final stellar yields for nuclei effected by the TDU. From Figure 7.6 we see that HG97 produce less 4He at intermediate masses compared to the other calculations. This result is independent of the metallicity, even though the mass at which the yields of HG97 become smaller decreases as a function of Z. There are probably two reasons for this, the first to do with the adopted mass-loss rates. Many of the calculations, including our own, use the VW93 mass-loss law, whereas both HG97 and FC97 use the Reimer’s law. How does this effect the results? We discussed earlier that the VW93 mass-loss law results in deeper TDU, and possibly HBB at lower masses, because the envelope is large for most of the TP–AGB. The main result of the Reimer’s law is more thermal pulses (and more TDU episodes) because there is no superwind to terminate the evolution. The Block¨ er (1995) mass-loss law (used by V02) is based on the Reimer’s formula but with the addition of a superwind phase, and probably results in behaviour somewhere in between the other two mass-loss formulae. HG97 reduce the mass-loss rate in their models (by reducing η) and find the yields to increase primarily because of the increased number of thermal pulses (and TDU episodes). The second reason that HG97’s yields are smaller at higher masses is because the growth rate of the H-exhausted core is larger in M01’s and I03’s models than in HG97. The growth rate of the H-exhausted core is calculated in the synthetic models using

dMH 1 q LH (M yr− ) = , (7.11) dt X where q is a constant that depends on the nuclear-burning eﬃciency. HG97 uses the value 12 1 1 of q given by Groenewegen & de Jong (1993) which is 9.55 10− M L − yr− , whilst both M01 and I03 use the metallicity dependent value of q giv×en by Wagenhuber & Groe- CHAPTER 7. STELLAR YIELDS 179

Figure 7.7: Weighted yield of 7Li as a function of the initial mass. The Z = 0.02 models are the black solid points, the Z = 0.008 models the blue open squares and the Z = 0.004 models the open red circles.

11 newegen (1998), which is q = (1.02 + 0.017 Z) 10− . I03 comment that this q value probably results in larger growth rates than found× in detailed models (which could be another reason why Marigo’s yields are larger than ours).

7.5.3 Lithium: 7Li Lithium is destroyed in stellar atmospheres at relatively low temperatures ( 2 106 K), and is therefore depleted by burning during the pre-main sequence phase. ∼The×first and second dredge-up further decrease the surface abundance of 7Li. During the TP–AGB phase, 7Li can be created by HBB, as described in Chapter 5, but is destroyed once the supply of 3He is exhausted. Thus the stellar yields of 7Li are negative in all but a few cases. The Z = 0.02 yields are all negative for all masses, but there is a narrow range of mass at Z = 0.008 and Z = 0.004 which have positive 7Li yields (see Figure 7.7). The narrow mass range represents models which experience HBB for only a short time during the TP– AGB. For these models (3.5M at Z = 0.004 and 4M , Z = 0.008 and Z = 0.004), HBB occurs later in the evolution so the 7Li is not destroyed before the cessation of HBB. Both FC97 and V02 provide yields of 7Li which we can compare our results too. At Z = 0.02, the FC97 yields are negative for the 3 and 4M models but positive for the 5 and 6M models with HBB. At Z = 0.005, the 7Li yields are negative for all three models. The results of FC97 are quite different to ours, and this possibly reflects the different choice of mass-loss law during the TP–AGB phase. The 7Li yields from V02 are similar to ours at solar-like metallicities (see the left panel of Figure 7.8). The main difference is that we observe a peak to the 7Li production at 5M , which is not seen in the yields of V02. At an SMC metallicity, the 7Li yields are

CHAPTER 7. STELLAR YIELDS 180

0 1e-08 Z=0.01 Ventura et al. (2002) Z=0.004 Ventura et al. (2002) Z=0.02 Z=0.004 -5e-09 0 -1e-08

-1.5e-08 -1e-08 ) ) sun sun -2e-08 (M (M

7 7 -2e-08 Li Li -2.5e-08

-3e-08 -3e-08

-3.5e-08 -4e-08 -4e-08 1 2 3 4 5 6 7 1 2 3 4 5 6 7

mass (Msun) mass (Msun)

Figure 7.8: 7Li yield as a function of the initial mass for the Z = 0.02 models (left panel) and the Z = 0.004 models (right panel), compared to the results of Ventura et al. (2002). quite different (see the right-panel of Figure 7.8). Whereas the trend of decreasing 7Li yield with increasing mass is similar, we observe a peak to the 7Li production at 4M , and our models destroy less 7Li at all masses. The combination of convective overshoot in the pre-AGB evolution plus a different convection model results in efficient HBB in V02’s 3.5M , Z = 0.01 and 3M , Z = 0.004 models. This possibly means that if there is a the peak to the 7Li production, it is found at lower mass. Furthermore, V02 use the Bock¨ er (1995) mass-loss law, which will significantly alter their 7Li yields compared to ours. The above comparison clearly demonstrates just how uncertain the 7Li yields are, and that they are highly dependent on many aspects of AGB modelling, such as the treatment of convection and the adopted mass-loss law.

7.5.4 Carbon: 12C The first and second dredge-up result in a decrease in the surface abundance of 12C. There- fore models with no TDU have negative 12C yields. During the TP–AGB phase, the intershell region is enriched in 12C from partial He-burning, so models that experience the TDU have positive 12C yields. The situation is more complex for models which experience HBB. The freshly synthesized 12C is burnt to 14N at the base of the envelope, thus prevent- ing carbon star formation. Even if the model becomes a carbon star after HBB ceases, the overall 12C yields for these models are much lower than for models with no HBB. In Figure 7.9 we show the 12C yield as a function of the initial mass. In the most massive models, HBB is active and there is good agreement between all calculations for M & 4M . We again find very good agreement between our yields and those of I03. This result is not surprising given that they use our prescription for the TDU. M01’s 12C yields are larger (by a factor of two or three) than all the other calculations, and is again related to her extra dredge-up, as described in 7.5.2. We observe larger 12C production in our models than HG97 for M & 2.5M at Z§= 0.02 and M & 1.5M at Z = 0.008 and Z = 0.004. These masses roughly correspond to the initial masses where efficient TDU first appears at each metallicity, and is the minimum mass for carbon star production in our models. CHAPTER 7. STELLAR YIELDS 181

Figure 7.9: Same as Figure 7.5 but for 12C. CHAPTER 7. STELLAR YIELDS 182

Thus, efficient TDU seems to outweigh a greater number of thermal pulses in the case of our yields compared to HG97. HG97 produce more 12C for models with M . 2.5M at Z = 0.02. This is not surprising, given that these Z = 0.02 models do not experience the TDU, whereas HG97 set λ = 0.75 when a minimum core mass of 0.58M is reached. The FC97 yields for 12C are very similar to HG97, which is not surprising because their TDU prescription is similar (λ = 0.6 in FC97; λ = 0.75 in HG97), and they both use the Reimer’s mass-loss law. V02 do not find the TDU in their models with Z = 0.01, and the TDU is only mildly efficient (where λ 0.5) for the most metal-poor cases. Combined with efficient HBB at all masses, it is not∼surprising that their 12C yields are negative in all cases. There is also the question of whether HBB actually occurs in stars with masses as low as 3M at an SMC composition, as found by Ventura et al. (2002). Observations of carbon stars in the SMC would suggest the answer to this question is no, because most of the C-rich stars in this system have masses between 1 to 3M (Wallerstein & Knapp 1997; also see Figure 3 in Lattanzio & Wood 2003).

7.5.5 Carbon: 13C The surface abundance of 13C increases after the first and second dredge-up. The TDU has only a very small effect on the surface abundance of 13C, because α capture in the He- burning shell destroys this isotope very quickly. Not all of the ashes of−the H-burning shell are engulfed by the convective pocket and there is a thin region in mass that is enriched in 13C. But overall, this region has little effect on the surface abundance of 13C. HBB converts the 12C mixed to the surface first to 13C, and later to 14N. When weighted by the IMF, we observe that low-mass stars contribute almost equally to the overall production of 13C at Z = 0.02 (see Figure 7.10). At lower Z, the production of 13C by HBB is very efficient and outweighs the contribution from lower mass stars. From Figure 7.10 we see there is a 13C peak at all metallicities, clearly shown by the synthetic models of M01 and I03. The position of the peak is at 5M at Z = 0.02, 4M at Z = 0.008 and at about 3.5M at Z = 0.004. For models that do not experience HBB, the agreement between all calculations is very good, except that HG97 produce more 13C at 1M at Z = 0.02. The reason for the peak at 1M is not clear, but it is probably related to the depth of the first dredge-up in the detailed models of Schaller et al. (1992). There is relatively good agreement between our HBB models and those of I03, M01 and FC97 at Z = 0.02. For the lower metallicities, the agreement between our yields and I03 remains good, which makes sense since they use the 13C surface abundance from our models to calibrate their synthetic HBB model. Marigo produces more 13C in the 4M case at Z = 0.008 and in the 3.5M at Z = 0.004, otherwise the agreement remains good. It is interesting that Marigo produces much more 13C at these masses compared to the other calculations. This could mean that HBB is very efficient for even the lowest mass that experiences it, unlike for our models where there is a varying level of efficiency that depends on the stellar mass. Neither HG97, V02 or FC97 produce very much 13C, and this is especially true for the lower metallicity models. The reason that HG97 do not produce as much 13C is because their parameterized HBB model is not as efficient compared to the other calculations. CHAPTER 7. STELLAR YIELDS 183

Figure 7.10: Same as Figure 7.5 but for 13C. CHAPTER 7. STELLAR YIELDS 184

However, it is not clear why the 13C yields from FC97 and V02 are much lower than ours. It cannot be related to the initial abundance, because HBB produces a mixture of primary and secondary 13C (the primary component is from proton capture on primary 12C, mixed into the envelope by the TDU). Could it be that the HBB is so eﬃcient that all the 13C created is destroyed to produce 14N? Comparing the 14N yields will help answer these questions.

7.5.6 Nitrogen: 14N Low to intermediate stars are believed to be the main producers of nitrogen in the Galaxy. Primary nitrogen is produced in AGB stars by HBB although smaller amounts of secondary 14N are also added to the surface from the first and second dredge-up. For this reason the yields of 14N are always positive, even for the lowest mass models. In Fig- ure 7.11 we see there is a sharp discontinuity between the small yields of 14N from the low-mass models and the large yields from the HBB models. At a solar composition, the contribution from low-mass models is important, producing as least as much 14N as the more massive models. At lower metallicity, the contribution from more massive AGB stars dominates, and there is a peak in the 14N production at about 5M at all metallicities. The situation for the 14N yields is similar to that for 13C. Again, there is good agreement between models that do not have HBB, and we once again observe a peak at 1M , Z = 0.02 in the HG97 yields. For models with HBB, the agreement between the various calculations is mixed. At Z = 0.02, our results agree very well with the results of I03, FC97 and M01, but HG97 do not produce as much 14N. The 14N yields from Marigo’s Z = 0.008 HBB models are smaller than ours, perhaps an indication that her HBB model is not as metallicity dependent compared to detailed calculations. At Z = 0.004, the minimum mass for HBB is 3.5M in M01, and consequently there is efficient 14N production at this mass. Indeed, the 3.5 and 4M , Z = 0.004 models of M01 produce more 14N than any other calculation. At lower metallicities, neither FC97 nor HG97 produce very much 14N. Both of these calculations do not produce very much 13C either, indicating that HBB is not as efficient in their models and/or the choice of the Reimers mass-loss law has a large effect on the HBB yields. If we now examine the 14N yields from V02, we observe that the yields decrease with an increase in the initial mass, and that there is no peak to the 14N production unlike in the other models (even HG97). The reason for this unusual trend is not clear. Perhaps the 14N peak is pushed to a lower mass, owing to a lower minimum mass for HBB? Or perhaps HBB is so efficient in these models that the 14N is also destroyed? A more complete set of stellar yields from V02 would help answer these questions.

7.5.7 Nitrogen: 15N The rare nitrogen isotope, 15N, is not produced in low to intermediate mass models. The ﬁrst and second dredge-up, as well as the TDU reduce the surface abundance of 15N so the overall yields are negative. HBB also destroys 15N, where the 14N/15N ratio is near the equilibrium value of 2.5 104 (Clayton 1983). In the left panel of Figure 7.12 we show × CHAPTER 7. STELLAR YIELDS 185

Figure 7.11: Same as Figure 7.5 but for 14N. CHAPTER 7. STELLAR YIELDS 186

Figure 7.12: (Left) same as Figure 7.5 but for the Z = 0.02 15N yields. (Right) the 15N yields for all our models . The black points refer to the Z = 0.02 models, the open blue squares to the Z = 0.008 models and the open red circles to the Z = 0.004 models. the 15N yields for the Z = 0.02 models. Note that HG97 and V02 do not calculate yields of 15N. We observe that our yields agree very well with Marigo’s for masses M > 1.5M ; and similarly, there is good agreement between our 15N yields and FC97, which are very similar in value and shape. The reason is that the initial 15N abundances are similarb, so the change to the surface 15N abundance from the first and second dredge-up is also similar. HBB efficiently destroys 15N so there is agreement between the most massive models. The 15N yields calculated by I03 are negligible owing to their approximate treatment of the CNO cycle. In right panel of Figure 7.12 we show the 15N yields from all our models. This diagram simply reflects the decrease in the initial abundance with decreasing metallicity. Because 15N is altered by secondary burning processesc, models with initially more 15N will consume more 15N via H-burning.

7.5.8 Oxygen: 16O AGB stars have not historically been associated with 16O production. The ﬁrst and second dredge-up decrease the surface abundance of 16O, but the TDU will increase the abundance, depending on the amount of 16O present in the intershell. Because the 16O intershell abundance is small (. 2%) in our models, the increase during the TP–AGB is also small. For most models this results in negative yields, but some of the Z = 0.004 models produce small positive 16O yields. HBB depletes 16O but because the ON cycle takes a long time to come into equilibrium, the surface depletions are small except in the most massive models. From Figure 7.13 we see that all the calculations produce very little 16O at a solar composition, except HG97, who produce positive 16O yields in their lowest mass, Z = 0.02

bthe initial 14N/15N ratio in our Z = 0.02 models is 272 compared to 253 in FC97 cby secondary we mean created or destroyed by proton/α±capture on nuclei present at the formation of the star, and not created from the initial H or He CHAPTER 7. STELLAR YIELDS 187

Figure 7.13: Same as Figure 7.5 but for 16O. CHAPTER 7. STELLAR YIELDS 188

Figure 7.14: (Left) same as Figure 7.5 but for the Z = 0.02 17O yields. (Right) 17O yields for all our models. The black points refer to the Z = 0.02 models, the open blue squares to the Z = 0.008 models and the open red circles to the Z = 0.004 models. models. In the top panel of Figure 7.13 the 16O yields from HG97’s 1 and 1.25M models are so large (after weighting with the IMF) that we do not include them on the graph for clarity. This result is unusual, because it means that the 16O abundance is enhanced by the first dredge-up. At Z = 0.008, we destroy more 16O than any other calculation, and M01 produce the most 16O at about 2M at this metallicity. I03 produce more 16O than we do min at most masses and metallicities, probably reflecting their lower values of Mc compared to the detailed calculations. At Z = 0.004 all the calculations produce little 16O except for M01. Both M01 and HG97 use the same intershell abundances, but M01 produce more 16O. Even though HG97 use a larger value of λ and their models experience more TDU min episodes (as a consequence of using the Reimers mass-loss law), the smaller value of Mc and a larger core growth rate in M01’s models results in larger 16O yields. The 16O yields of FC97 and V02 are consistent with our results, as their 16O yields are negative for the range of masses considered. In the case of V02, this is owing to efficient HBB, and no TDU at Z = 0.01.

7.5.9 Oxygen: 17O In the left panel of Figure 7.14 we show the Z = 0.02 17O yields from each of the authors as a function of the initial mass. HG97 do not provide yields of this isotope, and the 17O yields from I03 are zero for models without HBB. The surface abundance of 17O is enhanced by the first and second dredge-up, and for this reason there is a peak to the 17O yields at about 2.5M . Marigo produces more 17O than we do but finds the same peak at 2.5M . FC97’s 17O yields are similar to ours for the intermediate mass models. The 17O yields from V02 are not included because they are off the scale of Figure 7.14, and we wanted to show the peak at 2.5M . The reason that their 17O yields are larger is because of different initial oxygen isotopic ratios compared to us and FC97. For example, at Z = 0.01, 17 6 17 6 V02 use X( O) = 2.02 10− , compared to our value of X( O) = 3.876 10− . The initial 18O abundances are× remarkably different: the initial mass fraction used× by V02 is CHAPTER 7. STELLAR YIELDS 189

Figure 7.15: Same as Figure 7.14 but for 18O.

18 8 18 5 X( O) = 1.13 10− , compared to our value of X( O) = 2.16 10− . Note that our (and FC97’s) initial× values agree well with the meteoritic 17O/18O×value of 0.18 (Anders & Grevesse 1989). Similar to 15N, 17O is aﬀected by secondary burning processes, so the ﬁnal yields depend upon the initial abundance. This is illustrated in the right panel of Figure 7.14, where we show the 17O yields from all our models. From this diagram, we observe that the largest peak is for the Z = 0.02 models, and the 17O yields from the HBB models are small in all cases, with only a slight dependence on the initial metallicity.

7.5.10 Oxygen: 18O The ﬁrst and second dredge-up reduce the surface abundance of 18O, and HBB causes further depletions in the most massive models. For this reason, all our 18O yields are negative. In the right panel of Figure 7.15 we show the 18O yields from all our models. Analogous to both 15N and 17O, the Z = 0.02 models consume more 18O than the lower metallicity models. In the left panel of Figure 7.15 we show the Z = 0.02 18O yields from each author. Neither HG97 nor I03 compute yields for 18O. Our results agree very well with Marigo’s in shape and value. FC97s yields for 18O are similar to ours in shape, but we destroy more 18O in the most massive HBB models. We have included the 18O yields from V02, but they are almost zero and therefore negligible. The reason for this is twofold: 1) eﬃcient HBB at all masses destroys 18O and 2) their initial 18O abundance is much smaller than those used in the other calculations.

7.5.11 Fluorine The surface abundance of 19F is depleted by the ﬁrst and second dredge-up (there are a few cases where there is an increase, see Chapter 3), but the TDU results in a large increase in models with M 3M , resulting in positive yields. As discussed in 7.2.3, ∼ § CHAPTER 7. STELLAR YIELDS 190

-3.5 -4 19 -4.5 -5 -5.5 -6 -6.5 Z = 0.02 -7 Z = 0.008

intershell abundance: F -7.5 Z = 0.004 -8 1 2 3 4 5 6 7

mass (Msun) Figure 7.16: (Upper panel) weighted yield of 19F for the Z = 0.02, the Z = 0.008 and Z = 0.004 models. (Lower panel) the logarithm of the intershell 19F abundance (in mass fraction), taken from the last calculated thermal pulse. CHAPTER 7. STELLAR YIELDS 191 the 19F production mechanism is complex, and depends on many uncertainties, including the inclusion of a 13C pocket. We do not include a 13C pocket in our nucleosynthesis calculations, so we expect that the amount of 19F produced by our models will be smaller than models that do include this pocket. For the most massive models, HBB efficiently destroys 19F, so the yields are negative. From hereon, we only compare to the yields of FC97 unless specified. There is a peak to the 19F production between about 2 to 3M , depending on the metallicity (see the upper panel of Figure 7.16). The amount of 19F produced increases as the metallicity decreases, and the production is shifted toward lower mass. This is because the efficiency of the TDU increases with decreasing metallicity, plus for a given mass, the number of thermal pulses (and hence TDU episodes) also increases. FC97’s 19F yields are consistent with ours, especially in the most massive models where HBB destroys 19F in the envelope. In the lower panel of Figure 7.16 we show the 19F intershell abundances as a function of the initial mass. The abundances are in logarithm of the mass fraction, and are taken from the last thermal pulse calculated in detail (and are in fact the input for the synthetic algorithm). The peak to the 19F intershell abundances coincides with the peak observed in the yields, and is almost independent of the initial metallicity. The amount of 19F in the intershell of the most massive models has been depleted by H-burning before the pulse, so whilst there is an increase during He-shell burning, it is small compared to the amount produced in the less massive models.

7.5.12 The Neon Isotopes The yields of 20Ne are small in all cases as the abundance of this isotope does not change very much over the lifetime of the models. In the upper panel of Figure 7.17 we show the 20Ne yields as a function of the mass. There is a temperature (or mass) dependence to the stellar yields. For example, there are very small negative yields for the low-mass models. In these cases, the first dredge-up mixes material to the surface that is slightly depleted in 20Ne from H-burning (to produce 21Ne). For the more massive models (M > 2M ), the yield of 20Ne shows bi-modal behaviour with initial mass. This is because there is a small increase the surface 20Ne abundance during the TP–AGB in the M 3M models (more noticeable in the Z = 0.004 models). Thus the yields for these models∼are positi ve, because the temperature in the H-shell is hot enough for a small amount of 23Na (and possibly 19F) to be destroyed by proton capture. The TDU then mixes material into the envelope which is slightly enhanced in 20Ne. In the most massive models, the 20Ne abundance is enhanced by the more efficient destruction of 23Na, and possible leakage out of the CNO cycles via the reaction 19F(p,γ)20Ne, which is effective at temperatures over about 80 million K. The dip at 4M corresponds to the decrease in the 20Ne abundance from the second dredge-up. In the lo wer panel of Figure 7.17 we show the yields of 21Ne. In the low-mass models there is little change to the surface abundance of 21Ne from the first, second or third dredge-up, so the yields are very small in all cases. In the intermediate mass models, the surface abundance of 21Ne is enhanced by the first and second dredge-up. The TDU further increases the abundance, but only if the temperature is hot enough for the reaction CHAPTER 7. STELLAR YIELDS 192

Figure 7.17: (Upper panel) weighted yield of 20Ne for the Z = 0.02, the Z = 0.008 and Z = 0.004 models. The yields of Forestini & Charbonnel (1997) are also included. (Lower panel) same but for 21Ne. CHAPTER 7. STELLAR YIELDS 193

18O(α, n)21Ne to occur (T & 300 million K). We note there is also a small contribution from 17O(α,γ)21Ne. Thus in the intermediate mass models without HBB, there is a peak to the 21Ne yield at about 3 to 4M , depending on Z. At Z = 0.02 the peak is small and at 4M , but at Z = 0.004, the peak is larger and pushed to lower mass. In the most massive models, HBB efficiently destroys 21Ne, so the yields are negative. Both the first and second dredge-up result in a small decrease in the surface abundance of 22Ne, but the TDU results in a large increase. Consequently all models with efficient TDU have large positive yields of 22Ne, as shown in Figure 7.18. There is a peak to the 22Ne production at about 3M , depending on the composition. Both HBB and α capture in the He-shell destroys 22Ne, and consequently the yields from the most massiv−e models are almost zero or slightly negative. In the upper panel of Figure 7.18 we show the Z = 0.02 yields of 22Ne from our models (black solid points), I03 (blue crosses) and FC97 (solid green squares). Our yields are observed to agree very well with the results of I03. Note that yields of 22Ne from the most massive I03 models will be over-estimates, because they do not include 22Ne destruction rates in their synthetic model. We compare to FC97 below. In the lower panel of Figure 7.18 we present the yields of 22Ne from all our models, and observe that the lowest metallicity models produce the most 22Ne. The upper panel of Figure 7.17 shows that the 20Ne yields of FC97 are quite different to ours. Because the yields of 20Ne yields are very small, the reason for the difference could simply be a matter of different modelling used on the TP–AGB i.e. different mass- loss rates, different nuclear-reaction rates and less efficient TDU. In the lower panel of Figure 7.17 we see that the 21Ne yields agree quite well with each other, even though FC97 produce less 21Ne, and their peak of production is shifted toward lower mass. The lower panel of Figure 7.18 shows that the FC97’s 22Ne yields are consistent with our findings, except at 3M . At this mass, the 22Ne Z = 0.005 yield is smaller than the Z = 0.02 yield, contrary to our findings, which show increasing 22Ne production with decreasing metallicity. This is most likely related to FC97’s treatment of the TDU, the efficiency of which does not depend on the metallicity in their synthetic calculations. Our models show this trend because at M 3M , the Z = 0.004 models experience much deeper TDU than the Z = 0.02 models (see≈Table 7.1 on page 165).

7.5.13 Sodium The surface abundance of 23Na is enhanced by all mixing processes. The first and second dredge-up result in an increase for all models, with a further small increase from the TDU. The TDU results in an increase simply because 23Na is not altered by He-shell burning. In the most massive models, HBB creates 23Na at the expense of 22Ne, so the yields from these models are quite large. The efficient destruction of 22Ne by α–capture, and the destruction of 23Na by HBB results in smaller yields at 6M compared to the 5M models. From Figure 7.19 we see that the 23Na yields exhibit bi-modal behaviour as a function of the initial mass. The first peak, which shifts toward lower mass with a decrease in Z, is caused by the increase in the surface 23Na abundance from the first dredge-up. Note also that this peak has been emphasized by the IMF. The second peak, at about 5M , is caused by HBB. CHAPTER 7. STELLAR YIELDS 194

Figure 7.18: (Upper panel) weighted 22Ne yields from the Z = 0.02 models. The black dots are our results, the blue crosses are the results of Izzard et al. (2003) and the green squares are the results of Forestini & Charbonnel (1997). (Lower panel) weighted yield of 22Ne for the Z = 0.02, the Z = 0.008 and Z = 0.004 models. The yields of Forestini & Charbonnel (1997) are also shown. CHAPTER 7. STELLAR YIELDS 195

Figure 7.19: Same Figure 7.17 but for 23Na.

Figure 7.19 shows that the 23Na yields from FC97 are small compared to ours. FC97 do not find a 23Na peak at about 3M from the first dredge-up, but their peak could be at a lower mass. A more detailed set of stellar models would help understand this better. On the AGB, FC97 find that the 23Na abundance is virtually unchanged by the TDU, and they do not find efficient production in their HBB models. FC97 comment that they find 23Na is destroyed in the most massive models.

7.5.14 The Magnesium Isotopes The surface abundance of 24Mg is not effected by the first or second dredge-up. On the AGB, the TDU can decrease the surface abundance in models with M > 3M , owing to the dredge-up of material depleted in 24Mg from H-shell burning. When the temperature exceeds about 90 million K, HBB efficiently destroys 24Mg to produce 25Mg. This only occurs at 6M , depending on the composition. In Figure 7.20 we show the yields of 24Mg from∼all our models. The FC97 yields are very similar to ours at all masses, even for the Z = 0.005 models. This indicates that a similar amount of 24Mg is destroyed in the H-shell and/or by HBB. In Figure 7.21 we show the yields of 25Mg (upper panel) and the yields of 26Mg (lower panel). The heavy magnesium isotopes are produced in the He-burning shells of intermediate mass AGB stars. Thus the yields for 25Mg and 26Mg are large for models with M > 3M . There are small negative yields of 25Mg from the low-mass models. This is because the first dredge-up and the third dredge-up mix material to the surface that has been processed by H-burning and is depleted in 25Mg. In the 6M models, the destruction of 24Mg causes a large increase in the surface abundance∼of 25 Mg, resulting in large 25Mg yields at this mass. The yields of 26Mg are positive in all cases but small for the lowest mass models. There is a small contribution to the surface 26Mg abundance from CHAPTER 7. STELLAR YIELDS 196

Figure 7.20: Same as Figure 7.17 but for 24Mg. the β–decay of 26Al. There is a peak at 3M , notable in the lower Z models. This peak is caused by the 22Ne α–capture reactions becoming activated at about 3M (at Z < 0.02), whilst the dip at 4M corresponds to models with the second dredge-up. From Figure 7.21 we see that we produce more 25Mg and 26Mg than FC97, especially at low Z. This observation reinforces our earlier explanation for the small 22Ne yield at 3M , Z = 0.005, which is that FC97’s models experience shallower TDU in comparison to our models, and that the eﬃciency of the TDU does not depend on the metallicity in their synthetic calculations. FC97 also assume that the intershell abundances are constant from start of the synthetic evolution to the end of the TP–AGB phase. In our models, the 22Ne, 25Mg and 26Mg intershell abundances vary as a strong function of time for most of the TP–AGB; a factor which could help explain why our yields for these isotopes are diﬀerent.

7.5.15 The Aluminium Isotopes The radionuclide 26Al is produced by H-burning via the Mg–Al chain. Thus the surface is enhanced in 26Al by the first and second dredge-up, but because the half-life of 26Al is only τ = 7.1 105 years, it decays to 26Mg before the beginning of the TP–AGB phase. For this reason, ×the only contribution to the 26Al yields is from either the TDU or HBB. The TDU enhances the surface abundance of 26Al in the low-mass models because the 26Al in the intershell region is not efficiently destroyed by neutron capture. However, the increase is small, and owing to the long AGB lifetime of the low-mass models, this 26Al also mostly decays to 26Mg by the end of AGB. The most efficient 26Al production site in AGB stars is HBB. From the upper panel of Figure 7.22 we see that the 6M models produce the most 26Al as expected. The AGB lifetimes of these models are less or comparable to the half- life of 26Al, which is the second factor why there are such large yields from these models CHAPTER 7. STELLAR YIELDS 197

Figure 7.21: (Upper panel) same as Figure 7.17 but for 25Mg. (Lower panel) same as Figure 7.17 but for 26Mg. CHAPTER 7. STELLAR YIELDS 198

Figure 7.22: (Upper panel) same as Figure 7.17 but for 26Al. (Lower panel) same as Figure 7.17 but for 27Al. CHAPTER 7. STELLAR YIELDS 199

(even after weighting with the IMF). If these yields are to be used in a galactic chemical evolution model, the 26Al should be added to the 26Mg yields. Forestini & Charbonnel’s results for Z = 0.02 agree very well with our results, but we produce more 26Al at lower Z which, as indicated before for other species, implies more efficient HBB in our lower Z models plus the use of reaction rates that favour 26Al production. The stable aluminium isotope, 27Al is also produced via the Mg–Al chain, and is mixed to the surface by the first, second and third dredge-up events. In the lower panel of Fig- ure 7.22 we show the yields of 27Al from all our models. The amount of 27Al produced in our models increases with decreasing Z, but the peak production is found in the 4 and 5M , Z = 0.008 models. This is possibly related to the destruction of 27Al from proton capture above temperatures of about 90 million K. The NACRE rate for the reaction 27Al(p,γ)28Si shows that there can be some 27Al destruction at about 85 million K, which is why there is a dip at 6M and in the 5M at Z = 0.004 model. At a given initial mass, the temperature is hotter in a given nucleosynthesis site for the Z = 0.004 models. This means the peak of 27Al production is shifted toward lower mass (the 3.5M model) and that the rate of 27Al destruction is slightly more efficient in the 4, 5 and 6M , Z = 0.004 models than in the Z = 0.008 models of the same mass. For this reason we observ e that the total amount produced is overall less in the SMC intermediate mass models than the LMC models. The results of FC97 suggest there is a peak to the 27Al yields at about 3M , but the mass range is too small to say for sure. We produce more 27Al than do FC97 at all metallicities. The Z = 0.02 models agree reasonably well, and the Z = 0.005 models of FC97 show a large increase at 5M , consistent with the trend observed in our models. The differences between the Al yields are possibly explained by the reasons already put forward: 1) less efficient TDU at lower Z, 2) less efficient HBB, 3) constant intershell abundances for most of the TP–AGB and 4) different nuclear-reaction rates.

7.5.16 The Silicon Isotopes The surface abundance of 28Si does not change as a consequence of the first or second dredge-up (see the results for the first and second dredge-up in Appendix B). During the TP–AGB phase, 28Si is slightly destroyed by neutron capture for intermediate mass models with efficient TDU but no HBB i.e. M 3M . For these models, we observe small negative 28Si yields. HBB produces a small∼ amount of 28Si from proton capture onto 27Al, and overall the yields are small but positive for these models (see Figure 7.23). We observe in Figure 7.23 that the Z = 0.02 models produce very small amounts of 28Si, compared to the lower Z calculations. Forestini & Charbonnel’s results are similar except in the case of the 4M , Z = 0.02 model, where more 28Si has been destroyed than in any other case. This is most likely the result of neutron capture during a thermal pulse, where there will be more free neutrons in the 4M model than the 3M case, and no HBB to increase the 28Si abundance. The abundance of the neutron-rich Si isotopes, 29Si and 30Si, are not altered by the first or second dredge-up. During a thermal pulse, both of these isotopes can be created (in almost equal quantities) by neutron capture. In Figure 7.24 we show the 29Si (upper panel) and 30Si (lower panel) yields from our models. FC97 do not provide yields of these CHAPTER 7. STELLAR YIELDS 200

Figure 7.23: (Upper panel) Same as Figure 7.17 but for 28Si. isotopes. In this diagram, we observe that the behaviour of the 29Si and 30Si yields are almost exactly the same, and the low Z models follow the same bi-modal pattern with initial mass. There are two peaks: one at 3M and the other at 5M , and the Z = 0.02 models only have one peak, at about 3.5M . The ﬁrst peak, at about 3M (or 3.5M at Z = 0.02) is most likely the result of neutron capture in the He-shell, from the neutrons released by the 22Ne(α,n)25Mg reaction. The dip at 4M is possibly caused by extra dilution during the second dredge-up; although the 29Si and 30Si abundances are not changed, the mass of the envelope increases by a substantial amount. The peak at 5M is again the result of neutron capture in the He-shell, but the curve turns over at 6M because the proton capture reactions on both 29Si and 30Si become important at temperatures over about 90 million K.

7.6 Summary and Further Work

In this chapter we have calculated stellar yields from the mass-loss models, using a combination of detailed models for most of the TP–AGB, and synthetic models for the last few thermal pulses. This method of calculating yields is different to most other authors, where semi-analytic or fully analytic models are used to calculate nucleosynthesis during the TP–AGB phase. We give a discussion of the main sources of uncertainty effecting the yield calculations, which are mostly from the choice of mass-loss law used during the AGB and the treatment of convection which effects the operation of the TDU. The Vassil- iadis & Wood (1993) mass-loss law enhances the surface abundance of species effected by hot bottom burning, but the Reimers mass-loss law increases the number of thermal pulses (and therefore third dredge up episodes) experienced by a model. The uncertainties in the reaction rates, convection and opacities have a dramatic impact on the stellar structure and nucleosynthesis and hence affect the final stellar yields. The yields are also dependent upon the initial abundances, and are therefore highly metallicity dependent. CHAPTER 7. STELLAR YIELDS 201

Figure 7.24: (Upper panel) same as Figure 7.17 but for 29Si. (Lower panel) same as Fig- ure 7.17 but for 30Si. CHAPTER 7. STELLAR YIELDS 202

We present the yields in tabular form in Appendix D, and in graphical form in 7.5. We present stellar yields for hydrogen through to 30Si. We compare our results to a number§ of different authors, including van den Hoek & Groenewegen (1997), Forestini & Charbonnel (1997), Marigo (2001), Ventura et al. (2002) and Izzard et al. (2003). Relatively good agreement is reached for the CNO isotopes, even though it is evident that many aspects of the modelling greatly influences the final results. Both deep third dredge-up (where λ 0.9) and increasing the number of thermal pulses (where λ < 0.9) have similar effects upon∼ the stellar yields. This was the case when comparing our yields (with deep TDU at M > 3M ) to those of van den Hoek & Groenewegen (1997), who use the Reimers mass-loss rate but assume shallower TDU. We compare the yields of 7Li to those given by Ventura et al. (2002) and there is reasonable agreement at higher metallicity but poor agreement at Z = 0.004. The reason for the differences are most likely related to the mass loss and the treatment of convection, both of which differ significantly in our calculations. Only fair agreement is reached when comparing the yields of 19F through to 28Si to those provided by Forestini & Charbonnel (1997). For some isotopes, the agreement is excellent, but in many cases our yields are much larger. The reason for this seems to be deeper TDU and more efficient HBB in our lower metallicity models, as the agreement is better at Z = 0.02. Further work would be to re-run all the models with the addition of a 13C pocket. The 13C pocket would need to be added in manually, but this is the standard situation for detailed stellar models that do not include any sort of extra mixing (such as rotational shear or diffusion) during the AGB phase. Thus an extra source of uncertainty creeps in: how large do we make the 13C pocket, i.e. what fraction (in mass) of the intershell region is affected by the pocket? Another unknown is the 13C abundance in the pocket. For low mass models (1 to 3M ), Gallino et al. (1998) suggest that the mass of the 13C pocket 4 should be about 5 10− M , or about 1/20 of the typical mass involved in a thermal pulse. It is somewhat× easier to add in a proton profile, and let the 13C profile develop naturally from proton capture on the plentiful 12C present. The mass of the 13C pocket is smaller in intermediate mass stars, since 1/20 of the typical mass involved in a thermal pulse is reduced by a factor of 2 to 10. Regardless, it should be simple enough to add in a proton profile, and adjust the mass of the pocket until one obtains the required neutron flux. This work would be worthwhile, because we could estimate the effect of the 13C on the nucleosynthesis of the light elements in low to intermediate mass AGB stars. It would also be a worthwhile task to try variants upon the mass-loss law in our calculations. For example, we could use the Reimer’s (1975) or the Block¨ er (1995) mass-loss law, keep all the other uncertain variables (such as reaction rates, initial abundances and opacities) the same, and estimate the effect on the stellar yields. To do this thoroughly, we would have to re-run models over the same range in stellar mass and composition as studied here. It would also be useful to extend the range of mass and metallicity of the stellar yields presented in this chapter. For example, we could extend the range of mass from about 0.8M to 8M , as well as covering a larger range of initial metallicities, 0.01 6 Z 6 0.0001. Chapter 8

Conclusion

In this thesis we consider a diverse range of topics related to asymptotic giant branch stars. In Chapter 2, we apply a synthetic binary-evolution algorithm developed by Tout et al. (1997) to the task of explaining the orbital parameters of the chemically-peculiar barium stars. We substantially modify the binary-evolution algorithm to include a stellar- wind accretion model based on the formalism of Bondi & Hoyle, a tidal evolution model and an improved treatment for the AGB phase of evolution. The orbital parameters of the barium stars are, on the whole, consistent with our model, but the best fit requires a low accretion efficiency and much weaker tides than expected from either observation or theory (Zahn 1977). The model does not reproduce the orbital periods and eccentricities of the shortest-period barium stars, and tends to overproduce very-long period systems. One of the weaknesses of our wind-accretion model is the treatment of the third dredge- up (TDU), which brings s-process elements from deep in the interior to the surface. In the synthetic binary-evolution model, we parameterize the TDU using the results of Frost (1997) for stars with M > 4M , and Straniero et al. (1997) for stars with M 6 3M . No paper had a complete set of models ranging between 0.8M to 8M and there was a distinct lack of recently published models (using the OPAL opacities) with masses in the range 2 6 M(M ) 6 4. We remedy this situation by evolving two sequences of detailed stellar models, one with and one without mass loss. We calculate models with masses between 1M and 6M , because this range covers the behaviour we are interested in, with three different initial compositions, Z = 0.02, 0.008 and 0.004. We evolve the models from the pre-main sequence to near the end of the thermally-pulsing AGB phase, through all intermediate stages including the core He-flash for low-mass models. In Chapter 6 we present the results of the models without mass loss, and in Chapter 4 the results of the mass-loss models. The effect of mass loss is negligible for M & 3M , but has a dramatic impact on the evolution of low-mass models causing their third dredge-up efficiency to be greatly reduced. We compare our results to Straniero et al. (1997), and our models produce deeper dredge-up in a 3M , Z = 0.02 model but less efficient dredge-up in a 1.5M , Z = 0.02 model. Clearly the numerical differences between the codes, including the prescription for mass loss during the AGB and the treatment of convective boundaries, has a large impact on the stellar structure. The models without mass loss are used to parameterize the third dredge-up, and in

203 CHAPTER 8. CONCLUSION 204

Chapter 6 we present a set of fitting formulae for the core mass at the first thermal pulse, the core mass at the first TDU episode and for the TDU efficiency parameter, λ. We also include a simple scheme for modelling the increase of λ with pulse number. As they stand, we expect that these fitting formulae will not match the observed carbon-star luminosity functions in the Magellanic Clouds, a long-standing problem. Some adjustments to the fits may be necessary when used in synthetic evolutionary algorithms, but the changes must be consistent with the dependence on mass and composition presented here. This may constrain the adjustments and lead to a better understanding of where the detailed models can be improved. Further work would be to extend the parameterization to a much wider range of initial mass and composition. Models covering a larger range of initial mass, from 0.8M to 8M , with initial metallicities from Z = 0.0001 to Z = 0.03 would improve the usefulness of the fits by making them accessible to studies of Population II stars. The mass-loss models are ideally suited to study the contribution made by AGB stars to the interstellar medium. Hence we perform detailed nucleosynthesis calculations on these models, using a post-processing code with 74 species, 506 reactions and time- dependent mixing on all convective boundaries. Most of this thesis is spent discussing the details of these mass-loss models and presenting the results from this extensive study. In Chapter 3 we begin with a discussion of the stellar lifetimes and the surface abundance changes caused by the first and second dredge-up. We compare our results to the previously published calculations of Charbonnel (1994), El Eid (1995) and Boothroyd & Sackmann (1999). In Chapter 4 we discuss the AGB phase of evolution, and explore the structural features relevant to nucleosynthesis. In particular, we examine how the size and duration of the convective pocket, the temperature in the He-burning shell, the interpulse period and the pre-pulse surface luminosity maximum vary as a function of the H-exhausted core mass. We compare our results to fitting formulae from the literature, notably those by Iben (1977), Boothroyd & Sackmann (1988b,c), Wagenhuber & Groe- newegen (1998) and Izzard et al. (2003). The formulae of Iben (1977) and Boothroyd & Sackmann (1988b,c) only reproduce the broad qualitative features of our data whilst the formulae by both Wagenhuber & Groenewegen (1998) and Izzard et al. (2003) are quite successful at reproducing our results. Without modification, the formula of Wagenhuber & Groenewegen (1998) for the pre-pulse surface luminosity maximum is only moderately successful at reproducing the range of luminosities seen in the models, but by increasing the mixing-length parameter, α, we obtain an excellent fit. The formula of Izzard et al. (2003) is successful at reproducing the luminosities in the Z = 0.02 models, but less so for the lower metallicities cases. We conclude that the addition of a metallicity dependence would improve their fit. We extensively review the nucleosynthesis that occurs during the TP–AGB phase in Chapter 5, with a separate discussion for the nucleosynthesis resulting from thermal pulses and hot bottom burning. Efficient dredge-up results in large surface enhancements in 12C, 22Ne, and in the most massive models, enhancements in the heavy magnesium isotopes 25Mg and 26Mg. Hot bottom burning results in large enhancements in 14N, 23Na and 26Al. The surface abundance changes to sodium and the neon, magnesium and aluminium isotopes are studied in detail. These isotopes are particularly relevant when trying to understand the origin of the O–Na and Mg–Al anti-correlations observed in some globular CHAPTER 8. CONCLUSION 205 cluster stars (Kraft et al. 1998). For this reason, we compare our results to the latest observations of stars in the globular cluster NGC 6752 by Yong et al. (2003), but also to the observations of field stars by Gay & Lambert (2000) and to the dusty carbon star, IRC+10216, observed by Guelin´ et al. (2000). These observations share the common feature that they all measure the Mg isotopic ratios. Our models produce more 25Mg compared to 26Mg, a trend that is consistent with the observations of Gay & Lambert (2000) but not with the observations of Yong et al. (2003). The likely reason for the discrepancy is the effects of nucleosynthesis in low-metallicity stars, plus different initial conditions to those that we use. The initial Mg ratios are solar in our models but perhaps for low- metallicity stars it would be more appropriate to use non-solar Mg isotopic ratios, where the proportion of 24Mg to 25Mg and 26Mg is greater than that observed in the solar system (where about 80% of the Mg in the solar system is 24Mg). In the last part of Chapter 5 we compare the neon abundances predicted by the Z = 0.02 and Z = 0.008 models to the observations of elemental neon in planetary nebulae (PNe). We compare our results to the observations of Henry (1989), and to the more recent observations of Stasinska et al. (1998). All the Z = 0.02 and most of the Z = 0.008 models fit within the range of neon abundances observed in galactic and LMC PNe. The reason for the agreement is that the third dredge-up is not efficient enough in the low- mass models to significantly increase the elemental neon abundance. In the most massive models, the rate of 22Ne destruction from both hot bottom burning and α–capture causes a small increase in the total Ne abundance. Only the models with M 3M produce enough 22Ne to noticeably alter the Ne abundance. The largest increase is≈only about 0.2 dex for the 3M , Z = 0.02 model, but the increase is 0.6 dex for the 3M , Z = 0.008 model. The reason for the large increase is efficient TDU, very little 22Ne destruction from α–capture and no hot bottom burning. Whilst observations of LMC PNe might still be missing the few PNe with large neon abundances, we tentatively conclude that the third dredge-up is too efficient near the end of the TP–AGB for the 3M , Z = 0.008 model. This result (if confirmed by observations and future modelling) could have important implications for our knowledge of the third dredge-up. In Chapter 7 we calculate the stellar yields from the models with mass loss. We first present the method for estimating the enrichment from any remaining thermal pulses, and give a discussion of the uncertainties that affect the yields. We conclude that the largest uncertainties stem from the mass loss and the treatment of convection, and until we have a complete theory of these physical processes, they are likely to remain the largest sources of error. We present yields for hydrogen through to 30Si, in a form suitable for galactic chemical evolution models. We discuss our results for each nuclei, starting with hydrogen, and compare our results to the most recent calculations by van den Hoek & Groenewegen (1997), Forestini & Charbonnel (1997), Marigo (2001), Ventura et al. (2002) and Izzard et al. (2003). Most of these other studies, with the exception of Ventura et al. (2002), are calculated with semi-analytic or fully analytic models, whereas we use detailed stellar models for most of the TP–AGB phase. All the yields are, in general, very different from each other. This result is most likely related to the different modelling approaches used, including different mass-loss laws on the AGB. In conclusion, the differences between the various authors indicate that we still have a long way to go before we have an accurate CHAPTER 8. CONCLUSION 206 estimate of the contribution made by AGB stars to the chemical evolution of galaxies and star clusters.

8.1 Future Directions

It would be a worthwhile task to try variants upon the mass-loss law used in the current study. Because of the many numerical differences between our stellar models and those published in the literature, it is very difficult to accurately gauge what influence mass loss has on the nucleosynthesis. To do this, we would need to calculate an internally- consistent set of AGB models with the same input physics and initial compositions, but with variations on the mass-loss law. This study would be an invaluable resource as it would provide some idea of the errors associated with the mass loss upon the stellar yields. It is still unknown to what extent convective overshoot occurs inside real stars. There is a wealth of circumstantial evidence that suggests that convective overshoot occurs during the core burning phases of stellar evolution, but with improvements in the input physics, notably the opacity, it is still unclear to what extent overshoot is needed to match selected observable quantities. The inclusion of convective overshooting during the pre-AGB and AGB phase will also affect the evolution and nucleosynthesis that occurs during the final nuclear-burning phase of evolution, and a detailed study is required. Many of the nuclear-reaction rates used are still highly uncertain at the temperatures associated with AGB interiors. For this reason a study of how uncertainties in the reaction rates effect the stellar yields is necessary. Because we include over 500 reactions in the nucleosynthesis algorithm, we would need to focus on what reactions are the most important to the study at hand. For example, to probe the affect of the reaction rate uncertainties on the production of the magnesium isotopes, we would examine the upper and lower bounds on rates involved in the Mg–Al chain and the 22Ne α–capture reactions. To conclude, we have applied ourselves to wide variety of related problems: from the formation of the barium stars, to the dependence of the third dredge-up on mass and metallicity, and the production of the neon, magnesium and aluminium isotopes inside AGB stars. We also provide insight into the contribution made by AGB stars to the interstellar medium, and highlight some areas of uncertainty that would benefit from an in-depth analysis. Appendix A

HR Diagrams

Figures A.1 to A.3 show the HR diagrams for all the mass-loss models. All evolutionary tracks begin on the zero-age main sequence, though actual model sequences begin prior to this phase, near the Hayashi track. The TP–AGB phase is not included for clarity. The HR diagrams for the more massive models (M > 3M , depending on Z) show blue loops during the core He-burning phase, and core-breathing pulses are visible in some models.

207 APPENDIX A. HR DIAGRAMS 208

Figure A.1: Evolutionary tracks for the Z = 0.02 models. APPENDIX A. HR DIAGRAMS 209

Figure A.2: Same as Figure A.1 but for the Z = 0.008 models. APPENDIX A. HR DIAGRAMS 210

Figure A.3: Same as Figure A.1 but for the Z = 0.004 models. Appendix B

Surface Abundance Results From the First and Second Dredge-Up

In Tables B.1 to B.9 we present the surface abundance changes for hydrogen through to 30Si as a result of the first and second dredge-up, with a separate table for each metallicity. The results presented in these tables are discussed in detail in Chapter 3. Tables B.1, B.2 and B.3 detail the abundance changes made to hydrogen, 4He and the CNO isotopes for the Z = 0.02, Z = 0.008 and Z = 0.004 models, respectively. In each table we include the initial mass (in solar units), the H mass fraction, the 4He mass fraction, the C/Z, N/Z and O/Z ratios (by mass) and the surface isotopic ratios (by number) for 12C/13C, 14N/15N, 16O/17O and 16O/18O after the first and/or second dredge-up events. The first row of each table is the initial value of each quantity. In Tables B.4 to B.9 we present the fractional surface abundance changes made to 7Li, 19F, the neon isotopes, 23Na and the magnesium, aluminium and silicon isotopes, with a separate table for each metallicity. We also include the fractional changes to the elemental neon and magnesium abundances.

211 APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 212 solar ratios (in O The 18 / mass O 500.0 654.6 656.9 655.6 657.0 660.3 660.4 520.0 567.4 602.4 641.8 646.8 653.5 651.8 652.0 648.9 638.4 646.9 646.7 636.4 652.6 16 models. initial O 02 . 17 / the 0 O 2631 2597 2229 1504 is = 335.7 373.1 451.6 544.0 624.0 655.3 665.7 557.6 315.6 340.7 386.7 477.5 696.3 668.3 702.3 859.5 301.1 16 Z N the 15 column / for N 1468 1520 1570 1923 2291 2429 1124 1153 1333 1443 1481 1510 1470 1572 1592 1096 1262 272.8 508.6 741.8 937.1 14 ®rst The O . / dredge-up C 0.4775 0.3218 0.3222 0.3234 0.3252 0.3262 0.3267 0.4266 0.3821 0.3531 0.3304 0.3275 0.3232 0.3237 0.3252 0.3274 0.3338 0.3310 0.3316 0.3319 0.3231 number C second 13 by / C 28.0 and 89.87 23.40 22.96 22.69 22.00 21.52 21.38 31.95 25.78 24.50 24.30 23.78 23.66 23.27 23.08 23.01 22.71 22.63 25.02 24.18 are 12 ®rst ratios Z the / O O 18 / 0.48141 0.46569 0.46345 0.46065 0.44037 0.42147 0.41488 0.48137 0.48124 0.48121 0.48114 0.47240 0.46571 0.46362 0.46137 0.47321 0.45795 0.45685 0.48128 0.48121 0.47649 O after 16 Z and ratios / O N 17 0.05293 0.13669 0.13914 0.14187 0.16463 0.18618 0.19363 0.07438 0.10547 0.11507 0.11642 0.12839 0.13591 0.13771 0.13940 0.13886 0.14192 0.14294 0.09323 0.11446 0.12367 / O 16 undance Z ab N, / 15 C / ace N 0.17250 0.11275 0.11232 0.11208 0.10776 0.10346 0.10201 0.15437 0.12782 0.11961 0.11852 0.11484 0.11340 0.11342 0.11366 0.11886 0.11405 0.11400 0.13828 0.12013 0.11581 14 surf C, 13 and / He C 4 12 0.2928 0.30423 0.30455 0.30482 0.33398 0.35949 0.36788 0.31129 0.30592 0.30122 0.30034 0.30246 0.30421 0.30445 0.30435 0.29609 0.30420 0.30416 0.31070 0.30305 0.30133 undances. the ab fractions) and H 0.6872 mass 0.67577 0.67548 0.67523 0.64613 0.62067 0.61230 0.66860 0.67386 0.67861 0.67951 0.67748 0.67579 0.67558 0.67571 0.68398 0.67590 0.67595 0.66910 0.67675 0.67857 initial mass (in the by w ent v are ro SDU SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial Z / ) O undances ®rst

Ab the and (M 0 Z .1: / M 1.0 1.25 1.5 1.75 1.9 2.0 2.25 2.5 3.0 3.5 4.0 5.0 6.0 6.5 B and N , le Z / ab T units) C APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 213 O 18 / O 195.9 271.3 271.5 267.2 266.3 268.7 205.7 245.0 266.2 268.6 271.5 268.0 267.1 265.2 257.5 254.0 228.0 263.0 270.1 16 O 17 / O 1031 1022 271.3 294.5 343.8 367.9 367.7 628.5 267.8 230.7 225.1 285.5 313.1 357.9 525.2 603.5 907.7 355.7 190.8 16 N 15 / N 637.4 623.0 577.5 774.7 934.5 52.52 140.3 318.6 425.3 480.0 559.4 607.3 577.1 496.8 424.4 413.6 237.5 405.7 511.9 14 O / C 0.3163 0.3160 0.3227 0.3271 0.3264 0.5002 0.4409 0.3574 0.3243 0.3207 0.3168 0.3212 0.3227 0.3253 0.3356 0.3409 0.3895 0.3290 0.3182 008. . 0 C = 13 / Z C 16.78 16.58 16.73 16.61 16.48 36.43 21.53 18.67 17.71 17.51 17.24 17.06 16.98 16.94 16.92 16.85 19.71 17.95 17.41 12 for ut b Z / B.1 O 0.31478 0.31752 0.31958 0.29922 0.28666 0.33227 0.33218 0.33191 0.33178 0.32730 0.32097 0.31512 0.31847 0.32669 0.33184 0.33185 0.33201 0.33182 0.32502 able T Z for / N as 0.09172 0.08861 0.08439 0.10678 0.12152 0.01817 0.03550 0.06002 0.06984 0.07606 0.08447 0.08997 0.08568 0.07541 0.06643 0.06490 0.05060 0.06838 0.07941 Same Z / .2: B C 0.07497 0.07557 0.07765 0.07370 0.07047 0.12484 0.11017 0.08930 0.08099 0.07901 0.07654 0.07621 0.07737 0.08002 0.08387 0.08518 0.09732 0.08219 0.07786 le ab T He 4 0.26712 0.26288 0.26507 0.30289 0.32804 0.25510 0.27499 0.27252 0.26733 0.26708 0.26741 0.26697 0.26232 0.25751 0.25566 0.25566 0.27663 0.26954 0.26619 H 0.72499 0.72925 0.72708 0.68930 0.66417 0.73690 0.71693 0.71936 0.72462 0.72491 0.72464 0.72513 0.72981 0.73464 0.73651 0.73651 0.71527 0.72238 0.72582 ent v SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial )

(M 0 M 1.0 1.25 1.5 1.75 1.9 2.1 2.25 2.5 3.0 3.5 4.0 5.0 6.0 APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 214 O 18 / O 183.1 265.3 265.5 258.3 258.9 261.4 260.9 194.0 231.4 258.3 262.0 259.8 249.8 239.0 210.1 204.2 218.0 248.0 263.5 16 O 17 / O 963.7 164.2 200.2 256.0 251.6 256.8 259.7 953.0 477.1 178.8 168.1 212.4 385.0 580.0 895.0 917.7 786.4 252.7 137.8 16 N 15 / N 31.69 587.3 591.6 493.2 594.0 813.3 949.6 118.8 284.3 412.3 556.2 531.9 404.2 363.7 250.4 227.1 216.9 370.7 489.0 14 O / C 0.5122 0.3074 0.3111 0.3170 0.3182 0.3180 0.3205 0.4451 0.3614 0.3163 0.3126 0.3153 0.3226 0.3437 0.3913 0.4038 0.3886 0.3322 0.3106 004. . 0 C = 13 / Z C 34.85 16.39 16.06 16.28 16.27 16.03 15.97 20.67 17.71 17.04 16.75 16.58 16.53 16.47 16.45 16.37 19.33 17.14 16.90 12 for ut b Z / B.1 O 0.32313 0.30968 0.31113 0.31746 0.30647 0.28693 0.27555 0.32302 0.32275 0.32241 0.30987 0.31243 0.32264 0.32269 0.32288 0.32293 0.32283 0.32259 0.31743 able T Z for / N as 0.01155 0.08484 0.08339 0.07322 0.08558 0.10813 0.12065 0.03065 0.05458 0.06778 0.08319 0.07950 0.06400 0.05962 0.04597 0.04242 0.04680 0.06304 0.07510 Same Z / .3: B C 0.12434 0.07167 0.07182 0.07579 0.07343 0.06872 0.06652 0.10817 0.08782 0.07676 0.07294 0.07417 0.07981 0.08354 0.09516 0.09819 0.09442 0.08068 0.07423 le ab T He 4 0.2476 0.26016 0.25473 0.25234 0.27314 0.30928 0.33052 0.26734 0.26811 0.26327 0.26009 0.25421 0.24925 0.24849 0.24786 0.24775 0.27032 0.26480 0.25943 H 0.73577 0.74125 0.74365 0.72289 0.68678 0.66556 0.74840 0.72844 0.72766 0.73256 0.73584 0.74177 0.74674 0.74752 0.74818 0.74831 0.72545 0.73098 0.73646 ent v SDU SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial )

(M 0 M 1.0 1.25 1.5 1.75 1.9 2.25 2.5 3.0 3.5 4.0 5.0 6.0 APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 215 (in models. 6) − 02 . undances 0 Na ab = 23 1.39350 1.43063 1.44851 1.61853 1.76904 1.82286 1.00021 1.04771 1.17625 1.20451 1.32427 1.38759 1.41624 1.42339 1.42598 1.44673 1.45336 1.00794 1.12957 1.28315 Z 1.66046( initial the the 5) for 0 − w ro Ne / Ne 0.99266 0.99197 0.99164 0.98848 0.98569 0.98469 1.00000 0.99911 0.99671 0.99619 0.99395 0.99277 0.99224 0.99210 0.99206 0.99167 0.99155 0.99985 0.99758 0.99472 ®rst dredge-up 9.72244( the 6) and − second Ne and 22 units), 0.90097 0.89163 0.88713 0.84429 0.80698 0.79410 0.99995 0.98800 0.95567 0.94856 0.91841 0.90246 0.89525 0.89345 0.89280 0.88757 0.88590 0.99800 0.96741 0.92876 ®rst 6.60183( solar the (in 7) − after Ne mass 21 Na 1.01792 1.02390 1.03000 1.17695 1.30973 1.34856 1.00000 1.00003 1.00014 1.00024 1.00906 1.01788 1.02359 1.02819 1.03227 1.04244 1.04555 1.00000 1.00009 1.00441 23 2.20061( initial and the 5) is − Ne isotopes 20 0.99996 0.99995 0.99993 0.99958 0.99921 0.99906 1.00000 1.00000 1.00000 1.00000 0.99998 0.99996 0.99995 0.99994 0.99993 0.99990 0.99989 1.00000 1.00000 0.99999 column 9.04250( neon ®rst the 8) − F, The 19 F i, 19 L 7 0.97124 0.95657 0.94724 0.89340 0.85273 0.83841 1.01892 1.06110 1.04878 1.04388 1.00193 0.97168 0.95841 0.95159 0.94180 0.93495 0.93224 1.05228 1.05759 1.01922 2.44067( of changes. 9) − undance undances Li 7 ab ab 0.19798 0.18308 0.16811 0.12832 0.09558 0.04632 0.63487 0.37531 0.29220 0.27779 0.23338 0.20510 0.19172 0.17909 0.15229 0.13811 0.13917 0.50084 0.31794 0.26150 Ne 1.53042( ace surf total ent v the SDU SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial are )

Fractional (M .4: 0 B fraction). M included 1.0 1.25 1.5 1.75 1.9 2.0 2.25 2.5 3.0 3.5 4.0 5.0 6.0 6.5 le ab mole T Also APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 216 6) − Na 23 1.55206 1.52927 1.47701 1.72855 1.95698 1.00045 1.09993 1.30892 1.39268 1.49778 1.52744 1.49073 1.38174 1.22896 1.18806 1.01775 1.23815 1.43754 1.16523( 5) 0 − Ne / Ne 0.98971 0.99013 0.99112 0.98652 0.98238 0.99999 0.99814 0.99424 0.99268 0.99072 0.99017 0.99085 0.99288 0.99573 0.99649 0.99967 0.99556 0.99184 6.82434( 6) − Ne 22 0.86101 0.86678 0.88028 0.82313 0.78389 0.99989 0.97486 0.92230 0.90120 0.87470 0.86720 0.87648 0.90396 0.94241 0.95270 0.99554 0.94010 0.88991 008. 4.63286( . 0 = 7) Z − for Ne 21 ut b 1.05703 1.05031 1.10854 1.63352 1.70229 1.00000 1.00006 1.00041 1.00854 1.03076 1.05562 1.04276 1.01275 1.00017 1.00013 1.00001 1.00024 1.01252 1.54428( B.4 5) able − T as Ne 20 0.99987 0.99988 0.99972 0.99799 0.99648 1.00000 1.00000 1.00000 0.99998 0.99993 0.99987 0.99990 0.99997 1.00000 1.00000 1.00000 1.00000 0.99997 Same 6.34561( .5: B 8) − le F ab T 19 0.95624 0.95859 0.96610 0.89798 0.85525 1.02289 1.06705 1.03873 1.01120 0.98133 0.96000 0.96555 0.99091 1.01599 1.01507 1.06060 1.05469 0.99655 1.71275( 9) − Li 7 0.17850 0.14625 0.14547 0.12860 0.02145 0.62276 0.38570 0.28136 0.25072 0.20615 0.19431 0.16832 0.15951 0.15293 0.13994 0.49849 0.30811 0.23575 1.07398( ent v SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial )

(M 0 M 1.0 1.25 1.5 1.75 1.9 2.1 2.25 2.5 3.0 3.5 4.0 5.0 6.0 APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 217 7) − Na 23 1.62052 1.59913 1.47618 1.68014 2.15444 2.48376 1.00080 1.14828 1.38890 1.60212 1.55122 1.30270 1.17379 1.02263 1.01530 1.03462 1.29227 1.54178 6.05914( 5) 0 − Ne / Ne 0.98843 0.98883 0.99113 0.98746 0.97900 0.97312 0.99999 0.99723 0.99275 0.98877 0.98972 0.99435 0.99676 0.99958 0.99971 0.99999 0.99455 0.98990 3.54862( 6) − Ne 22 0.84378 0.84922 0.88043 0.84420 0.78510 0.75044 0.99980 0.96270 0.90218 0.84841 0.86127 0.92387 0.95629 0.99431 0.99615 0.99129 0.92649 0.86368 004. . 2.40906( 0 = 8) Z − for Ne ut 21 b 1.05225 1.04898 1.06678 1.47969 1.47570 1.29460 1.00000 1.00012 1.00085 1.05136 1.03916 1.00025 1.00011 1.00001 1.00001 1.00002 1.00047 1.01655 8.03019( B.4 able 5) T − for Ne 20 as 0.99988 0.99989 0.99982 0.99771 0.99330 0.99019 1.00000 1.00000 1.00000 0.99989 0.99991 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99996 3.29968( Same 9) .6: − B F le 19 ab T 0.96204 0.96359 0.98671 0.94461 0.87485 0.83454 1.02655 1.06647 1.03013 0.96414 0.97212 1.02105 1.02671 1.02138 1.01608 1.06666 1.04690 0.98598 8.90621( 10) − Li 7 0.18741 0.15523 0.15399 0.12389 0.11315 0.00003 0.60691 0.38036 0.26696 0.20798 0.18556 0.18570 0.15910 0.18580 0.17318 0.47977 0.29992 0.22279 5.58463( ent v SDU SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial )

(M 0 M 1.0 1.25 1.5 1.75 1.9 2.25 2.5 3.0 3.5 4.0 5.0 6.0 APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 218 7) − 02 i . S 0 initial 30 = 1.0000 1.0000 1.00000 1.00000 0.99999 0.99999 0.99955 0.99918 0.99999 0.99999 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 0.99999 1.00000 0.99999 1.00000 Z the 8.98249( w the ro 6) for − i ®rst S 29 the 1.00000 1.00000 1.00000 0.99999 0.99998 0.99997 1.00000 0.99999 0.99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 dredge-up 1.35037( and 5) − second units), i S and 28 solar 1.00000 1.00000 1.00000 1.00000 0.99999 0.99999 1.00000 0.99999 0.99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 ®rst 2.67074( (in the 6) mass − after Al 27 initial 1.00001 1.00002 1.00002 1.00031 1.00060 1.00072 1.00000 0.99999 0.99999 1.00000 1.00000 1.00000 1.00001 1.00002 1.00003 1.00004 1.00005 1.00000 0.99999 1.00000 2.46068( the isotopes is 5) 0 − silicon Mg / column and 1.00001 1.00002 1.00003 1.00011 1.00012 1.00007 1.00000 1.00000 1.00000 1.00000 1.00000 1.00001 1.00002 1.00002 1.00003 1.00005 1.00006 1.00000 1.00000 1.00000 Mg ®rst 3.11286( The 6) − aluminium Mg 26 1.00263 1.00431 1.00627 1.03683 1.06605 1.07615 1.00000 1.00000 1.00000 1.00001 1.00097 1.00263 1.00424 1.00584 1.00805 1.01172 1.01308 1.00000 1.00000 1.00037 changes. 3.42094( magnesium, 6) − the undance of ab Mg 25 0.99709 0.99524 0.99305 0.95876 0.92526 0.91307 1.00000 1.00000 1.00000 0.99999 0.99893 0.99709 0.99531 0.99354 0.99109 0.98703 0.98553 1.00000 1.00000 0.99959 Mg 3.10086( total undances 5) ab − the ace Mg are 24 surf 1.00000 1.00000 1.00000 1.00000 1.00002 1.00003 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 fraction). 2.46068( mole included ent (in Fractional v SDU SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial Also .7: B )

0 le undances M (M 1.0 1.25 1.5 1.75 1.9 2.0 2.25 2.5 3.0 3.5 4.0 5.0 6.0 6.5 ab ab models. T APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 219 7) − i S 30 1.00000 1.00000 1.00000 0.99998 0.99924 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 6.30349( 7) − i S 29 1.00000 1.00000 1.00000 0.99999 0.99999 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 9.47630( 5) − i S 28 1.00000 1.00000 1.00000 1.00000 1.00001 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 1.87420( 6) − 008. . 0 Al = 27 Z 1.00005 1.00005 1.00025 1.00238 1.00481 1.00000 1.00000 1.00000 1.00000 1.00002 1.00004 1.00004 1.00001 1.00000 1.00000 1.00000 1.00000 1.00001 1.72679( for ut b 5) 0 − B.7 Mg / able 1.00004 1.00004 1.00003 1.00005 0.99997 1.00000 1.00000 1.00000 1.00000 1.00002 1.00004 1.00004 1.00001 1.00000 1.00000 1.00000 1.00000 1.00001 Mg T 2.18505( for as 6) − Same Mg 26 .8: 1.00990 1.00987 1.01426 1.06428 1.09832 1.00000 1.00000 1.00000 1.00080 1.00422 1.00966 1.00874 1.00305 1.00000 1.00000 1.00000 1.00000 1.00138 B 2.40066( le ab 6) T − Mg 25 0.98904 0.98906 0.98394 0.92661 0.88643 1.00000 1.00000 1.00000 0.99911 0.99533 0.98931 0.99033 0.99663 1.00000 1.00000 1.00000 1.00000 0.99847 2.17604( 5) − Mg 24 1.00000 1.00000 1.00000 1.00002 1.00006 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.72679( ent v SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial )

0 M (M 1.0 1.25 1.5 1.75 1.9 2.1 2.25 2.5 3.0 3.5 4.0 5.0 6.0 APPENDIX B. SURFACE ABUNDANCE RESULTS FROM THE FIRST AND SECOND DREDGE-UP 220 7) − i S 30 1.00000 1.00000 0.99999 1.00000 0.99986 0.99924 0.99999 1.00000 0.99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 0.99999 3.27778( 7) − i S 29 1.00000 1.00000 0.99999 1.00000 0.99998 0.99998 0.99999 1.00000 0.99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 0.99999 4.92761( 6) − i S 28 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 9.74573( 7) − 004. . 0 Al = 27 Z 1.00004 1.00005 1.00016 1.00314 1.01122 1.01839 1.00000 1.00000 1.00000 1.00004 1.00003 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00001 8.97922( for ut b 5) 0 − B.7 Mg / able 1.00003 1.00004 1.00001 0.99989 0.99947 0.99908 1.00000 1.00000 1.00000 1.00003 1.00003 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00001 Mg T 1.13590( for as 6) − Same Mg 26 .9: 1.00786 1.00865 1.00711 1.03388 1.07911 1.10706 1.00000 1.00000 1.00001 1.00774 1.00741 1.00000 1.00000 1.00000 1.00000 1.00000 1.00001 1.00202 B 1.24833( le ab 6) T − Mg 25 0.99129 0.99041 0.99196 0.95985 0.90306 0.86635 1.00000 1.00000 0.99999 0.99144 0.99180 1.00000 1.00000 1.00000 1.00000 1.00000 0.99999 0.99776 1.13153( 6) − Mg 24 1.00000 1.00000 1.00000 1.00003 1.00013 1.00025 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 8.97921( ent v SDU SDU SDU SDU SDU SDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU FDU e initial )

0 M (M 1.0 1.25 1.5 1.75 1.9 2.25 2.5 3.0 3.5 4.0 5.0 6.0 Appendix C

Surface Abundance Results for the Asymptotic Giant Branch

In Figures C.1 to C.39 we show the time variation of selected species during the thermally- pulsing AGB phase. The results of the Z = 0.02 models are shown in Figures C.1 to C.14, the results of the Z = 0.008 models in Figures C.15 to C.27, and the results for the Z = 0.004 models in Figures C.28 to C.39. For each model we show the mass fraction of H and 4He, and the logarithm of the mole fraction for other species. For the low mass models that do not experience the third dredge-up we show the results of the ﬁrst dredge-up since there are no abundances changes on the TP–AGB. In each diagram, t = 0 corresponds to the time from the beginning of the TP–AGB phase. Note that we only show the results of the detailed nucleosynthesis calculations, and do not include the contribution from thermal pulses calculated synthetically, as described in Chapter 7.

221 APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 222

Figure C.1: The time variation of selected surface abundances, from the ®rst dredge-up to near the end of the TP-AGB phase for the 1M , Z = 0.02 model. The H and 4He abundances are given as mass fractions, and the abundances of all the other species as the logarithm of the mole fraction log Y. For this model, there is no third dredge-up on the TP±AGB. APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 223

Figure C.2: Same as Figure C.1 but for the 1.25M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 224

Figure C.3: Same as Figure C.1 but for the 1.5M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 225

Figure C.4: Same as Figure C.1 but for the 1.75M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 226

Figure C.5: Same as Figure C.1 but for the 1.9M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 227

Figure C.6: Same as Figure C.1 but for the 2M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 228

Figure C.7: Time variation of selected surface abundances during the TP-AGB phase for the 2.25M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 229

Figure C.8: Same as Figure C.7, but for the 2.5M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 230

Figure C.9: Same as Figure C.7, but for the 3M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 231

Figure C.10: Same as Figure C.7, but for the 3.5M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 232

Figure C.11: Same as Figure C.7, but for the 4M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 233

Figure C.12: Same as Figure C.7, but for the 5M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 234

Figure C.13: Same as for Figure C.7, but for the 6M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 235

Figure C.14: Same as for Figure C.7, but for the 6.5M , Z = 0.02 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 236

Figure C.15: Same as for Figure C.1, but for the 1M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 237

Figure C.16: Same as for Figure C.1, but for the 1.25M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 238

Figure C.17: Same as for Figure C.1, but for the 1.5M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 239

Figure C.18: Same as for Figure C.7, but for the 1.75M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 240

Figure C.19: Same as for Figure C.7, but for the 1.9M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 241

Figure C.20: Same as for Figure C.7, but for the 2.1M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 242

Figure C.21: Same as for Figure C.7, but for the 2.25M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 243

Figure C.22: Same as for Figure C.7, but for the 2.5M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 244

Figure C.23: Same as for Figure C.7, but for the 3M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 245

Figure C.24: Same as for Figure C.7, but for the 3.5M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 246

Figure C.25: Same as for Figure C.7, but for the 4M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 247

Figure C.26: Same as for Figure C.7, but for the 5M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 248

Figure C.27: Same as for Figure C.7, but for the 6M , Z = 0.008 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 249

Figure C.28: Same as for Figure C.1, but for the 1M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 250

Figure C.29: Same as for Figure C.1, but for the 1.25M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 251

Figure C.30: Same as for Figure C.7, but for the 1.5M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 252

Figure C.31: Same as for Figure C.7, but for the 1.75M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 253

Figure C.32: Same as for Figure C.7, but for the 1.9M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 254

Figure C.33: Same as for Figure C.7, but for the 2.25M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 255

Figure C.34: Same as for Figure C.7, but for the 2.5M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 256

Figure C.35: Same as for Figure C.7, but for the 3M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 257

Figure C.36: Same as for Figure C.7, but for the 3.5M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 258

Figure C.37: Same as for Figure C.7, but for the 4M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 259

Figure C.38: Same as for Figure C.7, but for the 5M , Z = 0.004 model.

APPENDIX C. SURFACE ABUNDANCE RESULTS FOR THE ASYMPTOTIC GIANT BRANCH 260

Figure C.39: Same as for Figure C.7, but for the 6M , Z = 0.004 model.

Appendix D

Stellar Yields

For each composition there are three tables containing stellar yields for every stable isotope from hydrogen to 30Si. We also include yields for the unstable aluminium isotope, 26Al, but note that the yield of 26Al should be added to the 26Mg yield. Each table also has the initial and final mass included in the first two columns. The Z = 0.02 yields are presented in Tables D.1 to D.3, the Z = 0.008 yields in Tables D.4 to D.6 and the Z = 0.004 yields in Tables D.7 to D.9. There are two sets of yields for the intermediate mass models with thermal pulses calculated synthetically. The first set of yields have λ = λmax and the second set λ = 0.3 for each remaining thermal pulse. Note that decreasing λ increases the final remnant mass, and this is reflected in the tables.

261 APPENDIX D. STELLAR YIELDS 262 N and 15 mass -1.6388E-05 -7.1417E-06 -6.0486E-06 -4.7005E-06 -3.5979E-06 -1.9229E-05 -2.0934E-05 -1.6439E-05 -7.1619E-06 -6.0595E-06 -4.7107E-06 -3.6018E-06 -1.9264E-05 -4.5846E-07 -2.0972E-05 -1.5894E-06 -2.4661E-06 -2.6785E-06 -1.0059E-06 -2.1677E-06 -3.1388E-06 initial N The 14 models. 1.3292E-02 5.5023E-03 4.3687E-03 3.3971E-03 2.5463E-03 3.7883E-02 3.8269E-02 1.3290E-02 5.4835E-03 4.3472E-03 3.3928E-03 2.5434E-03 3.7837E-02 1.9896E-04 3.8260E-02 9.6902E-04 1.5716E-03 1.7172E-03 5.6398E-04 1.3970E-03 2.1770E-03 02 . 0 C = 13 Z the for 2.1815E-03 1.8863E-04 1.5128E-04 1.1765E-04 9.8595E-05 5.6376E-04 5.5229E-04 2.1885E-03 1.8916E-04 1.5155E-04 1.1790E-04 9.8702E-05 5.6512E-04 2.6437E-05 5.5375E-04 5.6554E-05 7.6360E-05 8.1461E-05 4.2409E-05 6.6940E-05 8.9044E-05 C 12 nitrogen and 3.0430E-03 1.0034E-02 1.9183E-02 1.4168E-02 4.3109E-03 5.5689E-03 1.2680E-02 2.1429E-02 1.5155E-02 4.8532E-03 -1.0055E-02 -1.1251E-02 -7.7563E-03 -1.9273E-04 -8.3558E-03 -8.7616E-04 -1.4098E-03 -1.5364E-03 -5.1743E-04 -1.2527E-03 -8.1703E-04 i carbon L 7 lithium, -3.7635E-08 -8.9820E-09 -3.4023E-08 -3.2995E-08 -2.9559E-08 -1.8750E-08 -1.4212E-08 -3.7731E-08 -9.1040E-09 -1.9090E-09 -3.3102E-08 -3.4118E-08 -6.2552E-09 -2.9612E-08 -9.6880E-09 -1.8790E-08 -1.4067E-08 -1.4228E-08 -6.5028E-09 -8.4237E-09 -1.2276E-08 e helium, H 4 columns. hydrogen, 3.5764E-01 1.8602E-01 6.4426E-02 4.2982E-01 8.0976E-02 5.9786E-02 2.8512E-02 3.6139E-01 1.9051E-01 7.9372E-03 4.3340E-01 6.9986E-02 1.1889E-02 8.5934E-02 1.0551E-02 6.1891E-02 1.0123E-02 2.9508E-02 1.2140E-02 1.1507E-02 1.5455E-02 o of tw He 3 ®rst masses) the in 3.3788E-06 7.6430E-05 1.9009E-04 3.7575E-06 2.0421E-04 2.2230E-04 2.6158E-04 3.3804E-06 7.6656E-05 1.3009E-04 3.7591E-06 1.9062E-04 2.9948E-04 2.0458E-04 3.1735E-04 2.2278E-04 3.1374E-04 2.6187E-04 2.3319E-04 3.1681E-04 2.8926E-04 solar en v gi H (in k are M units) yields, -3.7514E-01 -1.9779E-01 -7.7477E-02 -4.4473E-01 -1.0377E-01 -7.6650E-02 -3.4447E-02 -3.8142E-01 -2.0502E-01 -7.9830E-03 -4.5148E-01 -8.5893E-02 -1.2084E-02 -1.1128E-01 -1.0754E-02 -7.9839E-02 -1.0317E-02 -3.6020E-02 -1.2273E-02 -1.1720E-02 -1.6654E-02 solar f Stellar (in M 0.938 0.885 0.800 0.972 0.721 0.687 0.664 0.929 0.872 0.573 0.962 0.791 0.600 0.716 0.636 0.682 0.640 0.662 0.578 0.620 0.650 .1: D mass 0 le M 6.0 5.0 1.0 6.5 4.0 1.5 1.25 3.5 1.9 1.75 3.0 2.0 2.5 2.25 ab T ®nal APPENDIX D. STELLAR YIELDS 263 Na 23 9.7853E-05 6.9399E-05 1.3984E-04 7.2603E-05 3.1028E-05 1.0041E-03 1.0724E-03 1.0496E-04 7.6220E-05 7.2760E-05 1.4185E-04 1.7634E-07 3.2015E-05 1.9785E-06 1.0606E-05 8.5582E-06 1.0738E-03 1.0069E-03 1.8123E-05 5.1127E-07 5.6582E-06 Ne 22 models. 2.6734E-03 1.2273E-03 9.0229E-04 1.8837E-03 5.3460E-04 2.9849E-03 2.0209E-03 1.5044E-03 1.1038E-03 5.8803E-04 1.2841E-04 5.9669E-05 02 -2.0200E-04 -1.2359E-05 -1.4751E-07 -1.8472E-06 -1.0072E-05 -8.1213E-06 -9.1862E-05 -4.5539E-07 -5.3550E-06 . 0 = Ne Z 21 the for 1.2636E-06 1.2749E-06 1.7439E-06 3.8065E-07 1.5827E-08 1.1937E-09 2.6002E-09 2.0314E-09 1.4688E-06 4.2128E-07 1.9941E-09 1.6124E-06 2.1760E-06 2.6503E-07 2.5086E-09 1.7640E-09 1.6130E-08 -2.3971E-05 -2.1855E-05 -2.3369E-05 -2.1323E-05 sodium Ne 20 and neon 8.7116E-06 8.5328E-06 9.4739E-06 2.1524E-05 1.8725E-05 2.1583E-07 4.4983E-07 6.7800E-07 1.1178E-05 8.3353E-08 6.3586E-07 1.1649E-05 1.2798E-05 2.4791E-05 2.1948E-05 3.3493E-07 5.6368E-07 3.0571E-07 -2.3842E-07 -1.1728E-06 -1.2971E-06 F 19 ¯uorine, oxygen, 5.2975E-06 3.6595E-06 9.0110E-07 1.6686E-06 5.0619E-07 3.6511E-09 9.9595E-07 2.5116E-08 2.7205E-08 5.9987E-06 3.9301E-06 2.8343E-08 2.0689E-06 6.1205E-07 1.5934E-08 3.0086E-08 1.2083E-07 -2.4514E-06 -2.2000E-06 -2.4502E-06 -2.1819E-06 of O 18 masses) solar -1.7173E-05 -1.3265E-05 -9.6784E-06 -1.9459E-05 -8.7945E-05 -1.1859E-04 -1.0860E-04 -4.1185E-07 -9.6890E-06 -3.3926E-06 -6.6462E-06 -1.7204E-05 -1.3294E-05 -6.0436E-06 -1.9514E-05 -8.8222E-05 -1.1880E-04 -1.0879E-04 -1.8030E-06 -5.2561E-06 -8.1420E-06 (in k O M 17 yields, 5.7327E-05 5.1768E-05 4.6499E-05 5.4978E-05 4.2527E-05 9.8681E-05 7.8610E-05 3.4263E-07 4.6549E-05 3.1920E-06 1.9333E-05 5.7430E-05 5.1879E-05 1.4419E-05 5.5132E-05 4.2657E-05 9.8829E-05 7.8730E-05 9.7672E-07 9.0623E-06 4.4896E-05 O Stellar 16 .2: D le ab T -1.7475E-03 -1.2059E-03 -4.7070E-04 -1.7328E-03 -3.6445E-03 -9.7749E-03 -8.3020E-03 -6.1933E-07 -4.8408E-04 -3.2904E-06 -1.7893E-05 -1.8282E-03 -1.2402E-03 -1.0482E-05 -1.8211E-03 -3.7088E-03 -9.8004E-03 -8.3521E-03 -2.2254E-06 -5.4827E-06 -2.0851E-04 f M 0.721 0.687 0.664 0.800 0.885 0.972 0.938 0.573 0.662 0.600 0.640 0.716 0.682 0.636 0.791 0.871 0.962 0.929 0.578 0.620 0.650 0 M 1.25 1.0 1.75 2.25 2.5 1.5 2.0 3.5 3.0 1.9 4.0 5.0 6.5 6.0 APPENDIX D. STELLAR YIELDS 264 i S 30 1.7888E-08 1.3690E-06 1.4932E-06 1.0175E-06 8.2961E-07 8.7015E-07 2.3273E-07 9.5024E-09 1.0168E-08 6.7048E-09 1.2159E-06 3.2442E-09 2.0871E-08 1.7949E-06 1.2783E-07 1.0740E-06 1.0128E-06 2.6754E-07 8.4146E-09 4.9986E-09 9.5133E-09 i S 29 models. 02 . 0 2.8998E-07 7.5404E-06 7.8068E-06 4.6928E-06 3.4739E-06 3.7864E-06 1.4846E-06 1.3704E-08 6.7845E-06 1.4759E-08 9.7061E-09 4.6221E-09 3.2126E-07 9.2396E-06 5.9267E-06 4.5153E-06 4.3643E-06 1.6410E-06 1.2365E-08 7.2923E-09 4.4012E-08 = Z Si 28 the for 1.1118E-07 2.7684E-06 6.7567E-07 2.6054E-07 3.2429E-06 2.8207E-07 1.8644E-07 8.8941E-08 7.7649E-08 2.3545E-07 1.3952E-07 3.0675E-07 -2.9712E-06 -3.1679E-06 -3.5304E-06 -1.0220E-06 -5.4017E-07 -4.3476E-06 -4.4215E-06 -4.1949E-06 -1.1874E-06 silicon Al and 27 4.9907E-07 1.5625E-04 1.6564E-05 6.8948E-06 2.4857E-06 6.4476E-06 1.2946E-06 2.3363E-08 1.7289E-05 2.5510E-08 1.6567E-08 7.8871E-09 5.3593E-07 1.8131E-05 7.9439E-06 2.9603E-06 1.4700E-05 1.3504E-06 2.0831E-08 1.2365E-08 1.2475E-07 aluminium Al 26 magnesium, 1.2106E-07 5.0298E-06 4.3747E-06 4.6286E-07 1.4436E-07 1.1317E-07 1.1703E-07 1.5599E-09 5.7594E-09 5.0050E-12 5.0300E-06 2.7354E-12 1.2106E-07 4.3744E-06 4.6324E-07 1.4430E-07 1.1297E-07 1.1706E-07 5.9899E-10 2.9923E-13 8.7344E-08 of g M 26 masses), 1.7693E-06 2.9609E-04 2.7382E-04 1.2425E-04 4.9032E-05 7.0783E-05 1.0205E-05 3.1832E-08 3.6031E-08 2.7198E-08 3.9090E-04 1.2467E-08 1.9633E-06 3.4595E-04 1.6605E-04 6.6976E-05 1.4432E-04 1.1384E-05 2.7925E-08 2.2723E-08 2.5237E-07 solar (in k Mg M 25 yields, 2.2849E-06 2.6528E-04 2.3787E-04 1.2200E-04 8.0617E-05 7.4243E-05 2.2059E-05 2.5029E-08 2.0926E-08 1.4552E-08 3.3485E-04 6.9267E-09 2.7225E-06 2.9160E-04 1.6461E-04 1.0841E-04 8.7515E-05 2.4999E-05 2.3487E-08 8.5238E-09 -4.6726E-08 g Stellar M 24 .3: D le 7.0431E-08 2.0658E-07 2.2294E-07 1.4738E-07 6.8947E-08 3.6554E-08 2.3528E-07 1.8475E-07 1.0993E-07 ab -1.0013E-04 -6.7354E-05 -2.4562E-05 -1.2674E-05 -4.0308E-06 -4.9174E-07 -1.0458E-04 -7.1693E-05 -2.9198E-05 -1.5926E-05 -4.3863E-06 -5.2270E-07 T f M 0.972 0.938 0.885 0.800 0.721 0.687 0.664 0.962 0.929 0.871 0.791 0.716 0.636 0.640 0.600 0.573 0.662 0.682 0.650 0.620 0.578 0 M 6.5 6.0 5.0 4.0 3.5 1.9 2.0 1.5 2.25 1.75 1.0 1.25 2.5 3.0 APPENDIX D. STELLAR YIELDS 265 N and 15 mass -8.9114E-06 -4.9951E-06 -1.2550E-05 -3.9617E-06 -9.9180E-06 -2.9275E-06 -8.9340E-06 -5.0062E-06 -1.2571E-05 -3.9700E-06 -9.9472E-06 -2.9356E-06 -1.9126E-06 -2.2695E-06 -3.3739E-07 -1.1704E-06 -2.4734E-06 -1.6697E-06 -7.3967E-07 initial N The 14 models. 1.7188E-03 1.4556E-03 4.4296E-02 1.2046E-03 5.3137E-02 8.8286E-04 1.7090E-03 1.4543E-03 4.4270E-02 1.2018E-03 5.3098E-02 8.8292E-04 5.2230E-04 6.5840E-04 6.7874E-05 3.1201E-04 7.2696E-04 4.6488E-04 1.8318E-04 008 . 0 C 13 = Z the 6.4985E-04 2.3608E-05 6.0409E-04 1.0380E-05 6.6636E-04 9.4957E-06 6.5169E-04 2.3663E-05 6.0559E-04 1.0400E-05 6.6865E-04 9.5212E-06 9.9569E-06 1.0045E-05 5.4745E-06 8.9131E-06 9.5585E-06 9.0162E-06 7.5442E-06 for C 12 nitrogen and 2.0082E-02 2.5997E-02 1.6476E-03 3.8287E-02 2.4432E-03 2.1396E-02 2.4536E-02 2.7284E-02 4.9049E-03 3.9739E-02 5.9865E-03 2.2181E-02 4.7467E-03 9.7211E-03 1.4325E-02 2.6721E-03 -6.0602E-05 -3.0622E-05 -1.5923E-04 i carbon L 7 lithium, 2.2873E-08 2.3043E-08 -1.4737E-08 -3.7620E-08 -1.6988E-08 -3.0487E-08 -1.0717E-08 -1.4760E-08 -3.7680E-08 -1.7024E-08 -3.0577E-08 -1.0747E-08 -1.3638E-09 -6.8475E-09 -8.1377E-09 -4.2122E-09 -8.8789E-09 -2.6661E-09 -5.9165E-09 e helium, H 4 columns. 8.5888E-02 4.2353E-01 7.6018E-02 1.2828E-01 2.9004E-01 7.4022E-02 8.5836E-02 8.9234E-02 4.2760E-01 1.3225E-01 2.9672E-01 7.5976E-02 8.2391E-03 2.5304E-02 3.8076E-02 1.6391E-02 5.0418E-02 1.4085E-02 2.2666E-02 hydrogen, o tw of He 3 ®rst the masses) in 2.3532E-04 1.7868E-06 1.8835E-04 2.4599E-04 1.9165E-06 2.7250E-04 1.8882E-04 2.3584E-04 1.7868E-06 2.4650E-04 1.9165E-06 2.7324E-04 1.2829E-04 3.2278E-04 3.1188E-04 2.9039E-04 2.9463E-04 2.0732E-04 3.1949E-04 solar en v gi H (in k are M units) yields, -1.1344E-01 -4.6298E-01 -9.7242E-02 -1.7058E-01 -3.4395E-01 -9.7258E-02 -1.1183E-01 -1.1821E-01 -4.7059E-01 -1.7828E-01 -3.5459E-01 -1.0008E-01 -8.2677E-03 -3.0747E-02 -4.8818E-02 -1.6731E-02 -6.6166E-02 -1.4143E-02 -2.5925E-02 solar f Stellar (in M 0.772 0.956 0.845 0.700 0.898 0.668 0.837 0.766 0.948 0.695 0.886 0.663 0.600 0.645 0.650 0.630 0.650 0.610 0.640 .4: D mass 0 le M 4.0 3.5 6.0 3.0 5.0 2.5 1.0 1.9 2.1 2.25 1.5 1.25 1.75 ab T ®nal APPENDIX D. STELLAR YIELDS 266 Na 23 1.4947E-04 6.3252E-05 5.0783E-05 6.7914E-05 1.7090E-03 8.7836E-04 3.6258E-06 1.5933E-04 6.4767E-05 5.3204E-05 4.1535E-07 7.0590E-05 2.9273E-05 1.7154E-05 1.7094E-03 4.2156E-05 8.7824E-04 9.7129E-07 1.1853E-05 Ne 22 models. 3.8018E-03 1.4958E-03 7.7643E-04 1.4942E-03 1.6172E-04 1.5384E-06 4.0503E-03 1.6114E-03 9.9601E-04 1.5774E-03 3.9344E-04 1.3315E-04 3.7375E-04 7.7059E-04 6.6980E-05 -3.0044E-04 -3.8241E-07 -2.3103E-04 -9.0487E-07 008 . 0 = Ne Z 21 the for 2.2409E-06 2.5617E-06 2.5659E-06 3.3232E-07 8.4216E-09 5.0863E-09 2.3994E-06 2.8261E-06 3.1678E-06 3.5559E-07 1.0206E-08 2.7544E-08 -1.1958E-05 -1.5364E-05 -1.7716E-08 -3.9298E-08 -1.4810E-05 -1.0994E-05 -8.1948E-09 sodium Ne 20 and neon 2.6275E-05 2.5017E-05 1.4496E-05 1.0163E-04 2.4046E-04 1.3149E-07 1.0012E-08 2.9077E-05 2.7671E-05 1.9380E-05 2.4346E-04 1.0895E-04 2.1368E-07 -2.2417E-06 -4.1515E-06 -2.4990E-06 -1.9022E-06 -1.5582E-06 -4.7339E-06 F 19 ¯uorine, 9.0373E-06 3.1581E-06 2.3649E-06 7.2029E-07 6.4150E-07 1.8667E-07 2.3717E-09 2.0168E-08 3.3585E-06 9.9766E-06 2.5256E-06 8.3297E-07 9.0125E-08 1.4941E-06 1.1401E-08 oxygen, -1.2135E-06 -1.6148E-06 -1.6163E-06 -1.1811E-06 of O 18 masses) solar -1.2651E-05 -9.0988E-06 -1.4123E-05 -2.3186E-05 -6.1748E-05 -7.5891E-05 -6.6141E-06 -5.3164E-06 -3.9665E-07 -2.8219E-06 -9.1238E-06 -1.2678E-05 -1.4154E-05 -2.3249E-05 -7.6012E-05 -6.1930E-05 -4.5889E-06 -7.4497E-06 -1.5151E-06 (in k O 17 M yields, 1.3748E-05 1.4993E-05 1.6121E-05 1.4724E-05 6.8292E-06 8.0104E-06 1.2662E-05 9.4807E-06 2.8610E-07 1.9727E-06 1.5033E-05 1.3776E-05 1.6156E-05 1.4761E-05 8.0154E-06 6.8448E-06 6.0267E-06 1.6553E-05 6.5704E-07 O Stellar 16 .5: D le 1.2310E-05 3.7930E-05 2.2020E-05 8.5696E-06 ab -4.1007E-04 -1.2595E-04 -3.7879E-04 -3.0156E-04 -2.4043E-03 -5.7461E-03 -2.6083E-06 -4.5432E-06 -1.2838E-04 -4.1554E-04 -3.8310E-04 -3.0125E-04 -5.7066E-03 -2.3885E-03 -4.0814E-06 T f M 0.700 0.668 0.772 0.845 0.898 0.956 0.600 0.630 0.663 0.650 0.645 0.695 0.766 0.837 0.948 0.886 0.610 0.640 0.650 0 M 1.25 1.0 1.5 1.75 2.25 2.5 2.1 1.9 3.0 3.5 4.0 6.0 5.0 APPENDIX D. STELLAR YIELDS 267 Si 30 1.3504E-06 4.0116E-06 3.5897E-06 1.3351E-06 1.1042E-06 1.3455E-07 2.0905E-09 4.1182E-09 1.7483E-06 4.4243E-06 3.9636E-06 1.4424E-06 1.1801E-06 1.5129E-07 3.3406E-09 5.3715E-09 -5.1041E-09 -7.5997E-09 -8.8403E-10 Si 29 models. 008 . 0 6.2158E-06 2.0237E-05 1.9032E-05 5.6396E-06 4.8045E-06 1.0708E-06 3.0486E-09 7.7907E-09 6.0409E-08 1.9504E-07 8.1022E-06 2.2077E-05 2.9914E-05 6.1225E-06 5.1120E-06 1.1490E-06 4.8749E-09 3.0946E-08 3.9860E-07 = Z Si 28 the for 2.6816E-06 1.6776E-05 2.9893E-05 5.7757E-08 1.2532E-07 1.4435E-07 5.5821E-08 2.3638E-06 1.5825E-05 2.9914E-05 9.2259E-08 1.4430E-07 -4.2296E-06 -4.7795E-06 -7.3930E-07 -4.7825E-06 -5.1463E-06 -8.1846E-07 -1.1147E-07 silicon Al and 27 2.2799E-05 4.7781E-05 3.0533E-05 1.2274E-05 5.0335E-06 2.7100E-06 5.4133E-09 9.2714E-08 8.3666E-07 1.4170E-06 2.5789E-05 1.2606E-05 5.1219E-06 2.7638E-06 5.0073E-05 3.2643E-05 8.6075E-09 5.1726E-07 1.9465E-06 aluminium l A 26 magnesium, 3.4975E-07 1.4145E-05 4.3195E-05 3.6715E-07 1.1607E-07 1.1121E-07 1.6790E-09 5.7168E-08 1.3404E-07 1.4331E-07 3.4837E-07 3.6712E-07 1.1593E-07 1.1130E-07 1.4135E-05 4.3171E-05 3.4407E-09 1.1946E-07 1.3154E-07 of g M 26 masses), 1.0245E-04 5.2646E-04 4.2005E-04 9.4886E-05 5.0617E-05 6.9610E-06 2.8616E-08 3.1920E-08 1.4680E-07 8.6305E-07 1.4358E-04 1.0616E-04 5.5068E-05 7.5526E-06 6.3113E-04 5.1036E-04 4.0294E-08 9.7418E-08 2.0867E-06 solar (in k Mg 25 M 1.3104E-04 5.3025E-04 1.0203E-03 1.4021E-04 1.0644E-04 1.5179E-05 7.7372E-07 1.7947E-04 1.5469E-04 1.1511E-04 1.6685E-05 6.0780E-04 1.0827E-03 3.3402E-06 yields, -1.6742E-08 -1.0535E-07 -2.9734E-07 -2.2217E-08 -3.9331E-07 g Stellar M 24 .6: D le 4.5824E-08 8.5740E-08 7.1013E-09 7.3545E-08 7.0984E-08 -3.4011E-05 -1.3270E-04 -7.5388E-04 -3.2312E-05 -2.2082E-06 -6.1339E-07 -9.8662E-08 -4.0771E-05 -3.3660E-05 -1.3544E-06 -6.0006E-07 -1.3719E-04 -7.5672E-04 -2.1077E-07 ab T f M 0.845 0.898 0.956 0.772 0.700 0.668 0.600 0.630 0.645 0.650 0.837 0.766 0.695 0.663 0.886 0.948 0.610 0.640 0.650 0 M 1.0 1.5 1.25 1.75 1.9 2.1 2.25 4.0 3.5 3.0 2.5 5.0 6.0 APPENDIX D. STELLAR YIELDS 268 N and 15 mass -3.9866E-06 -3.9367E-06 -4.8855E-06 -3.6844E-06 -3.9925E-06 -3.9466E-06 -4.8946E-06 -1.6347E-06 -2.1555E-06 -1.0659E-06 -6.1896E-07 -3.6911E-06 -1.7809E-07 -1.3320E-06 -8.9203E-07 -3.9047E-07 initial N The 14 models. 1.0859E-03 2.3344E-02 4.9844E-02 6.2688E-02 1.0845E-03 2.3303E-02 4.9865E-02 4.9418E-04 6.0475E-04 2.9556E-04 1.6382E-04 6.2633E-02 3.8021E-05 3.8175E-04 2.4406E-04 1.0046E-04 004 . 0 C 13 = Z the 2.9190E-04 1.2597E-03 6.6326E-04 7.7194E-04 2.9248E-04 1.2629E-03 6.6482E-04 2.4903E-06 4.1391E-06 3.8768E-06 4.2518E-06 7.7360E-04 2.4052E-06 2.9779E-06 4.1391E-06 3.0101E-06 for C 12 nitrogen and 2.3616E-02 8.9063E-03 5.0482E-03 5.4523E-03 2.5120E-02 1.2161E-02 7.0031E-03 3.4686E-02 3.8139E-02 8.7416E-03 1.4342E-03 8.3897E-03 2.3344E-02 5.3540E-03 5.6039E-05 -3.2400E-05 i carbon L 7 lithium, -1.9573-08 2.0894E-09 4.4362E-09 2.1255E-09 4.4288E-09 -1.9541E-08 -1.5931E-08 -6.1546E-09 -8.8219E-09 -3.7728E-09 -2.1600E-09 -6.8473E-10 -1.5958E-08 -4.6469E-09 -3.0847E-09 -1.3595E-09 e helium, H 4 columns. 7.1325E-02 1.3351E-01 5.2755E-01 3.8502E-01 1.1337E-01 1.1918E-01 7.4587E-02 3.9828E-02 2.2163E-02 7.8071E-03 1.3957E-01 5.3020E-01 3.8911E-01 7.5072E-02 3.1164E-02 1.5245E-02 hydrogen, o tw of He 3 ®rst the masses) in 5.5761E-05 2.1230E-05 1.4630E-06 1.7741E-06 2.5094E-04 2.4418E-04 5.5841E-05 3.4367E-04 2.6602E-04 1.1563E-04 2.1272E-05 1.4630E-06 1.7741E-06 2.6626E-04 3.2243E-04 1.8810E-04 solar en v gi H (in k are M units) yields, -9.6957E-02 -1.6732E-01 -5.7969E-01 -4.5462E-01 -1.5217E-01 -1.6146E-01 -1.0187E-01 -4.9417E-02 -2.3965E-02 -7.8823E-03 -1.7694E-01 -5.8450E-01 -4.6204E-01 -1.0071E-01 -3.7145E-02 -1.5517E-02 solar f Stellar (in M 0.818 0.860 0.986 0.913 0.678 0.730 0.814 0.650 0.646 0.630 0.852 0.978 0.906 0.660 0.650 0.640 .7: D mass 0 le M 2.5 3.0 3.5 1.9 2.25 1.5 1.75 1.0 4.0 1.25 6.0 5.0 ab T ®nal APPENDIX D. STELLAR YIELDS 269 Na 23 2.9463E-05 1.5583E-04 1.6311E-03 4.4722E-04 1.5642E-04 2.9970E-05 7.3008E-05 1.1360E-04 3.7298E-06 1.4616E-05 3.8845E-07 1.6303E-03 5.1411E-05 4.4722E-04 9.3537E-06 1.0673E-06 Ne 22 models. 8.6273E-04 7.4191E-04 2.9031E-04 8.9608E-04 9.4552E-04 3.1300E-03 3.2754E-03 1.6412E-05 2.5489E-04 3.9823E-04 1.6023E-03 1.0707E-04 4.3105E-07 -8.7156E-05 -3.6330E-07 -8.9613E-05 004 . 0 = Ne Z 21 the for 2.2716E-06 3.0527E-09 2.5538E-06 3.5860E-06 1.3113E-06 1.4570E-08 2.4568E-08 3.5734E-08 3.8128E-07 3.3827E-08 2.4489E-08 -6.5922E-07 -5.5184E-06 -7.2087E-06 -7.9362E-06 -4.6674E-06 sodium Ne 20 and neon 1.5406E-05 1.6315E-05 1.9079E-04 3.7908E-04 2.0075E-05 1.8541E-05 4.9182E-05 1.5363E-05 1.1278E-08 3.8220E-04 1.9746E-04 1.4317E-06 -7.3091E-07 -5.6375E-06 -2.0005E-06 -9.1153E-08 F 19 ¯uorine, 6.6449E-08 7.8303E-07 2.2912E-08 8.8972E-08 8.1671E-07 6.8954E-06 8.0991E-06 4.9608E-07 9.4462E-10 1.7284E-07 3.8720E-06 6.2277E-09 oxygen, -6.5107E-07 -8.3947E-07 -8.4068E-07 -6.4983E-07 of O 18 masses) solar -2.4560E-05 -1.9751E-05 -3.2006E-05 -3.9279E-05 -1.5635E-06 -2.4623E-05 -1.9781E-05 -6.9500E-06 -5.5011E-06 -3.2065E-06 -2.5334E-07 -3.9342E-05 -3.2061E-05 -2.5162E-06 -4.3546E-06 -8.9206E-07 (in k O 17 M yields, 1.7349E-06 4.3632E-06 2.0343E-06 1.0193E-06 1.5093E-06 1.7389E-06 4.3696E-06 1.0204E-05 9.9097E-06 7.5796E-06 2.2240E-07 1.0177E-06 2.0360E-06 4.4914E-06 1.1186E-05 4.9241E-07 O Stellar 16 .8: D le 2.2044E-05 6.7964E-05 1.9546E-05 5.4003E-05 7.8600E-05 1.6497E-05 9.3546E-05 9.5973E-05 7.5738E-05 1.4749E-04 ab -1.8811E-03 -3.9104E-03 -2.9158E-06 -3.8739E-03 -1.8314E-03 -2.0669E-06 T f M 0.860 0.913 0.818 0.986 0.630 0.646 0.852 0.978 0.906 0.814 0.730 0.678 0.650 0.640 0.650 0.660 0 M 1.25 1.0 1.5 4.0 6.0 5.0 3.5 3.0 2.5 1.9 1.75 2.25 APPENDIX D. STELLAR YIELDS 270 Si 30 9.5805E-07 1.4149E-06 3.6486E-06 3.0788E-06 4.3155E-07 4.3747E-10 1.0392E-06 1.6199E-06 3.8892E-06 3.2600E-06 1.0978E-06 1.4016E-07 6.2028E-10 -5.0344E-08 -6.3301E-10 -5.4042E-09 Si 29 models. 004 . 0 4.5823E-06 6.7411E-06 1.8132E-05 1.5437E-05 2.0424E-06 6.6984E-10 4.9597E-06 7.6070E-06 1.9031E-05 1.6116E-05 4.8413E-06 2.7287E-08 6.3255E-09 8.3850E-07 1.4179E-09 3.5858E-08 = Z Si 28 the for 8.5525E-06 4.4192E-06 1.9066E-05 2.4026E-05 1.2406E-08 5.3371E-07 3.9029E-06 1.9124E-05 2.4417E-05 2.7678E-08 2.0940E-08 1.4843E-08 -2.1551E-06 -5.4782E-06 -1.2633E-06 -7.8135E-07 silicon Al and 27 1.3232E-05 1.5292E-05 3.4233E-05 4.4232E-05 2.3164E-06 2.6794E-09 5.9637E-06 7.8811E-07 1.3660E-05 1.6469E-05 3.6019E-05 4.5324E-05 2.7975E-07 1.6715E-06 5.2588E-08 6.1386E-07 aluminium l A 26 magnesium, 1.6582E-07 3.2216E-07 2.3570E-05 9.2700E-05 6.1429E-08 3.1887E-09 1.4861E-07 6.8120E-08 1.6566E-07 3.2160E-07 2.3550E-05 9.2737E-05 7.0173E-08 5.6182E-08 3.0380E-08 6.5908E-08 of g M 26 masses), 1.3656E-04 2.4293E-04 9.3709E-04 6.3286E-04 2.5032E-05 2.8668E-08 1.1935E-04 1.7776E-07 1.5913E-04 2.9037E-04 1.0836E-03 6.9065E-04 1.5141E-08 6.7558E-06 4.0560E-08 8.3641E-08 solar (in k Mg 25 M 1.3321E-04 2.1669E-04 7.2470E-04 1.2149E-03 5.3362E-05 1.6524E-04 2.2344E-07 1.5057E-04 2.9263E-04 7.9235E-04 1.2512E-03 1.6223E-05 yields, -2.9012E-08 -1.9430E-07 -7.8991E-08 -2.0623E-07 g Stellar M 24 .9: D le 4.7162E-06 9.8444E-09 6.7722E-07 -1.9983E-05 -2.2444E-05 -1.9221E-04 -9.4558E-04 -8.1538E-06 -1.0467E-06 -4.5635E-08 -2.0965E-05 -2.4338E-05 -1.9321E-04 -9.4674E-04 -4.0600E-09 -2.0809E-08 ab T f M 0.818 0.860 0.913 0.986 0.678 0.630 0.730 0.650 0.646 0.814 0.852 0.906 0.978 0.660 0.640 0.650 0 M 2.5 1.0 3.0 1.9 2.25 1.25 1.5 1.75 3.5 4.0 5.0 6.0 APPENDIX D. STELLAR YIELDS 271

Table D.10: Initial abundances in mole fraction, Yk. species k Z = 0.02 Z = 0.008 Z = 0.004 H 6.8720E-01 7.3690E-01 7.4840E-01 3He 0.0000E+00 0.0000E+00 0.0000E+00 4He 7.3200E-02 6.3775E-02 6.1900E-02 7Li 1.5300E-09 1.0737E-09 5.5839E-10 12C 2.8400E-05 8.0799E-05 4.0191E-05 13C 3.1600E-06 2.2175E-06 1.1533E-06 14N 7.5300E-05 1.0172E-05 3.1925E-06 15N 2.7600E-07 1.9368E-07 1.0073E-07 16O 6.0000E-04 1.6497E-04 8.0192E-06 17O 2.2800E-07 1.6000E-07 8.3212E-08 18O 1.2000E-06 8.4210E-07 4.3796E-07 19F 2.4400E-08 1.7122E-08 8.9051E-09 20Ne 9.0400E-05 6.3439E-05 3.2992E-05 21Ne 2.2000E-07 1.5438E-07 8.0292E-08 22Ne 6.6000E-06 4.6316E-06 2.4088E-06 23Na 1.6600E-06 1.1649E-06 6.0584E-06 24Mg 2.4600E-05 1.7263E-05 8.9781E-06 25Mg 3.1000E-06 2.1754E-06 1.1314E-06 26Mg 3.4200E-06 2.4000E-06 1.2482E-06 26Al 0.0000E+00 0.0000E+00 0.0000E+00 27Al 2.4600E-06 1.7263E-06 8.9781E-07 28Si 2.6700E-05 1.8737E-05 9.7445E-06 29Si 1.3500E-06 9.4737E-07 4.9270E-07 30Si 8.9800E-07 6.3017E-07 3.2774E-07 Bibliography

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