synthesis of the elements in stars

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Synthesis of the elements in stars: forty years of progress

George Wallerstein

Department of Astronomy, University of Washington, Seattle, Washington 98195

Icko Iben, Jr.

University of Illinois, 1002 West Green Street, Urbana, Illinois 61801

Peter Parker

Yale University, New Haven, Connecticut 06520-8124

Ann Merchant Boesgaard

Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, Hawaii 96822

Gerald M. Hale

Los Alamos National Laboratory, Los Alamos, New Mexico 87544

Arthur E. Champagne

University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27594

and Triangle Universities Nuclear Laboratory, Duke University, Durham, North Carolina

27706

Charles A. Barnes

California Institute of Technology, Pasadena, California 91125

Franz Kappeler

Forschungzentrum, Karlsruhe, D-76021, Germany

Verne V. Smith

University of Texas at El Paso, El Paso, Texas 79968-0515

Robert D. Hoffman

Steward Observatory, University of Arizona, Tucson, Arizona 85721

Frank X. Timmes

University of California at Santa Cruz, California 95064

Chris Sneden

University of Texas, Austin, Texas 78712

Richard N. Boyd

Ohio State University, Columbus, Ohio 43210

Bradley S. Meyer

Clemson University, Clemson, South Carolina 29630

David L. Lambert

University of Texas, Austin, Texas 78712

(Received 25 June 1997)

Forty years ago Burbidge, Burbidge, Fowler, and Hoyle combined what we would now callfragmentary evidence from nuclear physics, stellar evolution and the abundances of elements andisotopes in the solar system as well as a few stars into a synthesis of remarkable ingenuity. Theirreview provided a foundation for forty years of research in all of the aspects of low energy nuclearexperiments and theory, stellar modeling over a wide range of mass and composition, and abundancestudies of many hundreds of stars, many of which have shown distinct evidence of the processessuggested by B 2FH. In this review we summarize progress in each of these fields with emphasis on themost recent developments. [S0034-6861(97)00204-3]

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CONTENTS

I. Preface by E. Margaret Burbidge, Geoffrey R.Burbidge, and Fred Hoyle 997

II. Introduction by George Wallerstein 998A. The cosmological foundations of B2FH 998B. The astronomical background in 1957 998C. The eight processes 998

1. Hydrogen burning 9982. Helium burning 9993. The process 9994. The e process 9995. The s process 9996. The r process 9997. The p process 9998. The x process 999

D. Neutrino astrophysics 10001. The solar neutrino problem 10002. Other aspects of neutrino astrophysics 1000

E. Related reviews 1001III. Stellar Evolution by Icko Iben, Jr. 1001

A. Historical preliminary 1001B. Evolution of single stars that become white

dwarfs 10011. Overview 10012. Nucleosynthesis and dredge-up prior to the

AGB phase 10033. Nucleosynthesis and dredge-up during the

AGB phase 10044. The born-again AGB phenomenon 10065. Other mixing processes, and wind mass loss,

which affect surface composition 1007C. Evolution of massive single stars that produce

neutron stars or black holes 1007D. Close binary star evolution 1009

1. Modes of mass transfer and of orbital angularmomentum loss 1009

2. Scenario modeling 1009a. Cataclysmic variables and novae 1010b. White dwarf mergers: R CrB stars and

type Ia supernovae 1010c. X ray binaries and pulsars 1012

IV. Hydrogen Burning in the pp Chain and CN Cycleby Peter Parker 1013

A. The p(p ,ee)d reaction 1013B. The 3He(3He,2p)4He reaction 1013C. The 3He(,) 7Be reaction 1014D. The 7Be(p,) 8B reaction 1015E. The 7Li(n ,)8Li reaction 1015F. The 14N(p,)15O reaction 1015G. The 17O(p,)14N reaction 1016H. The Hot CNO cycle 1016

V. The x Process by Ann Merchant Boesgaard 1016A. Introduction and retrospective 1016B. Abundances 1016

1. Lithium 10172. Beryllium 10183. Boron 1018

C. Nonlocal thermodynamic-equilibrium effects 1019D. Production mechanisms 1019

1. Big Bang 10192. Spallation 10193. Asymptotic giant branch stars 10194. Supernovae 1020

VI. Helium Burning by Gerald M. Hale 1020

A. Triple- capture 1020B. 12C capture 1021

1. E1 capture 1021a. Direct measurements 1021b. -delayed spectrum from the decay of

16N 10222. E2 capture 10223. Other analyses and recommended values 1023

VII. H Burning in the NeSi Region: Laboratory Studies

by Arthur E. Champagne 1024A. Introduction 1024B. Experimental approaches 1024C. The neon-sodium cycle at low temperatures 1025

1. Reaction rates 10252. Network calculations 1026

D. The magnesium-aluminum cycle at lowtemperatures 10271. Reaction rates 10272. Network calculations 1028

E. High-temperature behavior 10291. Reaction rates 10292. Network calculations 1029

F. Conclusion 1030VIII. Observational Evidence of Hydrogen Burning by

George Wallerstein 1030A. CN cycle and mixing 1031B. O depletion and the enhancement of Na and Al 1032C. Origin of the Na and Al enhancements 1033

IX. Carbon, Neon, Oxygen, and Silicon Burning byCharles A. Barnes 1034

A. Introduction 1034B. Carbon burning 1035C. Neon burning 1036D. Oxygen burning 1036E. Silicon burning 1036

X. s Process: Laboratory Studies and Stellar Models byFranz Kappeler 1038

A. The s process since B2FH 1038

B. Laboratory experiments 10381. Neutron capture cross sections 10382. Stellar decay 1039

C. The canonical s process 10401. The N curve 10402. Branchings 1041

XI. Observations of the s Process by Verne V. Smith 1042A. Brief history 1042B. Observations of nucleosynthesis and mixing in

CH, Ba, S, and C stars 1043C. The s process as a function of metall icity 1043D. Rubidium and the s-process neutron density 1045E. Recent models: Radiative burning of 13C during

the AGB interpulse phase 1046XII. The r Process by Robert D. Hoffman and Frank X.

Timmes 1047A. Introduction 1047B. A search for the astrophysical site 1047C. Early model results, from conflict to clarity 1048D. Twisting in the wind 1049E. Concluding remarks 1049

XIII. Observations of the r Process by Chris Sneden 1050A. Defining the r-process elements 1050B. Early r-process discoveries 1051C. Recent r-process surveys 1051D. Thorium and the age of the halo and disk 1053E. Filling out the picture 1054

XIV. The p Process by Richard N. Boyd 1054

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A. The p process 1054B. Early p-process models 1055C. The process 1055D. The rp process 1057E. The process 1058F. Recent developments 1058G. Summary 1059

XV. The e Process and the Iron-Group Nuclei byBradley S. Meyer 1059

A. Energetics and equilibria 1059B. Statistical equilibrium 1060C. A brief history of the ideas of iron-group

element synthesis 1063D. Significance for astrophysics 1065

XVI. Carbon Stars: Where Theory Meets Observationsby David L. Lambert 1066

A. Prologue 1066B. Carbon starsAn observers view 1066

1. What is a carbon-rich star? 10662. What makes a carbon-rich star? 10673. The principal types of carbon-rich stars 1068

a. R-type carbon stars 1068b. N-type carbon stars 1069c. Barium and related stars 1070

C. Epilogue 1071XVII. Conclusions 1071Acknowledgments 1072References 1072

I. PREFACE

It is curious that both the primordial and stellar theo-ries of the origin of the elements should have been pub-lished in the same year, 1946. Little notice was at firsttaken of the stellar theory, attention being directed atfirst overwhelmingly to Gamows suggestion of associat-ing nucleosynthesis with the origin of the Universe.

While it is true that the one theory had to do withcosmology, the other did not. The suggestion that stellarnucleosynthesis had a connection to cosmology was aninvention of Robert Oppenheimer and it never had anyreality to it, since the first paper on stellar nucleosynthe-sis appeared in 1946, two years before the steady-statecosmological model was published. Nor were the objec-tions to primordial synthesis by neutron addition by anymeans confined to the well-known gaps at A5 and

A8, as is sometimes stated. There was always thepragmatic objection that the elements were distributedwith too much spatial irregularity to be attributed to auniversal origin. And the well-known iron peak was an

obvious feature of solar system abundances, implyingthat at least in some places matter had been able toapproach its most stable form.

For matter to approach its most stable form, tempera-tures in stellar interiors would be needed of the order ofa hundred times those in main-sequence stars, a require-ment that it would have been difficult to accept if thebeginnings of an understanding of supernovae had notemerged in the early 1940s. In a rough kind of way it waspossible to compare supernovae with ordinary stars inthe same way that, in the mid 1940s, people were com-paring nuclear weapons with chemical ones, this suggest-

ing that temperatures vastly higher than those thatseemed plausible in Eddingtons day might be possible.

The fact that primordial nucleosynthesis ran far aheadof stellar nucleosynthesis in the years up to the early1950s turned out to be advantageous to the eventualemergence of the B 2FH paper in 1957, because it per-mitted facts to accumulate quietly without any frenziedcircus developing. In 1952, Salpeter discussed the stellar

formation of carbon from alpha particles, and a fewmonths later the relation of carbon synthesis to oxygensynthesis was shown to require a state in the carbonnucleus of about 7.65 MeV above ground level. Whenthis state was actually found, the laboratory discoverycarried a strong measure of conviction for all those whowere involved in it.

At the same time surveys of nearby stars by severalgroups, including some of us, were showing variations ofmetal abundances to hydrogen, the beginning of theconcept of metallicity, that seemed impossible to associ-ate with some form of universal synthesis. We were alsopuzzled by our 195354 analysis of high-dispersion spec-

tra obtained at McDonald Observatory, which showedstrange overabundances of some heavy elements andseemed to indicate that neutrons were involved. And in195354, carbon-burning and oxygen-burning werefound.

It was in the autumn of 1954 that the team of B 2FHcame together in the quiet ambience of Cambridge, En-gland, without, as mentioned above, any frenzied circusdeveloping.

During 195455 calculations involving neutron pro-duction in the interiors of evolved stars with helium-burning cores surrounded by a hydrogen-burning shellwere followed by calculations on neutron addition toelements from neon to scandium. The process of neu-tron addition was slow enough for each neutron captureto be followed by beta decay (the origin of what we laternamed the s process). In the same period, we becameaware that A.G.W. Cameron was working along similarlines in Canada. These calculations showed the possibil-ity of explaining the characteristic odd-even effect inabundance ratios.

In the autumn of 1955 we all moved, or returned, toPasadena, where there was still a relatively peaceful set-ting at both the Kellogg Radiation Laboratory atCaltech and the Mount Wilson and Palomar offices at813 Santa Barbara Street. We now also had the greatadvantage of being at the home of the large telescopes.

In 1952 the process of neutron liberation and capturehad been clinched by Paul Merrills discovery of the un-stable element technetium in S stars (evolved red gi-ants), and we were able to obtain spectra with theMount Wilson 100-inch telescope of an evolved star ofthe class known as Barium II stars, in which we deter-mined the overabundances of just those heavy elementsthat had at least one isotope with a magic neutron num-ber.

One could not be in the same town as Walter Baadewithout hearing a lot about his famous light curve of thesupernova in the galaxy IC 4182. With the interest of all

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four of us in supernovae as the spectacular death throesof stars at the end of their active evolution and thermo-nuclear energy production, we latched onto this. Theexponential decay of the energy output of this super-nova immediately suggested radioactive decay, follow-ing collapse, the release of a great burst of energy, neu-trons, and neutrinos, and the formation of heavyunstable nuclei.

Also in 1956, an improved solar-system abundancecurve became available from Suess and Urey. Values inthe upper half of the chart of the nuclides showed twothings: the association of magic neutron numbers withabundance peaks, and a separation of peaks correspond-ing to both slow and rapid neutron addition, the latterbeing fast enough not to allow time for the beta-decayinvolved in slow neutron addition. And released at thattime were hitherto classified data on (n ,) cross sectionsfor individual isotopes of elements, without which ameaningful analysis of what in B 2FH was called the sprocess would not have been possible.

B 2FH can be seen in retrospect to have been a highly

creative review article, putting together what the fourauthors had done previously, together with the facts onwhich the theory would now be based with considerableconfidence. We each brought ideas and data from verydifferent parts of physics to the table, and occasionallywe stopped arguing long enough to work things out andwrite them down. There were a number of original con-tributions, notably the calculations for the s and r pro-cesses. The rest consisted of an extensive updating ofprevious work. But one should not forget the introduc-tion of a lettering notation for the various nuclear pro-cesses: , e , s , r, p , and x, which may have donemore for the development of the subject than almost

anything else!

II. INTRODUCTION

As of 1957 enough evidence had been assembled for areview of what was known about nucleosynthesis instars. The data (which would now be called fragmentary,though it then appeared to be spectacular) allowed Bur-bridge, Burbridge, Fowler, and Hoyle, B 2FH,1 to com-bine progress in stellar and solar system abundanceswith laboratory nuclear physics data and stellar evolu-

tion calculations to show how stars can produce ele-ments and their isotopes from helium to uranium. Theirpaper provided a basis for nuclear astrophysics for thedecades that followed. How well did they do? In thisreview we will first outline the basic processes that theysuggested to be the sources of elements heavier thanhydrogen and evaluate progress in confirming and ex-tending their suggestion.

A. The cosmological foundations of B 2FH

In 1957 there were two basic cosmological models,though neither was sufficiently developed to deserve theterm theory. These were, of course, Big Bang andsteady state. The Big Bang obviously explained the ex-pansion of the Universe but failed to predict the originof the elements beyond the lightest species. Steady stateprovided for an understanding of the discrepancy be-

tween the expansion age and the apparent ages of theglobular clustersa problem that is still with usbutprovided no physical basis for the continuous creation ofmatter, the expansion, or the collection of diffusely cre-ated matter into galaxies.

One of the strengths of the B 2FH paper was its inde-pendence of the cosmological models then under discus-sion. The paper explained just how much stars couldcontribute to the synthesis of nuclei heavier than hydro-gen. Since they did not have the answers for D, Li, Be,and B they relegated them to the x process. Almostall other elements and isotopes could be produced in thestellar environment by one of their eight processes (with

their x process included as number eight).

B. The astronomical background in 1957

Starting in the late 1940s and early 1950s astronomerswere assembling data to show that all stars did not havethe same chemical composition. For 80 years it had beenknown that some red giants, referred to as carbon stars,showed molecular bands of carbon molecules in theirspectra while the vast majority of cool stars showed ox-ide bands. Due to blending of absorption lines and mo-lecular bands a quantiative comparison of carbon andoxygen red giants was not possible. More recently the

subdwarfs had been found to be metal-poor by factorsmore than 10 (Chamberlain and Aller, 1951), and theheavy element stars such as the Ba II stars and S starswere known to show abundance excesses of certain spe-cies by factors near 10 (Burbidge and Burbidge 1957).Of great importance was the discovery of Tc in S stars(Merrill, 1952) since the longest lived isotope of Tc has ahalf-life of 4106 years and Cameron (1955) had shownthat 99Tc, with a half-life of only 3105 years was themost easily produced isotope. This proved beyond anydoubt that nucleosynthesis took place within stars andthat the products could reach the stellar surface, withthe help of mass loss and mixing.

C. The eight processes

In order to produce (almost) all known nuclear spe-cies in stars B 2FH suggested eight separate processes.

1. Hydrogen burning

Following the fundamental papers by Bethe andCritchfield (1938) and Bethe (1939), B 2FH describedthe laboratory experiments and the derived reactionrates of the various proton captures of the pp chain andthe CNO cycle. The details of the rates of the individualreactions in the pp chain determine the energy spectrum

1This was probably the first astronomical paper to be referredto by the initials of its authors.




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of the resulting neutrinos, which is crucial in attempts tounderstand the solar neutrino problem. In addition theydiscussed the p capture by the neon isotopes to produce23Na (also mentioned in Bethes 1939 paper). In Secs.IV and VII, Parker and Champagne bring us up to dateon the laboratory rates of hydrogen burning reactionsfrom the pp reaction through the CNO cycle, the NeNacycle and the MgAl cycle. In addition the hot CNO cycleplays an important role in nova explosions, while the rpprocess discussed by Boyd in Sec. XIII may be respon-sible for the production of certain p-rich isotopes.

2. Helium burning

By the time of B 2FH helium burning to produce 12Chad been suggested by Opik (1951) and its rate esti-mated by Salpeter (1952). In Sec. VI. G. Hale describesthe present state of the experiments that located andmeasured the width of the vital 7.65 MeV level in 12C(predicted by Hoyle) and the complicated combinationof experiment and theory that is necessary to estimate

the rate of the 12C(,)16O reaction. The rate of the12C(,)16O reaction relative to the 3- process deter-mines the carbon/oxygen ratio in massive stars, and thisis crucial for the later evolution of such a star and itsresulting nucleosynthesis. Unfortunately, 40 years afterB 2FH, the rate of the 12C(,)16O reaction is still notwell determined. Fortunately the material that is re-turned to the interstellar medium by stars that are lessmassive than 11M and evolve into white drawfs hasbeen enriched by matter that has experienced only par-tial helium burning, so the uncertainty in the ratio ofreaction rates plays a minor role.

3. The process

B 2FH suggested that further captures could extendnucleosynthesis beyond 16O to 20Ne, 24Mg, etc., up tothe very stable, doubly magic nucleus, 40Ca. However,after experiments showed that the 16O(,)20Ne rate isvery slow in stellar interiors, it became evident that car-bon and oxygen burning are responsible for the origin ofspecies from Ne to S, with the nuclei consisting of inte-gral numbers of particles dominating the abundancecurve in this region.

4. The e process

At very high temperatures, about 4 or 5109 K, somany reactions take place that the nuclei settle down tostatistical equilibrium dominated by the most tightlybound nuclei around 56Fe. Such conditions are reachedonly in supernovae. The observation of rays fromSN1987A due to the deexcitation of 56Fe, resulting fromthe decay of 56Ni, has demonstrated the importance ofthe production of iron-peak species in supernova explo-sions. Modern calculations of the iron-peak abundancesare discussed by Meyer in Sec. XV.

5. The s process

Beyond the iron-peak, utilizing neutrons produced byreactions such as 13C(,n)16O and 22Ne(,n) 25Mg, nu-clei can be produced along or adjacent to the valley ofstability via a process in which sequential neutron cap-tures take place on a time scale that is slow compared tothe beta-decay lifetime of these nuclei. This process cancontinue all the way up to lead and bismuth; beyond

bismuth the resulting nuclei alpha decay back to Pb andTl isotopes. In Secs. X and XI Kappeler and Smith bringus up to date on laboratory measurements, stellar mod-els (also discussed in Sec. III by Iben), and abundancestudies of the s-process elements.

6. The r process

B 2FH showed that, in addition to the s process, theremust be another neutron capture process in which thesequential neutron captures take place on a time scalewhich is much more rapid than the beta decay of theresulting nuclei. This process produces the much more

neutron-rich progenitors that are required to account forthe second set of abundance peaks that are observedabout 10 mass units above the s-process abundancepeaks corresponding to the neutron magic numbers,N50 and 82. Historically, the r process has been asso-ciated with SN explosions, and in the past decade inter-est has focused more specifically on the neutrino-heatedatmosphere surrounding the newly formed neutron staras the r-process site. In Sec. XII Hoffman and Timmesreview both the physics and astrophysical scenario ofrapid neutron capture during the explosion of massivesupernovae.

7. The p process

There are some relatively rare proton-rich nuclei suchas 92Mo that are impossible to produce by n capturealone. They may be produced by p capture at highenough temperatures to overcome the huge coulombbarrier or by (,n) reactions during supernova explo-sions. Recent work on (p ,) and (,n) reactions includ-ing the rp process are reviewed by Boyd in Sec. XIV.

8. The x process

None of the above processes can produce D, Li, Be,or B, all of which are burned by p capture at low tem-

peratures in stars but hardly ever (except for 7Li) pro-duced in stars. B 2FH did not know how they were pro-duced so they ascribed their synthesis to the x process.Modern cosmological models of big bang nucleosynthe-sis are tuned to produce D, 3He, 4He, and some 7Li tofit observations of these species in very metal-poor starsand other astrophysical sources. The observations andtheories of production of 7Li, Be, and B in stars arereviewed by Boesgaard in Sec. V. For a brief review ofthe production of D, 3He, 4He, and 7Li in the earlyuniverse with further references see Olive and Schramm(1996).




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D. Neutrino astrophysics

1. The solar neutrino problem

In assembling this review we have decided to omit thesolar neutrino problem because it has been developingso rapidly and hence is reviewed so frequently that,within the limited space available, we could not add sub-stantially to the material already available. For the fun-

damentals of neutrino astrophysics see Bahcall (1989)and for a recent and thorough review see Haxton (1995).Some historical and summarizing remarks regarding thesolar neutrino problem follow. For a much more de-tailed and very interesting history of the solar neutrinoproblem see the Appendix of Bahcall (1989).

The important place of neutrinos in present day astro-physics could hardly have been anticipated by B 2FH,which predated neutrino physics as an experimental sci-ence and the modern understanding of neutrino interac-tions. The possibility of detecting solar neutrinos hadbeen suggested at the time of B 2FH, but not in the usualliterature. Pontecorvo (1946) and Alvarez (1949) inde-

pendently called attention to the possibility of the detec-tion of neutrinos. The basic papers calling attention tothe possibility of detecting solar neutrinos with a 37Cldetector were published in 1964 (Davis, 1964; Bahcall,1964). In 1968 Davis, Harmer, and Hoffman reportedthe first results of their Cl experiment in the HomestakeMine in South Dakota. Over the next 20 years or socontinued observations by Davis and his colleagues andimproved solar models by Bahcall and his colleagues (aswell as others) demonstrated a discrepancy of a factor of3 between the predicted number of 8B neutrinos and thedetection rate for neutrinos with energy above 0.814MeV. No likely solar models have been able to explainthe discrepancy. However, the suggestion by Mikheyevand Smirnov (1985) of matter-enhanced neutrino oscil-lations, following Wolfensteins (1978) discussion ofneutrino effective masses in matter, provides a verynatural explanation for the observations, assuming mas-sive neutrinos and flavor mixing. This possibility is com-monly known as the MSW mechanism.

Neutrino astrophysics was given a great boost in thelast decade by the results from three new solar neutrinoexperiments, one using a water Cerenkov detector tomeasure 8B neutrinos (Kamiokanda II/III) and two oth-ers using radiochemical 71Ga detectors (SAGE andGALLEX). The threshold of the 71Ga (e ,e)

71Ge re-action is sufficiently low, 0.233 MeV, that it is sensitive

to the flux of pp neutrinos produced in the first step ofthe pp chain. As a steady-state sun must burn enoughprotons to account for the measured solar luminosity,there is a minimum value for neutrino capture in thisdetector of about 77 SNU (solar neutrino units), pro-vided neutrinos behave as in the standard electroweakmodel. The measured value appears to be just consistentwith this minimum value. However, the combined re-sults of these three experiments are not fit very well byany combination ofpp , 7Be, and 8B neutrinos, and im-ply an almost total absence of neutrinos from the 7Be(e,e)

7Li reaction. This is very difficult to achieve in

solar models given that the Kamioka II/III detector seesa substantial number of the associated 8B neutrinos(about half the standard solar model value).

This discrepancy now constitutes a major problemwhose solution in terms of solar modeling, nuclear reac-tion rates, or neutrino physics has not yet become evi-dent. The most recent solar models by Bahcall, Pinson-nealt, and Wasserburg (1995) which include diffusion of

both helium and heavy elements actually predict slightlylarger fluxes than do their models without diffusion. Itappears that either new particle physics, such as theMSW mechanism (Haxton, 1995, Fig. 16), or rather des-perate changes in the astrophysics, such as the mixingmechanism suggested by Cummings and Haxton (1996),will be needed.

2. Other aspects of neutrino astrophysics

Neutrinos play additional roles in stellar evolutionand nucleosynthesis. At temperatures above about 109

K the energy of the photons is sufficiently great toachieve an equilibrium of electrons and photons de-scribed by ee. There remains a very smallprobability (about once in 1019 interactions) thateewhich is irreversible, so the neutrinos es-cape from the star. The consequent loss of energy canexceed the photon loss from the stellar surface, and thusgreatly accelerate the evolution of massive stars as theirtemperatures rise above 109 K. In addition, at the veryhigh densities achieved in stars that are evolving into

white dwarfs, the gas is both hot and degenerate. Elec-tromagnetic waves, when quantized into what are calledplasmons, can decay into a pair. These neutrinoscarry off energy and accelerate the cooling and evolu-tion of the central regions of those stars. For a full dis-cussion of these processes see Bahcall (1989).

As the core of a massive star collapses to form a neu-tron star, the flux of neutrinos in the overlying shells ofheavy elements becomes so great that, despite the smallcross section, substantial nuclear transmutation is in-duced. Neutrinos, especially the higher energy - and-neutrinos, can excite heavy elements and even heliumto particle unbound levels. The evaporation of a single

neutron or proton, and the back reaction of these nucle-ons on other species present, can significantly alter theoutcome of traditional nucleosynthesis calculations. Forexample, the large abundance of 20Ne in the neon-burning shell may be the source in nature of 19F, a rela-tively rare element, which is bypassed by most protonand -particle induced reactions Woosley et al. (1990),and Timmes, Woosley, and Weaver (1995). See alsoJorissen et al. (1993) and Forestini et al. (1992), who dis-cuss 19F production by n capture). This is now called theneutrino-, or -process, and should be added to theeight processes suggested by B2FH.




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E. Related reviews

There have been a number of reviews of nucleosyn-thesis in stars during the past 40 years. Trimble (1975)reviewed the situation very completely at about the half-way point between the publication of B 2FH and thepresent (see also Trimble, 1991, 1995). More recentlyWheeler, Sneden, and Truran (1989) reviewed stellarabundances and their nucleosynthetic origins. The

monograph by Arnett (1996) includes a substantialamount of material on stellar abundances and their ori-gins. We have not covered galactic nucleosynthesis butpoint to the excellent monograph by Pagel (1997). Theseare only samples of a substantial number of reviews thathave been published within the past 20 years.

III. STELLAR EVOLUTION

A. Historical preliminary

Tremendous progress has been made in the field of

stellar evolution since publication of the B

2

FH article in1957. Much of what the authors anticipated in the way ofelement and isotope synthesis in stars has been borneout by model calculations, certainly in conceptual terms,if not in precise detail. B 2FH noted, however, that thewhole problem of stellar evolution beyond the red giantstage is beset on the theoretical side by problems whichare very difficult to handle with the present computa-tional techniques. It is now apparent that much of theprogress in modeling that has occurred over the pastforty years is a consequence of the remarkable increasein the speed and memory of computers and of the intro-duction by Henyey, Forbes, and Gould (1964) of an im-plicit relaxation method for solving the equations of stel-lar structure.

On the other hand, all of the improvement in theworld in computational facilities and methods of solu-tion would have done little to advance the subject hadthe input physics remained inadequate. The absence ofmodels beyond the shell hydrogen-burning red giantstage about which B 2FH express concern was partly aconsequence of an incomplete quantitative descriptionof the physical conditions encountered by stars evolvingbeyond hydrogen-burning stages.

Although it was established by 1957 that the secondexcited state in the 12C nucleus plays a crucial role in theburning of 4He by the 3 process, the appropriate value

for the gamma width of this state was uncertain by atleast an order of magnitude. Strong neutrino losses inthe stellar interior during advanced stages of evolutionof intermediate mass stars play an absolutely crucial rolein determining the relationship between the initial massof the star and the mass of the white dwarf into which itevolves, but the current-current theory of weak interac-tions that predicts these losses was not invented until1958 (Feynman and Gell-Mann, 1958) and the corre-sponding energy loss rates were not fully worked outuntil the late 60s (Beaudet, Petrosian, and Salpeter,1967; Festa and Ruderman, 1969). The possibility of

neutral-current contributions to neutrino losses enteredphysics a decade later with the unified electroweaktheory (Weinberg, 1967, Salam, 1968), but calculation ofthe (1530 %) contributions of these currents did notbegin until several years later (Dicus, 1972, Dicus et al.,1976). Exploration of the full range of parameter spacerelevant for stellar physics is just now being completed(Itoh et al., 1996 and references therein).

Rosseland mean opacities in the continuum approxi-mation for mixtures of hydrogen and helium were avail-able (Keller and Meyerott, 1955), but the inclusion ofline transitions in a systematic way did not occur until adecade later (Cox, 1965; Cox et al., 1965), and opacitiesfor mixtures appropriate for more advanced stages andwith line transitions included did not become availableuntil even later (e.g., Cox and Stewart, 1970a, 1970b). Atabout the same time, accurate electron conductivitiesbecame available in the nonrelativistic regime (Hubbardand Lampe, 1969) and in the relativistic regime (Canuto,1970). More recently, advances in computer capabilities,the increasing volume of requisite atomic data, and thesophistication of the atomic and statistical physics em-

ployed have led to major improvements in the determi-nation of opacities in stellar envelopes (Iglesias, Rogers,and Wilson, 1990; Seaton et al., 1994), and opacities for awide range of conditions in the interior have becomeavailable (Rogers and Iglesias, 1992; Iglesias and Rog-ers, 1996, and references therein). Improvements have,of course, also been made in the equation of state.

Finally, the huge increase in the wavelength range ac-cessible to observers that has been brought about bytechnological advances in detectors and telescopes andby the move of telescopes and detectors into space hasvastly increased the contact between theoretical modelpredictions and the real world, permitting a test of those

phases of evolution in which most of the emitted light isoutside of the optical range.In the following, Sec. III.B will focus on the evolution

of single stars of low and intermediate mass evolvinginto white dwarfs (WDs) after passing through a plan-etary nebula (PN) stage; Sec. III.C will focus on the evo-lution of massive stars evolving into neutron stars (NSs)or black holes (BHs) after a type II supernova (SNII)explosion; and Sec. III.D will focus on the evolution ofclose binary stars that can evolve into (a) cataclysmicvariables (CVs) that experience nova outbursts, (b) apair of WDs that can merge to produce a type Ia super-nova (SNIa) or R Corona Borealis (R CrB) star, or (c)

an x-ray binary in which the accretor has experienced atype Ib or type Ic supernova explosion (SNIb,c) to be-come a NS or BH, and the mass donor is a low massmain-sequence star or subgiant (LMXBs) or a high massstar of spectral type OB (HMXBs).

B. Evolution of single stars that become white dwarfs

1. Overview

Some of what has been learned by modeling since thetime of B 2FH is summarized in the H-R diagram of Fig.




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1. In the rightmost portion of this figure (logTe4.7),evolutionary tracks of model single stars (of initial com-position close to that at the Suns surface and of initialmasses 0.2, 1, 5, and 25 M) are shown together with thepositions of a selection of bright and nearby optically

visible real stars. The tracks are defined by models thatare solutions of the equations of stellar structure, includ-ing nuclear transformations at rates based either onlaboratory cross-section measurements or on well-established weak interaction theory. Heavy portionsalong the tracks indicate where evolution proceeds on acore nuclear-burning time scale. The real stars are rep-resented by filled circles of four different sizes (1=smallest, 4=largest, etc.); most of them are from Allen(1973). Figure 1 is very busy and the reader might find ituseful to examine simultaneously Figs. 1 and 5 in Iben(1991a) and Fig. 1 in Chiosi et al. (1992).

One of the strongest arguments for the general notionthat heavy elements are made in stars is the existence ofthree different sequences defined by real stars that canbe understood in terms of the nuclear burning evolu-tionary models. Apparent discrepancies between the lo-

cation of real stars in a given sequence and the locusdefined by connecting the heavy portions of theoreticaltracks for the relevant nuclear burning stage can in mostcases be understood as consequences of uncertainties inobservational estimates or of inapproriate choices of theparameters (such as initial composition and the effi-ciency of convection) chosen for the theoretical models.

Size 2 filled circles in Fig. 1 define a main sequencethat is clearly coincident with the band defined by modelstars that are burning hydrogen in their cores (the firstheavy portion of the tracks). The main-sequence life-time is inversely proportional to roughly the 2.25 power

FIG. 1. Theory and observation compared in the theoretical H-R diagram. For guidance, lines are shown along which stellarradius is constant (dashed lines of negative slope). Solid curves in the right-hand portion of the figure are evolutionary tracks forsingle stars of mass 25, 5, 1, and 0.2 M . Heavy portions of the tracks denote phases of core nuclear burning. Dashed curveslabeled 0.6M and 0.8 5M are evolutionary tracks following the superwind state on the AGB during which model stars of initialmass 1M and 5M , respectively, lose most of their hydrogen-rich envelopes and evolve into white dwarfs; sunburst symbolsalong these tracks show where hard photons from the contracting central star are emitted sufficiently frequently to excite theejected matter into fluorescence as a planetary nebula. Solid circles are observed white dwarfs (smallest), core hydrogen-burningmain-sequence stars (next largest), core helium-burning stars (second to largest), and red giants or red supergiants (largest)burning hydrogen and/or helium in a shell (most data from Allen, 1973). Shown also is the track of a model of mass 0.6M which

experiences a final helium shell flash after having become a white dwarf and evolves into a born again AGB star. Thedash-dotted track describes the path of a model white dwarf of mass 1M accreting matter from a hydrogen-rich companion (Iben,1982); after accreting a critical mass, the model experiences a nova explosion and evolves to high luminosity before returning tothe white dwarf state. The solid star symbols describe ultra-soft x-ray binaries (USXRs, smallest), central stars of post classicalnovae (Post Novae, next largest), and central stars of planetary nebulae (PNNi, largest) (see Iben and Tutukov, 1996a forreferences). The maximum luminosity of low mass x-ray binaries for which distance estimates are available are shown in theleft-hand panel along a line of constant radius comparable to the radius of a 1.4 M neutron star or a 10M black hole (data fromvan Paradijs, 1995; van Paradijs and White, 1995; and Tanaka and Lewin, 1996, as used by Iben and Tutukov, 1977).




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of the stellar mass, decreasing from about a Hubble time(1010 years) for a star of mass 1M to 710

6

years for a star of mass 25M . A star less massivethan 0.81.0 M (depending on the composition) doesnot leave the main sequence in less than a Hubble time;this accounts for the shortness of the track displayed inFig. 1 for the 0.2M model.

Size 3 filled circles in Fig. 1 define a sequence that is

coincident with the band defined by model stars burninghelium in their cores and hydrogen in a shell (the secondheavy portion along the 1M and 25M tracks and thethird heavy portion of the 5M track). Hydrogen burn-ing provides most of the luminosity, but helium burningsets the time scale, which varies from about 25% of themain-sequence lifetime for intermediate mass stars toabout 10% for massive stars. Models less massive than2.3M have roughly the same helium core mass dur-ing this stage and therefore have nearly the same lumi-nosity (50L) and lifetime (10

8 years).Size 4 filled circles define a third, red giant, red super-

giant sequence that can be understood in terms of the

following: (a) models of initial mass

2.3M

with aninert electron-degenerate helium core and a hydrogen-burning shell (e.g., that portion of the 1M evolutionarytrack labeled RGB, the red giant branch); (b) models ofmass in the range 2.3 20 M during a first phase ofcore helium burning and shell hydrogen burning (e.g.,the second heavy portion of the 5M evolutionary trackpreceding the main core helium-burning phase); and (c)models that have a neutrino-cooled electron-degeneratecore composed of carbon and oxygen (CO core) (if ini-tial mass is in the range 19 M) or of oxygen and neon(ONe core) (if initial mass is in the range 911 M) andthat are burning alternately hydrogen and then heliumin shells. The case (c) models are known as asymptoticgiant branch (=AGB) stars because the track of amodel of low mass in this phase (e.g., the portion of the1M track in Fig. 1 labeled AGB) is asymptotic to thetrack during the RGB phase.

Another, clearly discernable sequence defined by op-tically selected stars is that of white dwarfs (size 1 filledcircles at low luminosity in Fig. 1). The formation rate ofwhite dwarfs (0.51 yr1) that is estimated by compar-ing the observed number-luminosity distribution ofnearby white dwarfs with models of cooling whitedwarfs is consistent with the formation rate of stars inthe mass range 111 M determined by comparing thenumber-luminosity distribution of main-sequence stars

with the model main-sequence lifetimes. This consis-tency supports the view (based on more detailed com-parisons between theory and observation) that all starsthat develop an electron-degenerate core along the giantbranch shed their hydrogen-rich envelope during theAGB phase to become, first, the central star of a plan-etary nebula (which is the envelope that has been shedand which is excited into fluorescence by hard photonsfrom the central star) and then a white dwarf.

The observation by Deutsch (1956) that demonstratedmass loss at a high rate from the M supergiant Hercu-lis and the speculation by Hoyle (1956) that, in the

words of B 2FH, mass loss may have a more importanteffect on the evolution of giants and supergiants of verylow surface gravity than have nuclear processes wasslow to be incorporated into theoretical models of giantsand supergiants. The first explicit calculation was that ofPaczynski (1971a) and the second was that of Harm andSchwarzschild (1975). Both calculations consisted of re-moving mass from the surface of a model AGB star at arate large compared to the rate at which matter is beingprocessed in the interior by nuclear reactions, and bothcalculations showed that, once the mass of thehydrogen-rich envelope is reduced below a very smallcritical value, the model rapidly evolves to the blue inthe H-R diagram. The dashed tracks in Fig. 1 labled 0.6M and 0.85M are the consequence of similar calcula-tions. The sunburst (rayed open circles) along thesetracks indicate where photons emitted from the surfaceof the contracting stellar remnant are hard enough toionize hydrogen atoms in the ejected material and toexcite this material into fluorescence. The resulting ob-ject is known as a planetary nebula. The location of tworelatively massive planetary nebula nuclei (PNNi) are

shown by the star symbols of intermediate size. A typi-cal PNN has a mass 0.550.6 M and burns hydrogenin a thin shell near its surface for a few times 104 years(Schonberner, 1981). This is consistent with the fact thatthe number-mass distribution of white dwarfs in the so-lar vicinity peaks somewhere in the 0.550.65 Mrange (Liebert and Bergeron, 1995).

Observation-based estimates of mass-loss rates fromMira variables and other acoustically pulsating AGBstars have shown that, once the pulsation period exceeds400 days, the rate of mass loss becomes of the order of105104 M yr

1, or typically 102103 times largerthan the rate at which nuclear burning processes matter

in the interior. This circumstance shows that a theoreti-cal mapping between initial mass and final white dwarfmass is possiblethe white dwarf mass is essentially themass of the electron-degenerate core of a model when itfirst becomes a double-shell-burning AGB star aug-mented by only several hundredths of a solar mass (pro-cessed by nuclear burning before the hydrogen-rich en-velope is removed by the stellar wind).

The AGB phase of mass loss at a high rate (some-times called the superwind phase [Renzini, 1981] andsometimes called a planetary nebula ejection event) isdue to a combination of effects (see, e.g., Bowen, 1988;Bowen and Willson, 1991): (1) pulsation (hard to calcu-

late because the envelope is convective) leads to shocks;(2) shock heating inflates the atmosphere; (3) dust grainsare formed and grow in low temperature regions of theatmosphere during the pulsation cycle; and (4) radiationpressure drives the grains to higher-than-escape velocity,and the grains drag the gas along.

2. Nucleosynthesis and dredge-up prior to the AGB phase

One of the recurrent themes in B 2FH is that the ex-otic abundances found at the surfaces of several groupsof evolved stars (e.g., R CrB stars, Wolf-Rayet stars,




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carbon stars) might be the consequences of nucleosyn-thesis, mixing, and mass-loss processes taking place insingle stars and that the different abundance patternsmight be correlated with the evolutionary stage in whicha star finds itself. Subsequent theoretical calculationshave shown this to be true in several instances, but, asrecognized by B 2FH, some exotic surface abundancescan be more easily understood in terms of binary star

evolution during which mass is lost and transferred,sometimes to the extent of a complete merger.There are four major dredge-up episodes which single

stars can experience. During each episode, turbulentconvection carries to the surface products of nucleosyn-thesis in the interior. The first episode occurs after hy-drogen is exhausted over a substantial fraction of theinterior (about 10% of the mass of the star if the mass ofthe star is less than 2.3M [Schonberg and Chan-drasekhar, 1942] or about 0.1M (M/M)

1.4 for largermasses M [Iben and Tutukov, 1984]), and the hydrogen-rich envelope expands in response to the contractionand heating of the helium core. The decreasing tempera-

tures and densities in the expanding envelope lead toincreasing opacities and a steepening of the temperatureprofile until turbulent convection is forced to carry theoutward flux of energy emerging from the hydrogen-burning shell (Hoyle and Schwarzschild, 1955). As thebase of the convective envelope moves inward in mass,it sweeps first into regions where fragile elements suchas lithium have been destroyed by reactions with pro-tons, next into a region where primordial 12C has beenconverted into 13C by the reactions12C(p ,)13N(e,)13C, then into a region where most ofthe primordial 12C has been converted into 14N, and, inthe most massive stars, into a region where 16O has beenconverted into 14N. Thus, the surface abundance oflithium drops first, followed by a decrease in the ratio12C/ 13C, a drop in the surface abundance of 12C, and anincrease in the abundance of 14N, and finally, in somestars, by a decrease in the surface abundance of 16O anda second increase in the abundance of 14N.

The predictions regarding changes in the surfaceabundances of CNO elements (Iben, 1964) are in rea-sonable agreement with the observations for field giants(Lambert and Ries, 1981). The predictions regardingchanges in the surface lithium abundance in metal-richstars (Iben, 1965, 1967b) are in reasonable agreementwith the observations for red giants more massive than1.3M (Lambert, Dominy, and Sivertsen, 1980; Pila-

chowski, 1986), but they underestimate the depletion oflithium by as much as two orders of magnitude for lowermass giants (Luck and Lambert, 1982), suggesting thatother mixing mechanisms involving convective over-shoot or differential rotation are much more importantthan the standard first dredge-up mechanism. Excellentmodern discussions of expected versus observed isotopicabundances of CNO elements are given by Dearborn(1992), El Eid (1994), and Prantzos et al. (1996).

In stars less massive than 2.3M , the helium corebecomes electron-degenerate before helium is ignited.The core is kept roughly isothermal by electron conduc-

tion, but plasma neutrino losses ensure that the maxi-mum temperature in the star is not at the center. As thecore grows in mass, gravitational potential energy is con-verted into heat and eventually, when the mass of thecore reaches 0.450.5 M , helium is ignited and burnsin a series of flashes that work their way to the center,lifting the degeneracy (Mengel and Sweigart, 1981). Thestar becomes a clump star (the heavy line segmentalong the 1M track at L50L in Fig. 1) and convertshelium into carbon under non-electron-degenerate con-ditions. In more massive stars, helium ignites in the he-lium core before electron degeneracy sets in.

Once helium is exhausted in model stars less massivethan 9M (for solarlike initial composition), the COcore becomes electron degenerate and, due to neutrinolosses, cools to the extent that carbon does not ignite.Helium burning in a shell provides most of the energyescaping from the surface. In models less massive than5M , when the helium-burning shell comes closeenough in mass to the hydrogen-helium discontinuity,hydrogen is reignited and the model enters a phase ofalternating hydrogen and helium burning. In models of

mass in the range 59 M , once the mass of the COcore exceeds 0.560.61 M , the hydrogen-exhaustedcore behaves like a red giant with a helium core: theelectron-degenerate CO core contracts and heats, whilethe helium rich layer above it expands and cools. Thebase of the convective envelope, which is initially abovethe hydrogen-helium discontinuity, moves inward inmass, extending eventually into the helium layer abovethe CO core. Fresh 4He and 14N (into which most of theprimordial CNO elements have been converted duringhydrogen burning) are dredged into the convective en-velope and appear enhanced at the surface (e.g., Iben,1972; Becker and Iben, 1980).

This second dredge-up episode occurs also in modelstars of mass in the range 911 M , but the phenom-enon is much more complex since the CO core is onlypartially electron degenerate and both carbon-burningand helium-burning shell flashes occur; the flux of en-ergy that reaches the base of the convective envelopeand forces this base to move inward in mass can have itsorigin in carbon burning, helium burning, the release ofgravothermal energy, or in some combination of thesesources. In a 9M model of solar initial composition, thedredge-up occurs in conjuntion with the first carbon-shell flash and gravothermal energy is responsible fordredge-up (Garca-Berro et al., 1997). In a 10M model,

dredge-up occurs near the end of the carbon-burningphase, and both helium burning and carbon burningcontribute to the dredge-up (Ritossa et al., 1996). In a10.5M model, dredge-up occurs also near the end ofthe carbon-burning phase, and all three energy sourcesplay a role in the dredge-up process (Iben, Ritossa, andGarca-Berro, 1997).

3. Nucleosynthesis and dredge-up during the AGB phase

Approximately 97% of all stars that can leave themain sequence in less than a Hubble time become AGB




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stars and experience thermal pulses due to helium shellflashes (Schwarzschild and Harm, 1965; Weigert, 1966)which periodically interrupt the normal, quiescenthydrogen-burning phase. Although their lifetime is only105106 years (depending on the initial mass), ther-mally pulsing AGB (TPAGB) stars produce much of the12C and most of the s-process isotopes in the Universe.They are also responsible for most of the nitrogen (the14

N entering the convective envelope during the firstand second dredge-up phases plus the 14N made by pro-ton burning at the expense of the 12C that enters theconvective envelope following helium-shell flashes in thethird dredge-up process described below).

The radius of the CO or ONe core of a TPAGB star issimilar to that of a white dwarf of the same mass intowhich the core eventually evolves. Helium shell flashesoccur for the same reason that white dwarfs accretingmatter at a rate smaller than is necessary to supportquiescent helium burning experience nova explosionswhen the mass of accreted helium reaches a criticalvalue which is larger, the smaller the accretion rate.During the quiescent hydrogen-burning phase in AGBstars, helium is deposited into a layer above the under-lying CO or ONe core at a rate which is an order ofmagnitude smaller than that necessary to maintain qui-escent helium burning. As the helium layer grows inmass, it becomes compressed and heated until eventu-ally helium ignites. The helium-burning rate is propor-tional to about the 40th power of the temperature andnuclear energy is released at a rate faster than the heatinto which it is converted can diffuse out radiatively. Abrief thermonuclear runaway continues until it isdamped out by expansion and cooling.

The large fluxes generated during a flash create a con-vective zone that extends from the base of the helium-burning region almost to the hydrogen-helium disconti-nuity. An entropy barrier, to which radiation pressurecontributes significantly, prevents the outer edge of thezone from reaching hydrogen-rich material (Iben, 1976,1977a). Prior to a flash, the helium layer that is built upcontains 14N at an abundance equal to that of primordialCNO elements. During the first part of a flash, this 14Nis converted completely into 22Ne by the reactions 14N(,)18F(,)18O(,)22Ne. In models in which thecore mass is 0.9M , temperatures at the base of theconvective zone can reach over 350106 K and the re-action 22Ne(,n)25Mg can be a potent source of neu-trons (Iben, 1975a, 1976, 1977a; Ritossa et al., 1996) for

the production ofs-process isotopes at abundances sev-eral hundred times larger than solar (Iben 1975b; Truranand Iben, 1977). It is interesting that B 2FH speculatedthat the 21Ne(,n)24Mg reaction might be an importantneutron source in stars and did not consider the 22Ne(,n)25Mg reaction (Cameron, 1961).

In AGB models with CO cores less massive than0.9M , temperatures do not become large enough atthe base of the convective envelope for the conversionof more than a percent or so of 22Ne into 25Mg (Becker,1981; Iben, 1983). However, it has been known since thediscovery of Tc in stars by Merrill (1952) that s-process

isotopes are made in abundance in such stars and thelikely neutron source is the 13C(,n)16O reaction (Cam-eron, 1955). The manner in which the 13C source is ac-tivated may not be unique and it is not completely un-derstood.

Schwarzschild and Harm (1967) found that, in one ofthirteen pulses computed, the convective shell actuallyreached the hydrogen-helium discontinuity and ingested

some hydrogen. Sanders (1967) pursued the conse-quences of this, finding that the ingested protons reactwith the 12C present at large abundance in the convec-tive shell to produce mostly 13C which diffuses convec-tively inward until reaching temperatures of the order of150106 K, whereupon -n reactions provide neu-trons for s-process nucleosynthesis. The difficulty withthis scenario is that the Schwarzschild and Harm (1967)models did not include radiation pressure, and the inges-tion mechanism has not been found in subsequent cal-culations which include radiation pressure. Another setof calculations (Iben and Renzini, 1982a, 1982b; Hollow-ell and Iben, 1989) shows that, after the convective shell

has died away in low mass AGB stars of low metallicity,semiconvective mixing forces 12C and 1H to overlap atcomparable number abundances in a region centered atthe location of the outer edge of the convective shell atits maximum extent during the flash. Ultimately, matterin this region contracts and heats, and a small pocket of13C is formed in consequence of proton capture on 12Cfollowed by a beta decay. When the next shell flash oc-curs, the convective shell ingests the pocket of 13C andmatters proceed as outlined by Sanders (1967). Thismechanism does not appear to work in AGB stars ofsolar metallicity (Iben, 1983).

It has recently been discovered that, even if a 13Cpocket is formed, it may act as a local neutron sourceduring the quiescent hydrogen-burning phase betweenflashes (Straniero et al., 1995), rather than as a distrib-uted source in a convective shell during flashes; surpris-ingly, the s-process abundance distributions producedare essentially the same in both cases (Straniero et al.,1995). Still more recently, Blocker et al. (1997) andHerwig et al. (1997) have shown that convective over-shoot beyond the base of the convective envelope dur-ing the third dredge-up phase leads to the formation of a13C pocket and this, in retrospect, could have been rec-ognized from calculations already in the literature (e.g.,Iben, 1976, Fig. 9).

The third dredge-up event occurs during the power

down phase of a shell flash, when energy produced byhelium burning leaks out of the carbon-rich region, in-creasing the flux of energy passing through the base ofthe convective envelope. This increase forces the base ofthe convective envelope to move inward in mass into theregion which contains the products of partial heliumburning. This event was first encountered in an AGBmodel of large CO core mass without invoking over-shoot at the base of the convective envelope (Iben,1975a), and later in AGB models of small core masswith the help of convective overshoot (Iben and Ren-zini, 1982a, 1982b; Iben, 1983). More on the history of




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this phenomenon may be found in reviews by Sackmannand Boothroyd (1991) and Iben (1991b).

Helium shell flashes and the third dredge up are re-sponsible for the formation of carbon stars (stars inwhich carbon is more abundant than oxygen) and forproducing overabundances ofs-process isotopes in suchstars. In addition to the enhancements which occur dur-ing the first and second dredge-up phases, a further en-

hancement of14

N can occur as a consequence of theburning of 12C into 14N at the base of the convectiveenvelope between flashes in massive AGB stars (e.g.,Iben, 1975a; Renzini and Voli, 1981). The freshly syn-thesized 14N, 12C, and s-process isotopes are returned tothe interstellar medium during the superwind phase thatconverts the AGB star into a planetary nebula. Thus,AGB stars are major contributors to the enrichment ofthe interstellar medium in these elements and isotopes(Iben and Truran, 1978), as well as in 4He (Renzini andVoli, 1981), although the Big Bang is by far the majorcontributor of this latter isotope. It is worth noting thatthe material which is dredged up during the TPAGBphase has experienced only partial helium burning, withthe abundance by mass of 12C in dredged-up matter be-ing only 0.150.25. This means that the uncertainty inthe cross section for the 12C(,)16O reaction relativeto the cross section for the triple alpha reaction does notproduce a similarly serious uncertainty in the abundanceof carbon in dredged-up matter.

The light isotopes 3He and 7Li are made in interme-diate mass stars in interesting quantities (relative to theBig Bang contribution of these isotopes) and returned tothe interstellar medium during the superwind phase.3He is made in the central regions of all main-sequencestars (Iben, 1967b) at an abundance by mass which,when averaged over the matter outside of activehydrogen-burning regions, is 4.6104M /MMS(Iben, 1977b). This 3He is preserved and returned to theinterstellar medium by AGB stars with initial masses atthe low end of the intermediate mass range (12 M), asis confirmed observationally by the high 3He/H ratio inplanetary nebulae (Balsar et al., 1997) which have pro-genitor masses less than 2M . However, the low12C/13C ratios in small-mass red giants of old galacticclusters, red giants in globular clusters, and extremelymetal-poor giants in the field (see Sec. VII) suggest that3He should be depleted in a substantial fraction of theirmass. Hogan (1995) has shown that this implies that theobserved 3He in the Galaxy may be less than the pri-

mordial 3He + D. In higher mass AGB stars, however,3He burns with 4He at the base of the convective enve-lope to form 7Be. Cameron (1955) and Cameron andFowler (1971) suggested that, if this 7Be could be mixedoutward to cooler regions where it would be destroyedby the 7Be(e,)7Li reaction rather than by the7Be(p ,)8B*(2 4He) reactions, this might explain theobserved superabundances of Li in some Galactic giants(Wallerstein and Conti, 1969; Boesgaard, 1970). Ofcourse, the mixing must also be on a long enough timescale to prevent the rapid destruction of 7Li by the7Li(p ,)4He reaction. Scalo et al. (1975) demonstrated

that this mechanism could work in the convective enve-lope of intermediate mass AGB stars. Smith and Lam-bert (1989, 1990) then showed that several of the mostluminous stars in the Large Magellanic cloud (LMC)which could be identified as AGB stars by an overabun-dance of ZrO in their spectra (Wood et al., 1983) aresuper lithium rich. This finding has been reinforced byfurther observations (Plez et al., 1993; Smith et al., 1995,and references therein). Finally, Sackmann and Boo-throyd (1992) have constructed evolutionary models ofstars in the 37 M range and find that (with appropri-ate choices of parameters in a time-dependent convec-tive mixing algorithm) the observed lithium abundancesand luminosities of the LMC super lithium rich stars canbe reproduced by models of initial mass in the range46 M .

4. The born-again AGB phenomenon

There are a number of observed stars and stellar sys-tems that can be understood in terms of a final heliumshell flash which a post-AGB star may experience afterit has ceased to burn hydrogen. The phenomenon waspredicted by Fujimoto (1977), encountered by Schon-berner (1979), and exploited by Iben et al. (1983) andIben (1984) in an attempt to explain the presence ofknots of helium-rich, nitrogen-rich clumps near to, butmoving with speeds of 20 to 30 km s1 away from, thecentral stars of the planetary nebulae Abell 30 (Hazardet al., 1980) and Abell 78 (Jacoby and Ford, 1983).

When the flash occurs, the mass of the helium layerbelow the hydrogen-helium discontinuity is slightlysmaller than that necessary for initiating a flash on theAGB. This is because the matter in the helium layer ispartially degenerate and is heated by adiabatic contrac-tion when the hydrogen-burning shell loses its power.The final flash is a strong one, and because the entropybarrier between hydrogen-rich matter and the heliumlayer is much smaller than on the AGB, the outer edgeof the convective shell forced by helium burning extendsinto the hydrogen-rich region of the star. Protons areingested and diffuse inward until reaching a point wherereactions with 12C inject so much entropy that the con-vective shell breaks into two parts, the lower shell beingforced by fluxes from helium-burning reactions and theouter one being forced by fluxes from hydrogen-burningreactions (Iben and MacDonald, 1995).

The outer convective shell consists primarily of 4He

(76% by mass) and 12C (20% by mass), with only a fewpercent by mass of 1H, and a trace of 16O. The hydro-gen abundance is so low because, prior to the flash, themass of the hydrogen-rich layer is about 20 times smallerthan the mass of the helium-rich layer with which it ismixed during the flash. As burning at the base of theouter convective shell (which extends to the photo-sphere) progresses, approximately 5% of the 12C is con-verted into 14N by proton burning. Thus, this mode ofevolution produces surface abundances that are the con-sequence of mixing products of partial helium burningwith a small amount of hydrogen-rich matter and sub-




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jecting the mix to another bout of hydrogen burning.The track of the model that produces the quoted abun-dances is shown by the dotted curve in Fig. 1. The factthat the track extends to large luminosities and low sur-face temperatures has led to the designation born againAGB stars. Perhaps 10% of all post-AGB stars mayexperience a final helium shell flash after hydrogenburning is completed (Iben, 1984). Another 15% mayexperience a final helium shell flash before hydrogenburning is finished. These too become born again AGBstars, but an entropy barrier prevents the ingestion ofhydrogen into the convective shell produced by fluxesfrom the helium-burning region. Wind mass loss duringthe born again phase and thereafter may remove most ifnot all of the hydrogen rich surface layer, and the starmay evolve into a non-DA white dwarf (surface abun-dance of hydrogen 104 by mass). There are nowabout a half dozen examples of stars that have beenidentified as passing through the born-again phase, thelatest being Sakurais object (Duerbeck and Benetti,1996).

5. Other mixing processes, and wind mass loss, whichaffect surface composition

Mixing processes other than the standard dredge-upprocesses thermal and gravitational diffusion, convec-tive overshoot at radiative-convective boundaries, androtation-induced mixing can in several instances af-fect the surface composition of single stars. Convectiveovershoot has already been discussed as an importantfactor in the third dredge-up episode.

A pronounced Li deficiency in Hyades main-sequencestars with surface temperatures in the 64007000 Krange (Boesgaard and Trippico, 1986) demonstrates thatlithium is diffusing into the interior of the gap stars. It ispossible that a combination of gravitational and thermaldiffusion inward through the base of a shallow convec-tive envelope combined with the effects of radiative levi-tation (Michaud, 1986) may account for a reduction inthe surface Li abundance, but the fact that the surfaceabundance of lithium in subgiants in the old cluster M67decreases with distance from the main sequence indi-cates that a fair fraction of the lithium diffusing inwardmust also be destroyed and this may imply rotation-induced mixing to temperatures large enough thatlithium is destroyed (Deliyannis, King, and Boesgaard,1996).

Wind mass loss can also influence surface composi-tion. For example, a radiative wind may remove thehydrogen-rich surface layer of a massive enough post-AGB star and possibly the remnant helium layer as well(Iben and Tutukov, 1996), and particle diffusion acts inwhite dwarfs of all masses to cause all but the lightestisotopes remaining in the white dwarf to settle below thephotosphere. On passing through a molecular cloud, awhite dwarf accretes hydrogen-rich material and thiscomplicates the interpretation of the observed surfaceabundance distribution. Several of these processes areprobably contributing to the observed variation in the

ratio of DA white dwarfs (hydrogen-rich spectrum) tonon-DA white dwarfs (hydrogen-deficient spectra) as afunction of luminosity (MacDonald, 1992).

C. Evolution of massive single stars that produce neutron

stars or black holes

Massive stars which ultimately explode as supernovae,leaving behind a relativistic remnant (NS or BH), areresponsible for most of the elements heavier than he-lium produced in the Galaxy. Lighter stars contributeprimarily carbon, nitrogen, and s-process isotopes. Oneof the major features of massive star evolution prior tothe supernova explosion is mass loss via a strong radia-tive wind (Cassinelli, 1979). The mass-loss rate increaseswith increasing initial mass and luminosity and continuesas the star evolves, with mass-loss rates becoming aslarge as several times 105 M yr

1. For the initiallymost massive stars, say 30M , the mass-loss timescale is comparable to or shorter than the nuclear-burning lifetime and, in constructing models to comparewith the observations, it is absolutely essential to takemass loss into account. Work along these lines prior to1986 is reviewed by Chiosi and Maeder (1986). Morerecently, extensive grids of theoretical models using Liv-ermore opacities (Iglesias and Rogers, 1996) and incor-porating mass loss according to various algorithms havebeen constructed by several groups for a variety of ini-tial compositions (e.g., Schaller et al., 1992; Schaereret al., 1992, 1993; Charbonnel et al., 1993; Bressan et al.,1992; Stothers and Chin, 1992, 1993a, 1993b).

The most massive stars eventually lose theirhydrogen-rich envelopes and expose, first, matter whichhas experienced complete hydrogen burning and, then,matter that has experienced partial helium burning.

These stripped stars are known collectively as Wolf-Rayet (WR) stars and are subdivided into the WN class(He and N in the spectrum) and the WC class (He, C,and O in the spectrum) (Abbott and Conti, 1987). In animportant study of the structure of WR stars, Langer(1989a, 1989b) compares models with the observationsto determine that the core mass MWR of WR stars fallsin the range 5 MMWR20M , independent of initialmass, that the mass-loss rate can be approximated byM WR(0.61.0)10

7(MWR /M)2.5 M yr

1, andthat the luminosity LWR satisfies 4.8logLWR /L5.5.The photosphere of WR stars lies in the escaping wind,and Langer finds the edge of the core to be typically at

optical depth 10. Finally, he finds that WN stars evolvefrom main-sequence progenitors of initial mass muchsmaller than the initial mass of WC progenitors.

The absence of all but a handful of stars in the Galaxywith luminosities greater than 106 L and with surfacetemperatures less than Te(13)10

4 (Humphreys,1978, Humphreys and Davidson, 1979, 1994) has beenargued by De Jager (1984) to be the consequence of aninstability in the photosphere, which sets in for starsbrighter than the observed luminosity limit. The instabil-ity leads to dissipation of mechanical energy and thedevelopment of supersonic turbulent motions, which re-




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sult in a dramatic increase in the mass-loss rate withincreasing luminosity. Stothers and Chin (1993a, 1993b,1996) suggest another mechanism of enhanced mass lossinvolving a classical ionization instability in the stellarenvelope. In both cases, enhanced mass loss inhibitsevolution into the Humphreys-Davidson forbiddenzone. Whatever the source of the instability, the obser-vations show that single stars initially more massive than

50M do not become giants before experiencing a su-pernova explosion, and this has important ramificationsfor massive close binary star evolution.

It is clear that massive stars interact with the interstel-lar medium both as energy sources and sources of iso-tope enrichment prior to the supernova explosion thatterminates their lives. Some aspects of this interactionare explored in a recent volume by Kunth et al. (1997).Models of Galactic chemical evolution (e.g., Pagel, 1989;Matteuchi, 1989; and Taylor, 1990) rely heavily on the-oretical estimates of the nucleosynthetic yields of mas-sive stars (e.g., Maeder, 1992; Timmes et al., 1996;Thielemann et al., 1986; Arnett 1996), but uncertainties

still remain (Arnett, 1995), particularly with regard tothe extent of mass loss prior to the supernova explosionand with regard to the dividing line in initial mass be-tween those stars which form BHs and therefore do notreturn iron-peak elements to the interstellar mediumand those which form NSs and do (Maeder, 1992;Timmes et al., 1996).

After the core carbon-burning phase, the chemicallyevolved interior of massive stars follows one of two evo-lutionary paths, depending on the initial stellar mass andcomposition. For solarlike initial composition, models ofinitial mass in the 1113 M range experience core col-lapse initiated by electron capture before they form anFe-Ni core (see below); in more massive models, corecollapse is initiated after the formation of the Fe-Nicore. A model star of mass 11M is just at the border-line between stars that become TPAGB stars with stableelectron-degenerate cores of ONe and stars that imme-diately form electron-degenerate ONe cores massiveenough that electron captures on 20Ne, 23Na, and 24Mgtrigger a rapid contraction that cannot be halted even byexplosive O and Ne burning (Miyaji et al., 1980; No-moto, 1984, 1987; Miyaji and Nomoto, 1987; Gutierrezet al., 1996). Nuclear reactions convert the compositionof the core into iron-peak isotopes, and the Fe-Ni corecollapses dynamically to nuclear matter densities. Thereal analog of the 11 M model and real analogs of mod-

els slightly less massive (say 10.75M) probably livelong enough as AGB stars that the ONe core becomesmassive enough and therefore dense enough to evolveinto a NS (Nomoto, 1987).

Model stars more massive than 13M burn carbon,neon, oxygen, and silicon quiescently and then form acore of iron-peak isotopes in statistical equilibrium (seeClayton, 1968; Arnett 1996; and Meyer, this review, Sec.XV). Beyond the core carbon-burning phase (e.g., thethird heavy portion along the 25M track in Fig. 1),there is essentially no motion in the H-R diagram; therapidly evolving core and the hydrogen-rich envelope

are essentially decoupled. Contraction and heating ofthe Fe-Ni core leads to partial photodisintegration ofiron-peak isotopes into alpha particles and neutrons (B2FH; Hoyle and Fowler, 1960; Fowler and Hoyle, 1964).Details of the subsequent dynamical collapse of thecore, neutronization, trapping of neutrinos, core bounce,and expulsion of the envelope in a type II supernova(SNII) explosion are described by many authors (e.g.,

Woosley and Weaver 1986; Arnett, 1996). The super-nova ejecta contains 56Ni, which beta decays into 56Co,which in turn beta decays into 56Fe. Associated gammaemission from the decay of excited nuclear levels helpspower the light curve. Analysis of the light curve ofSN1987a in the Large Magellanic Cloud (Arnett et al.,1989 and references therein) suggests that 0.1M of56Fe is ejected into the interstellar medium when a starof initial mass 20M explodes, showing that SNeII arepotent sources of iron in the Universe. That they mustalso be major sources of other heavy elements followsfrom detailed models of explosions set off in the enve-lopes of evolutionary models that have been carried to

the stage of Fe-Ni core collapse (e.g., Thielemann et al.,1996, and references therein).Despite much effort expended over the past 40 years,

an unambiguous theoretical picture of the detachmentof the neutronized, lepton-degenerate core has not yetemerged, although the evolution of the final detachedcore is reasonably well understood (Burrows and Lat-timer, 1986, 1987). The details of how energy is trans-ferred from the core in such a way as to cause expulsionof the stellar envelope and a quantitative estimate ofhow much matter falls back onto the core are not yetknown theoretically. Thus, a secure theoretical mappingbetween initial main-sequence mass and final NS or BHmass is not presently available (although, see Timmeset al., 1996 for an encouraging effort), and the criticalinitial mass Mcrit which separates stars that form NSremnants from those that form BH remnants is notknown. However, an understanding of massive close bi-nary evolution requires this information. In order tomake progress, a concrete choice must be made, and, inthe following, it will be assumed that Mcrit40M ,MNS1.4M , and MBH10M . The choice of 1.4 Mfor the gravitational mass of a typical neutron star is notinconsistent with the average value of 1.350.27 esti-mated by Thorsett et al. (1993) for 17 NSs in binary sys-tems.

Historically, supernovae have been classified prima-

rily on the basis of their spectral features, with type ISNe (SNeI) being hydrogen deficient and SNeII exhib-iting hydrogen lines. The SN light curve has in recentyears become important in identifying the nature of theexplosion. Comparison between spectral features andlight curves of theoretical models of explosions (see,e.g., Wheeler et al., 1995) suggest that SNeII are the endresult of the evolution of massive stars that retain theirhydrogen-rich envelopes and explode as red or blue su-pergiants (e.g., SN1987A), and that SNeIb,c are the re-sult of the evolution of stars that become WR stars be-fore exploding. In close binaries, the primary may




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become a WR star in consequence of Roche lobe over-flow, enlarging the range in initial mass of stars that pro-duce SNeIb,c. SNeIa may involve the explosion of COwhite dwarfs in close binary systems.

D. Close binary star evolution

Approximately half of all binary stars are at large

enough initial orbital separation that each componentevolves essentially as a single star for its entire life. Theother half interact by mass transfer and/or mass lossfrom the system and evolve into single stars (in conse-quence of mergers), sometimes with unusual surfacecompositions (e.g., R CrB and other hydrogen deficientstars), or into exotic binary systems with interesting andoften bizarre characteristics such as cataclysmic vari-ables, x-ray binaries, and close white dwarf pairs.

1. Modes of mass transfer and of orbital angular momentumloss

For a binary in a circular orbit, when the gravitational

potential is calculated in the rotating frame on the as-sumption that both components are point masses, thereis a unique equipotential surface consisting of twoRoche lobes (forming an hour-glass with a figure-eight cross section) which touch at a point along the linejoining the stellar centers. The radius of a sphere havingthe same volume as a Roche lobe is known as the Rochelobe radius. When, in the course of evolution, the radiusof a binary component approaches and exceeds itsRoche lobe radius, mass is transferred from the Rochelobe filling star to its companion.

When the two components are of comparable initialmass and the Roche lobe filling star does not possess adeep convective envelope, mass is transferred on a timescale which is comparable to or longer than the thermaltime scale of the accreting component. Under these con-ditions, mass is transferred conservatively (no mass islost from the system) and, when the mass of the accretorexceeds that of the donor, orbital angular momentumconservation requires that the orbital separation A in-creases with continued mass transfer.

When the donor is initially considerably more massivethan its companion (by, say, a factor of 2 or more)and/or if it possesses a deep convective envelope, mass istransferred initially so rapidly that the accretor cannotadjust its structure to accomodate the proferred mass,which forms an expanding hot blanket about the accre-

tor. The transferred matter soon fills the Roche lobe ofthe accretor and the situation can thereafter be de-scribed in terms of a common envelope (CE) (Pac-zynski, 1976) which consists of matter which has beensupplied by the donor. The matter in the CE is drivenaway from the system by an egg beater frictional in-teraction between the imbedded stars and the CE mate-rial. The energy required to drive off CE material de-rives from the orbital binding energy and the orbitshrinks. The efficiency of the CE mechanism may bedefined by CE=Eremove /Ebind , where Eremove is theenergy required to remove all of the CE matter from the

system and Ebind is the difference between the finaland the initial orbital binding energy. The smaller CE ,the greater is the degree of orbital shrinkage. The valueofCE appropriate for different situations has been de-bated for several decades (see, e.g., Iben and Livio,1993, and references therein), but recent three-dimensional smooth particle hydrodynamic calculationssuggest a value near unity (Razio and Livio, 1996; Yorke

et al., 1995).Roche lobe filling can be achieved in consequence ofthe radius increase brought about by the nuclear evolu-tion of a potential donor (e.g., Webbink et al., 1983;Taam, 1983a), or in consequence of orbital angular mo-mentum loss due to gravitational wave radiation (GWR)(Kraft et al., 1962; Paczynski, 1967; Faulkner, 1971), ordue to a magnetic stellar wind (MSW) (e.g., Verbuntand Zwaan, 1981; Taam, 1983b). In order for GWR tobe important, A must be less than 23 R (dependingon component masses). In order for a MSW to be effec-tive, the donor or the accretor must be a main-sequencestar in the approximate range 0.31 M . Reviews ofevolutionary processes in binary star evolution includePaczynski (1971b) and Bhattacharya and van den Heu-vel (1991).

2. Scenario modeling

With the help of the properties of evolving single starsdescribed in Secs. III.B and III.C, and using the prin-ciples just mentioned, one can construct scenarios forthe evolution of binary systems consisting initially ofmain-sequence stars for various choices of the mass M10of the primary, the mass ratio q0M20 /M10 , the orbitalsemimajor axis A0, and the eccentricity e0.

An approximation to the birthfunction for stars in the

Galactic disk which is based on an analysis of the prop-erties of many binary systems in the literature (Kra-icheva et al., 1978; Popova et al., 1982) is (Iben and Tu-tukov, 1984)

d3 yr1 0.2dlogA0dM10

M102.5 dq 0 , (1)

where M10 and A0 are in solar units. Integrating Eq. (1)over A01010

6, M100.8100, and q0=01 gives1 yr1 as the Galactic birthrate of stars that leavethe main-sequence in less than a Hubble time.

When mass transfer is deemed to be conservative,conservation of orbital angular momentum gives

JorbM1fM2f GA fMt1/2M10M20

GA 0

Mt 1/2, (2)

where the subscript f identifies system parameters afterthe mass transfer event is completed and MtM10M20= M1fM2f. When CE evolution is invoked, one mustadopt some approximations for Eremove and Ebind inthe definition ofCE . For example, for large q0, a roughapproximation is

CEGM10

2

A0 GM1fM20

A f M10

M20

M10

M1f

A f

A0. (3)




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Since both M20 and M1fare small compared to M10 , it isclear that orbital shrinkage can be considerable in a CEevent. Equations (1)(3), in conjunction with propertiesof evolving stellar models (including externally imposedmass loss at a high rate), have been used to reproduce(with CE 1) empirically estimated birthrates of a va-riety of evolved binary systems in the Galactic disk (e.g.,Yungelson and Tutukov, 1991; Tutukov and Yungelson,1992; Tutukov et al., 1992; Yungelson and Tutukov,1993; Iben, Tutukov, and Yungelson, 1997).

a. Cataclysmic variables and novae

Cataclysmic variables (CVs) are white dwarfs with alow mass companion that is typically a main-sequencestar that fills its Roche lobe (see Warner, 1995, and ref-erences therein). In systems with orbital periods in the1.32 h range, mass transfer is driven by angular mo-mentum loss due to GWR at a rate (12)1010 Myr1. In CVs with periods in the range 320 h, masstransfer is driven by a MSW at a rate in the range 109108 M yr

1. Systems born into the long period cat-egory evolve to shorter periods as mass is transferreduntil Porb3 h, whereupon the main-sequence compo-nent becomes completely convective and no longer sup-ports a MSW. The mass transfer rate drops abruptly,causing the donor to shrink within its Roche lobe andmass transfer to cease. Mass transfer begins again when,in consequence of GWR, the orbital separation has de-creased to the extent that the main-sequence star againfills its Roche lobe.

The hydrogen-rich matter supplied by the donor isstored in a disk and transferred from there to the whitedwarf, sometimes in a sporadic discharge that producesa flare up called a dwarf nova outburst. The light emit-

ted in such an outburst is due to the release of gravita-tional potential energy. In the absence of differential ro-tation, gravitational and thermal diffusion would mixhydrogen-rich accreted material with white dwarf mate-rial near the base of the accreted layer (Prialnik andKovetz, 1984). However, matter from the disk reachesthe surface of the white dwarf with velocities near theKeplerian velocity at the surface of the WD and angularmomentum diffuses inward through the accreted layerand beyond, creating a radial gradient in the angularvelocity. Differential rotation gives rise to a baroclinicinstability (Fujimoto, 1988, 1993) which causes evengreater mixing between accreted and white dwarf mate-

rial than does particle diffusion (Fujimoto and Iben,1997). When the white dwarf has accreted enough ma-terial (e.g., 105 M when M WD10

9 M yr1), hy-

drogen is ignited near the base of the accreted layer anda convective zone extends inward (to the point in themixing region where the abundance by mass of hydro-gen is initially 0.01) and outward (to the surface), mix-ing large quantities of CO or ONe material outward.Conversion of nuclear energy into heat lifts the electrondegeneracy in the hydrogen-rich layer, and enough en-ergy is injected to force this layer to expand to giantdimensions (e.g., to the right along the dash-dot curve in

Fig. 1 found in a model calculation that neglects thecompanion [Iben, 1982]). In the real analog, called aclassical nova, the expanding matter extends far beyondthe Roche lobe of the white dwarf and a combination ofCE action (e.g., MacDonald, 1980, 1986; Livio et al.,1990), wind mass loss (e.g., Kato, 1983; Kato and Ha-chisu, 1994), and, in some cases, dynamical acceleration(e.g., Starrfield et al., 1974, 1978) removes most of theexpanding layer. The remnant hydrogen-rich, heavy ele-ment rich layer contracts onto the white dwarf and, inmodel calculations, the star moves to the left in the H-Rdiagram in Fig. 1 along the dash-dot track at radiismaller than 1R .

Equations (1)(3), with CE 1, have been used toestimate that CVs are born at the rate 0.001 yr1

(Iben and Tutukov, 1984). When observational selectioneffects are taken into account, this is not inconsistentwith the birthrate estimated from the observed distribu-tion of CVs in the solar vicinity (Trimble, 1982). Therelative numbers of short period and long period CVscan be understood in terms of the time scales for masstransfer driven by GWR and a MSW and by observa-

tional selection effects (e.g., Iben et al., 1

synthesis of the elements in stars

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