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
27706Charles A. Barnes
California Institute of Technology, Pasadena, California 91125

Franz Käppeler
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 nuclear
experiments 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 processes
suggested by B 2 FH. In this review we summarize progress in each of these fields with emphasis on themost recent developments. [S0034-6861(97)00204-3]

Reviews of Modern Physics, Vol. 69, No. 4, October 1997

0034-6861/97/69(4)/995(90)/$23.50© 1997 The American Physical Society

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CONTENTS
I. Preface by E. Margaret Burbidge, Geoffrey R.
Burbidge, and Fred Hoyle
II. Introduction by George Wallerstein
A. The cosmological foundations of B 2 FH
B. The astronomical background in 1957
C. The eight processes
1. Hydrogen burning
2. Helium burning
3. The a process
4. The e process
5. The s process
6. The r process
7. The p process
8. The x process
D. Neutrino astrophysics
1. The solar neutrino problem
2. Other aspects of neutrino astrophysics
E. Related reviews
III. Stellar Evolution by Icko Iben, Jr.
A. Historical preliminary
B. Evolution of single stars that become whitedwarfs
1. Overview
2. Nucleosynthesis and dredge-up prior to the
AGB phase
3. Nucleosynthesis and dredge-up during the
AGB phase
4. The born-again AGB phenomenon
5. Other mixing processes, and wind mass loss,
which affect surface composition
C. Evolution of massive single stars that produce
neutron stars or black holes
D. Close binary star evolution
1. Modes of mass transfer and of orbital angularmomentum loss
2. Scenario modelinga. Cataclysmic variables and novaeb. White dwarf mergers: R CrB stars andtype Ia supernovaec. X ray binaries and pulsars
IV. Hydrogen Burning in the pp Chain and CN Cycle
by Peter Parker
A. The p(p,e 1 n e )d reaction
B. The 3 He( 3 He,2p) 4 He reaction
C. The 3 He( a ,g ) 7 Be reaction
D. The 7 Be(p,g ) 8 B reaction
E. The 7 Li(n, g ) 8 Li reaction
F. The 14N(p, g ) 15O reaction
G. The 17O(p, a ) 14N reaction
H. The Hot CNO cycle
V. The x Process by Ann Merchant Boesgaard
A. Introduction and retrospective
B. Abundances
1. Lithium
2. Beryllium
3. Boron
C. Nonlocal thermodynamic-equilibrium effects
D. Production mechanisms
1. Big Bang
2. Spallation
3. Asymptotic giant branch stars
4. Supernovae
VI. Helium Burning by Gerald M. HaleRev. Mod. Phys., Vol. 69, No. 4, October 1997

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A. Triple-a capture
B. a 1 12C capture
1. E1 capture
a. Direct measurements
b. b -delayed a spectrum from the decay of
16
N
2. E2 capture
3. Other analyses and recommended values
VII. H Burning in the NeSi Region: Laboratory Studies
by Arthur E. Champagne
A. Introduction
B. Experimental approaches
C. The neon-sodium cycle at low temperatures
1. Reaction rates
2. Network calculations
D. The magnesium-aluminum cycle at low
temperatures
1. Reaction rates
2. Network calculations
E. High-temperature behavior
1. Reaction rates
2. Network calculations
F. Conclusion
VIII. Observational Evidence of Hydrogen Burning by
George Wallerstein
A. CN cycle and mixing
B. O depletion and the enhancement of Na and Al
C. Origin of the Na and Al enhancements
IX. Carbon, Neon, Oxygen, and Silicon Burning by
Charles A. Barnes
A. Introduction
B. Carbon burning
C. Neon burning
D. Oxygen burning
E. Silicon burning
X. s Process: Laboratory Studies and Stellar Models by
Franz Käppeler
A. The s process since B 2 FH
B. Laboratory experiments
1. Neutron capture cross sections
2. Stellar b decay
C. The canonical s process
1. The s N curve
2. Branchings
XI. Observations of the s Process by Verne V. Smith
A. Brief history
B. Observations of nucleosynthesis and mixing in
CH, Ba, S, and C stars
C. The s process as a function of metallicity
D. Rubidium and the s-process neutron density
E. Recent models: Radiative burning of 13C during
the AGB interpulse phase
XII. The r Process by Robert D. Hoffman and Frank X.
TimmesA. Introduction
B. A search for the astrophysical site
C. Early model results, from conflict to clarity
D. Twisting in the wind
E. Concluding remarks
XIII. Observations of the r Process by Chris Sneden
A. Defining the r-process elements
B. Early r-process discoveries
C. Recent r-process surveys
D. Thorium and the age of the halo and disk
E. Filling out the picture
XIV. The p Process by Richard N. Boyd

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A. The p process
B. Early p-process models
C. The g process
D. The rp process
E. The n process
F. Recent developments
G. Summary
XV. The e Process and the Iron-Group Nuclei by
Bradley S. Meyer
A. Energetics and equilibria
B. Statistical equilibrium
C. A brief history of the ideas of iron-groupelement synthesis
D. Significance for astrophysics
XVI. Carbon Stars: Where Theory Meets Observations
by David L. Lambert
A. Prologue
B. Carbon stars—An observer’s view
1. What is a carbon-rich star?
2. What makes a carbon-rich star?
3. The principal types of carbon-rich stars
a. R-type carbon stars
b. N-type carbon starsc. Barium and related stars
C. Epilogue
XVII. Conclusions
Acknowledgments
References

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I. PREFACE

It is curious that both the primordial and stellar theories of the origin of the elements should have been published in the same year, 1946. Little notice was at firsttaken of the stellar theory, attention being directed at
first overwhelmingly to Gamow’s suggestion of associating 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 stellar
nucleosynthesis had a connection to cosmology was aninvention of Robert Oppenheimer and it never had any
reality to it, since the first paper on stellar nucleosynthesis appeared in 1946, two years before the steady-statecosmological model was published. Nor were the objections to primordial synthesis by neutron addition by anymeans confined to the well-known gaps at A55 andA58, 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, temperatures in stellar interiors would be needed of the order ofa hundred times those in main-sequence stars, a requirement that it would have been difficult to accept if the
beginnings of an understanding of supernovae had not
emerged 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 comparing nuclear weapons with chemical ones, this suggestRev. Mod. Phys., Vol. 69, No. 4, October 1997

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ing that temperatures vastly higher than those that
seemed plausible in Eddington’s day might be possible.
The fact that primordial nucleosynthesis ran far aheadof stellar nucleosynthesis in the years up to the early
1950s turned out to be advantageous to the eventual
emergence of the B 2 FH paper in 1957, because it permitted facts to accumulate quietly without any frenzied
circus developing. In 1952, Salpeter discussed the stellar
formation of carbon from alpha particles, and a few
months later the relation of carbon synthesis to oxygen
synthesis was shown to require a state in the carbon
nucleus of about 7.65 MeV above ground level. When
this state was actually found, the laboratory discovery
carried a strong measure of conviction for all those who
were involved in it.
At the same time surveys of nearby stars by several
groups, including some of us, were showing variations ofmetal abundances to hydrogen, the beginning of theconcept of metallicity, that seemed impossible to associate with some form of universal synthesis. We were also
puzzled by our 1953–54 analysis of high-dispersion spectra obtained at McDonald Observatory, which showedstrange overabundances of some heavy elements andseemed to indicate that neutrons were involved. And in
1953–54, carbon-burning and oxygen-burning were
found.
It was in the autumn of 1954 that the team of B 2 FH
came together in the quiet ambience of Cambridge, England, without, as mentioned above, any frenzied circus
developing.
During 1954–55 calculations involving neutron production in the interiors of evolved stars with heliumburning cores surrounded by a hydrogen-burning shellwere followed by calculations on neutron addition toelements from neon to scandium. The process of neutron addition was slow enough for each neutron capture
to be followed by beta decay (the origin of what we later
named the s process). In the same period, we becameaware that A.G.W. Cameron was working along similarlines in Canada. These calculations showed the possibility 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 setting at both the Kellogg Radiation Laboratory at
Caltech and the Mount Wilson and Palomar offices at
813 Santa Barbara Street. We now also had the great
advantage of being at the home of the large telescopes.
In 1952 the process of neutron liberation and capture
had been clinched by Paul Merrill’s discovery of the unstable element technetium in S stars (evolved red giants), and we were able to obtain spectra with theMount Wilson 100-inch telescope of an evolved star of
the class known as ‘‘Barium II stars,’’ in which we determined the overabundances of just those heavy elementsthat had at least one isotope with a magic neutron number.
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998

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four of us in supernovae as the spectacular death throesof stars at the end of their active evolution and thermonuclear energy production, we latched onto this. The
exponential decay of the energy output of this supernova immediately suggested radioactive decay, following collapse, the release of a great burst of energy, neutrons, and neutrinos, and the formation of heavy
unstable nuclei.
Also in 1956, an improved solar-system abundance
curve became available from Suess and Urey. Values in
the upper half of the chart of the nuclides showed twothings: the association of magic neutron numbers with
abundance peaks, and a separation of peaks corresponding to both slow and rapid neutron addition, the latterbeing fast enough not to allow time for the beta-decay
involved in slow neutron addition. And released at thattime were hitherto classified data on (n, g ) cross sections
for individual isotopes of elements, without which ameaningful analysis of what in B 2 FH was called the sprocess would not have been possible.
B 2 FH can be seen in retrospect to have been a highly
creative review article, putting together what the four
authors had done previously, together with the facts on
which the theory would now be based with considerable
confidence. We each brought ideas and data from very
different parts of physics to the table, and occasionally
we stopped arguing long enough to work things out andwrite them down. There were a number of original contributions, notably the calculations for the s and r processes. The rest consisted of an extensive updating ofprevious work. But one should not forget the introduction of a lettering notation for the various nuclear processes: a , e, s, r, p, and x, which may have done
more 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 Burbridge, Burbridge, Fowler, and Hoyle, B 2 FH,1 to combine progress in stellar and solar system abundanceswith laboratory nuclear physics data and stellar evolution calculations to show how stars can produce elements and their isotopes from helium to uranium. Their
paper provided a basis for nuclear astrophysics for the
decades 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 extending their suggestion.
1

This was probably the first astronomical paper to be referred
to by the initials of its authors.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

A. The cosmological foundations of B 2 FH

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 expansion of the Universe but failed to predict the originof the elements beyond the lightest species. Steady stateprovided for an understanding of the discrepancy between the expansion age and the apparent ages of theglobular clusters—a problem that is still with us—but
provided no physical basis for the continuous creation ofmatter, the expansion, or the collection of diffusely created matter into galaxies.
One of the strengths of the B 2 FH paper was its independence of the cosmological models then under discussion. The paper explained just how much stars could
contribute to the synthesis of nuclei heavier than hydrogen. Since they did not have the answers for D, Li, Be,
and B they relegated them to the ‘‘x process.’’ Almost
all other elements and isotopes could be produced in thestellar environment by one of their eight processes (withtheir ‘‘x process’’ included as number eight).
B. The astronomical background in 1957

Starting in the late 1940s and early 1950s astronomers
were assembling data to show that all stars did not havethe same chemical composition. For 80 years it had been
known 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 oxide bands. Due to blending of absorption lines and molecular bands a quantiative comparison of carbon andoxygen red giants was not possible. More recently the
subdwarfs had been found to be metal-poor by factors
more than 10 (Chamberlain and Aller, 1951), and theheavy element stars such as the Ba II stars and S stars
were known to show abundance excesses of certain species 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 a
half-life of 43106 years and Cameron (1955) had shownthat 99Tc, with a half-life of only 33105 years was themost easily produced isotope. This proved beyond any
doubt 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 species in stars B 2 FH suggested eight separate processes.
1. Hydrogen burning

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

Wallerstein et al.: Synthesis of the elementsof the resulting neutrinos, which is crucial in attempts tounderstand the solar neutrino problem. In addition they
discussed the p capture by the neon isotopes to produce23
Na (also mentioned in Bethe’s 1939 paper). In Secs.
IV and VII, Parker and Champagne bring us up to dateon the laboratory rates of hydrogen burning reactions
from the pp reaction through the CNO cycle, the NeNa
cycle and the MgAl cycle. In addition the hot CNO cycle
plays an important role in nova explosions, while the rp
process discussed by Boyd in Sec. XIII may be responsible for the production of certain p-rich isotopes.

2. Helium burning

By the time of B 2 FH helium burning to produce 12C
had been suggested by Öpik (1951) and its rate estimated by Salpeter (1952). In Sec. VI. G. Hale describes
the 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(a,g)16O reaction. The rate of the
12
C( a,g) 16O reaction relative to the 3-a process determines the carbon/oxygen ratio in massive stars, and thisis crucial for the later evolution of such a star and its
resulting nucleosynthesis. Unfortunately, 40 years afterB 2 FH, the rate of the 12C(a,g)16O reaction is still not
well determined. Fortunately the material that is returned to the interstellar medium by stars that are lessmassive than 11M ( and evolve into white drawfs has
been enriched by matter that has experienced only partial helium burning, so the uncertainty in the ratio ofreaction rates plays a minor role.

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5. The s process

Beyond the iron-peak, utilizing neutrons produced byreactions such as 13C(a,n) 16O and 22Ne( a ,n) 25Mg, nuclei can be produced along or adjacent to the valley ofstability via a process in which sequential neutron captures 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 Käppeler and Smith bringus up to date on laboratory measurements, stellar models (also discussed in Sec. III by Iben), and abundancestudies of the s-process elements.
6. The r process

B 2 FH showed that, in addition to the s process, there
must be another neutron capture process in which thesequential neutron captures take place on a time scale
which is much more rapid than the beta decay of theresulting nuclei. This process produces the much moreneutron-rich progenitors that are required to account forthe second set of abundance peaks that are observed
about 10 mass units above the s-process abundance
peaks corresponding to the neutron magic numbers,
N550 and 82. Historically, the r process has been associated with SN explosions, and in the past decade interest has focused more specifically on the neutrino-heatedatmosphere surrounding the newly formed neutron staras the r-process site. In Sec. XII Hoffman and Timmes
review both the physics and astrophysical scenario of
rapid neutron capture during the explosion of massive
supernovae.
7. The p process

3. The a process

B 2 FH suggested that further a captures could extend
nucleosynthesis beyond 16O to 20Ne, 24Mg, etc., up to
the very stable, doubly magic nucleus, 40Ca. However,after experiments showed that the 16O(a,g)20Ne rate isvery slow in stellar interiors, it became evident that carbon and oxygen burning are responsible for the origin of
species from Ne to S, with the nuclei consisting of integral numbers of a particles dominating the abundance
curve in this region.

4. The e process

At very high temperatures, about 4 or 53109 K, so
many reactions take place that the nuclei settle down tostatistical equilibrium dominated by the most tightly
bound nuclei around 56Fe. Such conditions are reachedonly in supernovae. The observation of g rays from
SN1987A due to the deexcitation of 56Fe, resulting fromthe b decay of 56Ni, has demonstrated the importance ofthe production of iron-peak species in supernova explosions. Modern calculations of the iron-peak abundances
are discussed by Meyer in Sec. XV.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

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 coulomb
barrier or by (g,n) reactions during supernova explosions. Recent work on (p, g ) and (g,n) reactions including 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 temperatures in stars but hardly ever (except for 7 Li) produced in stars. B 2 FH did not know how they were produced so they ascribed their synthesis to the x process.
Modern cosmological models of big bang nucleosynthesis are tuned to produce D, 3 He, 4 He, and some 7 Li tofit observations of these species in very metal-poor stars
and other astrophysical sources. The observations andtheories of production of 7 Li, Be, and B in stars arereviewed by Boesgaard in Sec. V. For a brief review ofthe production of D, 3 He, 4 He, and 7 Li in the early
universe 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 developing
so rapidly and hence is reviewed so frequently that,within the limited space available, we could not add substantially to the material already available. For the fundamentals 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 detailed and very interesting history of the solar neutrino
problem see the Appendix of Bahcall (1989).
The important place of neutrinos in present day astrophysics could hardly have been anticipated by B 2 FH,
which predated neutrino physics as an experimental science and the modern understanding of neutrino interactions. The possibility of detecting solar neutrinos had
been suggested at the time of B 2 FH, but not in the usual
literature. Pontecorvo (1946) and Alvarez (1949) independently called attention to the possibility of the detection of neutrinos. The basic papers calling attention to
the possibility of detecting solar neutrinos with a 37Cl
detector were published in 1964 (Davis, 1964; Bahcall,1964). In 1968 Davis, Harmer, and Hoffman reported
the first results of their Cl experiment in the Homestake
Mine in South Dakota. Over the next 20 years or so
continued observations by Davis and his colleagues andimproved solar models by Bahcall and his colleagues (aswell as others) demonstrated a discrepancy of a factor of
3 between the predicted number of 8 B neutrinos and thedetection rate for neutrinos with energy above 0.814
MeV. No likely solar models have been able to explain
the discrepancy. However, the suggestion by Mikheyev
and Smirnov (1985) of matter-enhanced neutrino oscillations, following Wolfenstein’s (1978) discussion ofneutrino effective masses in matter, provides a very
natural explanation for the observations, assuming massive neutrinos and flavor mixing. This possibility is commonly known as the MSW mechanism.
Neutrino astrophysics was given a great boost in thelast decade by the results from three new solar neutrino
experiments, one using a water Čerenkov detector tomeasure 8 B neutrinos (Kamiokanda II/III) and two others using radiochemical 71Ga detectors (SAGE and
GALLEX). The threshold of the 71Ga ( n e ,e) 71Ge reaction is sufficiently low, 0.233 MeV, that it is sensitive
to the flux of pp neutrinos produced in the first step of
the 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), provided neutrinos behave as in the standard electroweakmodel. The measured value appears to be just consistent
with this minimum value. However, the combined results of these three experiments are not fit very well by
any combination of pp, 7 Be, and 8 B neutrinos, and imply an almost total absence of neutrinos from the 7 Be(e, n e ) 7 Li reaction. This is very difficult to achieve inRev. Mod. Phys., Vol. 69, No. 4, October 1997

solar models given that the Kamioka II/III detector seesa substantial number of the associated 8 B neutrinos(about half the standard solar model value).
This discrepancy now constitutes a major problem
whose solution in terms of solar modeling, nuclear reaction rates, or neutrino physics has not yet become evident. The most recent solar models by Bahcall, Pinsonnealt, and Wasserburg (1995) which include diffusion ofboth helium and heavy elements actually predict slightly
larger fluxes than do their models without diffusion. It
appears that either new particle physics, such as the
MSW mechanism (Haxton, 1995, Fig. 16), or rather desperate changes in the astrophysics, such as the mixing
mechanism suggested by Cummings and Haxton (1996),
will be needed.

2. Other aspects of neutrino astrophysics

Neutrinos play additional roles in stellar evolution
and nucleosynthesis. At temperatures above about 109
K the energy of the photons is sufficiently great toachieve an equilibrium of electrons and photons described by g 1 g ↔e 1 1e 2 . There remains a very small
probability (about once in 1019 interactions) thate 1 1e 2 → n 1 n¯ which is irreversible, so the neutrinos escape from the star. The consequent loss of energy canexceed the photon loss from the stellar surface, and thus
greatly accelerate the evolution of massive stars as theirtemperatures rise above 109 K. In addition, at the very
high densities achieved in stars that are evolving into
white dwarfs, the gas is both hot and degenerate. Electromagnetic waves, when quantized into what are calledplasmons, can decay into a n 1 n¯ pair. These neutrinos
carry off energy and accelerate the cooling and evolution of the central regions of those stars. For a full discussion of these processes see Bahcall (1989).
As the core of a massive star collapses to form a neutron star, the flux of neutrinos in the overlying shells ofheavy elements becomes so great that, despite the small
cross section, substantial nuclear transmutation is induced. Neutrinos, especially the higher energy m- andt-neutrinos, can excite heavy elements and even helium
to particle unbound levels. The evaporation of a single
neutron or proton, and the back reaction of these nucleons on other species present, can significantly alter theoutcome of traditional nucleosynthesis calculations. For
example, the large abundance of 20 Ne in the neonburning shell may be the source in nature of 19F, a relatively rare element, which is bypassed by most proton
and a-particle induced reactions Woosley et al. (1990),
and Timmes, Woosley, and Weaver (1995). See also
Jorissen et al. (1993) and Forestini et al. (1992), who discuss 19F production by n capture). This is now called the‘‘neutrino-’’, or n-process, and should be added to theeight processes suggested by B2FH.

Wallerstein et al.: Synthesis of the elements

E. Related reviews

There have been a number of reviews of nucleosynthesis in stars during the past 40 years. Trimble (1975)
reviewed the situation very completely at about the halfway point between the publication of B 2 FH and thepresent (see also Trimble, 1991, 1995). More recently
Wheeler, Sneden, and Truran (1989) reviewed stellar
abundances and their nucleosynthetic origins. The
monograph by Arnett (1996) includes a substantial
amount of material on stellar abundances and their origins. We have not covered galactic nucleosynthesis but
point to the excellent monograph by Pagel (1997). These
are 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 ofstellar 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 2 FH noted, however, that ‘‘thewhole problem of stellar evolution beyond the red giantstage is beset on the theoretical side by problems which
are very difficult to handle with the present computational techniques.’’ It is now apparent that much of the
progress in modeling that has occurred over the past
forty years is a consequence of the remarkable increasein the speed and memory of computers and of the introduction by Henyey, Forbes, and Gould (1964) of an implicit relaxation method for solving the equations of stellar structure.
On the other hand, all of the improvement in theworld in computational facilities and methods of solution would have done little to advance the subject had
the input physics remained inadequate. The absence ofmodels beyond the shell hydrogen-burning red giantstage about which B 2 FH express concern was partly aconsequence of an incomplete quantitative descriptionof the physical conditions encountered by stars evolving
beyond hydrogen-burning stages.
Although it was established by 1957 that the second
excited state in the 12C nucleus plays a crucial role in theburning of 4 He by the 3a 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 it
evolves, but the current-current theory of weak interactions that predicts these losses was not invented until
1958 (Feynman and Gell-Mann, 1958) and the corresponding energy loss rates were not fully worked out
until the late 60’s (Beaudet, Petrosian, and Salpeter,
1967; Festa and Ruderman, 1969). The possibility ofRev. Mod. Phys., Vol. 69, No. 4, October 1997

1001

neutral-current contributions to neutrino losses enteredphysics a decade later with the unified electroweak
theory (Weinberg, 1967, Salam, 1968), but calculation ofthe (15–30 %) contributions of these currents did not
begin until several years later (Dicus, 1972, Dicus et al.,1976). Exploration of the full range of parameter space
relevant for stellar physics is just now being completed(Itoh et al., 1996 and references therein).
Rosseland mean opacities in the continuum approximation for mixtures of hydrogen and helium were available (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 opacities
for mixtures appropriate for more advanced stages andwith line transitions included did not become available
until even later (e.g., Cox and Stewart, 1970a, 1970b). At
about the same time, accurate electron conductivities
became available in the nonrelativistic regime (Hubbard
and 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 employed have led to major improvements in the determination 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 become
available (Rogers and Iglesias, 1992; Iglesias and Rogers, 1996, and references therein). Improvements have,of course, also been made in the equation of state.
Finally, the huge increase in the wavelength range accessible to observers that has been brought about bytechnological advances in detectors and telescopes andby the move of telescopes and detectors into space has
vastly increased the contact between theoretical model
predictions and the real world, permitting a test of thosephases of evolution in which most of the emitted light isoutside of the optical range.
In the following, Sec. III.B will focus on the evolutionof single stars of low and intermediate mass evolving
into white dwarfs (WDs) after passing through a planetary nebula (PN) stage; Sec. III.C will focus on the evolution 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) cataclysmic
variables (CVs) that experience nova outbursts, (b) apair of WDs that can merge to produce a type Ia supernova (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 become a NS or BH, and the mass donor is a low mass
main-sequence star or subgiant (LMXBs) or a high mass
star 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 2 FH is summarized in the H-R diagram of Fig.

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Wallerstein et al.: Synthesis of the elements

FIG. 1. Theory and observation compared in the theoretical H-R diagram. For guidance, lines are shown along which stellar
radius is constant (dashed lines of negative slope). Solid curves in the right-hand portion of the figure are evolutionary tracks for
single stars of mass 25, 5, 1, and 0.2 M ( . Heavy portions of the tracks denote phases of core nuclear burning. Dashed curves
labeled 0.6M ( and 0.85M ( are evolutionary tracks following the superwind state on the AGB during which model stars of initial
mass 1M ( and 5M ( , respectively, lose most of their hydrogen-rich envelopes and evolve into white dwarfs; ‘‘sunburst’’ symbols
along 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-burning
main-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. The
dash-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 to
the 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 1.4M ( 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).

1. In the rightmost portion of this figure (logTeu4.7),
evolutionary tracks of model single stars (of initial composition close to that at the Sun’s surface and of initial
masses 0.2, 1, 5, and 25 M ( ) are shown together with thepositions of a selection of bright and nearby opticallyvisible real stars. The tracks are defined by models thatare solutions of the equations of stellar structure, including nuclear transformations at rates based either onlaboratory cross-section measurements or on wellestablished weak interaction theory. Heavy portions
along the tracks indicate where evolution proceeds on acore nuclear-burning time scale. The real stars are represented 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 it
useful to examine simultaneously Figs. 1 and 5 in Iben
(1991a) and Fig. 1 in Chiosi et al. (1992).
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

One of the strongest arguments for the general notionthat heavy elements are made in stars is the existence of
three different sequences defined by real stars that canbe understood in terms of the nuclear burning evolutionary models. Apparent discrepancies between the location of real stars in a given sequence and the locus
defined 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 efficiency of convection) chosen for the theoretical models.
Size 2 filled circles in Fig. 1 define a ‘‘main sequence’’that 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 lifetime is inversely proportional to roughly the 2.25 powerWallerstein et al.: Synthesis of the elementsof the stellar mass, decreasing from about a Hubble time(;1010 years) for a star of mass ;1M ( to ;73106
years for a star of mass ;25M ( . A star less massive
than 0.8–1.0 M ( (depending on the composition) does
not 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 iscoincident with the band defined by model stars burning
helium in their cores and hydrogen in a shell (the second
heavy portion along the 1M ( and 25M ( tracks and thethird heavy portion of the 5M ( track). Hydrogen burning provides most of the luminosity, but helium burning
sets the time scale, which varies from about 25% of themain-sequence lifetime for intermediate mass stars toabout 10% for massive stars. Models less massive than
;2.3M ( have roughly the same helium core mass during this stage and therefore have nearly the same luminosity (;50L ( ) and lifetime (;108 years).
Size 4 filled circles define a third, red giant, red supergiant sequence that can be understood in terms of thefollowing: (a) models of initial mass u2.3M ( with aninert electron-degenerate helium core and a hydrogenburning shell (e.g., that portion of the 1M ( evolutionary
track 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 track
preceding the main core helium-burning phase); and (c)models that have a neutrino-cooled electron-degeneratecore composed of carbon and oxygen (CO core) (if initial mass is in the range 1–9 M ( ) or of oxygen and neon
(ONe core) (if initial mass is in the range 9–11 M ( ) andthat are burning alternately hydrogen and then heliumin shells. The case (c) models are known as ‘‘asymptotic
giant branch (=AGB)’’ stars because the track of amodel of low mass in this phase (e.g., the portion of the
1M ( track in Fig. 1 labeled AGB) is asymptotic to thetrack during the RGB phase.
Another, clearly discernable sequence defined by optically selected stars is that of white dwarfs (size 1 filled
circles at low luminosity in Fig. 1). The formation rate ofwhite dwarfs (;0.5–1 yr 21 ) that is estimated by comparing the observed number-luminosity distribution of
nearby white dwarfs with models of cooling whitedwarfs is consistent with the formation rate of stars in
the mass range 1–11 M ( determined by comparing thenumber-luminosity distribution of main-sequence stars
with the model main-sequence lifetimes. This consistency supports the view (based on more detailed comparisons between theory and observation) that all stars
that develop an electron-degenerate core along the giant
branch shed their hydrogen-rich envelope during the
AGB phase to become, first, the central star of a planetary nebula (which is the envelope that has been shed
and which is excited into fluorescence by hard photons
from the central star) and then a white dwarf.
The observation by Deutsch (1956) that demonstrated
mass loss at a high rate from the M supergiant a Herculis and the speculation by Hoyle (1956) that, in theRev. Mod. Phys., Vol. 69, No. 4, October 1997

1003

words of B 2 FH, ‘‘mass loss may have a more important
effect on the evolution of giants and supergiants of very
low surface gravity than have nuclear processes’’ wasslow to be incorporated into theoretical models of giants
and supergiants. The first explicit calculation was that of
Paczyński (1971a) and the second was that of Härm and
Schwarzschild (1975). Both calculations consisted of removing mass from the surface of a model AGB star at a
rate large compared to the rate at which matter is being
processed in the interior by nuclear reactions, and both
calculations showed that, once the mass of thehydrogen-rich envelope is reduced below a very small
critical value, the model rapidly evolves to the blue inthe H-R diagram. The dashed tracks in Fig. 1 labled 0.6
M ( and 0.85M ( are the consequence of similar calculations. 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 object is known as a planetary nebula. The location of two
relatively massive planetary nebula nuclei (PNNi) are
shown by the star symbols of intermediate size. A typical PNN has a mass ;0.55–0.6 M ( and burns hydrogenin a thin shell near its surface for a few times 104 years(Schönberner, 1981). This is consistent with the fact thatthe number-mass distribution of white dwarfs in the solar vicinity peaks somewhere in the ;0.55–0.65 M (range (Liebert and Bergeron, 1995).
Observation-based estimates of mass-loss rates from
Mira variables and other acoustically pulsating AGB
stars have shown that, once the pulsation period exceeds;400 days, the rate of mass loss becomes of the order of1025 –1024 M ( yr 21 , or typically 102 –103 times larger
than the rate at which nuclear burning processes matterin the interior. This circumstance shows that a theoretical mapping between initial mass and final white dwarf
mass is possible—the white dwarf mass is essentially themass of the electron-degenerate core of a model when it
first becomes a double-shell-burning AGB star augmented by only several hundredths of a solar mass (processed by nuclear burning before the hydrogen-rich envelope is removed by the stellar wind).
The AGB phase of mass loss at a high rate (sometimes 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 calculate because the envelope is convective) leads to shocks;(2) shock heating inflates the atmosphere; (3) dust grains
are 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 2 FH is that the exotic abundances found at the surfaces of several groupsof evolved stars (e.g., R CrB stars, Wolf-Rayet stars,

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Wallerstein et al.: Synthesis of the elements

carbon stars) might be the consequences of nucleosynthesis, mixing, and mass-loss processes taking place insingle stars and that the different abundance patterns
might be correlated with the evolutionary stage in whicha star finds itself. Subsequent theoretical calculations
have shown this to be true in several instances, but, as
recognized by B 2 FH, some exotic surface abundances
can be more easily understood in terms of binary starevolution 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, turbulent
convection carries to the surface products of nucleosynthesis in the interior. The first episode occurs after hydrogen 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 ( [Schönberg and Chandrasekhar, 1942] or about 0.1M ( (M/M ( ) 1.4 for larger
masses M [Iben and Tutukov, 1984]), and the hydrogenrich envelope expands in response to the contraction
and heating of the helium core. The decreasing temperatures and densities in the expanding envelope lead toincreasing opacities and a steepening of the temperature
profile until turbulent convection is forced to carry theoutward flux of energy emerging from the hydrogenburning 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 protons, next into a region where primordial 12C has been
13
converted
into
C
by
the
reactions
12
13
1
13
C(p, g ) N(e , n ) C, then into a region where most of
the primordial 12C has been converted into 14N, and, inthe most massive stars, into a region where 16O has been
converted into 14N. Thus, the surface abundance oflithium drops first, followed by a decrease in the ratio12
C/ 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 surface
abundances of CNO elements (Iben, 1964) are in reasonable agreement with the observations for field giants(Lambert and Ries, 1981). The predictions regarding
changes in the surface lithium abundance in metal-richstars (Iben, 1965, 1967b) are in reasonable agreementwith the observations for red giants more massive than
;1.3M ( (Lambert, Dominy, and Sivertsen, 1980; Pilachowski, 1986), but they underestimate the depletion oflithium by as much as two orders of magnitude for lower
mass giants (Luck and Lambert, 1982), suggesting that
other mixing mechanisms involving convective overshoot or differential rotation are much more important
than the standard first dredge-up mechanism. Excellent
modern discussions of expected versus observed isotopic
abundances 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 core
becomes electron-degenerate before helium is ignited.
The core is kept roughly isothermal by electron conducRev. Mod. Phys., Vol. 69, No. 4, October 1997

tion, but plasma neutrino losses ensure that the maximum temperature in the star is not at the center. As thecore grows in mass, gravitational potential energy is converted into heat and eventually, when the mass of thecore reaches ;0.45–0.5 M ( , helium is ignited and burns
in a series of flashes that work their way to the center,
lifting the degeneracy (Mengel and Sweigart, 1981). The
star becomes a ‘‘clump’’ star (the heavy line segment
along the 1M ( track at L;50L ( in Fig. 1) and converts
helium into carbon under non-electron-degenerate conditions. In more massive stars, helium ignites in the helium core before electron degeneracy sets in.
Once helium is exhausted in model stars less massive
than ;9M ( (for solarlike initial composition), the COcore becomes electron degenerate and, due to neutrino
losses, cools to the extent that carbon does not ignite.
Helium burning in a shell provides most of the energy
escaping from the surface. In models less massive than
;5M ( , 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 ofmass in the range 5–9 M ( , once the mass of the COcore exceeds ;0.56–0.61 M ( , the hydrogen-exhausted
core behaves like a red giant with a helium core: theelectron-degenerate CO core contracts and heats, while
the helium rich layer above it expands and cools. Thebase of the convective envelope, which is initially above
the hydrogen-helium discontinuity, moves inward in
mass, extending eventually into the helium layer above
the CO core. Fresh 4 He and 14N (into which most of theprimordial CNO elements have been converted duringhydrogen burning) are dredged into the convective envelope 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 9–11 M ( , but the phenomenon is much more complex since the CO core is only
partially electron degenerate and both carbon-burningand helium-burning shell flashes occur; the flux of energy that reaches the base of the convective envelope
and forces this base to move inward in mass can have its
origin in carbon burning, helium burning, the release of
gravothermal energy, or in some combination of thesesources. In a 9M ( model of solar initial composition, thedredge-up occurs in conjuntion with the first carbonshell flash and gravothermal energy is responsible fordredge-up (Garcı́a-Berro et al., 1997). In a 10M ( model,
dredge-up occurs near the end of the carbon-burning
phase, and both helium burning and carbon burning
contribute to the dredge-up (Ritossa et al., 1996). In a
10.5M ( model, dredge-up occurs also near the end of
the carbon-burning phase, and all three energy sourcesplay a role in the dredge-up process (Iben, Ritossa, andGarcı́a-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

Wallerstein et al.: Synthesis of the elements

stars and experience thermal pulses due to helium shell
flashes (Schwarzschild and Härm, 1965; Weigert, 1966)
which periodically interrupt the normal, quiescenthydrogen-burning phase. Although their lifetime is only
;105 –106 years (depending on the initial mass), thermally pulsing AGB (TPAGB) stars produce much of the
12
C 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 proton burning at the expense of the 12C that enters theconvective envelope following helium-shell flashes in the‘‘third 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 into
which the core eventually evolves. Helium shell flashes
occur for the same reason that white dwarfs accreting
matter at a rate smaller than is necessary to support
quiescent helium burning experience nova explosions
when the mass of accreted helium reaches a criticalvalue which is larger, the smaller the accretion rate.
During the quiescent hydrogen-burning phase in AGB
stars, helium is deposited into a layer above the underlying CO or ONe core at a rate which is an order ofmagnitude smaller than that necessary to maintain quiescent helium burning. As the helium layer grows in
mass, it becomes compressed and heated until eventually helium ignites. The helium-burning rate is proportional to about the 40th power of the temperature andnuclear energy is released at a rate faster than the heat
into which it is converted can diffuse out radiatively. A
brief thermonuclear runaway continues until it isdamped out by expansion and cooling.
The large fluxes generated during a flash create a convective zone that extends from the base of the heliumburning region almost to the hydrogen-helium discontinuity. An entropy barrier, to which radiation pressure
contributes 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 up
contains 14N at an abundance equal to that of primordial
CNO elements. During the first part of a flash, this 14N
is converted completely into 22Ne by the reactions 14N( a , g ) 18F( b 1 , n ) 18O( a , g ) 22Ne. In models in which thecore mass is i0.9M ( , temperatures at the base of theconvective zone can reach over 3503106 K and the reaction 22Ne( a ,n) 25Mg can be a potent source of neutrons (Iben, 1975a, 1976, 1977a; Ritossa et al., 1996) forthe production of s-process isotopes at abundances several hundred times larger than solar (Iben 1975b; Truran
and Iben, 1977). It is interesting that B 2 FH speculated
that the 21Ne( a ,n) 24Mg reaction might be an importantneutron source in stars and did not consider the 22Ne
( a ,n) 25Mg reaction (Cameron, 1961).
In AGB models with CO cores less massive than
;0.9M ( , temperatures do not become large enough at
the 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
Rev. Mod. Phys., Vol. 69, No. 4, October 19971005

isotopes are made in abundance in such stars and thelikely neutron source is the 13C( a ,n) 16O reaction (Cameron, 1955). The manner in which the 13C source is activated may not be unique and it is not completely understood.
Schwarzschild and Härm (1967) found that, in one ofthirteen pulses computed, the convective shell actuallyreached the hydrogen-helium discontinuity and ingested
some hydrogen. Sanders (1967) pursued the consequences of this, finding that the ingested protons reactwith the 12C present at large abundance in the convective shell to produce mostly 13C which diffuses convectively inward until reaching temperatures of the order of;1503106 K, whereupon a -n reactions provide neutrons for s-process nucleosynthesis. The difficulty withthis scenario is that the Schwarzschild and Härm (1967)
models did not include radiation pressure, and the ingestion mechanism has not been found in subsequent calculations which include radiation pressure. Another set
of calculations (Iben and Renzini, 1982a, 1982b; Hollowell 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 1 H to overlap atcomparable number abundances in a region centered at
the location of the outer edge of the convective shell at
its maximum extent during the flash. Ultimately, matterin this region contracts and heats, and a small pocket of13
C is formed in consequence of proton capture on 12C
followed by a beta decay. When the next shell flash occurs, 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 13C
pocket is formed, it may act as a local neutron source
during the quiescent hydrogen-burning phase between
flashes (Straniero et al., 1995), rather than as a distributed source in a convective shell during flashes; surprisingly, the s-process abundance distributions produced
are essentially the same in both cases (Straniero et al.,1995). Still more recently, Blöcker et al. (1997) and
Herwig et al. (1997) have shown that convective overshoot beyond the base of the convective envelope during the third dredge-up phase leads to the formation of a
13
C pocket and this, in retrospect, could have been recognized 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, increasing the flux of energy passing through the base of
the convective envelope. This increase forces the base of
the convective envelope to move inward in mass into theregion which contains the products of partial helium
burning. This event was first encountered in an AGB
model of large CO core mass without invoking overshoot at the base of the convective envelope (Iben,
1975a), and later in AGB models of small core mass
with the help of convective overshoot (Iben and Renzini, 1982a, 1982b; Iben, 1983). More on the history of1006

Wallerstein et al.: Synthesis of the elements

this phenomenon may be found in reviews by Sackmann
and Boothroyd (1991) and Iben (1991b).
Helium shell flashes and the third dredge up are responsible for the formation of carbon stars (stars inwhich carbon is more abundant than oxygen) and for
producing overabundances of s-process isotopes in suchstars. In addition to the enhancements which occur during the first and second dredge-up phases, a further enhancement of 14N can occur as a consequence of theburning of 12C into 14N at the base of the convective
envelope between flashes in massive AGB stars (e.g.,
Iben, 1975a; Renzini and Voli, 1981). The freshly synthesized 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 of
the interstellar medium in these elements and isotopes(Iben and Truran, 1978), as well as in 4 He (Renzini andVoli, 1981), although the Big Bang is by far the major
contributor of this latter isotope. It is worth noting thatthe material which is dredged up during the TPAGB
phase has experienced only partial helium burning, withthe abundance by mass of 12C in dredged-up matter being only ;0.15–0.25. This means that the uncertainty inthe cross section for the 12C( a , g ) 16O reaction relative
to 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 3 He and 7 Li are made in intermediate mass stars in interesting quantities (relative to the
Big Bang contribution of these isotopes) and returned tothe interstellar medium during the superwind phase.
3
He 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 ;4.631024 M ( /M MS(Iben, 1977b). This 3 He is preserved and returned to theinterstellar medium by AGB stars with initial masses atthe low end of the intermediate mass range (1–2 M ( ), as
is confirmed observationally by the high 3 He/H ratio inplanetary nebulae (Balsar et al., 1997) which have progenitor masses less than ;2M ( . However, the low
12 13
C/ C ratios in small-mass red giants of old galactic
clusters, red giants in globular clusters, and extremely
metal-poor giants in the field (see Sec. VII) suggest that3
He should be depleted in a substantial fraction of theirmass. Hogan (1995) has shown that this implies that theobserved 3 He in the Galaxy may be less than the primordial 3 He + D. In higher mass AGB stars, however,3
He burns with 4 He at the base of the convective envelope to form 7 Be. Cameron (1955) and Cameron andFowler (1971) suggested that, if this 7 Be could be mixed
outward to cooler regions where it would be destroyedby the 7 Be(e 2 , n ) 7 Li reaction rather than by the7Be(p, g ) 8 B* (2 4 He) reactions, this might explain theobserved superabundances of Li in some Galactic giants(Wallerstein and Conti, 1969; Boesgaard, 1970). Of
course, the mixing must also be on a long enough timescale to prevent the rapid destruction of 7 Li by the7
Li(p, a ) 4 He reaction. Scalo et al. (1975) demonstrated
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

that this mechanism could work in the convective envelope of intermediate mass AGB stars. Smith and Lambert (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 overabundance 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 Boothroyd (1992) have constructed evolutionary models ofstars in the 3–7 M ( range and find that (with appropriate choices of parameters in a time-dependent convective mixing algorithm) the observed lithium abundancesand luminosities of the LMC super lithium rich stars canbe reproduced by models of initial mass in the range
4–6 M ( .
4. The born-again AGB phenomenon

There are a number of observed stars and stellar systems that can be understood in terms of a final helium
shell flash which a post-AGB star may experience after
it has ceased to burn hydrogen. The phenomenon waspredicted by Fujimoto (1977), encountered by Schönberner (1979), and exploited by Iben et al. (1983) and
Iben (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 s21 away from, thecentral stars of the planetary nebulae Abell 30 (Hazard
et al., 1980) and Abell 78 (Jacoby and Ford, 1983).
When the flash occurs, the mass of the helium layer
below the hydrogen-helium discontinuity is slightly
smaller than that necessary for initiating a flash on the
AGB. This is because the matter in the helium layer ispartially degenerate and is heated by adiabatic contraction when the hydrogen-burning shell loses its power.
The final flash is a strong one, and because the entropy
barrier between hydrogen-rich matter and the helium
layer is much smaller than on the AGB, the outer edgeof the convective shell forced by helium burning extends
into 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 convective shell breaks into two parts, the lower shell being
forced by fluxes from helium-burning reactions and theouter one being forced by fluxes from hydrogen-burning
reactions (Iben and MacDonald, 1995).
The outer convective shell consists primarily of 4 He
(76% by mass) and 12C (20% by mass), with only a few
percent by mass of 1 H, and a trace of 16O. The hydrogen abundance is so low because, prior to the flash, themass of the hydrogen-rich layer is about 20 times smaller
than 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 photosphere) progresses, approximately 5% of the 12C is converted into 14N by proton burning. Thus, this mode ofevolution produces surface abundances that are the consequence of mixing products of partial helium burningwith a small amount of hydrogen-rich matter and sub-

Wallerstein et al.: Synthesis of the elements

jecting the mix to another bout of hydrogen burning.
The track of the model that produces the quoted abundances is shown by the dotted curve in Fig. 1. The factthat the track extends to large luminosities and low surface temperatures has led to the designation ‘‘born again
AGB’’ stars. Perhaps 10% of all post-AGB stars may
experience a final helium shell flash after hydrogenburning is completed (Iben, 1984). Another 15% may
experience a final helium shell flash before hydrogenburning is finished. These too become born again AGB
stars, but an entropy barrier prevents the ingestion ofhydrogen into the convective shell produced by fluxes
from the helium-burning region. Wind mass loss during
the born again phase and thereafter may remove most if
not all of the hydrogen rich surface layer, and the star
may evolve into a ‘‘non-DA’’ white dwarf (surface abundance of hydrogen u1024 by mass). There are now
about a half dozen examples of stars that have been
identified as passing through the born-again phase, thelatest being Sakurai’s object (Duerbeck and Benetti,1996).
5. Other mixing processes, and wind mass loss, which
affect surface composition

Mixing processes other than the standard dredge-up
processes — thermal and gravitational diffusion, convective overshoot at radiative-convective boundaries, androtation-induced mixing — can in several instances affect the surface composition of single stars. Convective
overshoot 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 6400–7000 K
range (Boesgaard and Trippico, 1986) demonstrates thatlithium is diffusing into the interior of the gap stars. It is
possible that a combination of gravitational and thermal
diffusion inward through the base of a shallow convective envelope combined with the effects of radiative levitation (Michaud, 1986) may account for a reduction in
the surface Li abundance, but the fact that the surface
abundance of lithium in subgiants in the old cluster M67
decreases with distance from the main sequence indicates that a fair fraction of the lithium diffusing inward
must also be destroyed and this may imply rotationinduced mixing to temperatures large enough thatlithium is destroyed (Deliyannis, King, and Boesgaard,
1996).
Wind mass loss can also influence surface composition. For example, a radiative wind may remove thehydrogen-rich surface layer of a massive enough postAGB 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 lightest
isotopes 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 surface
abundance distribution. Several of these processes areprobably contributing to the observed variation in theRev. Mod. Phys., Vol. 69, No. 4, October 19971007

ratio of DA white dwarfs (hydrogen-rich spectrum) to
non-DA white dwarfs (hydrogen-deficient spectra) as afunction of luminosity (MacDonald, 1992).
C. Evolution of massive single stars that produce neutronstars 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 helium produced in the Galaxy. Lighter stars contributeprimarily carbon, nitrogen, and s-process isotopes. Oneof the major features of massive star evolution prior to
the supernova explosion is mass loss via a strong radiative wind (Cassinelli, 1979). The mass-loss rate increases
with increasing initial mass and luminosity and continues
as the star evolves, with mass-loss rates becoming aslarge as several times 1025 M ( yr 21 . For the initiallymost massive stars, say i30M ( , the mass-loss timescale is comparable to or shorter than the nuclearburning 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 Livermore opacities (Iglesias and Rogers, 1996) and incorporating mass loss according to various algorithms havebeen constructed by several groups for a variety of initial compositions (e.g., Schaller et al., 1992; Schaerer
et 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 which
has experienced complete hydrogen burning and, then,matter that has experienced partial helium burning.
These ‘‘stripped’’ stars are known collectively as WolfRayet (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 an
important study of the structure of WR stars, Langer(1989a, 1989b) compares models with the observations
to determine that the core mass M WR of WR stars falls
in the range 5M ( ,M WR,20M ( , independent of initialmass, that the mass-loss rate can be approximated byṀ WR5(0.621.0)31027 (M WR /M ( ) 2.5 M ( yr 21 , andthat the luminosity L WR satisfies 4.8,logLWR /L ( ,5.5.
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 Galaxy
with luminosities greater than ;106 L ( and with surface
temperatures less than T e;(1–3)3104 (Humphreys,
1978, Humphreys and Davidson, 1979, 1994) has beenargued by De Jager (1984) to be the consequence of an
instability in the photosphere, which sets in for stars
brighter than the observed luminosity limit. The instability leads to dissipation of mechanical energy and thedevelopment of supersonic turbulent motions, which re-1008

Wallerstein et al.: Synthesis of the elements

sult in a dramatic increase in the mass-loss rate withincreasing luminosity. Stothers and Chin (1993a, 1993b,
1996) suggest another mechanism of enhanced mass loss
involving a classical ionization instability in the stellar
envelope. In both cases, enhanced mass loss inhibits
evolution into the Humphreys-Davidson forbiddenzone. Whatever the source of the instability, the observations show that single stars initially more massive than
;50M ( do not become giants before experiencing a supernova explosion, and this has important ramifications
for massive close binary star evolution.
It is clear that massive stars interact with the interstellar medium both as energy sources and sources of isotope enrichment prior to the supernova explosion thatterminates their lives. Some aspects of this interaction
are 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 theoretical estimates of the nucleosynthetic yields of massive stars (e.g., Maeder, 1992; Timmes et al., 1996;
Thielemann et al., 1986; Arnett 1996), but uncertainties
still remain (Arnett, 1995), particularly with regard to
the extent of mass loss prior to the supernova explosion
and with regard to the dividing line in initial mass between those stars which form BHs and therefore do not
return iron-peak elements to the interstellar medium
and those which form NSs and do (Maeder, 1992;
Timmes et al., 1996).
After the core carbon-burning phase, the chemically
evolved interior of massive stars follows one of two evolutionary paths, depending on the initial stellar mass andcomposition. For solarlike initial composition, models of
initial mass in the 11 – 13 M ( range experience core collapse initiated by electron capture before they form an‘‘Fe-Ni’’ core (see below); in more massive models, corecollapse is initiated after the formation of the Fe-Ni
core. A model star of mass 11M ( is just at the borderline between stars that become TPAGB stars with stableelectron-degenerate cores of ONe and stars that immediately form electron-degenerate ONe cores massive
enough that electron captures on 20Ne, 23Na, and 24Mg
trigger a rapid contraction that cannot be halted even by
explosive O and Ne burning (Miyaji et al., 1980; Nomoto, 1984, 1987; Miyaji and Nomoto, 1987; Gutierrez
et al., 1996). Nuclear reactions convert the compositionof the core into iron-peak isotopes, and the Fe-Ni corecollapses dynamically to nuclear matter densities. The
real analog of the 11M ( model and real analogs of models slightly less massive (say i10.75M ( ) probably live
long enough as AGB stars that the ONe core becomes
massive enough and therefore dense enough to evolve
into 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
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

are essentially decoupled. Contraction and heating of
the 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 supernova ejecta contains 56Ni, which beta decays into 56Co,
which in turn beta decays into 56Fe. Associated gamma
emission from the decay of excited nuclear levels helps
power the light curve. Analysis of the light curve of
SN1987a in the Large Magellanic Cloud (Arnett et al.,1989 and references therein) suggests that ;0.1M ( of56
Fe is ejected into the interstellar medium when a starof initial mass ;20M ( explodes, showing that SNeII are
potent sources of iron in the Universe. That they mustalso be major sources of other heavy elements follows
from detailed models of explosions set off in the envelopes of evolutionary models that have been carried tothe 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 yet
emerged, although the evolution of the final detachedcore is reasonably well understood (Burrows and Lattimer, 1986, 1987). The details of how energy is transferred from the core in such a way as to cause expulsionof the stellar envelope and a quantitative estimate of
how much matter falls back onto the core are not yet
known theoretically. Thus, a secure theoretical mapping
between initial main-sequence mass and final NS or BH
mass is not presently available (although, see Timmes
et al., 1996 for an encouraging effort), and the critical
initial mass M crit which separates stars that form NS
remnants from those that form BH remnants is notknown. However, an understanding of massive close binary evolution requires this information. In order tomake progress, a concrete choice must be made, and, inthe following, it will be assumed that M crit540M ( ,M NS51.4M ( , and M BH510M ( . The choice of 1.4M (for the gravitational mass of a typical neutron star is notinconsistent with the average value of 1.3560.27 estimated by Thorsett et al. (1993) for 17 NSs in binary systems.
Historically, supernovae have been classified primarily on the basis of their spectral features, with type I
SNe (SNeI) being hydrogen deficient and SNeII exhibiting hydrogen lines. The SN light curve has in recent
years become important in identifying the nature of the
explosion. Comparison between spectral features andlight curves of theoretical models of explosions (see,e.g., Wheeler et al., 1995) suggest that SNeII are the end
result of the evolution of massive stars that retain theirhydrogen-rich envelopes and explode as red or blue supergiants (e.g., SN1987A), and that SNeIb,c are the result of the evolution of stars that become WR stars before exploding. In close binaries, the primary may

1009

Wallerstein et al.: Synthesis of the elements

become a WR star in consequence of Roche lobe overflow, enlarging the range in initial mass of stars that produce SNeIb,c. SNeIa may involve the explosion of CO
white 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 component
evolves essentially as a single star for its entire life. The
other half interact by mass transfer and/or mass loss
from the system and evolve into single stars (in consequence of mergers), sometimes with unusual surface
compositions (e.g., R CrB and other hydrogen deficientstars), or into exotic binary systems with interesting andoften bizarre characteristics such as cataclysmic variables, x-ray binaries, and close white dwarf pairs.
1. Modes of mass transfer and of orbital angular momentum
loss

For a binary in a circular orbit, when the gravitational
potential is calculated in the rotating frame on the assumption that both components are point masses, there
is a unique equipotential surface consisting of two‘‘Roche’’ lobes (forming an ‘‘hour-glass’’ with a figureeight cross section) which touch at a point along the line
joining the stellar centers. The radius of a sphere having
the same volume as a Roche lobe is known as the Roche
lobe radius. When, in the course of evolution, the radiusof a binary component approaches and exceeds its
Roche lobe radius, mass is transferred from the ‘‘Roche
lobe filling’’ star to its companion.
When the two components are of comparable initial
mass and the Roche lobe filling star does not possess a
deep convective envelope, mass is transferred on a timescale which is comparable to or longer than the thermal
time scale of the accreting component. Under these conditions, mass is transferred ‘‘conservatively’’ (no mass islost from the system) and, when the mass of the accretor
exceeds that of the donor, orbital angular momentum
conservation requires that the orbital separation A increases with continued mass transfer.
When the donor is initially considerably more massive
than 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 cannot
adjust its structure to accomodate the proferred mass,
which forms an expanding hot blanket about the accretor. The transferred matter soon fills the Roche lobe of
the accretor and the situation can thereafter be described in terms of a ‘‘common envelope’’ (CE) (Paczyński, 1976) which consists of matter which has been
supplied by the donor. The matter in the CE is driven
away from the system by an ‘‘egg beater’’ frictional interaction between the imbedded stars and the CE material. The energy required to drive off CE material derives from the orbital binding energy and the orbit
shrinks. The efficiency of the CE mechanism may be
defined by a CE=DE remove /DE bind , where DE remove is theenergy required to remove all of the CE matter from theRev. Mod. Phys., Vol. 69, No. 4, October 1997

system and DE bind is the difference between the final
and the initial orbital binding energy. The smaller a CE ,the greater is the degree of orbital shrinkage. The valueof a CE appropriate for different situations has been debated for several decades (see, e.g., Iben and Livio,
1993, and references therein), but recent threedimensional smooth particle hydrodynamic calculations
suggest a value near unity (Razio and Livio, 1996; Yorkeet al., 1995).
Roche lobe filling can be achieved in consequence ofthe radius increase brought about by the nuclear evolution of a potential donor (e.g., Webbink et al., 1983;
Taam, 1983a), or in consequence of orbital angular momentum loss due to gravitational wave radiation (GWR)(Kraft et al., 1962; Paczyński, 1967; Faulkner, 1971), or
due to a magnetic stellar wind (MSW) (e.g., Verbunt
and Zwaan, 1981; Taam, 1983b). In order for GWR to
be important, A must be less than 2–3 R ( (depending
on component masses). In order for a MSW to be effective, the donor or the accretor must be a main-sequencestar in the approximate range 0.3–1 M ( . Reviews of
evolutionary processes in binary star evolution include
Paczyński (1971b) and Bhattacharya and van den Heuvel (1991).
2. Scenario modelingWith the help of the properties of evolving single stars
described in Secs. III.B and III.C, and using the principles just mentioned, one can construct scenarios forthe evolution of binary systems consisting initially of
main-sequence stars for various choices of the mass M 10of the primary, the mass ratio q 0 5M 20 /M 10 , the orbital
semimajor axis A 0 , and the eccentricity e 0 .
An approximation to the birthfunction for stars in the
Galactic disk which is based on an analysis of the properties of many binary systems in the literature (Kraicheva et al., 1978; Popova et al., 1982) is (Iben and Tutukov, 1984)
d 3 n ~ yr21 ! 50.2dlogA 0

dM 10
M 2.5
10

dq 0 ,(1)

where M 10 and A 0 are in solar units. Integrating Eq. (1)over A 0 5102106 , M 1050.82100, and q 0 =0–1 givesn ;1 yr 21 as the Galactic birthrate of stars that leave
the main-sequence in less than a Hubble time.
When mass transfer is deemed to be conservative,
conservation of orbital angular momentum gives

S D

S D

GA f 1/2
GA 0 1/2
J orb5M 1fM 2f
5M 10M 20
,
Mt
Mt

(2)

where the subscript f identifies system parameters after
the mass transfer event is completed and M t5M 101M 20= M 1f1M 2f . When CE evolution is invoked, one must
adopt some approximations for DE remove and DE bind in
the definition of a CE . For example, for large q 0 , a roughapproximation is

a CE;

S DYS
GM 210
A0

D

GM 1fM 20
M 10 M 10 A f
5.
Af
M 20 M 1f A 0

(3)

1010

Wallerstein et al.: Synthesis of the elements

Since both M 20 and M 1f are small compared to M 10 , it isclear that orbital shrinkage can be considerable in a CE
event. Equations (1)–(3), in conjunction with propertiesof evolving stellar models (including externally imposed
mass loss at a high rate), have been used to reproduce(with a CE ;1) empirically estimated birthrates of a variety 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 references therein). In systems with orbital periods in the
1.3–2 h range, mass transfer is driven by angular momentum loss due to GWR at a rate (1–2)310210 M (yr21. In CVs with periods in the range 3–20 h, masstransfer is driven by a MSW at a rate in the range 1029 –1028 M ( yr 21 . Systems born into the long period category evolve to shorter periods as mass is transferred
until P orb;3 h, whereupon the main-sequence component becomes completely convective and no longer supports 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 decreased to the extent that the main-sequence star again
fills 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 produces
a flare up called a dwarf nova outburst. The light emitted in such an outburst is due to the release of gravitational potential energy. In the absence of differential rotation, gravitational and thermal diffusion would mix
hydrogen-rich accreted material with white dwarf material near the base of the accreted layer (Prialnik andKovetz, 1984). However, matter from the disk reaches
the surface of the white dwarf with velocities near the
Keplerian velocity at the surface of the WD and angularmomentum diffuses inward through the accreted layer
and beyond, creating a radial gradient in the angularvelocity. Differential rotation gives rise to a baroclinic
instability (Fujimoto, 1988, 1993) which causes evengreater mixing between accreted and white dwarf material than does particle diffusion (Fujimoto and Iben,
1997). When the white dwarf has accreted enough material (e.g., ;1025 M ( when Ṁ WD;1029 M ( yr 21 ), hydrogen is ignited near the base of the accreted layer anda convective zone extends inward (to the point in the
mixing region where the abundance by mass of hydrogen is initially ;0.01) and outward (to the surface), mixing large quantities of CO or ONe material outward.
Conversion of nuclear energy into heat lifts the electron
degeneracy in the hydrogen-rich layer, and enough energy is injected to force this layer to expand to giant
dimensions (e.g., to the right along the dash-dot curve inRev. Mod. Phys., Vol. 69, No. 4, October 1997

Fig. 1 found in a model calculation that neglects thecompanion [Iben, 1982]). In the real analog, called aclassical nova, the expanding matter extends far beyond
the Roche lobe of the white dwarf and a combination of
CE action (e.g., MacDonald, 1980, 1986; Livio et al.,1990), wind mass loss (e.g., Kato, 1983; Kato and Hachisu, 1994), and, in some cases, dynamical acceleration(e.g., Starrfield et al., 1974, 1978) removes most of the
expanding layer. The remnant hydrogen-rich, heavy element rich layer contracts onto the white dwarf and, inmodel calculations, the star moves to the left in the H-R
diagram in Fig. 1 along the dash-dot track at radii
smaller than ;1R ( .
Equations (1)–(3), with a CE ;1, have been used toestimate that CVs are born at the rate ;0.001 yr 21(Iben and Tutukov, 1984). When observational selection
effects are taken into account, this is not inconsistent
with the birthrate estimated from the observed distribution of CVs in the solar vicinity (Trimble, 1982). The
relative numbers of short period and long period CVs
can be understood in terms of the time scales for masstransfer driven by GWR and a MSW and by observational selection effects (e.g., Iben et al., 1991).
The maximum brightness of a classical nova is directly
related to the mass of the white dwarf according to therelationship L max /L ( 546,000(M WD /M ( 20.26) (Iben
and Tutukov, 1989, 1996a), which, for M WDiM ( , issimilar to the Paczyński-Uus (Paczyński, 1970; Uus,
1970) relationship, L max /L ( 559,000(M WD /M ( 20.52),
that is applicable to central stars of PNe. The difference
between the two relations is due to the fact that theunderlying white dwarf in the nova case is colder andtherefore has a smaller radius than in the PN case.
The dependence of L max on M WD provides a strong
observational bias for novae with massive white dwarf
cores and explains why the progenitor of a typical observed nova is a main-sequence pair with a fairly massive primary. The requirement that the main-sequence
donor be less massive than the white dwarf by a factor of;2/3 (to ensure stable mass transfer) means, then, thatq 0 !1. This, in turn, means that mass transfer in the progenitor binary is such that a CE is surely formed andthat orbital shrinkage will, in many instances, force themain-sequence component to be drawn by a MSW or
GWR into Roche lobe contact in less than a Hubble
time.
b. White dwarf mergers: R CrB stars and type Ia
supernovae

R Corona Borealis (R CrB) stars and SNeIa are mosteasily explained in terms of the merger of two whitedwarfs. In the case of SNeIa, one of the most compellingarguments for implicating binary stars is the fact that
SNeIa occur in elliptical galaxies that are primarily
made up of old stars that are not by themselves massive
enough to become SNe. SNeII do not occur in ellipticalgalaxies, showing that the formation of massive stars is
not now taking place, and various observations tell us
that, in our own Galaxy, single stars less massive than
;10M ( evolve into white dwarfs.

Wallerstein et al.: Synthesis of the elements

A natural way of accounting for these facts is to suppose that, during an earlier period of active star formation, two fairly massive intermediate mass mainsequence stars evolve through two CE episodes into apair of white dwarfs of total mass larger than the Chandrasekhar mass (;1.4M ( ) and at a separation smallenough that angular momentum loss due to GWR
brings the two together in less than a Hubble time until
a merger followed by an explosion occurs (Iben and Tutukov, 1984; Webbink, 1984). The details of how thiscomes about are not resolved (see, e.g., Iben, 1997). If,
when the less massive white dwarf overfills its Rochelobe, the mass ratio of the white dwarfs is less than 2/3,
mass transfer is stable (Tutukov and Yungelson, 1979).
Otherwise, the less massive white dwarf is transformed
on a dynamical timescale into a thick disk around themore massive white dwarf (Tutukov and Yungelson,
1979; Iben and Tutukov, 1984; Webbink and Iben, 1987;
Benz et al., 1990; Mochkovich and Livio, 1990). If theangular momentum in the disk cannot diffuse outward
and be carried away in a wind rapidly enough (Mochkovich and Livio, 1990), carbon burning may convert theinterior into an ONe composition (Nomoto and Iben,
1985; Saio and Nomoto, 1985) and the final outcome
may be collapse into a neutron star. If angular momentum can diffuse outward fast enough, a collapse will befollowed by a star disrupting explosion, which converts
much of the star into iron-peak elements moving withvelocities of ;104 km s 21 , similar to those observed for
SNeIa explosions.
There is an abundance of theoretical calculations
showing that, when a thermonuclear explosion is set off
at the center of a CO white dwarf model, the resultant
spectral distribution reproduces many of the observed
features of SNIa spectra (e.g., Nomoto et al., 1984,1997),
the theoretical light curves can be made to match observed light curves by varying parameters in the theory(e.g., Höflich and Khokhlov, 1996; Höflich et al. 1996),
and the isotope mix produced in the explosion can help
explain the abundances of several isotopes that are notproduced in supernovae of other types (e.g., Thielemannet al., 1997). Most extant simulations of SNeIa explosions do not address the physics of the merger process,but assume rather that the CO WD accretes matter quiescently from a presumably hydrogen-rich companion.
However, the merger configuration is highly nonspherical, carbon may ignite off center in many places at thesame time, and the physics of burning under turbulent
conditions is complex (e.g., Niemeyer et al., 1996). It will
be interesting to see what new features are revealed
when three-dimensional models of the merging process
are constructed with adequate resolution and realistic
physics.
On the scenario front, the estimate of 0.003 yr 21 forthe formation frequency of double CO white dwarfs in
our Galaxy, which have a total mass larger than 1.4M (and which are close enough to merge in a Hubble time
(Tutukov et al., 1992), is almost identical with empiricalestimates of the SNIa birthrate (Baade and Minkowski,
1957; van den Bergh et al., 1987; van den Bergh andRev. Mod. Phys., Vol. 69, No. 4, October 1997

1011

Tammann, 1991). On the observational front, eight systems consisting of two close white dwarfs have been discovered and at least three of these pairs are at an orbital
separation such that a merger will occur in less than a
Hubble time. In most of the observed systems, the visible star is a helium WD. The distribution in number
versus period of the observed systems is consistent withscenario predictions, and it is predicted that the observed sample of close binary WDs must be increased byat least a factor of 4 before the merging CO WD scenario for SNeIa begins to be seriously challenged or possibly strengthened (Iben, Tutukov, and Yungelson,
1997).
Other possible progenitors of SNeIa include supersoft
x-ray sources that are thought to be white dwarfs accreting matter at a high rate from a Roche lobe filling mainsequence star of mass 1–2 M ( (van den Heuvel et al.,
1992; Di Stefano et al., 1997). The high rate of masstransfer prevents nova explosions, but wind mass loss
from the system is important and the hydrogen-rich matter in the system must somehow be hidden if these systems evolve into SNeIa.
R CrB stars may also be explicable in terms of binary
star evolution (Webbink, 1984; Iben and Tutukov,
1984a, 1985; Iben and Tutukov, 1996a, 1996b). They arebright ( ^ L & ;73103 L ( ) and cool (average T e;7,000K) with extreme hydrogen-deficient spectra and strong
overabundances of carbon (Bidelman, 1953) and nitrogen (Rao and Lambert, 1996, and references therein).
Single star models of the born-again variety, while developing appropriately exotic surface compositions(Renzini, 1990; Iben and MacDonald, 1995), possibly do
not live long enough as cool bright stars to account formost of the estimated 200–1000 R CrB stars in the Galaxy (Lawson et al., 1990). The problem is that theamount of fuel available for nuclear burning is inversely
proportional to a high power of the mass of the underlying white dwarf core but the luminosity and radiusachievable are directly proportional to core mass.
In the binary scenario, the immediate precursor of an
R CrB star is a CO WD and a He WD at a separation
small enough that a merger due to GWR will occur inless than a Hubble time. The idea is that, when it fills its
Roche lobe, the lighter He WD will spill over onto the
CO WD, and turbulent convective motions will mix carbon and oxygen from the CO WD into the helium andnitrogen of which the He WD is composed. Helium
burning at the base of the helium-rich convective envelope of the merger product increases the mass of the COcore and, if the mass of the merger product is large
enough (i0.78M ( ), the star may evolve into the domain in the H-R diagram occupied by R CrB stars when
the CO core mass exceeds (;0.56–0.61 M ( ).
The fuel available in the helium envelope for burning
at high luminosity is ;20 times larger than in the bornagain scenario, and the predicted presence of 200–600
bright and cool systems in the Galaxy at any given time
is consistent with the observational estimate of 200–
1000. The most important task in pursuing the binary
scenario is the construction of 3-D hydrodynamic mod-

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Wallerstein et al.: Synthesis of the elements

els of the merging process to see if compositional mixing
occurs to the degree necessary to explain the observed R
CrB compositions. The technology for exploring thisquestion is advancing rapidly (e.g., Bazan and Arnett,
1994; Burrows et al., 1995; Livne and Arnett, 1995; Glasner and Livne, 1995) and one may anticipate resolutionof this question in the not too distant future. On theobservational and scenario fronts, there are two known
white dwarf pairs (WD 2331+290 and WD 0957-666)
that will merge in less than a Hubble time and that havemass estimates that do not preclude evolution into theconfigurations envisioned here (Iben, Tutukov, andYungelson, 1997).
c. X-ray binaries and pulsars

Approximately half of all massive stars are in binaries
close enough that the primary fills its Roche lobe andloses its hydrogen-rich envelope before developing an
Fe-Ni core that evolves into a NS or BH. The nucleosynthetic yield of such stars is quite different from thatof single massive stars, and this difference must be taken
into account in constructing models of Galactic nucleosynthesis. Furthermore, much of what we know about
NSs and BHs comes from a study of close binaries inwhich at least one of the components is an accreting NS
or BH (Novikov and Zeldovich, 1966; Shklovski, 1967).
Low mass x-ray binaries (LMXBs) consist of a NS or
BH and either a low mass main-sequence donor or a
subgiant donor with an electron-degenerate helium core
(Bhattacharya and van den Heuvel, 1991, and references
therein). The donor transfers matter to the relativistic
component because it either fills or is close to filling its
Roche lobe. Conventional wisdom has it that the donor
fills its Roche lobe and that mass transfer is driven by
angular momentum loss due to a MSW (if the donor is amain-sequence star) or by radial expansion of the donor
due to interior nuclear transformations (if the donor is a
subgiant). In this picture, the mechanisms of mass transfer in a CV and in an LMXB are identical and differences between the two types of system stem primarily
from the huge difference (factor of ;1000) in the potential energy of a particle at the surface of the relativistic
component in an LMXB relative to that of a particle atthe surface of the WD in a CV. In an LMXB, thenuclear energy liberated in a thermonuclear outburst isa small fraction of the potential energy liberated in theprocess of accretion between thermonuclear outbursts;
in a CV, the reverse is true. Thus, an LMXB is mostconspicuous when an accretion disk discharges onto therelativistic component, whereas a CV is most conspicuous during a classical nova outburst when the luminosity
is due to the release of nuclear energy.
In Fig. 1, the maximum luminosity (van Paradijs,
1995) of a number of LMXBs with NS accretors forwhich distances have been estimated (van Paradijs andWhite, 1995) are shown along a line of constant radius(1025 R ( ) to provide an indication of the large temperatures characterizing the observed spectrum. Shownalso are the maximum luminosities of LMXBs with BH
accretors according to Tanaka and Lewin (1995).
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

The progenitor of an LMXB is distinguished fromthat of a CV progenitor by the magnitude and nature of
the hurdles that it must surmount (van den Heuvel,
1985; Webbink and Kalogera, 1994). The net result isthat the theoretical birthrate of LMXBs with mainsequence or subgiant donors and NS accretors is quite
small. For example, one estimate of this birthrate isn ;231025 yr 21 when a CE=1 and ;231026 yr 21 whena CE=0.5 (Iben and Tutukov, 1995). With the choices
M BH510M ( and 40,M 10 /M ( ,50, systems with BH
accretors are born approximately 20% as frequently as
the corresponding systems with NS accretors.
The NS in an LMXB accretes mass from the base ofan accretion disk at essentially the Keplerian velocity at
the surface of the NS (Alpar et al., 1982), causing the NS
to spin up until the counter torque exerted by a magnetic field coupled to ionized accreted matter balances
the torque supplied by the accreted matter (Gosh et al.,
1977; Chen and Ruderman, 1993). Accretion of only
;0.01M ( is sufficient to decrease the spin period into
the millisecond range (u1031023 s). In LMXBs withmain-sequence donors, the donor is probably destroyedby absorption of particle and photon emission from therapidly spinning NS, which then evolves into a single
millisecond pulsar (MSP). In LMXBs with subgiant donors, the electron-degenerate core survives as a helium
WD and the final system is a binary MSP. Comparison
between a measured beat period and the Keplerian period permits one to infer the spin period of the NS in an
LMXB (Alpar and Shaham, 1985; Lamb et al., 1985;
Gosh and Lamb, 1992), and this has been accomplishedin three LMXBs (Strohmayer et al., 1996; Morgan andSmith, 1996; Ford et al., 1997).
High mass x-ray binaries (HMXBs) consist of a NS or
BH which accretes from the radiative wind emitted by amassive OB star which does not fill its Roche lobe (vanden Heuvel and Heise, 1972; Tutukov and Yungelson,
1973). In many instances, the main-sequence components in the primordial binary are of comparable mass,
so that the first mass transfer event is conservative, withthe orbital separation of the binary increasing during thetransfer and the secondary being spun up in the process(Rappaport and van den Heuvel, 1982).
Because the secondary in the progenitor system ismassive to begin with, and the primary evolves into ahelium star much less massive than the secondary, massloss during the SNIb,c explosion of the helium star does
not in general disrupt the binary system. The duration of
the bright x-ray stage is but a small fraction of the mainsequence lifetime of the donor (Massevich et al., 1979;
Savonije, 1979), and this accounts for the observed paucity of bright HMXBs (there may be only ;100 with
L X.103 L ( in the Galaxy).
Eventually, the OB component in an HMXB evolves
to fill its Roche lobe, lose most of its hydrogen-rich envelope in a CE event, and explode as a SNIb,c. The
binary system is, in general, disrupted. The typical final
result is two high velocity NSs with spin rates of ;3 s or

1013

Wallerstein et al.: Synthesis of the elements

less. Of course, some few remain bound, as is demonstrated by the famous binary pulsar 1913+16 (Hulse andTaylor, 1975).
NSs formed in SNIb,c explosions in close binaries may
be the origin of most pulsars with spin period in the
0.01–3 s range. The first formed NS has a modest spin
rate because it was spun up by accretion from its OB
companion. The second NS has a modest spin rate because its progenitor was spun up during the first masstransfer episode. The birthrate of all SNeIb,c explosions
in the Galaxy is predicted to be ;0.007 yr 21 (Tutukov
and Yungelson 1993), compared with a theoretical birthrate of ;0.005 yr 21 for NSs formed by systems whichpass through the HMXB stage. These estimates are consistent with a semiempirical estimate of the pulsar birthrate of 0.004–0.008 yr 21 (Lorimer et al., 1993). On theother hand, the predicted birthrate of NSs formed in
SNII explosions of massive single stars (including thoseformed by a merger in a first CE event) and massive
stars in wide binaries is ;0.021 yr 21 (Tutukov and Yungelson, 1993), which agrees with the empirical estimateof the SNII birthrate (Baade and Minkowski, 1957; vanden Bergh and Tammann, 1991). Thus, one could inferthat pulsars are produced only by close binaries.
IV. HYDROGEN BURNING IN THE pp CHAIN AND CN
CYCLE1

S 11~ 0; d !
5S 11~ 0;0 ! BM69
3 ~ 11 d ! 2 ,A. The p(p,e 1 n e )d reaction

The rate of the p(p,e 1 n e )d reaction is determinedfrom the ratio of the weak interaction coupling constants, G A /G V , and may be related to the results of
Bahcall and May (1969) as
Rev. Mod. Phys., Vol. 69, No. 4, October 1997(4)

where L is the overlap integral between the two protons
and the deuteron (e.g., Bahcall and May, 1969);(GA /GV ) is the ratio of the axial-vector and vector weakRinteraction coupling constants; (f pp) is the phase-spacefactor for the pp reaction, (f pp ) 69=0.142 andR(f pp)50.144 (INT Workshop, 1997); @ (ft) 0 1 →0 1 # corresponds to the strength of 0 1 to 0 1 superallowed beta
decays; and d is the fractional correction to the nuclear
matrix element due to exchanges of p and r mesons.
Since 1980, there has been a marked increase in the precision of measurements of the neutron decay parameters(the neutron lifetime, as well as the b asymmetry parameter in the decay of polarized neutrons, and the e-n angular correlation parameter) (e.g., Freedman, 1990; andBarnett et al., 1996).
By making use of recent values for these parameters,
L 2 56.92 ~ 61% !
~ Kamionkowski and Bahcall, 1994),~ G V /G V ! 51.265460.0042

4

In 1957, the energetics of 4 H→ He fusion were wellestablished, and the two sequences for accomplishingthat fusion (the carbon-nitrogen cycle and the protonproton chain) were laid out. The rates of each of thefour reactions of the CN cycle had been measured directly (down to energies of '100 keV) to determine extrapolated values of S(0) with reported precisions of 15
to 30 %. But from the perspective of 1997, it may be
hard to realize what a mere thimbleful of information
about the rates of the reactions in the pp chain wasavailable to B 2 FH 40 years ago. The rate of thep(p,e 1 n e )d reaction (and hence the rate of the pp
chain) was 30% too small, as a result of uncertainties in
the weak-interaction coupling constants. (Measurementsof the related lifetime of the neutron were 25% toolong.) Initial measurements of the rate of the termination reaction, 3 He( 3 He,2p) 4 He, would turn out to be afactor of 4 too small (due to a problem with the 3 Hetarget), and the possibility of other terminations of the
pp chain via 7 Be and 8 B were not yet even discussed.
All of that changed very rapidly during the next 10
years.
Now, 40 years later, we present a summary of the current status of the most important of the nuclear reaction
rates in 41H → 4He fusion, emphasizing recent important developments and still open questions.

R! @~ ft ! 0 1 →0 1 # 69
L 2 ~ G A /G V ! 2 ~ f pp
2
2 ~f !
L 69 ~ G A /G V ! 69 pp 69 @~ ft ! 0 1 →0 1 #(Barnett et al., 1996;
INT Workshop, 1997),@~ ft ! 0 1 →0 1 # 53073.363.5 ~ Hardy et al., 1990),

d 50.01~ 0.004< d <0.03!
~ Bahcall and Pinsonneault, 1992! ,we find
S 11~ 0; d ! 53.78310225
3

S

D

S DS
6.92
7.08

DS D

1.2654 2 0.144
1.23820.142

3074~ 11.01! 2 MeV b.
3073.3(5)

Currently, the most important source of uncertainty inthis value is in d . The current range in d corresponds toa range in S 11(0;d ) from 3.94310225 to 4.15310225
MeV b (i.e., a +4.0%/21.3% uncertainty).
B. The 3 He( 3 He,2p) 4 He reaction

Measurements of the 3 He( 3 He,2p) 4 He reaction haverecently been extended down to E c.m.=24 keV (Krauss
et al., 1987) and E c.m.=21 KeV (Arpesella et al., 1996),
well into the Gamov peak corresponding to the center ofthe sun; for this reaction, a temperature of T 6 =15.6 corresponds to a Gamov peak at E o =22 keV with a width
DE o =12 keV. The most recent of these experiments was
carried out using the Gran Sasso tunnel as a cosmic-ray
shield. These two data sets were measured at energies
that are low enough so that interacting nuclei have a1014Wallerstein et al.: Synthesis of the elements

TABLE I. Experimental measurements of the 3 He + 4 He reaction.
Reference

S 34(0) (keV b)
0.47 6 0.05
0.58 6 0.07
0.45 6 0.06
0.52 6 0.03
0.47 6 0.04
0.53 6 0.03

Measurement of capture g rays
Parker and Kavanagh (1963)
Nagatani et al. (1969) a
Kräwinkel et al. (1982) b
Osborne et al. (1982,1984)
Alexander et al. (1984)
Hilgemeier et al. (1988)
Weighted mean=0.507 6 0.016

0.55 6 0.05
0.63 6 0.04
0.56 6 0.03

Measurement of 7 Be activity
Osborne et al. (1982,1984)
Robertson et al. (1983)
Volk et al. (1983)
Weighted mean=0.579 6 0.024

a

As extrapolated using the direct-capture model of Tombrello and Parker (1963).
As renormalized by Hilgemeier et al. (1988).

bclassical turning point which is near or outside theatomic radius R a and therefore electron-screening corrections are necessary in order to relate the laboratory(screened) measurements ( s s ) to cross sections which
would be measured for bare nuclei ( s b ) (e.g., Assenbaum et al.., 1987).

s b 5 s s /exp~ p h U e /E c.m.! ,(6)

been proposed as an explanation for the solar neutrino
problem; while there have already been a number ofnegative indirect searches (e.g., McDonald et al., 1977,
and references therein), the direct low-energy results ofKrauss et al. (1987) and Arpesella et al. (1996) now set
an upper limit on the energy (E c.m.,21 keV) for any
such resonance.

where
U e 5Z 1 Z 2 e 2 /R a ,(7)

and h is the Coulomb parameter. For example, for the3
He( 3 He,2p) 4 He reaction (U e =240 eV) at E c.m.=22
keV, this amounts to a 20% correction, while forE c.m..150 keV the correction is <1%. [This correction
to the laboratory measurements should not be confusedwith the plasma screening correction (Salpeter, 1954)
which must be applied in relating laboratory measurements of bare-nuclear cross sections to reaction rates inplasmas.] The results of Krauss et al. (1987) and Arpesella et al. (1996) are in agreement with each other andare consistent with earlier results at higher energies (seeBahcall and Pinsonneault, 1992, and references therein).
A combination of all of these results yields a value of
S33(0)=5.1560.35 MeV b, where there are comparable
contributions to this uncertainty from the statistical andsystematic uncertainties and the uncertainty in the energy dependence of the extrapolation. It is important tonote that as the Gran Sasso experiment continues, weare now rapidly approaching the time when the crosssection will be measured across the entire Gamow peak;at that time the rate of this reaction in the energy rangeof the Gamow peak will be determined directly from themeasured cross sections, and the issue of how to extrapolate from the higher-energy cross sections will nolonger be important.
A low-energy 3 He + 3 He resonance has occasionallyRev. Mod. Phys., Vol. 69, No. 4, October 1997

C. The 3 He( a ,g ) 7 Be reaction

The relative rates of the 3 He( a ,g ) 7 Be andHe( 3 He,2p) 4 He reactions determine what fraction of
pp-chain terminations result in high-energy ( 7 Be or 8 B)
neutrinos. There are six sets of measurements of thecross section for the 3 He( a ,g ) 7 Be reaction which are
based on detecting the capture gamma rays (Table I).
The weighted average of those six results provides avalue of S 34(0)=0.5076.016 keV b, based on extrapolations using the calculated energy dependence for thisdirect-capture reaction. Separate calculations of this energy dependence based on the resonating group method(Kajino et al., 1987) and on a direct-capture clustermodel (Tombrello and Parker, 1963) agree to within
61.25% and are in good agreement with the energy dependence of the measured cross sections. The energy
dependence of this reaction seems to be well determinedby resonating group models (as noted above) so that its
extrapolation to E50 seems well defined, and the only
free parameter is associated with the normalization ofsuch models to the measured data sets. While the energy
dependence of these models is in good agreement withthe energy dependence of the measured cross sections, itwould be useful to carry out a theoretical study of how
robust this energy dependence is for a wider range ofmodels.
3Wallerstein et al.: Synthesis of the elements

In addition to the six data sets of capture gamma ray
measurements, there are three sets of cross-section measurements for this reaction based on the activity of thesynthesized 7 Be (Table I). These decay measurements
have the advantage of determining the total cross section directly, but have the disadvantage that (since thesource of the residual activity cannot be uniquely identified) there is always the possibility that some of the7
Be may have been produced in a contaminant reaction
in some way that evaded tests for it. The three activity
measurements (when extrapolated in the same way as
the capture gamma ray measurements) provide a valueof S 34(0)=0.5796.024 keV b which disagrees with thevalue based on the direct-capture gamma rays at the 3s
level.
When the nine experiments are combined, theweighted mean is S 34(0)=0.53360.013 keV b, withx 2 513.4 for 8 degrees of freedom (d.o.f.). The probability of such a distribution arising by chance is 10%, andthat, together with the apparent grouping of the results
according to whether they have been obtained from activation or prompt-gamma yields, suggests the possible
presence of a systematic error in one or both of thetechniques. An approach that gives a somewhat moreconservative evaluation of the uncertainty is to form theweighted means within each of the two groups of data(the data show no indication of nonstatistical behavior
within the groups), and then determine the weighted
mean of those two results. Then an arbitrary but standard presciption can be adopted in which the uncertainties of the means of the two groups (and hence the overall mean) are increased by a common factor of 3.7 (inthis case) to make x 2 50.46 for 1 degree of freedom,
equivalent to making the estimator of the weighted
population variance equal to the weighted sample variance. The result is an overall weighted mean of S 34(0)=0.5360.05 keV b.
D. The 7 Be(p,g ) 8 B reaction

The 8 B termination of the pp chain is so weak(;.01%) that its details do not contribute significantly
to the energy generation of the sun. However, the relatively high-energy neutrino resulting from the 8 B beta
decay (E max
n '14 MeV) plays a key role in several of thesolar neutrino experiments (e.g., Homestake, Kamiokande, Super-Kamiokande, and SNO), so that the rateof the 7Be(p, g ) 8B reaction is a key to understanding theresults of those experiments. There have been six direct
measurements of this reaction using radioactive 7 Be targets. A reanalysis of the five most accurate experiments(Johnson et al., 1992; Langanke, 1994) and a renormalization to a 7 Li(d,p) cross section of 146 mb (Strieder
et al., 1996) determined a best value of S17(0)=20.362.0
eV b where the uncertainty includes a correction ofAx 2 / d.o.f.=1.32 to take into account the disagreement
among the five experiments (apparently arising fromunidentified systematic errors) and includes the uncertainty in the 7 Li(d,p) 8 Li cross section that was used as anormalization for all of those 7 Be(p, g ) 8 B experiments.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1015

More recently, at the INT Workshop on Solar Nuclear
Fusion Rates (1997), a more selective analysis was used
to cite the Filippone et al. (1983a, 1983b) experiment as
the most reliable of the direct measurements, resulting
in a recommended value of S17(0)519 14
22 eV b.
We now appear to be on the threshold of a series ofpromising new experiments to measure the rate of thisreaction using a variety of new techniques, such as 7 Be
beams, new 7 Be target strategies, and the Coulomb dissociation of 8 B. Measurements with these techniques
will have the advantage of avoiding some of the systematic uncertainties inherent in the previous (p, g ) measurements [e.g., questions relating to target uniformity
and the determination of the 7 Be content of the target
through the 7 Li(d,p) 8 Li reaction]. At the same timethese new techniques will have systematic uncertainties
of their own which will need to be understood in order
to interpret the results of these new experiments together with the previous (p, g ) results. Initial measurements of the Coulomb dissociation of 8 B have alreadybeen reported (Motobayashi et al. 1994), indicating that
their preliminary results are consistent with the (p, g )results of Filippone, et al. (1983a, 1983b) and are encouraging for further studies to provide an independent
determination of S 17(0). Recently, the NABONA collaboration has measured initial data utilizing a 7 Be
beam and a hydrogen gas target (Gialanella et al., 1996).
E. The 7 Li(n, g ) 8 Li reaction

This reaction is of interest both as the isospin mirrorof the 7 Be(p, g ) 8 B reaction and as the entrance to
A>12 nucleosynthesis in an inhomogeneous big bang.
Measurements by Wiescher et al. (1989) had indicatedthat the cross section for this reaction was a factor of 2
smaller than reported by Imhof et al. (1959), thereby reducing the rate of heavy element synthesis in any primordial r process. However, a number of more recent
measurements (Blackmon et al., 1996, and references
therein) are in good agreement with the Imhof (1959)
results supporting the reaction rate used in the calculation of heavy element production by Malaney andFowler (1987). It is perhaps interesting to note that thethree-cluster generator coordinate method calculationsof Descouvemont and Baye (1994), which are consistent
with the Imhof/Blackmon 7 Li(n, g ) 8 Li results, predict
S17(0)>24 eV b for the isospin mirror, 7 Be(p, g ) 8 B reaction.
F. The 14N(p, g ) 15O reaction

Recent measurements of the 14N(p, g ) 15O reaction(Schröder et al., 1987) show clear evidence for interference effects due to the high-energy tail of a subthresholdresonance (E c.m.52504 keV; J p 53/21 ), which accounts
for nearly half of the total S(0) for this reaction. As theslowest piece of the CNO cycle, the rate of this reaction
governs the rate of the cycle, and as such plays an important role in determining the evolution of massive

1016Wallerstein et al.: Synthesis of the elements

stars and the ages of globular clusters. Combining the
Schröder et al. (1987) measurements with the earlier
low-energy (93,E c.m.,126 keV) measurements of
Lamb and Hester (1957), determines S(0)=3.5+0.4
21.6 keV b(INT Workshop, 1997), where the uncertainty arises primarily from uncertainty (because of the interference effects due to the subthreshold resonance) in how to extrapolate the laboratory measurements to zero energy.
This uncertainty in the rate of the 14N(p, g ) 15O reaction
corresponds to an uncertainty of ;1 Gy in the age ofglobular clusters (e.g., Chaboyer et al., 1996; and Demarque, 1996).

of ;4310 25 produced in the CNO cycle), indicatingthat in the solar system ( @ 15N/ 14N]'4310 23 ) most of
the observed 15N was produced in nova explosions. At
still higher temperatures, the reaction sequence
15
O(a,g) 19Ne(p, g ) 20Na may provide a breakout pathfrom the Hot CNO Cycle to the rp process (Wallace andWoosley, 1981), which can provide contributions to the
nucleosynthesis of proton rich isotopes up through 65As
(e.g., Mohar et al., 1991), to as high as 68Se (e.g., Blank
et al., 1995), or even to 96Ru (Schatz et al., 1997).

G. The 17O(p, a ) 14N reaction

A. Introduction and retrospective

The 17O(p, a ) 14N reaction provides an important closure for the CNO Tri-Cycle (e.g., Rolfs and Rodney,1975). Observations of the 16O/ 17O ratio in the envelopes of red giants can be combined with information
about the production and destruction of 17O in the stellar interior to determine the depth of convective mixing
between the surface and the interior of such stars (e.g.,
Dearborn, 1992). Until recently there was a factor of;100 uncertainty/disagreement in the rate of the17O(p, a ) 14N reaction, which is a key to determining the16
O/ 17O ratio in stellar nucleosynthesis. For temperatures below T 9 ;0.1, the rate of the 17O(p, a ) 14N reaction is determined by the properties of the J p 51 2 resonance at E c.m.=70 keV, and a recent direct measurementof the strength of that resonance (Blackmon et al., 1995),
vg'5.5 neV, has reduced its uncertainty from a factor of;100 to '6 20%. This much more accurate result will
now allow refined observations of the 16O/ 17O ratio(particularly in stars with masses greater than 5 solar
masses) to place significant constraints on models of
convective mixing.

It is quite remarkable how correct most of the basic
ideas and the data were that were presented by B 2 FH
on the x process. In particular, the solar Li and Be abundances determined by Greenstein and Richardson(1951) and Greenstein and Tandberg-Hanssen (1954)
are very close to the contemporary values. And whereas
the meteoritic Suess-Urey (1957) value for Be is near
the the solar value, the solar Li was found to be 100
times smaller than the meteoritic, still the situation today.
B 2 FH point out that Li, Be, and B are extremely rarecompared to the elements near them on the periodic
table (H, He, C, N, O), but they are 100 times moreabundant than the majority of the heavy elements.
B 2FH give the nuclear reactions that destroy the rare
light elements, and that occur at temperatures found instellar interiors, in the hydrogen-burning zone. These reactions convert the light elements into He isotopes
through fusion with protons. The authors point out that
before a star evolves off the main sequence Li, Be, andB are destroyed in the vast preponderance of a star’s
mass. Given this circumstance, a setting other than stellar interiors is required for the synthesis of these lightelements. Possibilities such as spallation reactions instellar atmospheres and supernovae shells were discussed by B 2 FH, as was the preservation of 7 Li as 7 Be
in late-type stars.

H. The Hot CNO cycle

One wrinkle, which was not anticipated in B 2 FH, wasthe Hot CNO cycle that can become important whentemperatures and densities are sufficiently high so thatthe 13N(p, g ) 14O reaction is fast enough to bypass thenormal beta decay of 13N. This possibility was first described by Caughlan and Fowler (1962), and direct measurements of the rate of the 13N(p, g ) 14O reaction haverecently been made with radioactive beams; using 13N
beams to measure the reaction directly (Decrock et al.,1991), and using 14O beams to measure the inverse reaction, 14O( g ,p) 13N (Motobayashi et al., 1991; Kiener
et al., 1993). These three experiments are all in good
agreement and measure a strength v g =2.360.5 eV for
the E r =529 keV, l =0, resonance that determines therate of this reaction at nova temperatures. These results
indicate that at the temperatures characteristic of a nova
explosion (0.2<T 9 <0.4), the Hot CNO cycle will be thedominant energy generation process. One importantconsequence of the Hot CNO cycle is the production of
an @ 15N/ 14N] isotope ratio of ;1 (compared to the ratio
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

V. THE x PROCESS

B. Abundances

Because Li, Be, and B are so readily destroyed instars, their ‘‘cosmic’’ or ‘‘initial’’ abundances need to be
determined in environments that are unaffected by
nuclear processing in the interiors. The interstellar gas?
The atmospheres of the youngest stars? Primitive solar
system material? The first generation of stars?
Atomic and spectroscopic properties of these elements conspire to make all these species difficult to observe. Because the abundances are low, it is best to observe these elements through their absorption in theresonance lines of their various ions. For Li, the resonance doublet of Li I at 6707 Å is readily observed, buteven in the coolest stars, Li is predominantly ionizedbecause it has a low ionization potential as it is an alkali
metal. And Li II is like He I with two electrons in aWallerstein et al.: Synthesis of the elements

closed shell and has its resonance line at 199 Å. The
resonance line of Be I is at 2348.6 Å, unobservable withground-based telescopes. The resonance doublet of Be
II is near 3130 Å where only about 30% of the flux
penetrates the Earth’s atmosphere. In solar-type stars,
where the dominant ion is Be II, this region of the spectrum is crowded with lines of other elements. For B,
both B I and B II resonances lines are in the ultraviolet
at 2497.7 and 1362.5 Å, respectively. Nonetheless, theimportance of these elements—as tracers of stellar structure, galactic evolution, and cosmology, and as part ofthe picture of the origin of the elements—has led to ahuge amount of research work on determining theirabundances in many astrophysical environments.
1. Lithium

With the exception of the hot O and B stars containing no ions of Li I in their atmospheres, Li abundances
have been determined for stars of every spectral typeand luminosity class. Among the first of a veritable blizzard of papers surveying stellar Li contents was by Herbig (1965). More recent surveys of solar-type field stars
include Duncan (1981), Boesgaard and Trippico (1986),
Balachandran (1990), and Lambert et al. (1991). The
Pop I maximum for field stars is about log N(Li)=3.1 –3.3. There have been several studies on Li abundances inthe youngest stars, T Tauri stars, which include Zapala(1972), Padgett (1990, 1991), Basri et al. (1991), King
(1993), Lee et al. (1994), Cunha et al. (1995), and Duncan and Rebull (1996). Cunha et al. (1995) find a mean
Li abundance for stars in the Orion association, as corrected for nonlocal thermodynamic equilibrium effects(see below), to be log N(Li)=3.2 (60.1) which is very
close to the meteoritic value of 3.3 and conclude that
there has been little Li enrichment in the Galaxy in thelast 5 Gyr.
There has been considerable work done on Li abundances in open clusters, partly with the goal of obtainingan ‘‘initial’’ Li abundance for Pop I stars. See Thorburn
et al. (1993 and references therein) for a partial reviewof the subject. The maximum Li abundances for stars ina given cluster can be found in the temperature region
near 6200 K with the youngest clusters having the mostLi (Boesgaard, 1991). Deliyannis et al. (1994) show thatshort-period tidally locked binaries (SPTLB) near thattemperature in the Pleiades, age ;80 Myr, have Li
abundances on the mean trend, while in the Hyades (age; 700 Myr) and M 67 (age ;5 Gyr) they have somewhat higher Li abundances than their respective clustermates. This suggests that Li depletion has been inhibitedin the SPTLB by the lack of rotationally induced mixing
and indicates that the Li-age relation of Boesgaard(1991) results from a slow mixing process such as rotation. The maximum abundances for clusters are measured or inferred to be log N(Li)=3.1–3.3.
It is well known that evolved stars dilute their surface
Li content as their convective zones deepen (Iben, 1965,
1967a). More recently Deliyannis, Demarque, andKawaler (1990) have made dilution calculations forRev. Mod. Phys., Vol. 69, No. 4, October 1997

1017

metal-poor stars. Although there has been much observational research on Li in such stars, it is not relevant tothe ‘‘initial’’ Li content and will not be discussed here.
The abundance of Li in the oldest stars may reveal theprimordial (big bang) value of Li. In a study of Al in
halo dwarfs, Spite and Spite (1982a, 1982b) discovered a
Li line present in their spectra and found an approximately uniform value for the Li abundance that was
about a factor of 10 lower than the Pop I star maximum.
Several surveys since then have confirmed and extended
these results: Spite et al. (1984), Rebolo et al. (1988),
Hobbs and Duncan (1987), Spite and Spite (1993), Thorburn (1994), and Molaro et al. (1995). Several papers
address the constancy of the Li abundance in the lowmetallicity Li plateau, for example Deliyannis et al.
(1993), Ryan et al. (1996), Spite et al. (1996). These papers reach different conclusions about the amount of the
intrinsic spread in Li and its dependence on effective
temperature and [Fe/H]. But, as the rotational models
predict (Deliyannis, 1990; Pinsonneault et al., 1992;
Chaboyer and Demarque, 1994), even a small dispersion
could be the result of Li depletion by an order of magnitude. The most comprehensive review of all the data,
put on a uniform temperature scale, is by Ryan et al.
(1996). They find that the correlation between Li abundance and temperature and Li abundance and [Fe/H] isreal. They point out three turn-off stars that have similar
temperatures and very low metallicities (between –3.0
and –3.5 in [Fe/H]) for which 3–5 Li observations havebeen made. The Li abundances are 1.8960.05, 2.01
60.04, and 2.2960.05, indicative of an intrinsic spread(of a factor of 2.5) due to influences other than the uniform nucleosynthesis of Li in the Big Bang.
The halo star, BD +23 3910, has a Li abundance thatis a factor of 2 to 3 above the plateau (King, Deliyannis,
Hiltgen, et al., 1997). This high Li may result from alower than average depletion in Li from a higher (primordial) Li value or may result from some type of galactic Li enrichment mechanism. These possibilities are
explored in a search of other spectroscopic signatures ofenrichment, but since none of the predicted anomalies
are found, it seems more likely that Li in this star has
been preserved from a high initial value.
Recently, Li has been detected in stars near the main
sequence turn-off in globular clusters—in NGC 6397 by
Molaro and Pasquini (1994) and Pasquini and Molaro(1996) and in M 92 by Deliyannis et al. (1995) and Boesgaard et al. (1997). For NGC 6397 the three turn-offstars have log N(Li)=2.2860.10, higher than the halo
field stars. From the Keck study of seven similar M 92
stars there is clear evidence for a dispersion in Li abundance of 0.5 dex. The Li abundance in one of the M 92
stars is nearly as high as that in BD +23 3910. The M 92
results were attributed to differences in the amount of Li
depletion in otherwise identical (age, temperature, mass,
luminosity, metallicity) stars. The differing degrees ofdepletion could result from differences in the initial angular momentum of the individual stars and the spindown rates. When the M 92 Li dispersion is coupled witha possible dispersion in Li in the field star plateau, it1018Wallerstein et al.: Synthesis of the elements

indicates that there may be some Li depletion there too.
The interesting discovery of 6 Li in a halo star (Smithet al., 1993) further complicates the interpretation that
halo stars contain pristine Big Bang 7 Li only. However,see Lambert (1995) and Deliyannis and Malaney (1995)
for discussions of stellar self-synthesis of 6 Li.
Boesgaard and Tripicco (1986) examined their sample
of Pop I and old disk stars for a relation between Li and[Fe/H]. They found the full range of Li abundances atsolar metallicity, but an absence of low metalicity, Lirich stars. Surveys of Rebolo et al. (1988), Balachandran(1990), and Lambert et al. (1991) extend that result andinclude the Li abundances of the Pop II stars. There isan upper envelope to the Li abundance with a decreasein Li abundance as [Fe/H] goes from –0.4 to –1.4. Interpretations differ: if there is an age-metallicity relationand thus a Li-age relation, the upper envelope shows theincrease in Li content over the evolution of the galactic
disk, while the same relation with age could mean thatthe older stars have had longer to deplete Li by various‘‘slow’’ processes from a higher or even much higher Li
abundance.
The abundance of Li in the sun, log N(Li)=1.16(Steenbock and Holweger, 1984), is depleted relative toother Pop I dwarfs and relative to meteorites at 3.3
60.04 (Anders and Greveese, 1989). King, Deliyannis,
Hiltgen, et al. (1997) derive a solar Li abundance of 1.05
from three separate sets of high-resolution, high signalto-noise spectra of the sky and/or the moon. The Li content and isotope ratio has been determined in the interstellar medium, most recently by Lemoine et al. (1995).
They conclude that there must be an extra source of 7 Liin the Galaxy.
It is likely that both stellar processing of Li, which
depletes Li in the oldest stars, and galactic processing ofLi, which contributes to the increased amount of Li in
the youngest stars, exist together. These processes makeit difficult to assess the true value for primordial Li.
However, since all the stellar processes are known todeplete Li, the Pop II star Li probably sets a lower limit
to the primordial Li. The best estimates of that cover therange from log N(Li)=2.2 to 2.6.

2. Beryllium

The study of Be abundances in Pop I began in the
1960’s and was reviewed by Wallerstein and Conti(1969). A survey of Be in solar-type stars was done by
Boesgaard (1976) who found that while most such stars
had normal (i.e., solar) Be, some of the hotter ones were
deficient in Be. Boesgaard and King (1993) presented
Be results for Li-normal, solar-metallicity stars andfound a range in abundances of a factor of about three.
A survey of Be in some 60 Li-deficient F and G dwarfs(Pop I) by Stephens et al. (1997) shows Be deficiencies
in the hotter stars, analogous to the Li dip at those temperatures, and near solar Be abundances for stars cooler
than 6000 K. Stars in which both Li and Be are depleted
but present in detectable amounts show a depletion patRev. Mod. Phys., Vol. 69, No. 4, October 1997

tern very close to the predictions of models by Pinsonneault, Kawalar, and Demarque (1990) of rotationallyinduced mixing.
There have been several recent studies of Be in halostars: Gilmore, Edvardsson, and Nissen (1991), Gilmore
et al. (1992), Rebolo et al. (1988), Ryan et al. (1992),
Boesgaard and King (1993), Garcia-Lopez et al. (1995),
a Keck Be project reported by Boesgaard (1996), Thorburn and Hobbs (1996), Primas (1996), and Deliyannis
et al. (1997). All show an increase in Be in stars as themetallicity increases. The slope of log N(Be) vs [Fe/H] isclose to 1.0. There is no evidence of a Be plateau at thelowest [Fe/H] values; the lowest Be detection is log
N(Be)/N(H)=–13.5 for BD –13 3442 at [Fe/H]=–3.0. Thisevidence points to a galactic-wide production mechanism and no signature of any primordial Be.
The solar Be/H value has been well determined by
Chmielewski, Muller, and Brault (1975) as Be/H=1.4(60.6)310211. The meteoritic value given by Anders
and Grevesse (1989) is 2.6310211. Recent work by King,
Deliyannis, and Boesgaard (1997), who include the solar
Be spectrum in their study of Be in a Cen A and B, also
indicates that the sun is depleted in Be relative to meteorites by a factor of 1.4 to 3.2. The sun appears to be
mildly depleted in Be while still retaining some Li,
which suggests a slow-mixing process below the surface
convection zone.

3. Boron

Under the observational evidence section on the x
process, B 2 FH say only ‘‘Boron is spectroscopically unobservable.’’ In fact the resonance lines of both B I and
B II are in the ultraviolet region of the spectrum, unobservable from the ground. However, with the advent of
the Copernicus satellite in 1972, the IUE in 1978, andthe Hubble Space telescope (HST) in 1990, B has finallybeen seen.
Vega was discovered to have the resonance line of B
II at 1362 Å by Boesgaard et al. (1974) with Copernicus
observations at 0.2 Å resolution. The B/H abundance
derived, with an estimate made for departures from local thermodynamic equilibrium, was 1.0310210. Additional observations by Praderie et al. (1977) confirmed
this result from spectra with 0.05 Å resolution. In a
study of 16 normal A and B stars, Boesgaard and Heacox (1978) made Copernicus observations to determine
B/H abundances from the B II resonance line. After correction for non-LTE effects, they give a mean value forcosmic B/H of 2 (60.9)310210.
Recent results for B in stars come from HST observations. Examples of papers on B abundances for Pop I
stars are Lemke et al. (1993), Duncan et al. (1994),
Cunha et al. (1997), and for Pop II stars are Duncan
et al. (1992), Edvardsson et al. (1994), Duncan et al.
(1997). The papers on Pop II stars probe the relationship between the B abundance and the Fe abundance
and assess the B/Be ratio. The slope of the relation between log N(B LTE ) and [Fe/H] is ;1 and between log

Wallerstein et al.: Synthesis of the elements

N(B NLTE ) and [Fe/H] is ;0.7 and the B/Be ratio is typically ;15 (Duncan et al., 1997).
The solar abundance of B has been determined fromthe resonance line of B I at l2496.8 by Kohl, Parkinson,
and Withbroe (1976) with a rocket-borne photoelectricdetector with a spectral resolution of 0.028 Å; they derived a B/H abundance of 1.6310210. Zhai and Shaw
(1994) find meteorites to have 16.9 62.2 B atoms per 106
Si atoms or B/H=6.0310210. Anders and Grevesse(1989) report the 11B/ 10B to be 4.05. Boron has beenfound in the interstellar gas from HST observations, e.g.,
Federman et al. (1993), Jura et al. (1996), with the abundance of B/H of 0.9310210. Federman et al. (1996) findthe isotope ratio, 11B/ 10B, in the interstellar medium tobe 3.4 11.3
20.6 , probably not different from the meteoritic
value.
C. Nonlocal thermodynamic-equilibrium effects

The above analyses were based on the assumption of
local thermodynamic equilibrium. At the time of the
publication of the B 2 FH paper, the actual solutions forthe corrections needed were years in the future. For Li
the most complete results are presented by Carlsson
et al. (1994). For B similar results can be found in Kiselman (1994) and Kiselman and Carlsson (1996). An interesting situation prevails for Be: neutral Be atoms suffer overionization and the ground state of Be II is over
populated, but bound-bound pumping overexcites Be II
which depopulates the ground state. For solar-type stars
the overionization is roughly balanced by the overexcitation and the effects of NLTE, which exist, cancel eachother (Kiselman and Carlsson, 1995; Garcia-Lopez et al.,1995).
D. Production mechanisms
1. Big Bang

Big Bang nucleosythesis of light isotopes has been
predicted by ‘‘standard’’ models by Wagoner et al.
(1967), Wagoner (1973), and Thomas et al. (1993) andby inhomogeneous models by Thomas et al. (1994).
Only the isotope 7 Li can be produced in substantial
quantities in the standard models. At low values of thenucleon to photon ratios (or baryonic mass densities) it
is produced mainly by 4 He( 3 H,g ) 7 Li. As the ratio
increases, that 7 Li is destroyed by 7Li(p, a ) 4He
reactions. At still higher ratios, 7 Li can be produced by
4
He( 3 He,g ) 7 Be(e 2 ,n e ) 7 Li. The two processes result ina dip for the Li abundance at intermediate values of thenucleon to photon ratio (127310210). Predicted 7 Li/H
ratios cover the range of 10 210 to 10 29 . See Boesgaard
and Steigman (1985) and Krauss and Romanelli (1990)
for more discussion of the synthesis of 7 Li during the
Big Bang.
2. Spallation

There is the suggestion in B 2 FH that Li, Be, and B
could be produced by spallation reactions on abundant
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1019

elements such as C, N, O, Fe with energies . 100 MeV
per nucleon. Although the site was thought to be stellar
atmospheres, it was shown by Ryter et al. (1970) that
production of light elements in the atmospheres of T
Tauri stars was energetically unfeasible. Reeves et al.
(1970) suggested that energetic galactic cosmic rays
could bombard C, N, O atoms, breaking them up into
the light isotopes. This concept was expanded by Meneguzzi et al. (1971). Their model was successful in many
aspects, but it underproduced 7 Li and could not matchthe observed isotope ratios for Li and B. A low-energy‘‘carrot’’ of cosmic ray flux was added to help these
problems. However, some additional 7 Li is produced in
the Big Bang and in the envelopes of AGB stars. Aninteresting addition to spallation production is that spallation in the halo would be more efficient than a + areactions (Steigman and Walker, 1992).
In addition to accounting for the present day light element ratios, spallation models can be constrained better if they are required to explain the evolution of lightelements over the history of the Galaxy. Information has
become available on the evolution of both Be and B (asrevealed by the use of [Fe/H] as a chronometer) and theapparent constancy of Li. Models of galactic chemical
evolution have been made to interpret the evolution of
the light elements (Vangioni-Flam et al., 1990; Ryanet al., 1992; Prantzos, Cassé, and Vangioni-Flam, 1993;
Feltzing and Gustafsson, 1994; Tayler, 1995; Lemoine
et al., 1997). Duncan et al. (1997) interpret the most recent observations of B and Be evolution to favor asomewhat different version of the classical spallation
production: it is the C and O atoms that bombard theambient protons and alpha–particles. Vangioni-Flam
et al. (1996) show that spallation by energetic C and O
atoms can account for the nearly constant increase in[Be] and [B] with [Fe/H]. The source of C and O is fromwinds from massive stars, such as Wolf-Rayet stars, andmassive supernovae. However, Prantzos et al. (1993)
and Malaney and Butler (1993) argue that this mechanism has serious problems.
3. Asymptotic giant branch stars

It was pointed out in B 2 FH that Li-rich red giants had
been discovered and that Cameron (1955) had proposed
preserving 7 Li in the form of 7 Be in the hydrogenburning region. This mechanism was rediscussed by
Cameron and Fowler (1971). There is very interesting
modern data to support the production of 7 Li in AGB
stars, first presented by Smith and Lambert (1989, 1990)
for AGB stars in the Magellanic clouds. This work has
been extended by Plez et al. (1993) and by Smith et al.
(1995). The stars in the Magellanic clouds with Li excesses of 100–1000 times over the predicted (diluted)
values range between bolometric magnitudes M bol=–7.2
to –6.5 corresponding to 4–8 M ( . Predictions have been
made by Scalo et al. (1975) and Sackmann and Boothroyd (1992) that hot-bottom convective envelope
burning would produce and then transport 7 Be to theobservable surface of AGB stars where it becomes 7 Li

1020

Wallerstein et al.: Synthesis of the elements

through electron capture. The predictions and the observations are in good agreement. This mechanism may
produce a substantial amount of enrichment in the galactic Li abundance over the Big Bang production (unless, of course, it is the case that Big Bang 7 Li was ashigh as the Pop I star value of 3.1–3.3). In any event, it isprobable that this production mechanism exists inasmuch as there are some very Li-rich late-type giants,
particularly carbon stars, many of which had been discovered by the time of the B 2 FH review and are referenced there.
4. Supernovae

In 1973 Audouze and Truran suggested that Li and B
could be produced in supernovae shells from high energy particles that were produced in the shock waves;this was another component of spallation. However, today’s theories for supernovae production of the light elements are very different. Models of type II supernovae
by Woolsey et al. (1990) and Timmes et al. (1995) suggest a ‘‘n process’’ that produces B and Li in supernovae
envelopes. As a massive star undergoes core collapse toform a neutron star, it generates a huge flux of neutrinos. Although the cross sections for collision are small
there are so many neutrinos that they excite heavy elements. Since deexcitation by particle emission is almost
always favored over radiative deexcitation, the affectednuclei are likely to emit a proton, neutron or alpha particle. Among the products created are 11B and 7 Li.
There is some evidence that indicates that this mechanism does not play a dominant role. Duncan et al. (1997)
conclude that spallation by low energy C and O atoms
onto protons and a particles in the vicinity of massive
supernovae produce sufficient B and Be in the early history of the Galaxy and that the n process would produce
only a small part of the total B. Further evidence thatthe n process would play only a minor role comes from
Cunha et al. (1997) who found that two of their fourstars in the Orion Association are O rich (by 60%) andB deficient (by 250%). If supernova ejecta produce theincrease in O, there should be an increase in B, not thedecrease observed.
VI. HELIUM BURNING

As hydrogen burns in a star, a hot, dense core of helium is formed that fuels the nucleosynthesis of theheavier elements. The first stage of this process is theso-called ‘‘triple-a ’’ capture to form 12C, followed by thesubsequent capture of a particles to form 16O. In essence, helium burning terminates there, because furthera captures (to form 20Ne, for example) occur too slowly
at these temperatures and densities to be significant.
The termination of helium burning at 16O was notrealized at the time of B 2 FH, because of uncertainty inthe level structure of 20Ne near the a 1 16O threshold.
However, it was later discovered that the levels closest
to the threshold had the wrong parity and/or angularmomentum to make a large resonant contribution to theRev. Mod. Phys., Vol. 69, No. 4, October 1997

FIG. 2. The astrophysically important energy levels of 12C.

rate. A calculation by Fowler et al. (1975) showed thatthe rate for 16O( a , g ) 20Ne is far below that for
12
C(a,g)16O for T 9 <0.2. Therefore, the blocking of further a captures for normal helium-burning conditions
allows us to concentrate on triple-a and a 1 12C capturein the following discussion. Although the formulas used
to describe the rates of these reactions are essentiallyunchanged from forty years ago, the major advance, as
already pointed out earlier, has been the experimental
determination, with great precision in some cases, of thenuclear parameters upon which the expressions depend.
A. Triple-a captureThis reaction actually occurs in two stages: first, two aparticles resonate in the low-lying (but unbound) statethat forms the ground state of 8 Be. This state issufficiently long-lived ( t 1/250.968310216 s) that there isa non-neglible probability that a third a particle will
be captured before it disintegrates, forming
12
C** (E x 57.6542 MeV, J p 50 1 ). A level diagram
showing the relevant states and thresholds for 12C isgiven in Fig. 2. Because of its quantum numbers, there is
only a small probability that this excited state will deexcite (rather than decay back into three a particles), either by e 1 -e 2 pair production, or by a g ray cascade
through the first excited state, leaving 12C in its 0 1
ground state. The prediction (Hoyle et al., 1953), andsubsequent experimental verification, of the propertiesof 12C ** in order to account for the observed abundance of 12C remains one of the most impressive accomplishments of nuclear astrophysics.
The rate per unit volume for the deexcitation process
at temperature T is given by the resonance form (Rolfs
and Rodney, 1988)
r 3a5

S

D

S D

N a3 3/2 2 p \ 2 3 G a G rad
Q
3
exp 2
,
2
M a kT\G
kT(8)with N a the a particle number density, M a its mass,
Q5 ~ M 12C 23M a ! c 2 5379.560.3 keV,**

(9)and G a and G rad the decay widths for a and (e 6 1 g )emission, respectively, which sum to the total width G.
Since G rad!G a 'G, the triple-a rate depends only
on G rad5G g 1G pair . The current values (Ajzenberg-

Wallerstein et al.: Synthesis of the elements

1021

Selove, 1990) of these widths are G g 5(3.6460.50) meV
and G pair5(60.563.9) m eV. The underestimation of thisrate by B 2 FH due to their value of G g 51 meV wascompensated in part by having at that time a smaller Q
value (372 keV) in the exponential factor of Eq. (8).
B. a 1 12C capture

The rate of this reaction is responsible for one of themost important uncertainties in nuclear astrophysics today. Its rate at stellar temperatures, relative to that of‘‘triple-a ’’ capture, determines how much of the 12C
formed is converted to 16O, and thereby the carbon/
oxygen abundance ratio in the ejecta of massive stars
and in the interiors of white dwarfs formed by stars initially less massive than ;11M ( . The relative amountsof carbon and oxygen at the end of the helium-burning
phase set the initial conditions for the next phase in massive stars, which is heavy-ion burning. In the process,many of the heavier elements up through iron are synthesized. All of these processes have been found to bequite sensitive to the a 1 12C capture rate (Weaver andWoosley, 1993), so that the abundances of the mediummass elements, and even the final evolution of massive
stars that explode as supernovae, depend critically on its
determination.
The rate per unit volume for a 1 12C capture is givenby the familiar expression involving the number densities N a and N 12C, and the Maxwellian-averaged ^ s v &for temperature T,
r a 1 12C5N a N 12C ^scapv & T .

(10)

The cross section is parametrized ass cap~ E ! 5

S cap~ E !
exp~ 22 p h !
E(12)where h 5Z 1 Z 2 e 2 /\ v .
The ^ s v & integral is mainly determined by the valueof the cross section (or S factor) at the Gamow energy,
which for helium-burning temperatures (T 9 '0.2–0.6) isE 0 50.3–0.9 MeV. Since present measurements cannot
be extended to such low energies in the presence ofCoulomb barriers as large as that for a 1 12C, the rate
must be found by theoretical extrapolation. Due to theirdifferent energy dependencies, the E1 and E2 multipole
components of the cross section are extrapolated separately to E5E 0 , both being influenced by the presenceof subthreshold levels in 16O having J p 51 2 and 2 1 , respectively. The levels of 16O near the a 1 12C threshold
are shown in Fig. 3.
1. E1 capture

Much effort has been devoted to extracting just theE1 part of the capture cross section, which is dominatedby a broad 1 2 resonance at E52.4 MeV (E x 59.585
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

MeV; all level energies and widths for 16O are takenfrom Tilley et al., 1993). The importance of the subthreshold 1 2 state at E5245 keV (E x 57.11685 MeV)in fitting and extrapolating the E1 cross section was first
demonstrated by Dyer and Barnes (1974). Most subsequent analyses have fit S factors extracted from themeasurements along with the P wave phase shift obtained from a 1 12C elastic scattering measurements(Plaga et al., 1987), using either R-matrix or K-matrix
theory. More recently, several groups have acted on thelongstanding suggestion of Barker (1971) to measure theb -delayed a spectrum from the decay of 16N with thehope of better determining the contribution of the subthreshold state, which is evident in the spectrum as asecondary maximum at low energies.

(11)

in terms of Sommerfeld’s Coulomb parameter h fora 1 12C, and the S factor for the capture reaction,
S cap~ E ! 5S E1 ~ E ! 1S E2 ~ E ! ,FIG. 3. The astrophysically important energy levels of 16O.

a. Direct measurements

Several direct measurements (Dyer and Barnes, 1974;
Redder et al., 1987; Kremer et al., 1988; Ouellet et al.,1992) of the E1 capture cross section have been made in
the c.m. energy range E51.0–3.0 MeV. With the recently reported corrections in the data of Ouellet et al.
(1996), these measurements are in relatively good agreement at energies between 1.3 and 3.0 MeV. They now all
imply constructive interference of the subthreshold levelwith the positive energy resonance [originally, Ouellet
et al. (1992), had found destructive interference], leading
to extrapolated S factors at E50.3 MeV that range from10 to 200 keV b. The extrapolated S factors obtained
from these measurements, along with their assigned uncertainties, are given in Table II. The table also notes
briefly the methods used to make and analyze the measurements. Most of the analyses involved doing standard
R-matrix or K-matrix fitting, or using the ‘‘hybrid’’
R-matrix method (Koonin, Tombrello, and Fox, 1974),
in which a potential is used to represent the resonances
at 2.4 MeV and above, in order to reduce the uncertainty in the ‘‘background’’ contribution to the R matrix.

1022

Wallerstein et al.: Synthesis of the elements

TABLE II. Extrapolated values of S E1 (E 0 50.3 MeV) obtained from direct measurements of thedifferential capture cross section.
Reference

Methods

S E1 (E 0 ) (keV b)

measured s (90°); corrected for E2 withdirect-capture calculations
3-level R matrix
hybrid R matrix
measured s ( u ); separated E1 and E2
3-level R matrix
hybrid R matrix
measured s (90°), using g -recoil coincidence
3-level R matrix, constrained by l51 a 1 12C
phase shiftmeasured s ( u ); separated E1 and E2
R-matrix, K-matrix fits; includes b -delayed aspectraDyer and Barnes (1974)
140 1140
2120
80 150240
Redder et al. (1987)
200 12702110
140 1120280
Kremer et al. (1988)
10 1130
210
Ouellet et al. (1996)
79616

b. b -delayed a spectrum from the decay of 16N

At least three groups (Buchmann et al., 1993; Zhao
et al., 1993; Zhao et al., 1995) have measured the delayed a spectrum from the b decay of 16N down to energies low enough to see the secondary maximum attributed to the presence of the subthreshold 1 2 state. Someof these new spectral measurements have been includedin the fitting along with direct measurements of the E1
cross section, in order to better constrain the parameters(in particlular, the reduced a width) of the subthresholdstate. These constrained analyses give extrapolated values for S E1 (E 0 ) in the range 80295 keV b, as favoredby the sensitivity study (Weaver and Woosley, 1993) ofthe dependence of calculated elemental abundances on
the extrapolated a 1 12C capture cross section.
The functional form of the spectrum can be obtained
from the usual R-matrix relation of the scattering statesto the level matrix, giving

U( (

dN a
5f b ~ E ! P a ~ E !
dE

l

Bl

l8

U

2

A ll 8 ~ E ! g l 8 a ,(13)in which f b is Fermi’s integrated b decay function for
Z58, P a is the a 1 12C penetrability, B l is the dimensionless b -feeding amplitude for level l, A is the level
matrix, and g l 8 a is the reduced-width amplitude in thea 1 12C channel for level l 8 . The product of the first two
terms in the expression gives the phase-space behaviorof the spectrum, while the third term gives the structure.
A similar expression results from the K-matrix formalism (Humblet, Filippone, and Koonin, 1991).
The interference minimum between the two maxima
in the spectrum cannot fix the interference of the levelsin the cross section, due to the undetermined relative
phases of the B l . However, as was mentioned in theprevious section, all the direct cross-section measurements indicate that the interference is constructive. The
magnitude of the low-energy peak in the spectrum issensitive to the value of the reduced-width amplitude in
the a 1 12C channel for the subthreshold level, assumingthat the one in the g 1 16O channel, and the b -feeding
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

amplitude for the state have been fixed by g decay data.
When the spectral data were included in some of therecent analyses (Azuma et al., 1994; Buchmann et al.,1996), this sensitivity exaggerated the chi-square differences between the constructive- and destructiveinterference solutions for the cross section enough torule out the latter solution.
The spectra measured by the TRIUMF (Buchmann
et al., 1993; Azuma et al., 1994) and Yale/Connecticut(UConn) (Zhao et al., 1993) groups are generally ingood qualitative agreement. However, questions havebeen raised on both sides about important experimental
details, such as the shape of the low-energy peak in the
Yale/UConn measurement, the shape of the high-energy
peak in the TRIUMF data compared to an earlier measurement (Neubeck, Schober, and Wäffler, 1974), andthe effect of target-thickness corrections on comparisonsof the two measurements.
Two new measurements of the spectrum have recently
been done at the University of Washington in Seattle(Zhao et al., 1995), and at Yale/UConn (France et al.,1997). These new measurements are said to agree wellwith each other, being as much as a factor of 2 higher
than the TRIUMF data in the minimum between thetwo peaks. This difference could imply that a larger
F-wave contribution to the spectrum is required in order
to fill in the minimum, thus decreasing the P-wave contribution (and the extrapolated value of S E1 ) correspondingly. However, Hale (1997) has shown that allowing the b -feeding amplitude of the positive energy levelto have a small imaginary part also has the effect offilling in the minumum between the peaks. Therefore
the impact of these new measurements will depend on
how they are analyzed.
2. E2 capture

The extrapolated E2 capture cross section is also determined by the interaction of positive energy levels(and ‘‘direct capture’’ contributions) with a subthreshold
2 1 state at E52245 keV (E x 56.9171 MeV). In thisWallerstein et al.: Synthesis of the elements

1023

TABLE III. Extrapolated values of S E2 (E 0 50.3 MeV) obtained from direct measurements of thedifferential capture cross section.
Reference

S E2 (E 0 ) (keV b)

Redder et al. (1987)
96 124230
80625
Ouellet et al. (1996)
3666
Trautvetter et al. (1996)
14.5 196
214

case, the only broad 2 1 level in the region is quite far
above the threshold, at E54.36 MeV (E x 511.52 MeV).
In addition, there is a narrow resonance (G50.625 keV)
at E x 59.8445 MeV that is not even visible in many ofthe measurements. The approximate centroid of the E2
strength appears to be at E x 515 MeV, with a spread of
about 15 MeV. Therefore, one expects the dominant E2contribution above threshold to come from distant levels
and direct capture terms.
The direct measurements lead to extrapolated valuesof S E2 (0.3 MeV) ranging from 14.5 to 96 keV b, as areshown in Table III. Most of the analyses used some formof microscopic or potential model (usually of Gaussianform) to do the extrapolation, since a pure R-matrix or
K-matrix fit would have involved too many parameters
for the number of available data points. In some cases,
the potential-model amplitudes were combined withsingle-level parametrizations of the subthreshold level.
However, the present E2 capture data are simply notgood enough to allow an unambiguous separation of theresonant and direct effects, even when considered simultaneously with the elastic scattering data.
It is not known with certainty, for example, whether
the interference between the subthreshold 2 1 state at
E52245 keV and the distant-level (or ‘‘direct’’) contribution is constructive or destructive. The recent analysis
of Trautvetter et al. (1997) is one of the few that has
attempted to include the two positive-energy 2 1 resonances at E x <11.52 MeV. The uncertain nature of theinterferences of these levels was a major contributing
factor to the rather large uncertainty on the extrapolated S factor listed in Table III. Thus the uncertain roleof resonances in the predominantly direct reaction
mechanism, and the increased reliance on theoreticalmodels, introduce even more uncertainty into the E2
extrapolation than that for the E1 contribution.
An attempt has been made recently (Kiener et al.,1997) to extract the E2 cross section from the inverse
process, Coulomb dissociation of an 16O beam by a
208
Pb target. The data analysis to obtain the equivalentcapture cross sections from the a -12C coincidence spectra is heavily dependent on theoretical calculations, but
the qualitative agreement with the capture measurements achieved at this stage shows the promise thismethod may hold for obtaining new information about
the E2 cross section.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

Methods
measured s ( u ); separated E1 and E2
direct capture + single-level Bréit-Wigner form
hybrid R matrix
measured s ( u ); separated E1 and E2
microscopic cluster model
measured s ( u ); separated E1 and E2
R matrix, including 2 1 levels below E x 511.52
MeV

3. Other analyses and recommended values

A number of analyses not connected with any particular measurement have been done in recent years(Barker, 1987; Filippone et al., 1989; Barker and Kajino,
1991; Humblet et al., 1991) that consider several data
sets simultaneously, including the earlier measurement(Neubeck et al., 1974) of the a particle spectrum fromthe b decay of 16N. These analyses often included theoretical refinements of the usual R-matrix and K-matrix
fitting procedures, such as taking into account thechannel-region contributions to the photon widths in
R-matrix theory, and explored the parameter spaces of
the representations used. The extrapolated S factors obtained range from 30 to 260 keV b for S E1 , and from 7 to
120 keV b for S E2 .
Two recent analyses considered only the available primary data in order to avoid the correlations introduced
by using derived quantities (such as elastic-scattering
phase shifts) in determining the extrapolated E1 and E2
S-factor values. The analysis of Buchmann et al. (1996)
included the differential cross-section measurements for
12
C( a , a ) scattering and 12C( a , g ) capture, along withthe b -delayed a spectrum of Azuma et al. (1994). Theirresults for the E1 part of the cross section essentiallyconfirmed
theirearlier
extrapolatedvalue
S E1 (E 0 )5(79621) keV b, and for the E2 part, established the upper limit S E2 (E 0 ), 140 keV b.
Hale (1996) recently reported an R-matrix analysis of
the E1 cross section alone that also used the elasticscattering angular-distribution data of Plaga et al. (1987)
rather than the P wave scattering phase shift. This
analysis indicated that a rather low value of S E1 (E 0 ), 20
keV b, was consistent with all the measured data, including those of the b -delayed a spectrum. However, afterthat analysis was done, the paper of Ouellet et al. (1996)
appeared, reporting corrections to earlier data (Ouellet
et al., 1992) that significantly raised their low-energy
cross sections. Because the uncertainties were so small,their earlier data had been a determining factor in theextrapolated value of S E1 (E 0 ) obtained by Hale (1996).
Therefore, the recent changes in those data will likely
raise Hale’s extrapolated E1 S factor above 20 keV b.
At this point, it appears that the best values to recommend for the extrapolated S factors for a 1 12C capture
are the recent ones of Ouellet et al. (1996). The energy-1024Wallerstein et al.: Synthesis of the elements

FIG. 6. The NeNa and MgAl cycles for T 9 =0.05 and r =100g/cm 3 . Arrows show integrated fluxes with the heavy lines indicating the strongest flows and the dashed lines representing
the weakest. Shaded boxes denote stable nuclei and double
boxes indicate the presence of an isomeric state.
FIG. 4. Experimental s factors of the E1 component of
12
C( a,g)16O.

dependent S factors from their analysis are shown compared with various direct measurements in Fig. 4 for theE1 component, and in Fig. 5 for the E2 contribution.
The curves extrapolate at E 0 =300 keV to the values
S E1 (E 0 )=(79616) keV b and S E2 (E 0 ) =(3666) keV b,giving in round numbers S cap(E 0 )=(120640) keV b. To
this needs to be added the cascade contributions to E2
capture, which Redder et al. found to be (2069) keV b.
These values are quite consistent with the best estimatesobtained by the TRIUMF group, with many of the earlier direct measurements, and with the results of the sensitivity study (Weaver and Woosley, 1993). However, it
should be remembered that some of the recent data andanalyses continue to indicate lower values of the E1S
factor. While the uncertainties of these important parameters are gradually decreasing with time, they remain well outside of the 10–15 % level that is desirable
for astrophysical applications. Clearly, more work remains to be done on a 1 12C capture, especially concerning the extrapolation of the E2S factor.

VII. H BURNING IN THE NeSi REGION: LABORATORY STUDIES
A. Introduction

Hydrogen burning beyond the CNO cycles can begin
in the NeNa cycle and continue into the MgAl cycle(Fig. 6). The former cycle was described by B 2 FH, primarily because they were interested in 21Ne as a potential neutron source. They recognized that since the23Na(p, a ) 20Ne reaction is stronger than the competing
23
Na(p, g ) 24Mg reaction, a direct flow to heavier nuclei
would be very weak. For this reason, their considerationof hydrogen burning stopped at Na. The years following
the publication of B 2 FH have seen (a) the discovery of
the 22Ne isotopic anomaly in meteorites (Black, 1972;
Lewis et al., 1979; Eberhardt et al., 1979), (b) observations of anticorrelations of Na and Al with O in globularclusters (reviewed by Kraft, 1994 and Kraft et al., 1997),
and (c) observations of Na enhancements in red giant
atmospheres (Takeda and Takada-Hidai, 1994). All ofthese effects are thought to arise from some form of a
NeNa cycle, and so it has remained a topic of interest(additional details concerning Na and Al in globularclusters can be found in Sec. VIII of this review). Similarly, the discovery of 26Mg abundance anomalies in meteorites (Lee et al., 1977; Ireland and Compston, 1987;
Gallino et al., 1994, Hoppe et al., 1994; Russell et al.,1996), and the discovery of 26Al in the ISM of our galaxy (Mahoney et al., 1984; Share et al., 1985; MacCallum
et al., 1987; von Ballmoos et al., 1987, Diehl et al., 1995)
have focused a great deal of attention on the MgAl
cycle. The reactions in these two cycles have been subjected to careful experimental scrutiny. Nonetheless,some uncertainties remain as a challenge to nuclear
physics. In the following, we review what is known, andwhat is not known about hydrogen burning in the Ne-Si
region.
B. Experimental approaches

FIG. 5. Experimental s factors of the E2 component of
12
C(a,g)16O.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

Unlike the situation found in the CNO cycles, for almost all astrophysical scenarios the cross sections in theWallerstein et al.: Synthesis of the elements

NeNa and MgAl cycles are determined primarily by thecontributions of resonances. Although the procedure formeasuring these cross sections is rather straightforward
in concept, the experiments are difficult in practice because they involve energies well below the Coulomb
barrier [see, e.g., Rolfs and Rodney (1988)]. At the lowest energies one must simultaneously produce high quality, high-current proton beams, targets that can withstand the high currents, and efficient detection systems;
in many cases, that is a very daunting task.
A large number of resonance strengths has been published for the reactions of interest at lower temperatures. Many of these values result from relative yieldmeasurements that are normalized to the strengths ofvarious standard resonances. This has led to some discrepancies in the literature for cases where the absolutestrengths have been in disagreement. For example, several studies have normalized their results to the absolutestrengths of Engelbertink and Endt (1966), some ofwhich have since been called into question. A number ofmore recent absolute measurements have been reported, the most precise being those of Paine and Sargood (1979), Keinonen and Brandenburg (1980), andAnderson et al. (1982), that all quote uncertainties on
the order of 10%. These (and other) measurements haveprovided a consistent and accepted set of absolutestrengths. Corrected resonance strengths have beentabulated by Endt (1990, hereafter Endt90), but caremust be exercised when referring to the original sourcesof these data and to collections of reaction rates [e.g.,
Caughlan and Fowler (1988), hereafter CF88].
The lowest-energy resonance accessible in a direct
measurement is ultimately determined by count-rate
limitations imposed by the Coulomb barrier. Most of thereactions of the low-temperature NeNa and MgAl cycles
have been measured directly down to energies on theorder of E c.m.=150–200 keV which corresponds to stellar
temperatures of about T 9 =0.1 (where the subscript denotes temperature in units of 10 9 K). It does appear
possible to extend these measurements to lower energies
with the use of modern detection techniques. However,at present the information pertaining to lower-energy
resonances has been obtained from indirect nuclear
spectroscopy. A simplifying feature at low energies isthat the proton partial width, G p , becomes much smaller
than those for gamma or alpha emission. Consequently,
the resonance strength is proportional to G p , which canbe extracted from stripping data via the relation
G p 5C 2 S G sp . Here C 2 S is the proton spectroscopic factor and G sp is the proton width for a (ficticious) puresingle-particle state. Since these are model-dependent
quantities, potentially large systematic errors (factors of
10 or more) could enter into the final results. However,
it is possible to check the reliability of this technique by
also applying it to measured resonances.
At the high temperatures encountered during explosions, reactions can compete with b decay. Consequently, the reaction flow moves to the proton-rich sideof stability (Fig. 7) and radioactive beam techniquesmust be employed to measure the relevant reactions.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1025

FIG. 7. Integrated fluxes for T 9 =0.3 and r =6000 g/cm 3 .

Because reactions will typically occur on time scales thatare short when compared with the duration of the explosion, the primary experimental focus is on measuring therates of reactions that impede the flow to higher masses.
The reactions of interest can be identified on the basis of
their (p, g ) Q values (Iliadis, 1996; see also Rembges
et al., 1996). For Q (p, g ) '2 – 6 MeV, the level density is
low enough that the reaction rate is composed of thecontributions of individual resonances and nonresonant
direct capture. Although this in itself does not imply thatthe reaction is slow, it does mean that the rate is governed by the properties of only a few states. For Q (p, g )below this range, the system can establish a (p, g )-( g ,p)equilibrium, in which case the reaction rate is less important. For higher Q values, a large number of resonances may be populated and the reaction can take on a
statistical character. Using these criteria, the reactions
that are candidates for direct measurements are
21
Na(p, g ), 23Mg(p, g ), and 24,25Al(p, g ).
C. The neon-sodium cycle at low temperatures
1. Reaction rates

All of the reactions of the NeNa cycle have been measured directly and the experimental results may befound in the tabulations by Endt (1990). Reaction rates
derived from these measurements are listed in CF88,
and little in the way of new experimental information
has been presented since then. However, there are significant uncertainties in several of the rates caused by
the possible existence of states that could correspond to
low-energy resonances. The existing data were recently
reevaluated by El Eid and Champagne (1995; hereafter
EEC95) who recommended some modifications of the
CF88 rates. The experimental situation may be briefly
summarized as follows.
(1) The 20Ne(p, g ) 21Na reaction
At low temperatures, the 20Ne(p, g ) 21Na reaction includes an important contribution from the tail of a subthreshold resonance located at E c.m.=26.5 keV. The S
factors for this state, for the low-energy resonances, andfor nonresonant direct capture have been determined by
Rolfs et al. (1975). Although Q (p, g ) is now 0.5 keV lower(Endt90), EEC95 found that it had little effect on the1026Wallerstein et al.: Synthesis of the elements

reaction rate. However, this reaction is currently being
remeasured (Wiescher, 1996).
(2) The 21Ne(p, g ) 22Na reaction
Resonances in the 21Ne(p, g ) 22Na reaction have beenmeasured (Berg et al., 1977; Keinonen et al., 1977;
Görres et al., 1982; Görres et al., 1983; Becker et al.,1992) down to E c.m.=121 keV. Two lower-energy states
exist and uncertainties concerning their single-particle
strengths led to large uncertainties in the rate tabulated
in CF88. Limits on the proton spectroscopic factors forthese states, based upon available data, were deduced by
EEC95. They predict that the two low-energy statesshould have an almost negligible effect on the rate. The
current lower limit on the rate is very nearly the same as
the upper limit appearing in CF88.
(3) The 22Na(p, g ) 23Mg reaction
Although 22Na is radioactive, it could provide a pathway for leakage out of the NeNa cycle in situations
where the (p, g ) lifetime is comparable to the lifetime
for b 1 decay (3.75 y). Schmidt et al. (1995) have usedthe 22Na( 3 He,d) reaction to determine the strengths of
low-energy resonances and direct (p, g ) measurements
have also been performed by Seuthe et al. (1990) and by
Stegmüller et al. (1996). For pure hydrogen at a densityof 10 3 g/cm 3 , the b 1 decay and (p,g ) lifetimes are equal
for temperatures T 9 =0.039–0.068, where the range intemperature reflects the uncertainty in the reaction rate.
Note that this density is high for some low-temperaturesites. At 10 g/cm 3 , the transition occurs for T 9 =0.08–0.1.
(4) The 22Ne(p, g ) 23Na reaction
Unfortunately, the rate of the 22Ne(p, g ) 23Na reaction
is rather poorly known at low temperatures because it
appears that 13 states exist in the region between threshold and the first measured resonance (Endt90). Two ofthese states have been observed in ( 3 He,d) studies andthe appropriate resonance strengths have been calculated by EEC95. Further (p, g ) studies by Görres et al.
(1982) and Görres et al. (1983) have indicated that perhaps two of the remaining eleven states might be astrophysically significant, but it is not clear that these states
exist, and it is certainly not possible to place stringent
limits on their resonance strengths. This situation leads
to an uncertainty in the reaction rate of a factor of 2–
3400 for T 9 =0.03–0.08. Although it is most likely thatthe actual rate is near its current lower limit, it is presently the most uncertain rate in the NeNa cycle.
(5) The 23Na(p, g ) 24Mg and 23Na(p, a ) 20Ne reactions
Resonances in the 23Na1p system have been measured down to E c.m.=171 keV (Zyskind et al., 1981;
Görres et al., 1989), but again, the possibility exists that
there are other resonances between this level andthreshold that may be important at low temperatures.
Crude estimates of spectroscopic factors were given by
EEC95. In the case of the (p, a ) reaction, large uncertainties exist, but only for T 9 < 0.03, whereas the (p, g )reaction is uncertain by about a factor of 2 for a narrow
range of temperatures near T 9 =0.08. Although better
experimental information is desirable, it is clear that the(p, a ) is much faster than the (p, g ) reaction at low temperatures and therefore the NeNa cycle will in fact cycle.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

FIG. 8. Time evolution of mass fractions in the NeNa and
MgAl cycle for T 9 =0.05, r =100 g/cm 3 , and solar initial composition. Shaded areas indicate the range of values allowed by
the upper and lower limits on the individual reaction rates.

The situation is reversed for temperatures T 9 =0.18–0.6,
but as the temperature increases, the reaction flow will
begin to leave the NeNa cycle via the 22Na(p, g ) reaction, bypassing 23Na.
2. Network calculations

Shown in Fig. 8 are the results of network calculations
performed for T 9 =0.05, assuming a constant density of
100 g/cm 3 , and solar initial abundances [other calculations may be found in Arnould et al. (1995)]. Although,these are artificial and rather generic parameters, theresults should illustrate what might be expected fromthe NeNa cycle in red giants, for example. The moststriking features of Fig. 8 are the almost negligible
depletion of 20Ne (5%) and the strong destruction of
22
Ne which together lead to an enrichment in 23Na by
about a factor of 7. Because 21Ne is rapidly destroyed,
the uncertainty in the 21Ne(p, g ) reaction rate has little
effect on the final abundances. The first branch point inWallerstein et al.: Synthesis of the elements

the cycle appears at 22Na. At its upper limit, the22Na(p, g ) rate is about 3 times faster than the rate forb 1 decay. This leads to a gradual production of 23Mg,
but since this decays rapidly to 23Na, the net effect ofthe 22Na(p, g ) reaction is just to bypass 22Ne(p, g ).
Since the latter reaction is also comparatively fast, thishas little effect on the final abundance of 23Na. For similar reasons, the large uncertainty in the 22Ne(p, g ) rate
does not translate into a comparable uncertainty for
23
Na. The overall cycle time (9160 y for these conditions) is determined by the rates of the slowest reactions, namely 20Ne(p, g ) and 23Na(p, a ), and at T 9 =0.05
these rates are known rather precisely. Because the23Na(p, a ) reaction is about 18 times faster than thecompeting 23Na(p, g ) reaction, the NeNa cycle is essentially closed (though this will not be the case at higher
temperatures and densities, as will be seen below).
D. The magnesium-aluminum cycle at low temperatures
1. Reaction rates

The origin of the radioactive isotope 26Al became aninteresting problem in astrophysics with the discovery ofan excess of its daughter species, 26Mg, in the Allendemeteorite (Lee et al., 1977). The presence of a g ray line
due to the decay of 26Al to 26Mg in the Galaxy pointed
to the production of 26Al in the Galaxy at the present
time, since the half-life of 26Al is 7.23105 years. In addition the discovery of an excess of Al (presumably
27Al) in globular cluster stars that are deficient in oxygen and show an excess of Na (Kraft et al., 1997) added
impetus to efforts to understand the MgAl cycle. Although uncertainties in the MgAl cycle are somewhat
smaller than those for the NeNa cycle, some further investigation is warranted. Unfortunately, most rate compilations predate some of the more definitive experimental results. Therefore in the following we havesummarized the current status of these reactions.
(1) The 24Mg(p, g ) 25Al reaction
The 24Mg(p, g ) 25Al reaction has been studied rather
extensively by Trautvetter and Rolfs (1975) and Trautvetter (1975). The reaction rate found in CF88 is based
upon these results and appears to be reliable. However,it has been suggested (Cavallo et al., 1996; Shetrone,
1996a, 1996b) that the rate must be higher than believedin order to account for trends of Mg vs Al observed in
globular clusters. Further work on this reaction is inprogress (Iliadis, 1996a).
(2) The 25Mg(p, g ) 26Al reaction
Surveys of resonances in the 25Mg(p, g ) 26Al reaction
have been reported (Elix et al., 1979; Anderson et al.,
1980; Endt et al., 1986; Iliadis et al., 1990) and results aretabulated in Endt90. A search for possible low-energy
resonances was undertaken by Champagne et al. (1983)
using data obtained via the 25Mg( 3 He,d) reaction. In
order to eliminate the model dependence associatedwith calculations of single-particle widths, they obtained
proton widths by normalizing to higher energy resonances, where both proton widths and spectroscopic facRev. Mod. Phys., Vol. 69, No. 4, October 1997

1027

tors could be measured. Unfortunately, some of the conclusions that they drew from their data were in error.
Motivated by the 25Mg(p, gg ) studies of Endt et al.
(1986), improved, higher-resolution ( 3 He,d) measurements were performed (Champagne et al., 1986a) that
produced a more reliable set of spectroscopic factors
and reaction rates. A notable outcome of this work was
the identification of a strong s-wave resonance at an energy of E c.m. = 58 keV which substantially increased the
production of 26Al at low temperatures. However, Endt
and Rolfs (1987) calculated reaction rates that were in
disagreement with these results, primarily because of
their use of square-well single-particle widths, but also
because they overestimated several important spectroscopic factors. The situation was clarified by Champagne
et al. (1989) and Rollefson et al. (1990) who produced
rates consistent with one another and with Champagne
et al. (1986a) while also extending their analyses to include weakly populated states. Finally, Iliadis et al.
(1996) derived reaction rates from the existing data byemploying a realistic diffuse optical model potential in
the calculation of single particle widths. Although theresults obtained were virtually identical to those ofChampagne et al. (1989), the treatment of single particle
widths by Iliadis et al. (1996) has the advantage of selfconsistency. Though laborious, this approach appears tobe the best method for extracting resonance parametersfrom stripping data.
Proton capture will produce 26Al in either its longlived ground state ( 26Al 0 , T 1/257.23105 y) or in its
short-lived first excited state ( 26Al 1 , 6.35 s) and thesestates will not equilibrate at low temperatures (Wardand Fowler, 1980). Therefore, gamma ray branching ratios (Endt and Rolfs, 1987) must be employed in order
to separate the total reaction rate into separate contributions for each state. The most recent evaluations ofthese rates is given by Iliadis et al. (1996) and the results
are similar to the tabulation found in Iliadis et al. (1990).
However, they are significantly smaller (by factors of 3
to 6 for T 9 =0.05–0.1) than the rates found in CF88.
(3) The 26Mg(p, g ) 27Al reaction
Low-energy resonances are also found to make important contributions to the rate of the 26Mg(p, g ) 27Al
reaction. States corresponding to resonances below 105
keV have been detected using the 26Mg( 3 He,d) reaction (Champagne et al., 1990). Unfortunately, only an
upper limit was obtained for the strength of a possible
90 keV resonance. The resulting uncertainty in the reaction rate is a factor of 37 at T 9 =0.04 and grows to afactor of 74 at T 9 =0.06 before falling at higher temperatures. This is particularly troublesome because it directly
affects predictions of the the 26Al/ 27Al production ratio.
Further measurements are certainly called for. Direct(p, g ) measurements have also been carried out (Buchmann et al., 1980; Keinonen and Brandenburg, 1980; Iliadis et al., 1990), and in contrast to the situation at lower
temperatures, the rate at higher temperatures (i.e.,
T9>0.1) is well established. The rate given in CF88 wasbased upon preliminary results for the low-energy reso-

1028

Wallerstein et al.: Synthesis of the elements

nances and differs by a factor of 0.45–2 from the present
upper limit appropriate for temperatures T 9 =0.05–0.1.
(4) The 26Al0(p, g ) 27Si reaction
Initial measurements of the 26Al0 (p, g ) 27Si reaction
were carried out by Buchmann et al. (1984) and a more
complete investigation was made by Vogelaar (1989).
However, the rate for T9<0.15 is quite uncertain, again
because of the possibility of several unobserved lowenergy resonances. None of these states were detectedin ( 3 He,d) measurements (Vogelaar, 1989, Vogelaar
et al., 1996), but they have been treated theoretically byChampagne et al. (1993). Although the results of thesecalculations seem justified, there is little in the way ofexperimental information (in particular, spins, parities,
or spectroscopic factors) that would support them. Improved ( 3 He,d) studies, using more highly enriched26
Al targets will be needed before uncertainties in thisrate can be reduced.
(5) The 27Al(p, g ) 28Si and 27Al(p, a ) 24Mg reactions
Once 27Al is synthesized, a (p, a ) reaction may act toclose the MgAl cycle, or alternatively, the reaction flow
will proceed to higher masses via a (p, g ) reaction.
Resonances at E c.m.=72 and 85 keV (Endt90) could playa role in both reactions at low temperatures, but again
they lie too low in energy to be observed in proton capture. The corresponding states in 28Si were studied via( 3 He,d) and ( a ,t) spectroscopy (Champagne et al.,1986b) and their particle decays have also been measured (Champagne et al., 1988). Although the original
study (Champagne et al., 1986b) questioned whether either state possessed any single particle strength, themagnitude and shape of the ( 3 He,d) angular distribution for the lower state may imply a single particle contribution. Consequently, the rate used in the review ofArnould et al. (1995) includes a lower limit for thestrength of the 72-keV resonance. As a result, the possible 85-keV resonance was found to be too weak to beof astrophysical significance.
Between these low-energy states and the known(p, a ) resonance at E c.m.=487 keV lie several states thathave been observed in both the 27Al(p, g ) 28Si and 24Mg
( a ,g ) 28Si reactions. This means that they must be resonances in the 27Al(p, a ) 24Mg reaction. Limits on theirstrengths from direct (p, a ) measurements (Timmermann et al., 1988) and complementary information frommeasurements of particle decays (Champagne et al.,1988) indicate that all can be ignored except for resonances at 196 and 215 keV. However, the strengths ofthese two resonances are unknown and this is a major
source of uncertainty in the reaction rate. Subthresholdresonances may also be ignored [though note that theupper limits quoted by Timmermann et al. (1988) are infact lower limits]. The strengths of higher energy resonances are listed in Endt90.
The reaction rate is uncertain by about a factor of six
for T9<0.07, primarily because of the upper and lower
limits on the strength of the 72-keV resonance. At
higher temperatures, the uncertainty grows dramaticallyas the 196 and 215 keV resonances come into play. As
was discussed by Champagne et al. (1988), the rate apRev. Mod. Phys., Vol. 69, No. 4, October 1997

FIG. 9. Time evolution of mass fractions for T 9 =0.3, r =6000
g/cm 3 , and O-Ne initial abundances.

pearing in CF88 is based in part on erroneous data andhence bears little resemblance to the rate derived fromcurrent information.
From these same data sets it is possible to obtain the27
rate for theAl(p, g ) 28Si reaction. Beyond
T9=0.07–0.08, the (p, g ) reaction is much faster than thecompeting (p, a ) reaction. For lower temperatures, the(p, a ) reaction is faster by about a factor of 4, but thecycle time is significant. Hence the MgAl reactions will
behave more like a reaction chain than a reaction cycle.
2. Network calculations

Network calculations for the MgAl cycle have been
performed for the same conditions discussed above for
the NeNa cycle and results are displayed in Fig. 9. The
main effect of the MgAl cycle at low temperatures is the
production of 26Al at the expense of 25Mg. This also
leads to 26Mg production following the b 1 decay of
26
Al1 . If the rate for the 26Mg(p, g ) reaction is near its
present upper limit, then 26Mg suffers a net depletion ofa factor of 2.3 while 27Al increases by a similar factor.
Neither the 27Al(p, g ) nor the 27Al(p, a ) reaction has
much effect on the final abundance of 27Al since bothreaction times are longer than the time required to deplete hydrogen (in this simulation). Hence the MgAl

Wallerstein et al.: Synthesis of the elements

cycle acts as individual reactions terminating in 26,27Al.
Rather high 26Al/ 27Al ratios are achieved (0.38–0.95).
The uncertainty in this ratio stems primarily from theuncertainty in the 26Mg(p, g ) rate and, to a lesser extent, from 26Al(p, g ).

E. High-temperature behavior
1. Reaction rates

For temperatures in excess of T 9 =0.2, nuclear burning
proceeds in more or less a continuous flow through the
Ne-Si region to higher masses. The distinguishing feature of this scenario is that the reaction path is moved tothe proton-rich side of stability. As was shown in Fig. 7,
a number of (p, g ) reactions involve short-lived nuclei,
namely 21Na, 22,23Mg, and 23,25Al. Since these reactions
have not been measured directly, we presently rely uponestimated reaction rates.
A number of authors (Wiescher and Langanke, 1986;
Wiescher et al., 1986; Wiescher et al., 1988, van Wormer
et al., 1994; Herndl et al., 1995; Iliadis et al., 1996b) haveexamined the reactions in this mass region and haveused similar procedures for obtaining resonance parameters. All make use of the known properties of states instable isobars. In some cases, analog assignments werecomplicated by the fact that the states of interest, expected on the basis of the level structure in the analog
nucleus, had not been observed and shell-model calculations were used in order to predict their locations. Proton widths were calculated from both experimental andtheoretical single-particle spectroscopic factors. In thecase of the 21Na(p, g ) reaction, the magnitude of theenergy shift between analogous states in 22Mg and 22Na
was used to calculate single-particle reduced widths(Wiescher and Langanke, 1986). This procedure assumes that the states in question are primarily puresingle-particle states and this must be justified on a state
by state basis. For the 22Mg(p, g ) reaction, estimatedspectroscopic factors were initially employed (Wiescher
et al., 1986), but were later improved upon by usingshell-model predictions (Wiescher et al., 1988). With the23
Mg(p, g ) and 25Al(p, g ) reactions, it was possible to
make use of measured spectroscopic factors, obtained
from 23Na(d,p) and 25Mg(d,p), respectively (Wiescher
et al., 1986, Iliadis et al., 1996b). Finally, in the case of
the 23,24Al(p, g ) reactions, spectroscopic factors were estimated from assumed nuclear structure systematics(van Wormer et al., 1994), and from shell-model calculations (Herndl et al., 1995).
Values for g widths were obtained from values fortypical g ray transition strengths. The latter resulted
from shell-model calculations of B(E2) factors as well
as from simple estimates based upon the known lifetimes of the analog states. Because changes in excitationenergies or widths may easily change a reaction rate byorders of magnitude, it is clear that the rates for thesereactions must be periodically reexamined as new experimental information becomes available.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1029

2. Network calculations

Network calculations were again performed for constant temperature and density, in this case T 9 =0.3 andr =6000 g/cm 2 . These conditions might be encounteredat the peak of a nova outburst on an O-Ne white dwarf.
Initial abundances were taken from Politano et al.
(1995). Abundances were tracked until just after hydrogen exhaustion, which occurred after about 200 s. Notethat the time scale is set by the rate of hydrogen consumption in the CNO region rather than in the Ne-Si
region. Also, it should again be emphasized that thiscalculation represents an artificial situation. Nonetheless, it should illustrate some of the behavior that would
be found in a more realisitic treatment. The results areshown in Fig. 9.
Burning starts with a rapid decline in 21Ne. This isfollowed by a period of near equilibrium between 21Ne
production [from 20Ne(p, g ) 21Na( b 1 ) 21Ne] and destruction via the 21Ne(p, g ) reaction. This situation persists
until the abundance of 20Ne starts to drop (at about 10s) which slows the production of 21Ne. The equilibrium
at 21Ne results in a slow increase in the 22Na abundance.
As 21Ne falls out of equilibrium and is destroyed, the22Na abundance also falls as its production can nolonger keep pace with the 22Na(p, g ) reaction. Similar
behavior is observed for 23Na. Its increased abundance
after 1 s is a consequence of the b 1 decay of 23Mg following the 22Na(p, g ) reaction. The decrease in 22Na
following 20Ne slows production of 23Na while it is being
destroyed by 23Na(p, g ) and 23Na(p, a ). After hydrogen
exhaustion, the fast b 1 decays of 21Na and 22,23Mg produce rapid increases in the abundances of 21Ne and
22,23
Na.
The abundance of 24Mg displays a rapid initial dropfrom 24Mg(p, g ) and later production after b 1 decay of
24
Al. The 25Mg abundance is initially nearly flat as the25
Mg(p, g ) is almost offset by production of 25Mg
through the b 1 decay of 25Al. However, this causes significant production of 26Al which for a short time is thesecond-most abundant nuclide in the MgAl region (after
24Mg). At these temperatures and densities, the timerequired for 26Al 0 to come into equilibrium with 26Al 1
is long compared with the overall burning time and thus
these two levels may still be considered independently(Ward and Fowler, 1980). Following this period, 26Al
evolves in parallel with 25Mg. The drop in 24Mg diminishes production of 25Mg and results in a decrease inboth 25Mg and 26Al. This trend is reversed for a shorttime following the production of 24Mg, as a result ofmaterial moving out of the NeNa region. The 27Al abundance shows the effects of competition between the26
Mg(p, g ) reaction, and the 27Al(p, g ) and 27Al(p, a )reactions. The b 1 decay of 26Al 1 produces a slow rise in26
Mg followed by a sharp increase after burning hasceased. Similar effects are observed for 24,25Mg and
26,27
Al.
As was mentioned above, under these conditions thereaction network is characterized by a flow through the
Ne-Si mass region rather than by reaction cycles. Only

1030

Wallerstein et al.: Synthesis of the elements

about 0.2% (by mass) of the initial NeNa material survives, primarily as 20Ne. About 0.04% of the MgAl material remains, mostly as 27Al.
F. Conclusion

Many of the developments that have lead to our continuing interest in the NeNa and MgAl cycles could nothave been anticipated at the time of B 2 FH, but arenonetheless direct descendents of this initial framework.
It is possible that there are aspects to nucleosynthesis in
this mass region that remain to be uncovered. It is clearthat there is still a significant amount of experimental
and theoretical nuclear astrophysics to be done before
we will understand such things as the origin of 26Al andthe mechanism behind the Na/Al abundance anomalies
in meteorites. Many reaction rates are still uncertain,
particularly at the extremes of temperature. For low
temperatures, the problems seem tractable. However,calculations of nucleosynthesis at high temperatures require a wealth of information, such as masses, lifetimes,
and decay schemes, in addition to reaction rates. Collecting these data requires the development of new experimental techniques, for example, the use of radioactive beams. Gaining insight from them will entail theconstruction of increasingly sophisticated models.
VIII. OBSERVATIONAL EVIDENCE OF HYDROGEN
BURNING

There can be no denying that hydrogen burning provides the primary source of energy in the sun. No otherprocess could maintain the sun’s luminosity of 431033
ergs sec21 over a time scale of 4.53109 years (Bethe,
1939). Although the neutrino spectrum from the sun has
shown certain discrepancies when compared to predictions (see Introduction), these problems are likely to besolved as our understanding of both solar physics andneutrino physics improves (Haxton, 1995).
Since hydrogen burning dominates the energy generation of main-sequence stars and their early evolution it isworthwhile to mention some aspects of their evolution
to supplement Sec. III of this review. Stars of more thanabout 20M ( suffer mass loss at a rate that depends sensitively on the mass, reaching 1025 M ( yr21 for masses
near 50M ( (Cassinelli, 1979). The mass loss is so substantial that the radiative outer layers may be lost before
the core has converted its H to He. As the products ofnucleosynthesis reach the stellar surface the star first becomes an OB-peculiar star with excessive N (Walborn,1987) and then a Wolf-Rayet star whose surface composition is dominated by He and N. The latter species isgreatly enhanced by the conversion of C and O during
the CNO bi-cycle (Abbott and Conti, 1987). During
their further evolution with an increasing mass-loss rate
their He and N enriched envelope is lost and they become Wolf-Rayet stars of the carbon type, revealing the
products of helium burning reactions.
Further down the main sequence (near 0.8M ( ) are
the stars in globular clusters that are evolving off theRev. Mod. Phys., Vol. 69, No. 4, October 1997

main-sequence after spending about (11–13)3109 years
burning H in their radiative cores (Reid, 1997). Our understanding of the evolution of globular cluster stars has
been vastly developed since the groundbreaking paperof Hoyle and Schwarzschild (1955) and has most recently been reviewed by Vandenberg et al. (1996). Afterthe evolving star has crossed the subgiant branch its luminosity increases rapidly as the outer convection zone
penetrates to greater depths and begins to dredge up
material that had been hot enough for the CN cycle toconvert C to N (Iben, 1967a). This process is commonly
called the ‘‘first dredge-up.’’As the star evolves up the red giant branch, additional
mixing occurs (to be discussed later in this section) until
the now degenerate core reaches about 803106 K andthe triple-alpha reaction triggers the ‘‘helium flash.’’ The
star then settles down on the horizontal branch of the
HR diagram burning helium in its core and hydrogen ina shell that surrounds the core. When the helium in thecore is depleted the star brightens along the asymptotic
giant branch as described by Iben in Sec. III of this review.
Hydrogen burning has successfully predicted many
phenomena that have been seen in the changes ofchemical composition of the observable surface layers ofstars. Two examples of the comparison of stellar observations with the theory of stellar structure, nuclear reaction rates, and mixing processes are the ratio of 12C/13C
and the observed excesses of Na and Al in globular cluster stars. We first discuss the change of the 12C/13C ratios
in stars as they evolve from the main sequence to thefirst-ascent giant branch. As stars advance up the red
giant branch the convection zone deepens until it
dredgesupmaterial
in
which
both
12
C (p, g ) 13N ( b +n ) 13C and 13C(p, g ) 14N had been active (Iben, 1967a; Dearborn et al., 1976; Sweigert et al.,1989). The ratio of rates of these reactions is insensitive
to temperature so that the calculated 12C/13C ratio at any
depth in a stellar model is very robust. The depth of
mixing in the outer hydrogen convection zone increases
as the star evolves to become a red giant, thereby bringing material with enhanced 13C to the stellar surface. For
small mass stars this occurs rather suddenly as the starleaves the subgiant branch and starts to evolve to higher
luminosities.
The theory of mixing in the outer regions of stars is
not simple. If we look at stellar models of cool stars
from the outside inward we see that in the outermost
layers, say with an optical depth less than unity, heat istransferred by radiation with most of the radiation escaping the star. For temperatures starting around 7000–
8000 K, photoionization of hydrogen from the first andsecond excited states causes the opacity to rise rapidly inthe wavelength regions of maximum flux. This causes
the temperature gradient, which is proportional to theopacity in a radiative atmosphere, to rise beyond thepoint of convective instability (Schwarzschild, 1906). So
long as the radiative temperature gradient exceeds thelimit set by

Wallerstein et al.: Synthesis of the elements

dlogT
;0.4
~ dlogP ! ad

1031

(14)

convection will take place. The fraction of energy carried by convection will depend on the density and themixing length, defined as the distance over which a convective element travels before losing its identity. Theoretical estimates of the mixing length are difficult to calculate but we know enough about the sun, i.e., its mass,
radius, luminosity, helium content, and vibration modes
to establish a ratio of mixing length to scale height ofl/h.1.75.
The application of the solar mixing length to red
giants is questionable. The low density in a red giant’s
atmosphere means that convection will be less efficient.
Hence radiation will still carry a substantial fraction of
the flux and convective velocities will be higher than in
the sun to compensate for the low density. Furthermore,
the contribution to the opacity by bound-bound transitions in the sun in the 105 –106 K region (Rogers andIglesias, 1992) depends on the metal abundance, whichis reduced by a factor of as much as 200 in globularclusters and metal-poor fields stars, thus reducing theiropacity and radiative temperature gradient. Hence theapplicability of the solar ratio of l/h to red giants ofradius 10–50 R ( (and hence a surface gravity reduced by
102 to 2.53103 ) is questionable.
Convective overshooting at the bottom of the formal
convection zone may be significant. The higher densities
at those depths and positive density gradient tend tolimit the overshooting to a fraction of a scale height.
Overshooting is not sufficient to extend it to depths at
which the 12C once on the surface will be convected
down to where it will be subjected to proton capture,i.e., where the temperature is more than about 1.53107
K.
The problem is how to mix material from the bottomof the convection zone to deeper layers. This problem isalso of importance in the sun and other main-sequencestars of similar mass and age to the sun because the low
Li abundance in the solar atmosphere cannot be explained by standard convection theory. Mixing must be
induced by another effect. Two possibilities have beensuggested. These are turbulence associated with a gradient in solar rotation and ionic diffusion (Richard et al.,1996).

FIG. 10. Theoretical predictions of 12C/13C ratio for a 1.4M(
star by Sweigert et al. (1989) for two different compositions.
(Courtesy of the Astrophysical Journal.)

The real stars follow the models up the giant branchwith just about the predicted 12C/13C ratio until theyreach T eff.4200 at which point additional mixing togreater depths further lowers the 12C/13C ratio to near
10. Models with standard convection theory fail to predict this additional mixing. Recent progress on the problem of mixing in small mass red giants and their 12C/13C
ratios has been made by Charbonnel (1995) and by Boothroyd and Sackmann (1997).
The analysis of stars in many open clusters of differing
ages and hence differing turn-off masses by Gilroy(1989) shows that the standard theory holds well formasses greater than about 2.0M ( but fails to predict thelower 12C/13C ratios in the smaller mass stars. The extreme example is the metal-poor red giants analyzed by

A. CN cycle and mixing

A good example in which the models may be compared to theory is the rather old open cluster, M67. In
Fig. 10 we show the results of calculations of the 12C/13C
ratio for evolving stars in the theoretical HR diagram.
The initial ratio was assumed to be 89, the solar system
value. The ratio drops rapidly as the star evolves off thesubgiant branch and remains near 25 for its entire lifetime on the giant branch as well as on the ‘‘clump’’ ofpost-helium-flash models (Sweigert et al., 1989). Observations of the 12C/13C ratio by Gilroy and Brown (1991)
are shown in Fig. 11 in the observational HR diagram.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

FIG. 11. Member stars of the cluster M67. The 12C/13C ratio
for each star is shown beside the corresponding star. The color
index, B-V, transforms to logT eff with B-V=1.0 and 1.3 representing logT eff=3.70 and 3.65, respectively. (Courtesy of the
Astrophysical Journal.)

1032

Wallerstein et al.: Synthesis of the elements

Sneden et al. (1986). Those stars, with metallicities similar to globular cluster stars, probably have masses near
0.8M ( . They show 12C/13C ratios near four, which isnearly the equilibrium abundance ratio under proton
burning by the CN cycle. In globular clusters the ratio
can be measured only for stars near the red giant tip ofthe color-magnitude diagram, where the ratio is found tolie in the range of 4 to 6. Such small ratios indicate that
not only has fresh 13C been scooped up but that most of
the visible material on the stellar surface has been carried down to burning depths so there has been almost no
dilution of material processed at depth. Further evidence for depletion of 12C by proton capture is provided
by the analysis of giants from M v 510.8 to 22.2 in theglobular cluster NGC6397 showing steady C depletionas stars evolve up the red giant branch (Briley et al.,1996).
B. O depletion and the enhancement of Na and Al

Within the past few years proton captures by various
isotopes from oxygen through magnesium have been invoked to explain the low oxygen combined with excess
Na and Al in globular cluster red giants (Kraft, 1994;
Kraft et al., 1997 and references therein). At the time of
the writing of B 2 FH no member of a globular cluster
had been analyzed in detail for its chemical composition.
Multicolor photometry and low resolution spectroscopy
had strongly suggested that globular cluster stars weremetal deficient, probably similar to the subdwarfs analyzed by Chamberlain and Aller (1951). Using spectra
obtained by Greenstein with the then new 200 inch telescope and exposures of up to three full nights, Helfer
et al. (1959) found metal deficiencies of a factor of 20
and 200, as compared with the sun for stars in M13 andM92. Of the various elements studied no substantial discrepancies of their ratios to iron (as compared to thesolar ratio) were found among the elements studied except for a deficiency of Ba and the rare earths in their
M92 star whose [Fe/H] value is 22.3. Oxygen, sodium,and aluminum were not among the species for whichuseful data were available. High resolution spectroscopyof numerous globular cluster stars was not possible until
the 4 m telescopes with their echelle spectrographs andimage tube detectors became operational in the mid
1970’s. That combination made globular cluster stars accessible in the spectral region extending to about 8000 Å
at high resolution (but usually with lower signal to noise
than the observers realized). By the mid 1980s CCD detectors replaced the image tubes and their photographic
plates.
The first hint of unexpected proton captures by 16O
and heavier species was reported by Pilachowski et al.
(1980) from the study of five red giants in the globularcluster M13, where a maximum difference of 1.5 dex in[O/Fe] was reported for the most extreme stars in theirsample. The large spread of O/Fe in M13 was successively confirmed with improving accuracy by Leep et al.
(1986), Hatzes (1987), and Brown et al. (1991). In definitive studies Kraft et al. (1993; also Kraft et al., 1997)
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

found a spread from [O/Fe]=+0.46 to 20.71 among thered giants in M13. They attributed the O-poor stars to
ON cycling and mixing of processed matter to the surface.
The evidence for Na enhancements in the O-poor
star, II-67, in M13 also dates from 1980 (Peterson, 1980).
She found [Na/Fe]=+0.69 in the O-poor star II-67 while[Na/Fe]=20.34 and +0.13 in two O-normal stars. A followup large scale study by Kraft et al. (1993) showed aclear correlation of Na excess with O deficiency in manystars in M13 and a few stars in M3.
In M10, a cluster with a very blue horizontal branch,like M13, Kraft et al. (1995) have found [O/Fe] to range
from +0.5 to 20.35 in an analysis of 14 stars. For six
stars with [O/Fe]>O, [Na/Fe]=20.6560.14, while forseven stars with [O/Fe]<O, [Na/Fe]=0.2360.08. A variety of other species from Si to Ni show no correlation
with [O/Fe]. Aluminum lines were not observed. In
none of these clusters was a significant spread in [Fe/H]
detectable.
Perhaps the most interesting database is that of Norrisand Da Costa (1995, NDC) for Omega Centauri (seealso Zucker et al., 1996), the most massive globular cluster in our Galaxy. Omega Cen is the only globular toshow a wide spread in metallicity, certainly due to selfenrichment. NDC analyzed 40 red giants with a range in[Fe/H] from 21.9 to 20.8. For species such as Si, Ca,
and Ti the spread in [X/Fe] is no more than might be
expected from observation and analytical uncertainties
and no trend with [Fe/H] is discernible. The same holds
for four iron-peak elements. The [O/Fe] value ranges
from +0.5 (as commonly seen in metal-poor stars) down
to ,20.4 throughout the entire range of [Fe/H]. Moststars are depleted in carbon and enhanced in nitrogen,as is expected by CN and CNO cycling. The sum of
C+N+O appears to be constant, near +0.3 dex, as is usually the case for metal-poor stars. Of course this points
to a rearrangement of the CNO nuclei during proton
captures. There are no trends with [Fe/H] to indicatethat the cause of the changes is related to the general,and presumably gradual, metallicity enhancement in thecluster.
When the abundances of Na, Mg, and Al are examined the plot thickens. For Na the correlation of [Na/Fe]
with [O/Fe] is similar to that in M13, running from [Na/Fe]'+0.6 for stars with [O/Fe] between 20.1 and,20.4 to [Na/Fe] ranging from 20.3 to +0.3 for thestars with [O/Fe] between 0.0 and 0.6. For aluminum therange is larger. The O-poor stars show [Al/Fe] as large
as +1.0 while the O-normal stars show [Al/Fe]=+0.4 toabout 20.2, with seven Al abundances being only upper
limits. These trends were confirmed and strengthened by
Zucker et al. (1996) who observed some additional Al
lines not covered by NDC. In Omega Cen the correlation of Mg and Al is really strange. There is a clear
break in [Al/Fe] at +0.5 dex. The low-Al stars range
from [Al/Fe]=+0.4 to &20.2. All of them show
[Mg/Fe].+0.45 which is normal for metal-poor stars.
The Al-rich stars with [Al/Fe]=+0.7 to +1.1 show a rangeof [Mg/Fe] values from +0.5 to 20.1 with no correlation

Wallerstein et al.: Synthesis of the elements

TABLE IV. Abundances on the logN scale with logN(H)512
for O-rich (i.e., normal) and O-poor (i.e., depleted) stars in
M13 and Omega Centauri.
logN
ObjectSun a
M13, O-rich b
M13, O-poor b
Omega Cen, O-rich c
Omega Cen, O-poor c
Omega Cen, O- and Mg-poor c

16

20

O
Ne
8.9 8.1:
7.6 6.8:
6.5?
7.6 6.8:
6.9
?
6.8
?

23

Na
6.3
4.5
5.1
4.6
5.5
5.2

24

Mg
7.6
6.3
6.0
6.4
6.4
6.0

27

Al
6.5
4.9
6.1
4.9
5.8
5.8

a

From Anders and Grevesse (1989).
Assuming [Fe/H]=21.6.
cAdjusted to [Fe/H]=21.6 using individual [Fe/H] values
from NDC.
bwith [Al/Fe]. If Al is produced by p captures on 24Mg
(as will be discussed below) this distribution is very
strange because it indicates that the Mg abundance isunchanged as the Al abundance increase by a factor of
10, except in five stars in which Mg was depleted by afactor 3 as Al increased by the same amount as it did in
the Mg-normal stars. The Mg-poor stars show moderate
metal deficiencies with [Fe/H] ranging from 21.67 to
21.35, [Na/Fe] excesses near +0.5 and large oxygen
depletions. One problem is that there is no really securebaseline for [Al/Fe] which can be described as ‘‘normal’’for a metal-poor globular. The Mg-deficient stars aresome of the brightest red giants in Omega Cen. In moderately metal-poor main sequence stars, the [Al/Fe] ratio is slightly positive, about +0.2 dex, but ranging from
0.0 to +0.4 (Edvardsson et al., 1993). For dwarfs with
[Fe/H] between 21 and 22, [Al/Fe] is about +0.1
(Francois, 1986).
In M13 there are similar correlations (Kraft et al.,1997, and references therein). As [O/Fe] descends from
+0.4 to 20.2, [Na/Fe] rises from 20.2 to +0.4 and remains near +0.4 for the most O-poor stars at [O/Fe]=
20.8. Metal-poor field giants do not show a significant
spread in [O/Fe] but show the usual [O/Fe].0.3, though
Na ranges from [Na/Fe]=20.6 to 0. In M13 the ratio of
Al/Fe shows a tighter correlation with oxygen in thesame sense as does Na with [Al/Fe] ranging from 0.0 at[O/Fe]=+0.4 to +1.2 for [O/Fe] between 20.2 and 20.8.
Magnesium shows the standard excess of [Mg/Fe]=+0.3
for 0.4*[O/Fe]*20.1 and then drops to 0.0 for [O/Fe]&20.2. At this point it is best to revert to absolute abundances. In Table IV we show the logarithmic abundances of the critical species, O, Na, Mg, and Al on theusual arbitrary astronomical scale of logN(H)512.
The oxygen-rich stars show abundances similar tometal-poor field stars and similar to metal-poor stars inmany other globular clusters. Hence the O-poor stars
are the abnormal ones.
C. Origin of the Na and Al enhancements

If the excesses of Na and Al seen in the O-poor stars
are due to proton captures and subsequent mixing
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1033

within the observed stars themselves, Table IV tells us
what must have happened. First we must look at thenuclear physics and secondly we will look at the problemof mixing. In 1957 Marion and Fowler suggested that
under the right conditions a NeNa cycle could occur by
successive proton captures by the Ne isotopes followedby b decays and ending with 23Na(p, a ) 20Ne. The analogy to the CN cycle is obvious. The NeNa cycle is not
likely ever to be a major energy source but it can readily
convert significant amounts of 22Ne into 23Na (provided
that the star’s original 22Ne/ 23Na ratio was similar to thesolar ratio of 4.1) or 20Ne to 23Na by three p captures
and a sufficiently low temperature or low cross section
for p capture by 23Na. Denissenkov and Denissenkova(1990) were the first to suggest that the NeNa cycle may
be effective in cool stars. There have been many follow
up reviews such as Briley et al. (1994), Kraft (1994), andKraft et al. (1997), the first concentrating on CNO andthe latter two on the Na-Mg-Al problem.
The tough question is the production of the excess Al.
In M13, Al must rise from logN54.9 to logN56.1, while
in Omega Cen the rise is from 4.9 to 5.8. If the source is24
Mg, approximately half of the original 24Mg must havebeen converted to 27Al in M13, leaving the other half as
Mg in accordance with the changes shown in Table IV.
This solution excludes any dilution by unprocessed material which would reduce the surface Al abundance andrestore the surface Mg toward its original value. For
Omega Cen the rise in Al must be from logN54.9 to 5.8
according to NDC and to 6.0 according to Zucker et al.
who measured four lines, rather than one or two by
NDC. In that case, a third of the original 24Mg musthave been processed. If the Mg-poor stars in Omega
Cen really are depleted from logN56.4 to 6.0, then conservation of nuclei implies that their Al abundance must
be logN56.2. These severe requirements strain credulity. To overcome them, Cavallo et al. (1996) suggestthat 20Ne, which presumably is present with logN56.9(assuming its ratio to iron is enhanced, as are O and Mg,
in most globular cluster stars), may be processed up to
Al. The trouble with their solution is the tendency for
the NeNa sequence to cycle because the reaction rate of23
Na(p, a ) is larger than the rate for 23Na(p, g ). The
rates are fully discussed in Sec. VII of this review by A.
Champagne. The most successful effort to date to explain the excess of Al (Cavallo et al., 1997) requires that24
Mg be the source of the Al excess. For M13, as shownin Table IV, the ratio of 24Mg in O-rich stars to 27Al in
O-poor stars is only 0.2 dex. Hence much of the original
24
Mg must be processed to 27Al; it must not cycle backto 24Mg and must not capture an additional proton to
produce 28Si.
If the observed stars did not enhance their Al (andperhaps the Na as well), then their observed abundances
must have been inherited from inhomgeneous enrichment very early in the lifetime of the cluster as suggested by Cottrell and Da Costa (1981) for 47 Tuc andNGC 6752. Additional evidence for an inhomogeneityof Na has been presented by Briley et al. (1996) for thecluster 47 Tucanae, the second most massive globular in1034

Wallerstein et al.: Synthesis of the elements

our Galaxy. They have measured the sodium D lines insix stars near the turnoff of the main sequence whereconvection cannot possibly mix material from the center
to the surface and where central temperatures are near
23107 K, too cool to run the NeNa cycle. The derived
Na abundances run from logN55.4 to 5.8, while the derived Ca abundances show no correlation with the Naabundances. Furthermore the Na abundances correlatewith the strength of the CN band at 3883 Å showing acorrelation of Na with the N abundance. Current models
have so far been unable to explain this quantitatively, soeither the models are seriously in error or the Na andCN excesses were present at the time of star formation.
Detailed calculations by Denissenkov et al. (1997) showthat metal-poor AGB stars of 5–8 M ( are able to produce (and presumably eject by a stellar wind) an excess
of 23Na but fail to produce the observed excess of 27Al.
Self-enrichment by globular clusters during their earlyyears has been shown to be very likely (Morgan andLake, 1989). With much interstellar matter still in thecluster, gas and dust ejected by massive stars may be
confined within a small fractional volume of the cluster.
After the intracluster matter has been blown away,
probably by supernova explosions, further ejecta fromstars will easily escape from the relatively weak gravitational well of the cluster. Massive stars are known tolose material at a prodigeous rate, especially as WolfRayet (WR) stars. The latter type show an excess of N,
C, O, and presumably 22Ne in their atmospheres. Linesof Na are not likely to be present in WR stars but the Al
III resonance lines at 1855 and 1863 Å were readily observable with the International Ultraviolet Explorer andmay easily be observed with the Hubble Space Telescope. Their analysis to derive the Al abundance in WR
stellar winds would be extremely valuable. The same
holds for the analysis of other hot stars with strong nitrogen lines known as OBN stars (Walborn, 1987).
Clearly we are seeing proton captures by 16O, and by
the Ne isotopes as first suggested by Marion and Fowler(1957)2 to enhance Na and by the Mg isotopes to produce the excess Al. The uncertain question is whether
the observed star is responsible for polluting its own atmosphere or did the star inherit its unusual composition
while still forming from a nearby rapidly evolving neighbor? For 16O there is no question but that CNO cycling
is responsible because the sum of C+N+O is the same in
the O-rich and O-poor stars in M13 (Brown et al., 1991);
however, that does not tell us where or when it happened.
IX. CARBON, NEON, OXYGEN, AND SILICON BURNING
A. Introduction

In addition to producing 12C and 16O, helium burning
was identified by B 2 FH as the source of 20Ne and posThe possibility of the reaction 22Ne(p, g ) 23Na was noticed
by Bethe (1939, Table VI).
2

Rev. Mod. Phys., Vol. 69, No. 4, October 1997

sibly some 24Mg. It is now known that there is no suitable excited state of 20Ne near the a 1 16O threshold toserve as a resonance for a capture on 16O. (See Sec. VIof this review.) Formation of nuclei heavier than 16O
then proceeds successively by core carbon, neon, oxygen, and silicon burning. (Of these processes, only neonburning was included in B 2 FH.)
An important new feature of the carbon burning andensuing stages is that the star’s dominant luminosity(i.e., its energy loss) is by neutrino emission directlyfrom the core of the star, rather than electromagnetic
radiation from the star’s surface. A natural result of thisprodigious luminosity is that the star’s core temperature
is increased, evolution is greatly sped up, and carbon
burning and the following stages have successively
shorter time durations.
The exhaustion of carbon in the core is followed at
T9;1.3 by a process called neon burning, in which theambient blackbody radiation photodisintegrates thematter present with the lowest a threshold energy, 20Ne,
with a ( g , a ) threshold of only 4.7 MeV. The a particles
liberated in this way can react with 20Ne to form 24Mg.
Although the photodisintegration is endoergic, the exoergic reaction that forms 24Mg has a higher energy release (Q value), and the overall combination of the tworeactions is exoergic.
Following neon burning, the core contracts and heats
to T9;2 when oxygen can begin to burn. The principal
immediate products are 28Si, 31P, and 31S, but a large
range of nuclei are also built up to A;40, with the A=4n nuclei favored by their higher binding energies. Together, neon burning and oxygen burning are the modern version of the a process of B 2 FH.
The temperature increases throughout oxygen burning and eventually, at T9;3–3.5, oxygen burning merges
into silicon burning, which can best be described as thephotodisintegration of nuclei to produce a particles,
protons, and neutrons from the nuclei present, beginning with those with the lowest thresholds, and finallyconsuming the most ‘‘refractory’’ nuclei like 28Si. The
freed alphas, protons, and neutrons build nuclei up to
the region of the most stable nuclei, the iron abundance
peak. If the buildup occurs rapidly, with little increase of
the neutron excess, h 5 ( i (N i 2Z i )/A i , the most prominent nucleus will be 56Ni. On the other hand, if thebuildup occurs slowly, h will increase during siliconburning from its initial value of about 1.531023 to 1022
or even higher, and the most abundant nucleus will shift
to 54Fe, 56Fe, or 58Fe for larger h (see Sec. XIV of thisreview).
The observation in g detectors carried by both satellites and high altitude balloons of g rays from SN1987a,
which
arise
from
the
positrondecays56
56
Ni→ 56Co1e 1 1 n e ,Co→ 56Fe1e 1 1 n e ,and57
Co→ 57Fe1e 1 1 n e , confirms the general correctnessof this picture (Matz et al., 1988; Rester et al., 1988; Mahoney et al., 1988; Teegarden et al., 1989; Sandie, et al.,1988).
A more detailed discussion of the scenarios for thevarious burning processes than space allows would dis-Wallerstein et al.: Synthesis of the elements

1035

tinguish between core burning and shell burning, andalso whether the burning is hydrostatic or explosive. In
the case of explosive nucleosynthesis and the comparison of its products with the spectrum of radiation fromstellar surfaces, or with solar system matter, such as terrestrial material and meteorites, it is always necessary toremember that much of the stellar material that has
evolved through the most advanced explosive evolutionary stages may remain ‘‘locked up’’ in some form ofsupernova remnant, such as a neutron star or black hole.
Other matter that may not have gone through advanced
evolution may be locked up in low mass white or browndwarfs.

B. Carbon burning

The suggestion that carbon burning would occur after
helium burning was made by Salpeter (1952). For carbon burning, the ‘‘Gamow Window’’ peaks at E 0 52.42
5/6
T 2/3
9 MeV and has a width D51.054 T 9 ; from stellar
models the temperature range for hydrostatic carbon
burning is 0.8&T 9 &1.2. The principle reactions are12

C1 12C→ 20Ne1 4 He14.617 MeV(15)→ 23Na1p12.241 MeV(16)→ Mg1n22.599 MeV.

(17)

23

Clearly, the neutron producing reaction will be theweakest of these, especially at lower temperatures.
Other 12C + 12C reactions are possible but are believed
to be much less important for a variety of reasons. The
freed a particles, protons, and neutrons can interact
with the nuclei present, including CNO nuclei from hydrogen and helium burning, to build a range of nuclei
including 20,21,22Ne, 23Na, 24,25,26Mg, 26,27Al, 28,29,30Si, and31
P, but the main products are 20Ne and 24Mg, the latterfrom a capture on 20Ne, or 23Na(p, g ).
After early studies at the Chalk River Laboratories(Almqvist et al., 1960), 12C+ 12C studies were resumed at
Caltech at lower energies (Patterson, Winkler, and Zaidins, 1969). By this time it was known that a state in20
Ne that had appeared to be a possible resonance forradiative a capture by 16O had a combination of J andparity that cannot be formed by 16O + 4 He, and thatformation of 20Ne by helium burning would not proceedwithout a much higher temperature than helium burning
would provide. It was then realized that carbon burning
would follow the production of 12C and 16O in helium
burning.
Numerous other studies of the 12C+ 12C reactions havesince been published (Dayras et al., 1977; Mazarakis andStephens, 1972; High and Cujec, 1977; Kettner et al.,
1980; Erb et al., 1980; Becker et al. 1981). The moststriking revelation of these studies is that the yield vsenergy curve (‘‘excitation function’’) displays a strongly
resonant structure, the origin of which remains obscure,notwithstanding many suggestions regarding this intermediate structure (see Fig. 12). The various experiments
also reveal rather poor agreement at the lowest energies,
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

FIG. 12. 12C+12C S factor vs center-of-mass energy. The data
shown are from Patterson et al. (1969), black dots and 3’s;
Mazarakis and Stephens (1972), plus signs; High and Cujec(1977), open squares; and Kettner et al. (1980), open circles.
The dotted curve is a rough sketch of the intermediate structure resonances. The dashed curve is an incoming wave boundary condition model, and the solid curve is a statistical model
fit of Caughlan and Fowler (1988).

and even the trend of the s (E) curve, or the S(E)curve, remains somewhat uncertain at the lower energies. The results of Mazarakis and Stephens led Michaud(1972) to postulate a new phenomenon called ‘‘penetration under the barrier,’’ attributable to a nucleusnucleus potential that had its imaginary (absorptive)
part extending to radii larger than the real part of thepotential. This was assumed in order to produce an enhancement of the cross-section curve above the conventional barrier penetration expectation because it would
effectively result in a thinner Coulomb barrier. However, later studies of 12C+ 12C and other heavy-ion reactions such as 12C+ 16O (Cujec and Barnes, 1976) did notsupport such low-energy enhancements. It seems possible that the observed effect was produced by a slightly
erroneous effective beam energy, as the s (E) curve is avery steep function of E at low energies. A review of the12
C+ 12C (and other heavy-ion reactions) has been published by Barnes et al. (1985); the average trend of thedata (ignoring the resonances) has been fitted with avariety of optical models that give somewhat different
values of the extrapolated cross sections. Fortunately,
the measured S(E) values cover the range E 0 6D/2 for
explosive carbon burning, and reach almost down to E 0
for hydrostatic carbon burning.
The 12C+ 16O reactions show some hints of resonant
structure at the lowest energies. These reactions only
become significant near 12C exhaustion, and thus play
only a small role, but they do help to improve agreement
between predicted and measured abundances.

1036

Wallerstein et al.: Synthesis of the elements

C. Neon burning

At the end of carbon burning, at T9;1.2, the remaining core nuclei are mainly 16O, 20Ne, and 24Mg. Before
oxygen can react with oxygen, the temperature becomeshigh enough to photodisintegrate 20Ne, because of its
unusually low a threshold energy (4.73 MeV). The freed
a particles can combine readily with 20Ne to form 24Mg.
Thus the reactions involved are
Ne1 g → 16O1 4 He,

20

Ne1 4 He→ 24Mg1 g

20

(18)

and are equivalent, in an energetic sense, to20

Ne1 20Ne→ 16O1 24Mg.

(19)

The overall products are rather similar to those fromcarbon burning, with the obvious exception that neonburning destroys neon, whereas carbon burning produces neon.
D. Oxygen burning

After neon burning has destroyed most of the coreneon, the temperature rises to T 9 ;2, where oxygen can
begin to fuse with oxygen. For oxygen burning, E 0 53.91
5/6
T 2/3
9 MeV and D51.34 T 9 MeV. The principal reactionsareHowever, the various data sets do not agree well with
one another at the lowest energies (Spinka and Winkler,
1972; Hulke et al., 1980; Wu and Barnes, 1984; Thomas
et al., 1986). From the point of view of explosive oxygenburning where T9;3.1–3.9, the experimental data covermost of the needed region of energies. For hydrostatic
oxygen burning, at T 9 ;2, less than half the needed energy region is covered by experiment, but the covered
region includes the larger cross-section portion. Amongthe many model fits that have been made, we cite thetime-dependent Hartree-Fock model calculations by Reinhard et al. (1984) and the parametrized statisticalmodel fit of Caughlan and Fowler (1988). These fitsagree within a factor of 2 over the energy range coveredby experiment as well as the range of energies needed
for both explosive and hydrostatic oxygen burning. Although this is reassuring, the experiments disagree
among themselves by more than a factor of 5 at thelowest energies, and exhibit different trends at these energies [see, for example, Barnes et al. (1985)]. It is understandable that the different experimental methods
pursued may suffer from different systematic errors, but
a substantial improvement in the reproducibility of thelow-energy data is clearly desirable.
E. Silicon burning16

O1 16O→ 28Si1 4 He19.594 MeV→ 31P1p17.678→ 31S1n11.500→ 30Si12p10.381→ 30P1d22.409,(20)

as well as several other branches with less favorable Q
values. The released a particles, protons, and neutronsinteract with the nuclei present to produce a large
spread of nuclei including 28Si, 32,33,34Si, 35,37Cl, 36,38Ar,
39,41
K, 40,42Ca, 46Ti, and 50Cr, of which 28Si and 32S are
the main results (Woosley et al., 1972, 1973; Woosley,
1986; Arnett, 1996). The calculation of Arnett (1996)
indicates that the neutron excess, h , increases fromabout 231023 to ;631023 during core oxygen burning.
Since such a large value of h would give a nucleosynthetic result that disagrees with observation, it seems
likely that the products of such core oxygen burning
must be only rarely, if ever, ejected into the interstellar
medium (ISM). Oxygen shell burning can proceed atlower density and higher temperature, up to T9;2.6–3,
and yields a distribution in better accord with solar system abundances (Arnett, 1996).
The burning process is accompanied by quasiequilibrium nuclear clusters that link together at higher temperatures; at the end of oxygen burning the nuclei forma cluster from 24Mg to the mass region 40&A&50,
where the closed nuclear shells at Z=20 and N=20 and
28 inhibit propagation of the cluster to the iron-peaknuclei (Woosley, 1986; Arnett, 1996).
The curve of S(E) vs E for the 16O+ 16O reactions
does not exhibit the resonant structure seen in 12C+ 12C.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

The silicon-burning process refers to the large groupof nuclear reactions that start with the products of theprevious burning processes and build the nuclei up to
the iron peak in the abundance curve. Photodisintegration of the nuclei present plays a crucial role as it liberates the a particles, protons, and neutrons needed tocarry out this buildup, first photodisintegrating the nuclei with the lowest thresholds and proceeding to themost refractory nuclei. An oversimplified but instructive
picture of silicon burning would be to consider the process of breaking down 28Si by ( g , a ), ( g ,p), and ( g ,n)processes, and building heavier nuclei by adding theseparticles in reactions such as ( a , g ), (p, g ), (n, g ),( a ,p), and ( a ,n) (Truran et al., 1966; Bodansky et al.,
1968; Clayton, 1968). Core silicon burning begins at
about T 9 =2.7, allowing time for the neutron excess per
nucleon, h , to increase while the temperature rises toabout T 9 =4 or higher. When the buildup reaches thesituation where most of the core has reached the ironpeak, the most abundant nuclei are likely to be 54Fe,
56
Fe, and 58Ni. Here we see the residual effect of the eprocess of B 2 FH. However, not only would this produce
too much iron-peak material if ejected into space, it isalso not a good fit to isotopic abundance observations.
Thus, this material must remain part of the ensuing supernova remnant if current theory is correct. As we havenoted earlier, explosive silicon burning produces significant amounts of the N5Z nucleus 56Ni, which decays to56
Co and then to 56Fe. These and other radioactivities
power at least some supernova light curves (e.g., SN
1987a). (See, for example, Woosley et al., 1981.)
During core silicon burning, helium, carbon, and oxygen shell burning can occur in massive stars, and theWallerstein et al.: Synthesis of the elements

elements found by optical or x-ray observations to beenhanced in many supernova nebular remnants fit rather
well to oxygen-shell oxygen burning, enhanced somewhat by shock wave induced explosive burning.
As the number of nuclear reactions involved in siliconburning is so large, and as many would involve very
short-lived radioactive targets, the response of the experimental nuclear astrophysics community is to measure as many of the reactions as resources permit, chosen both for their own intrinsic importance, andespecially as guides for theoretical efforts to devise models that can predict nuclear reaction cross sections withacceptable precision. Such models rely on a high densityof states in the compound nucleus, and try to determine
the way the particle decays of these states distributethemselves among the energetically allowed final states(channels). These statistical models, usually referred toas Hauser-Feshbach models, work best for A*30 nuclei
interacting with protons or alpha particles, but work
quite well also for a few lighter nuclei with unusuallyhigh compound nucleus excitation energy, i.e., where thecompound nuclear level density is high.
Briefly, the aim of these models is to average overcompound nucleus levels. The cross section for the reaction I m 1j→L n 1k, where j and k are particles with noexcited states, can be written as (Hauser and Feshbach,1952; Vogt, 1968; Woosley, 1986)
mns̄ jk~ E mj ! 5p l” 2Ij~ 2J Im 11 !~ 2J j 11 !

3

( ~ 2J c 11 !

JcP

T mj ~ J c P ! T kn ~ J c P !
T tot~ J c P !

.

(21)

T mj (J c P) is the transmission coefficient for the incident
channel to form the compound nuclear state with angular momentum J c and parity P. T tot(J c P) is the sum of
all transmission coefficients, including that of the entrance channel. The summation indicated is over all J c
and P in the compound nucleus. The transmission coefficients are calculated from a Schrödinger equation with
proton, neutron, or a particle potentials provided as input. The calculation also requires a level density formulain order to carry out the indicated summation (or integration). If temperatures are high enough that many nuclei are in exicted states, partition functions are alsoneeded. There are many other fine points that we cannot
discuss here for reasons of length. A few of the various
models for reaction rates are those of Truran et al.
(1966); Arnett et al. (1971); Holmes et al. (1976), Woosley and Howard (1978); Mann (1976); Thielemann et al.
(1986); Rauscher et al. (1997). The results of the mostrecent models seem to be roughly comparable.
As far as experiment is concerned, we can also only
cite a sampling of the many papers from a variety ofsources, including Caltech (Zyskind et al., 1977), theUniversity of Colorado (Roughton et al., 1979, Roughton et al., 1983), and the University of Melbourne (Sargood, 1982).
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1037

FIG. 13. Excitation functions for the reactions 54Cr(p, g ) 55Mn
(black dots), and 54Cr(p,n) 54Mn (3’s). The rapid decrease in
the g ray yield beginning near 2.3 MeV is caused by the rapid
opening of the neutron channel for breakup of the compound
nucleus.

Figure 13 shows an example of a phenomenon that
Fowler and Caltech colleagues have named competition
cusps. The gamma-ray yield from the reaction
54
Cr1p→ 55Mn1 g shows an abrupt decrease (by abouta factor of 10 from the level it would otherwise havereached) beginning at the threshold for the reaction
54
Cr1p→ 54Mn1n (the structure is not a true mathematical cusp, and the name is purely descriptive).
When such structures were discovered, it was immediately clear that they would provide sensitive tests of themodels of the decay of the compound nucleus, requiring
all transmission coefficients to be essentially correct.
Some of the most recent work of the Melbourne
group is summarized in a paper by Hansper et al. (1997)
who have made a new search to improve the nuclear
potentials used in the Hauser*4 model they have chosen
for comparison (Mann, 1976). In the cases studied, they
have reduced the deviations from unity of the ratios th / s exp to less than a factor of 2. For some of the reactions at least, the deviations appear to be systematic incompound nucleus excitation energy; further improvements in the models may thus be possible.
To some extent, the tendency of silicon burning towards quasiequilibrium means that the needed data often reduce to nuclear masses, angular momenta, andother structural properties. However, there is little
doubt that the need to include nuclear reaction rates fornuclei far from the locus of the stable nuclei in the Z-N
plane will lead to continuing demands for both experimental and theoretical improvements in silicon-burning
reaction rates, and for other reactions among still1038

Wallerstein et al.: Synthesis of the elements

heavier nuclei as well. We can anticipate that the new
radioactive beam facilities now on line, under construction, or planned will lead to further clarification of thereaction rates for radioactive or isomeric nuclei.
As noted in Sec. VI of this review, the carbon to oxygen ratio produced in helium burning significantly affects the subsequent nucleosynthesis, and possibly theratio of black hole to neutron star supernova remnants
for massive stars (Weaver and Woosley, 1993). If the
carbon to oxygen ratio is less than a few percent, corecarbon and neon burning will essentially be skipped, andcore oxygen burning involving more core material ensues, leading to a larger iron core, and lowering the initial stellar mass required to form black-hole remnants.
X. s PROCESS: LABORATORY STUDIES AND STELLAR
MODELS
A. The s process since B 2 FH

Essentially all relevant features of the slow neutron
capture process were already considered by B 2 FH. They
included the stellar (n, g ) cross sections as the importantnuclear physics input, the canonical s-process model, theneutron sources in the helium-burning zones and the observations of s-process enhancements in stellar atmospheres.
This section concentrates on the nuclear physics input
and its consequences for the closely related canonical orclassical model, which is based on the empirical assumption that a certain fraction of the original iron was exposed to an appropriate neutron irradiation at constanttemperature. It is impressive to remember the surprisingly detailed conclusions derived by B 2 FH despite thevery uncertain input data available 40 years ago. The
product of the stellar cross section and the resulting
abundance, ^ s & (A) N s(A) , was clearly identified as thecharacteristic quantity of the s process. Two different
s-process components were inferred with the ‘‘local approximation’’ being valid between magic neutron numbers. Finally, the information that can be deduced frombranchings in the synthesis path was outlined. Morequantitative conclusions could not be obtained at thattime, mostly due to severe uncertainties in the stellar(n, g ) cross sections.
This major uncertainty in s-process studies has been
greatly reduced. Neutron physics in the astrophysically
relevant energy regime from about 0.1 keV to a few
hundred keV has reached a relatively mature stage. The
advent of suitable accelerators and detector technologies led to the present situation where the stellar (n, g )rates for s-process applications start to be sufficiently
reliable for interpreting the observed abundance patterns as a critical test for models of stellar helium burning. If the current efforts in this field are pursued foranother decade, then the quality of the data will reach asatisfactory level.
One problem, however, that was not anticipated by
B 2FH was the enhancement of b decay rates at stellar
temperatures. In typical s-process environments, such
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

enhancements result from transitions between individual, low-lying nuclear states that are populated by the
hot photon bath. So far, theoretical estimates for theseenvironments can be tested by only a limited number ofexperiments. In view of its importance for the interpretation of s-process branchings, this topic should be reexamined.
At present, the stellar rates for (n, g ) reactions and b
decays allow one to draw an increasingly quantitative
picture of the s process. This holds even for the canonical model despite the simple and schematic assumptions
on which it is based. Within the uncertainties of the input data this model is capable of describing thes-process abundance distribution as well as the various
branchings. Only recently have accurate cross-section
measurements led to significant inconsistencies with certain abundance features. At the same time, refined astronomical observations and the discovery of a variety
of isotopic anomalies in meteoritic material confirmed
the limited applicability of this approach. Further
progress, therefore, has to be based on more realistic
s-process modeling with advanced stellar codes.
B. Laboratory experiments
1. Neutron capture cross sections

Though produced with high initial energies predominantly in the exothermic ( a ,n) reactions on 13C and22
Ne, stellar neutrons are quickly thermalized unders-process conditions. According to the range of thermalenergies from '8 keV in low mass stars to 80 keV during carbon burning in massive stars, laboratory studies
of stellar (n, g ) cross sections cover the neutron energy
range up to a few hundred keV.
Neutrons in the keV range are produced in the laboratory in several ways. Nuclear reactions, such as
7
Li(p,n) 7 Be in connection with low-energy particle accelerators offer the possibility of tailoring the neutron
spectrum exactly to the energy range of interest; this has
the advantage of low backgrounds. Therefore, comparably short neutron flight paths can be used in these cases
to compensate for the limitations in neutron flux. Measurements with this approach in Oak Ridge by Macklin
and Gibbons (1967) provided the first comprehensive setof stellar cross sections for s-process studies. Presently,
this neutron source is extensively used, for example, atthe pulsed electrostatic accelerators in Karlsruhe, Germany (e.g., Käppeler, Beer, and Wisshak, 1989) and Tokyo, Japan (e.g., Nagai et al., 1991).
Much higher intensities can be achieved at linear accelerators via ( g ,n) reactions by bombarding heavy
metal targets with electron beams of typically 50 MeV.
When these very energetic neutrons are slowed down by
a moderator the resulting spectrum contains all energies
from thermal to near the initial electron energy. With
respect to s-process data, the ORELA accelerator in
Oak Ridge was the most productive facility where numerous (n, g ) measurements were carried out by Macklin and collaborators (see, for example, the compilation

Wallerstein et al.: Synthesis of the elementsof Bao and Käppeler, 1987). Another intense neutron
source is provided by spallation reactions induced byhigh energy particles. The very high neutron fluxes
achieved at the LANSCE facility in Los Alamos allow
even time-of-flight (TOF) studies on radioactive samples(e.g., Koehler et al., 1994).
The experimental methods for measuring (n, g ) crosssections fall into two groups, TOF techniques and activations. TOF techniques are applicable to all stable nuclei. Such measurements require a pulsed neutron
source for determining the neutron energy via the flighttime between target and detector. Capture events in thesamples are identified by the prompt g -ray cascade in
the product nucleus. The differential data, s (E n ), obtained in TOF experiments are then folded with theMaxwellian neutron spectra for various stellar temperatures to determine the effective stellar cross sections. In
general, the Maxwellian averaging for the ‘‘standard’’
thermal energy of kT530 keV requires data in the energy range from a few keV up to ;200 keV. Recent
stellar models suggest, however, that the s process may
partly operate at much lower temperatures corresponding to kT58 keV, which means that stellar cross sections are also affected by contributions from lower energies.
A completely different approach for the determination of stellar (n, g ) rates is by activation in a quasistellar neutron spectrum. Compared to the techniques
based on the prompt capture g rays, this method offers
the advantages of superior sensitivity (which means thatmuch smaller samples can be measured reliably—an important aspect for the investigation of radioactive isotopes on the s-process path), and of selectivity (whichmeans that samples of natural composition can be studied instead of expensive enriched samples required by
the TOF techniques). However, it is restricted to thosecases where neutron capture produces an unstable
nucleus.

2. Stellar b decay

The idea that isotopes, that are exposed to high temperatures and densities in a stellar plasma may experience a dramatic enhancement of their decay rate originated in the forties (Daudel et al., 1947), but moredetailed investigations started much later (Bahcall, 1961;
Bahcall, 1964a, 1964b). The enhancement results fromdifferent effects related to the thermal population of excited nuclear states and to the high degree of ionization.
In 1987, a comprehensive tabulation of stellar decay
rates for s-process studies by Takahashi and Yokoi(1987) provided the necessary basis for a systematic discussion of s-process branchings. Such branchings result
from the competition between neutron capture and b
decay whenever the s-process reaction path encountersan unstable isotope with an effective half-life comparable to the respective neutron capture time. In addition
to previously studied branchings, the work of Takahashi
and Yokoi (1987) revealed a number of unexpected
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1039

FIG. 14. The stellar enhancement factors (stellar decay rate/
terrestrial rate) for some important branch point nuclei on thes-process path at kT530 keV.

cases where terrestrially stable isotopes such as 163Dy
and 179Hf become unstable under s-process conditions.
Figure 14 shows the stellar enhancement factors, i.e.,
the ratios of the stellar decay rates and the terrestrial
rates, for the most important branch point nuclei. Obviously, there are two groups, one showing almost no enhancement at s-process temperatures and a second witha significant acceleration of the decay. Accordingly, theabundance patterns of the first type of branchings issuited for the determination of the s-process neutron
density, while the second group can be interpreted interms of the s-process temperature. The enhancement
factors differing by large factors provides an additional
test for the consistency of the branching analyses and,hence, for the investigated model.
Thermal enhancement effects are difficult to study inthe laboratory since in practically all cases excitednuclear states decay much faster via g emission than byb decay. The only experimentally investigated example
for a b decay from an excited state is 79Se. In this isotope, the enhancement is determined by the b decaybranch from the 96 keV isomer (t 1/2=4 min) which could
be detected in a reactor experiment (Klay and Käppeler,
1988).
An indirect determination of a stellar lifetime was carried out for 176Lu where thermal excitation of the shortlived, b -unstable isomer was quantified by a detailed
study of the level scheme by means of several reactions.
In this case, it was shown that the half-life of the potential s-process chronometer 176Lu (t 1/2=36 Gyr) reduces
to a few years at s-process temperatures. Accordingly,
the observed 176Lu abundance can be interpreted as athermometer for the s-process site (Klay et al., 1991).
The capability to accelerate and to store relativistic
heavy ions at GSI Darmstadt, Germany, gave us the opportunity to demonstrate the bound state b decay. If
these terrestrially stable isotopes are completely
stripped of their electrons, the electron binding energy
adds to their marginally negative Q b values, thus enabling the emission of electrons into the unoccupied orbits. In the GSI measurements, stripped ions were accumulated in the storage ring and their decay followed
over a period of several hours. In this way, the half-lives

1040

Wallerstein et al.: Synthesis of the elements

for bound state decay could be measured for 163Dy(Jung et al., 1992) and 187Re (Bosch et al., 1996), andwere found in reasonably good agreement with the calculated values of Takahashi and Yokoi (1987). In allother cases, the stellar enhancement was calculated byusing b -decay matrix elements for the excited states thatwere derived via the systematics of known analogoustransitions. The detailed discussion of these problems
and uncertainties by Takahashi and Yokoi (1987) should
be considered as the basis for further theoretical andexperimental improvements.
C. The canonical s process
1. The s N curve

The first attempt at a quantitative description of the sprocess dates back to B 2 FH, and was later improved
mainly by Seeger, Fowler, and Clayton (1965). In thisapproach it is empirically assumed that a certain fraction
G of the original stellar 56Fe was irradiated by an exponential distribution of neutron exposures. In this case,an analytical solution can be obtained if a possible timedependence of the neutron capture rates, l n 5n n ^ s & v T ,is neglected. In other words, it is assumed that temperature and neutron density, n n , are constant. Then the
product of stellar cross section and resulting s abundance, which is the characteristic s-process quantity, canbe given by^ s & ~ A !N s~ A !5

G•N (
56

A

S

1

) 11 t 0 ^ s & i
t 0 i556

D

21

,

(22)

where G is the fraction of 56Fe abundance required as
56
the s-process seed, N (Fe abundance, and
56 is the solart 0 is the mean neutron exposure in units of mb21 .
Apart from the two parameters G and t 0 (which areadjusted by fitting the abundances of the s-only nuclei),the only remaining input for this expression is the stellar(n, g ) cross sections ^ s & . Further details may be foundin Käppeler et al. (1989).
Given the very schematic nature of this classical approach, it was surprising to see that it provides an excellent description of the s-process abundances. Figure 15
shows the calculated ^ s & N s values compared to the corresponding empirical products of the s-only nuclei (symbols) in the mass region between A556 and 209. The
error bars of the empirical points reflect the uncertainties of the abundances and of the respective cross sections. One finds that equilibrium in the neutron capture
flow was obtained between magic neutron numbers,
where the ^ s & N s curve is almost constant. The small
cross sections of the neutron magic nuclei around
A;88, 140, and 208 act as bottlenecks for the capture
flow, resulting in the distinct steps of the s N curve.
The global parameters G and t 0 , which determine theoverall shape of the ^ s & N s curve, represent a first constraint for the stellar s-process site with respect to therequired seed abundance and total neutron exposure. It
is found that 0.04% of the observed 56Fe abundance area sufficient seed, and that on average about 15 neutronsRev. Mod. Phys., Vol. 69, No. 4, October 1997

FIG. 15. The characteristic product of cross sections times
s-process abundance plotted as a function of mass number.
The solid line was obtained via the classical model, and thesymbols denote the empirical products for the s-only nuclei.
Some important branchings of the neutron capture chain are
indicated as well.

are captured by each seed nucleus (Käppeler et al.,1990). These numbers refer to the main s-process component given by the thick line in Fig. 15. For A,90, thisline falls below the empirical points, and these discrepancies require an additional, the so-called weak, component. Given the large number of nuclei between Fe andBi that are described with very few parameters, the excellent agreement between the empirical points and thedata obtained with the classical model is a unique feature of the s process. In fact, those s-only nuclei that are
not affected by branchings are reproduced by the modelwith a mean square deviation of only 3% (Käppeler
et al., 1990).
In terms of stellar sites, the main component can be
attributed to helium shell burning in low mass stars(Iben and Renzini, 1983; Hollowell and Iben, 1989). In
this stage of evolution, enery is produced in a narrowdouble shell is produced on top of an inert core consisting of 12C and 16O. Hydrogen burning at the bottom of
the deeply convective envelope produces helium that accumulates in a thin layer around the core. As soon as
this layer exceeds a critical mass, helium burning is ignited leading to a highly unstable situation because of
the large associated energy production. As a result,strong convective motions cause practically instantaneous mixing in the helium-burning layer, and eventually freshly synthesized matter is dredged up to the surface (third dredge-up, see Sec. III). While helium
burning lasts for only about 200 yr, it takes about 53104
yr for hydrogen burning to replenish the consumed helium. These helium-burning episodes can repeat up to
about 20 times.
Neutron production and concordant s processing occur in two steps: by the 13C( a ,n) 16O reaction during thehydrogen-burning stage at relatively low temperaturesof T 8 ;1 and during the subsequent helium burning by
the 22Ne( a ,n) 25Mg reaction (T 8 ;3). The resulting
s-process abundances have been investigated by numerous papers from the Torino group (e.g., Gallino et al.,

1041

Wallerstein et al.: Synthesis of the elements

volved branch point nuclei as well as by the ^ s & N s values of the involved s-only isotopes,f b5~ ^ s & N s ! 148Sm
lb'l b 1l n ~ ^ s & N s ! 150

(23)

Sm

(Clayton, 1968). Inserting the equations for the decay
rates, this expression can be solved for the neutron density n n . If, for simplicity, only the branching at 148Pm isconsidered, one obtains
n n5
FIG. 16. The s-process reaction path in the Nd/Pm/Sm region
with the branchings at A5147, 148, and 149. Note that 148Sm
and 150Sm are shielded against the r process. These two isotopes define the strength of the branching.

1988; Gallino, 1989; Straniero et al., 1995; Gallino et al.,1996).
As a consequence of the comparably short time scale
and the strong impact of convection, stellar models forthe red giant phase are very different from the assumption of a steady situation made in the classical approach.
In order to account for the more complex stellar scenario, the canonical model was recently complemented
by a network code (Jaag, 1990) and by a parametrized
model (Beer et al., 1997) for describing the combined
action of the pulsed 13C and 22Ne sources.
The weak component has been ascribed to core helium burning in massive stars (Peters, 1968; Couch,
Schmeidekamp, and Arnett, 1974; Lamb et al., 1977;
Langer et al., 1989; Prantzos et al., 1990; El Eid and Baraffe, 1990; Raiteri et al., 1993) when the temperatures
become sufficiently high for neutron production via the22
Ne( a ,n) 25Mg reaction. The resulting neutron density
is comparatively low and lasts for about 33104 yr, resulting in the modest neutron exposure of the weaks-process component.
The models for the scenarios suggested for the main
and weak components can best be tested by detailed
analyses of the s-process branchings that occur in therespective mass regions.
2. Branchings

A branching in the reaction flow occurs when an unstable nucleus that exhibits comparable neutron capture
and b decay rates is encountered. The resulting abundance pattern can be used to determine information on
the physical conditions during the s process. The example of Fig. 16 shows the s-process flow in the mass
region between neodymium and samarium, with thepossible branchings at 147Nd and 1472149Pm. Note that148
Sm and 150Sm are shielded against the r process by
their isobars in neodymium. As the result of a significant
branching at A5147–149, the ^ s & N s value of 150Sm will,
therefore, always be larger than that of 148Sm.
The strength of a branching can be expressed in termsof the rates for b decay and neutron capture of the inRev. Mod. Phys., Vol. 69, No. 4, October 1997

1
12f b
f b v T ^ s & 148

ln2
t*

.

(24)

148
Pm 1/2~ Pm!

This equation demonstrates the input data that areimportant for reliable branching analyses.
The first term depends on the cross sections for thes-only nuclei, which define the branching factor f b .
Since the neutron density is required to be ;10%, thebranching factor, and hence the cross sections for thes-only nuclei need to be known to about 1% in manycases. While conventional techniques are limited to uncertainties .4%, new developments such as a 4p BaF 2
detector (Wisshak et al., 1990) or—in favorable cases—
the activation technique allow measurements of the required accuracy.
The second term contains the stellar cross sections of
the radioactive branch point isotopes. Since there arepractically no measurements for the unstable branch
point nuclei, only calculated cross sections are available
at present. Even the most careful statistical model calculations are limited to uncertainties of 20% to 30%, notsufficient for deducing the entire information containedin the abundance patterns. For some of these short-lived
nuclei, experimental cross section studies aiming at a 5%
to 10% uncertainty were recently suggested (Jaag andKäppeler, 1995; Käppeler, 1992).
The last term denotes the stellar decay rate of the
branch point isotope. While there is no difference between the stellar and the terrestrial rate for some of the
branch points, a variety of examples sometimes exhibit
drastic changes under the high temperatures and densities of the stellar plasma (Fig. 14).
Since the b decay rates of the branch points at
A5147–149 in Fig. 16 are not significantly affected bytemperature, these branchings can be used for a determination of the s-process neutron density. Compared tothe previous result of Winters et al. (1986)(f b 50.9260.04), the measurement with a 4p BaF 2
detector yields f b 50.87060.009, with a four times
smaller uncertainty (Wisshak et al., 1993). This branching
factorimpliesaneutrondensityofn n 5(4.160.6)3108 cm 23 (Toukan et al., 1995), inagreement with the previously estimated (3.461.1)3108
cm23 (Käppeler et al., 1990). An example of a branchingthat is readily observable in stellar atmospheres occurs
at 85Kr as described by V. V. Smith in Sec. XI.
Along the s-process path there are about 15 to 20
significant branchings which can be studied with respect
to the physical conditions at the stellar site. In a first1042

Wallerstein et al.: Synthesis of the elements

TABLE V. Results from various branching analyses of relevance for the main s-process component.
Branch point isotope

Deduced s-process parameter

Reference147

Nd/ 147Pm/ 148Pm
151
Sm/ 154Eu
163
Dy/ 163Ho
176
Lu

n n 5(4.160.6)3108 cm 23
T 8 =3.5 6 0.4
r s 5(6.563.5)3103 gcm 23
T 8 =3.1 6 0.6

Toukan et al. (1995)
Wisshak et al. (1995)
Jaag and Käppeler (1996)
Klay et al. (1991)

121

Sn/ 122Sb
134
Cs

T 8 . 2.4
T 8 =1.9 6 0.3
T 8 =1.7 6 0.5

185

8
23
n n 5(3.511.7
21.1)310 cm

Käppeler et al. (1993)
Voss et al. (1994)
Koehler et al. (1996)
Käppeler et al. (1991)

W/ 186Re

step, the neutron density must be obtained from thosebranchings that are not affected by temperature. With
this information, the effective branching factor of theremaining cases can be derived and the mean stellar decay rates be determined. Eventually, the dependence of
these rates on temperature and/or electron density
yields the s-process temperature and mass density (Käppeler et al., 1990). The present status of branching analyses with the classical approach is summarized in Table
V. Prominent examples in this list are from the lanthanide region, which is particularly important in thiscontext because the chemical similarity of these elements implies that their relative abundances are well defined. Therefore, the total reaction flow can be definedby 150Sm, analogously to the situation of Fig. 16. Such
branchings occur at A5141, 151, 154, 163, and 169 andare defined by the s isotopes 142Nd, 152Gd, 154Gd, 164Er,
and 170Tm, respectively.
As far as the neutron density is concerned the bestvalue in Table V, n n 5(4.160.6)3108 cm 23 , is in good
agreement with the estimate obtained from the 185W/
186
Re branching. The various estimates of the s-processtemperature are also compatible with each other, except
for the result from the 134Cs branching. This discrepancy, however, may originate from an uncertainty in thetemperature-dependent decay rate of 134Cs which had
been noted explicitly by Takahashi and Yokoi (1987).
Hence, this problem should be investigated further before the 134Cs branching can be discussed in any quantitative way.
A first attempt to determine the mass density in the sprocess was carried out by Beer et al. (1985) by analyzing a small branching to 164Er. Their value,r s 5(865)3103 g cm 23 , is in reasonable agreementwith the recent analysis cited in Table V (Jaag and Käppeler, 1996).
So far, the results of the different branching analyses
in Table V seem to be compatible with each other,pointing to the physical conditions typical of the22
Ne( a ,n) 25Mg neutron source. In a way, this apparent
success also marks the limitation of the classical model:
it can only interpret the final abundance pattern. Being
thus restricted to the last modification of the s abundances, the dominance of the 13C( a ,n) 16O source ismissed, in clear contradiction to observation (see Sec.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

XI). Though there are a few features from the 13C era
surviving the 22Ne( a ,n) episode—especially near magic
neutron numbers where cross sections are small—thesehints are too weak for reconstructing the true stellar s
process. This limitation is common to all attempts ofreconstructing the s process from the observed abundance pattern alone.
Accordingly, progress in understanding the physics ofthe s process can only be achieved by means of stellar
models. But since these models are still plagued by anumber of open questions, further laboratory studies
have to concentrate on the aspect of constraining criticalparameters in these calculations. In particular, theoretical concepts of convection and mixing may be tested via
their effect on s-process branchings. In any case, themeasurements have to account for the lower temperatures during the 13C( a ,n) phase.
XI. OBSERVATIONS OF THE s PROCESS
A. Brief history

At the time of the writing of B 2 FH, one of the key
observations—that many of the elements heavier thanHe might be produced in stars—was Merrill’s (1952)classic, and often cited, spectroscopic detection of neutral technetium in certain red giant stars of spectral type
S. As technetium consists only of cosmically short-lived
isotopes (t 1/2;105 –106 yrs), the Tc I observed by Merrill
had been ‘‘recently’’ synthesized in the interiors of the S
stars and mixed to their surfaces. The Tc in the S stars,
and the other neutron-rich heavy elements observed to
be overabundant in many types of red giants (e.g., Zr,
Ba, or La), was created at low neutron densities and is a
product of B 2 FH’s s process.
It has been known since the work of Schwarzschildand Härm (1967) and Sanders (1967) that the preferred
astrophysical site for the production of many of thes-process nuclei was in a region associated with the Hand 4 He-burning shells found in asymptotic giant
branch (AGB) stars. Much of the early work which connected the AGB stellar models to the details of thes-process nucleosynthesis was carried out by Iben andcollaborators and is summarized nicely in the review byIben and Renzini (1983).

Wallerstein et al.: Synthesis of the elements

This is a brief review of a few of the recent observational results of the s process in AGB stars.
B. Observations of nucleosynthesis and mixing in CH, Ba,
S, and C stars

Of the various nuclear processes characterized by
B 2FH, the s process is perhaps one of the easiest to
study using direct observations of stars. Because the sprocess has the greatest effect on the relatively minor
abundance elements heavier than Fe, such as Y, Zr, Ba,or La (all of which have readily accessible spectral
lines), rather small abundance changes of a factor of 2,
or more, can be quickly detected by straightforward
spectroscopic techniques. Also, the s process is activatedduring the relatively stable phase of stellar evolution
along the AGB, thus there are a variety of stars that canbe studied which bear the spectroscopic signs of the mixing of freshly synthesized s-process elements to theirsurfaces. The result of s-process nucleosynthesis in AGB
stars has left its mark on a number of stellar classes
which exhibit overabundances of the classic s-process
nuclei, although of course it has only been since the linking of the s process to evolution along the AGB thatthese rather diverse stellar types have been found tohave common origins.
As quasiperiodic shell 4 He burning occurs during
thermal pulses on the AGB, mixing episodes increase
the atmospheric C/O ratio in these red giants, giving riseto the spectral sequence of M to MS to S to C. This
increase in C/O is accompanied, in general, by an increasing s-process overabundance, thus the MS, S, andC stars are heavy element rich AGB stars. Carbon stars
have been known for well over a century, while the intermediate C/O ratio S stars were first isolated as a specific stellar type by Merrill (1922) and their spectral
cousins the MS stars were discussed by Keenan (1954).
Other warmer, heavy element rich stellar classes are
known; Bidelman and Keenan (1951) identified thebarium (Ba) giants as being heavy element (Ba and Sr)rich, and soon after it was realized that the CH stars
discovered and classified by Keenan (1942) were metalpoor halo relatives of Ba stars. In more recent times,
Bond (1974) found subgiant and near main-sequence
relatives of the giant CH and Ba stars and dubbed them
CH subgiants. Since Bond’s discovery, many more true
main-sequence s-process rich stars of spectral types F
and G have been identified: Tomkin et al. (1989) orNorth et al. (1994). Observationally defined s-processrich stars thus span the range in luminosity from mainsequence to extremely luminous AGB giants.
Since the s process was identified with AGB evolution
in the 1970’s, the lower luminosity s-process rich Ba andCH giants and subgiants defied explanation for many
years. McClure et al. (1980) broke the logjam of understanding these diverse groups when they discovered thatall Ba stars were binaries with white dwarf companions.
The solution to the s-process nature of these stars thus
centers on mass transfer; when the white dwarf now inhabiting these binary systems was a luminous AGB starRev. Mod. Phys., Vol. 69, No. 4, October 19971043

and had mixed s-process rich material to its surface, ittransferred some of this processed matter to its companion, which is now observed as an s-process rich star.
Since the pioneering paper of McClure et al. (1980),
follow-up studies (McClure, 1984; McClure and Woodsworth, 1990) have confirmed the mass-transfer scenario. In addition, it has been shown that the 40% of S
stars which do not show the lines of Tc I in their spectra
are the cooler relatives of the Ba stars and result frommass transfer (Iben and Renzini, 1983; Smith and Lambert, 1988; Brown et al., 1990; Jorissen et al., 1993). Most
recently, a number of the active symbiotic binaries havebeen identified as C rich (Schmid and Nussbaumer,
1993; Schmid, 1994), while the yellow symbiotic stars
AG Dra and BD-21° 3873 have both been shown to be
s-process rich and thus technically Ba stars as well as
symbiotics (Smith et al., 1996, 1997). There is now a coherent picture of the s-process rich classes of stars: theN-type carbon stars, along with the MS and S stars withTc, are bonafide AGB stars that have recently dredged
fresh s-process material to their surfaces. The other variety of s-process stars (the Ba and CH giants, subgiants,
and main-sequence stars, the carbon star dwarfs, andnow the C and s-process rich symbiotic stars) result frommass-transfer in certain binary star systems.
With so many varieties of stars exhibiting clear signatures of enhancements of the s process in their abundance patterns, it is not surprising that a great deal ofeffort has been expended by observational spectroscopists in analyzing the details of s-process nucleosynthesisin the heavy-element rich stars. A number of detailed
abundance analyses have been conducted on the various
heavy-element rich stars over the last 15 years and canbe summarized as follows: for the barium giants we have
Tomkin and Lambert (1983, 1986), Smith (1984), Kovacs (1985), Smith and Lambert (1984), Malaney andLambert (1988), and Busso et al. (1995), while the mainsequence Ba stars have been studied by Tomkin et al.
(1989) and North et al. (1994). The CH giants have beenstudied by Vanture (1992a, 1992b). The MS and S stars
have been studied by Smith and Lambert (1990), Plez
et al. (1993), Busso et al. (1995), and Lambert et al.
(1995), while the most detailed analysis of the s process
in the carbon stars is still Utsumi (1985). Recent abundance studies have uncovered s-process rich post-AGB
stars in the globular cluster v Cen (Gonzalez andWallerstein, 1994), as well as s-process enhancements inthe yellow symbiotics (Smith et al., 1996, 1997).
Taken together, these abundance studies provide theraw material with which to probe our current understanding of the details of s-process nucleosynthesis in
AGB stars and in the next sections we summarize theseresults.
C. The s process as a function of metallicity

One question that is basic to an understanding of thes process in AGB stars is the nature of the neutron
source. Since the realization that the slow capture ofneutrons is a distinct stellar nuclear process, two candi-

1044

Wallerstein et al.: Synthesis of the elements

date neutron producing reactions have been suggested:13
C( a ,n) 16O and 22Ne( a ,n) 25Mg (Cameron, 1955;
Greenstein, 1954). These two separate sources have very
different characters; the s process in AGB stars is associated with 4 He-burning temperatures of T>108 K, withthe 13C source being active near 13108 K, while the22
Ne source requires higher temperatures nearer to(2–3)3108 K. The different temperatures required todrive the s process via the different neutron sources will
occur in different mass AGB stars, or at different points
along the AGB. Historically, the 22Ne source wasfavored as the most likely s-process candidate because
the production of neon-22 can occur quite naturally
as the result of succesive a captures onto 14N intheHe-burning
shell
of
an
AGB
star
@ 14N( a , g ) 18F(e 1 , n ) 18O( a , g ) 22Ne], with the initial 14N
itself coming from CNO-cycle H burning (Iben, 1975a,
1975b, 1976, 1977a, 1977b). The carbon-13 neutron
source required an extra mixing mechanism to moveprotons into the He-burning shell to produce 13C via
12
C(p, g ) 13N(b+,n)13C, and it was not clear as to thephysical nature of this extra mixing. However, the expected behavior of these neutron sources with metallicity is different and can be tested by the observations.
A convenient parameter to characterize the relative
abundance distribution of the s process is the neutron
exposure t defined as

t5

E

t8

0

N n ~ t ! V ~ t ! dt,(25)

where N n is the neutron density and V(t) the relative
velocity of neutrons and nuclei, with the integral takenover the interval of the s-process episode. A large neutron exposure leads to a larger amount of heavier nuclei
produced (e.g., Ba, La, or Ce) relative to the lighter nuclei, such as Y or Zr.
As pointed out by Clayton (1988), the nature of thedominant s-process neutron source can be tested by using the heavy- to light-element abundance distributionof the s process (a measure of t ) as a function of metallicity. For example if, during the s-process episode, theneutrons are in local equilibrium between their production and destruction, then the neutron density will be
N n 5 ~ SN i N j ^ s & i,j ! / ~ SN k ^ s & k,n ! ,(26)

where the numerator is summed over all neutronproducing reactions between species i,j, with
Maxwellian-averaged cross sections of ^ s & i,j , and thedenominator has the neutron destruction (absorbing) reactions, summed over all neutron absorbers, k, andagain a Maxwellian-averaged neutron absorption crosssection of ^ s & k,n . Making the simplifying assumptionthat the neutron producing reactions come from a captures, and that the dominant neutron absorbers are theseed nuclei of Fe, the above expression can be approximated as
N n } ~ N HeN j ! / ~ N Fe! ,(27)

where N j 5N( 13C) for the carbon-13 source andN( 22Ne) for the neon-22 source. Since the 22Ne in theRev. Mod. Phys., Vol. 69, No. 4, October 1997

FIG. 17. Published abundance results for the heavy s-processelements (Ba, La, and Ce) relative to the light s-process species (Y and Zr) for the MS/S stars (filled circles, with Tc, andopen circles, without Tc), Ba giants (open triangles), Ba dwarfs(open pentagons), CH giants (open squares), CH subgiants(crosses), and yellow symbiotics (six-pointed stars). The
middle panel shows the same quantity, [hs/ls], as in the top
panel, except that the abundances come from models ofs-process nucleosynthesis from Malaney (1987). This is plottedversus the neutron exposure, t , for a single exposure model. In
the bottom panel are shown the observed values of [hs/ls]
transformed to t from Malaney’s models, plotted versus [Fe/H]. The increase of neutron exposure with decreasing metallicity is clear.

He-burning layers comes from 14N which, itself, comesfrom the initial SCNO which is }N Fe , the expected
neutron density (and neutron exposure) would be expected to be roughly independent of metallicity.
Carbon-13, on the other hand, must result from the mixing of protons into a 12C-rich region, where thecarbon-12 comes from 4 He, which is almost independentof metallicity. Thus if the structures of He shells in AGB
stars are not strong functions of metallicity the numerator of the above expression is constant and N n (and t ) is
}N 21
Fe .
The neutron exposure can be estimated using published abundance results and we show the results of suchan exercise in Fig. 17. In the top panel is plotted thequantity [hs/ls] versus [Fe/H] from a number of studies.
The value of [hs/ls] stands for heavy-s to light-s and is anaverage overabundance of representative heavy

Wallerstein et al.: Synthesis of the elements

1045

s-process nuclei (Ba, La, and Ce) relative to an averageoverabundance of the light s-process species Y and Zr,
measured in standard spectroscopic bracket notation.
There is rather large scatter, although this is not surprising as we are comparing results from a large number ofinvestigators using somewhat different techniques anddifferent quality data. Nonetheless, there is a clear andsignificant trend of increasing [hs/ls] with decreasing metallicity. The quantity [hs/ls] can be mapped into a neutron exposure using the theoretical calculations from
Malaney (1987), which we illustrate in the second panel:here we plot [hs/ls] defined in exactly the same way as
for the stellar results, but using Malaney’s model abundances, versus the single-exposure neutron exposure t .
In the bottom panel, the observed t derived from Malaney’s ‘‘calibration’’ is plotted versus [Fe/H]. There is asignificant increase of neutron exposure with decreasing
metallicity as predicted from Clayton’s (1988) simple argument for the 13C( a ,n) neutron source. Based on theobservations, it would appear that the current best candidate for the s-process neutron source in the AGB starsof low mass is carbon-13.

D. Rubidium and the s-process neutron density

In an s-process environment, for the limiting casewhere the neutron density goes to zero, the s process
follows a unique path along the valley of stability because the first b -unstable nucleus will decay rather thancapture a neutron. As N n increases, however, certainb -unstable nuclei will either decay or capture anotherneutron depending on how the neutron capture rate
(which increases with increasing neutron density) compares to the b decay rate for that particular nucleus. As
described in detail in Sec. X, at these critical branch
points, the s-process path can follow somewhat different
routes, and certain branch points can result in measurable abundance differences, depending on which paththe s process travels (which will depend on the neutron
density). One such branch point that can have observable consequences occurs at 85Kr, which controls the
s-process path either through 85Rb or 87Rb. Krypton-85
has a ground-state half-life of 10 years and b decays tothe stable nucleus 85Rb. If 85Kr captures a neutron,
however, it leads to the stable isotope 86Kr which will
neutron capture to 87Kr, which will then b decay (in justover an hour) to stable 87Rb. Depending on whether
85
Kr b decays or captures a neutron, the s-process pathwill travel to either 85Rb or 87Rb; the half-life of 86Rb is
18 days (much less than 85Kr), so the low-density branchof the s process bypasses 87Rb, while the high-density
branch bypasses 85Rb. These two Rb isotopes have very
different neutron capture cross sections @ s ( 85Rb)=240
mb and s ( 87Rb)=21 mb at kT530 keV (Beer and
Macklin, 1989), so the resultant Rb abundance will vary
according to the ratios of their respective cross sections.
Both the high- and low-density s-process paths go from
Rb on to the nearby elements Sr, Y, and Zr, so theelemental abundance ratio of Rb to Sr, or Y, or Zr canRev. Mod. Phys., Vol. 69, No. 4, October 1997

FIG. 18. The [Rb/Zr] s-process abundances versus [Fe/H] (toppanel) in the Ba giants (filled triangles), the MS/S stars (opensquares), the yellow symbiotic stars (filled circles), and the v
Cen giants (filled squares). The increase of [Rb/Zr] with decreasing [Fe/H] is striking. In the bottom panel, [Rb/Zr] hasbeen transformed to neutron density using calculations presented in Malaney (1987) and Malaney and Lambert (1988). A
linear least-squares fit to the MS/S, Ba, and yellow symbiotic
stars is shown: this line has a slope of 21. The v Cen stars are
not included in this fit because the s-process enrichments result
from more complicated chemical evolution (see the text for a
discussion).

be used as a neutron density monitor. The Rb abundance changes most rapidly between neutron densities
of about 107 –1010 cm 23 .
As Rb is a fairly low abundance species and represented only by Rb I, whose ionization potential is 4.18
eV, it is difficult to analyze and must be attacked using
the resonance line at 7800 Å. In the last 15 years or so a
number of results for Rb have become available for Ba
stars (Tomkin and Lambert, 1983; Smith and Lambert,
1984; Malaney and Lambert, 1988), the MS and S stars(Lambert et al., 1995), s-process rich giants in the globular cluster v Cen (Vanture et al., 1994), and thes-process rich yellow symbiotic stars (Smith et al., 1996,1997). In addition, Wallerstein (1992) has analyzed the
Rb I resonance line in emission in the S star, R Andromeda, and estimated the Rb/K abundance ratio (using theK I resonance line also in emission). Wallerstein assumed that the collisional excitation and deexcitation
cross sections are the same for both Rb I and K I. Thus
there is now a rather substantial set of data on the be-1046

Wallerstein et al.: Synthesis of the elements

havior of Rb, and hence the neutron density, over arange in stellar metallicity. The discussion of the previous section holds here and, for the case of the 13C( a ,n)neutron-source, one might expect an increasing neutron
density with decreasing metallicity. Results for Rb aresummarized in Fig. 18: the top panel presents the observational quantities [Rb/Zr] versus [Fe/H] for the barium
star references listed above (filled triangles), the MS andS stars (open squares), the v Cen giants (filled squares),and the yellow symbiotics (filled circles). The large increase of the Rb/Zr abundance ratio with decreasing[Fe/H] is obvious. The observed quantity [Rb/Zr] can beconverted to a rough neutron density for the s processusing the results from Malaney and Lambert (1988) and
Malaney (1987) and this is shown in the bottom panel ofFig. 18. The dashed line in the bottom panel is a leastsquares fit to the MS/S, Ba star, and yellow symbiotic
points, while the v Cen stars (filled squares) clearly fallabove this trend. The MS/S, Ba, and symbiotic stars exhibit s-process abundance enhancements that result
from either the dredge up of processed material in asingle star, or mass transfer from a single AGB star (inthe case of the Ba and symbiotic stars). The v Cen stars,
on the other hand, owe their s-process enhancements tothe material from which they formed, as first suggested
by Lloyd Evans (1983) and shown recently by Vanture
et al. (1994). The s-process elements in v Cen presumably result from processing by a number of AGB stars
covering a range in mass and metallicity, as well as Fe
abundances ejected from SN II. It may not be surprisingthat they do not fall on a relation defined by single AGB
stars; clearly Rb in v Cen deserves further attention in
the future. Concentrating only on the s process isolated
from single AGB stars, the linear least-squares fit from
Fig. 18 has a slope of 20.97, or, N n }N 21
Fe . This is the
expected result from the simple arguments discussed
above for the 13C neutron source. If this result holds fora larger sample of stars covering a range in metallicity, it
may suggest that the internal structure of the Heburning shells in AGB stars is largely independent ofmetallicity.
E. Recent models: Radiative burning of 13C during
the AGB interpulse phase

Based upon the observational results provided by theheavy element s-process rich stars, there is strong evidence that 13C( a ,n) 16O is the dominant neutron sourcein the AGB stars. Both the increasing neutron exposuresand neutron densities support the picture of a neutron
source whose main reactant is a primary element ( 13Cfrom 12C from 4 He). The production of sufficient quantities of 13C at the H-He interface remains a difficult
problem for theorists to model, but much progress has
been made in recent years in reconciling model predictions from AGB stars with the observations (see Sec. III
by I. Iben in this review).
Based on models of intermediate mass stars (sayM;4–8 M ( ), neutrons were predicted to be generated

Rev. Mod. Phys., Vol. 69, No. 4, October 1997

by the 22Ne( a ,n) 25Mg reaction (see Iben and Renzini,
1983, for references to these earlier models). In suchmodels, peak neutron densities at solar metallicities
reached 109 –1010 cm 23 ; as can be seen from the bottom
panel of Fig. 18, such densities are not fit by the Rb/Zr
abundance ratios observed in solar metallicity AGB
stars. Various modifications were tried by a number ofauthors and Lambert (1991) and Lambert et al. (1995)
give nice discussions of these efforts. The basic result isthat intermediate mass AGB models with the neon-22
neutron source do not provide the low neutron densities
needed for the s process.
Investigations using lower mass AGB models with the13
C neutron source require some form of extra mixing at
the H-He interface. Using refinements to semiconvective mixing, originally used for Population II AGB models by Iben and Renzini (1982a, 1982b) and Hollowelland Iben (1988, 1989), it was found that thermal pulses
in such models give rise to a double burst of neutrons. In
the first burst, neutrons are generated at T;108 K by
13
C( a ,n) 16O, while in a second, weaker burst (later inthe evolution of the thermal pulse when temperatures
are higher) 22Ne( a ,n) 25Mg is activated, but contributes
much less to s-process production. Gallino (1989) andKäppeler et al. (1990) explored such models; however,
the total amount of 13C found in the He shell prior toignition was basically an adjustable parameter. Busso
et al. (1992, 1995) continued such parametrized calculations with the s-process neutron density most strongly
dependent on how much 13C is synthesized by H diffusing into the shell between thermal pulses and the rate of
injection of the 13C into the convective He-burning
shell. In all cases, however, the predicted neutron densities were still higher than indicated by the observations.
A recent potential breakthrough in understanding thediscrepancies between the neutron densities in the AGB
models versus the observations has been found by
Straniero et al. (1995). These authors followed the evolution of a 3M ( stellar model from the ZAMS up to
thermal pulses on the AGB. They find that any 13C synthesized at the H-He interface on the AGB in between
thermal pulses (by any mechanism) is burned via ( a ,n)reactions before the next thermal pulse develops. This
results in two major modifications to the s process: thefirst is that most of the s-process nucleosynthesis occurs
at temperatures of less than 108 K. At these lower temperatures, the rate at which 13C burns is much slower,with lower resultant neutron densities of ;(122)3107
cm 23 at solar metallicity. Inspection of the lower panelof Fig. 18 shows that this value is in excellent agreementwith the observationally derived N n ’s. The second important result is that much of the s process will now
occur in a radiative, instead of a convective, environment. Because the temperature and density varythroughout the radiative He shell, each radiative layer ischaracterized by different conditions, such that the resulting distribution of neutron exposures is not an exponential distribution. Further modeling may result in bet-

Wallerstein et al.: Synthesis of the elements

ter fits to the abundance distributions in low metallicity
s-process rich stars which, traditionally, seem to havebeen better fit by single exposure models (e.g., Vanture,
1992a; Smith et al., 1996, 1997).

XII. THE r PROCESS
A. Introduction

Of the eight nucleosynthetic processes described by
B 2FH, the theoretical quest to explain the production of
the r-process isotopes, and in particular to identify aplausible astrophysical site capable of synthesizing themin amounts that satisfy observational constraints, has entertained perhaps the widest range of possibilities. Manyof the basic nuclear systematics of rapid neutron capture
on preexisting seed nuclei discussed in the first and pioneering explorations of B 2 FH remain true in the modern context. Some of the details have changed, especially in the light of new observations andmeasurements. For example, the accepted values of thesolar r abundances, scarcity of reliable nuclear data inthe vicinity of the neutron drip line, and the observed
r-process abundance levels as a function of the Milky
Way’s evolutionary history each determine to some extent the important details that the origin site must satisfy. The difficulty is not a lack of theoretical models—there are numerous ones and the large number ofmodels is a measure of the complexity of the problem—
but deciding which one (or ones) are the most promising. The ultimate goal of a successful model for the origin site of the r process must be explaining the variety ofstellar observations with the nuclear data that are available, and to provide unambiguous predictions that canbe tested by future observations.
B 2FH and Cameron (1957) suggested that during ther process a large flux of neutrons (whose source wasunspecified) would be captured by seed nuclei (typicallyiron) on time scales that were short compared to the b 2
decay time scales of the newly created nuclei. Neutron
capture on nuclei with a given number of protons would
proceed to very neutron-rich isotopes until the competition between b 2 decay and photodisintegration reactions force the nuclear flows to move onwards to nuclei
with a larger number of protons. With a suitable neutron
exposure, the r process could proceed to the heaviest
elements (thorium and uranium), and achieve a steady(n, g )↔( g ,n) flow equilibrium. As the neutron flux diminished, b 2 decays would then populate the first stableisobar encountered for a given mass number A. Comprehensive reviews of the r process by Hillebrandt(1978), Schramm (1982), Mathews and Ward (1985),
Cowan et al. (1991), and Meyer (1994) provide very
thorough and complete background material. The purpose of this survey is to confront the search for the astrophysical origin site of the r process with the observational and theoretical information that has accumulated
over the last few years.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1047

B. A search for the astrophysical site

All proposed origin sites for the r process can be classified as either primary or secondary production sites. In
the primary production scenarios, neutrons are captured
rapidly onto seed nuclei that are made in situ. Notable
examples would include inhomogeneous primordial nucleosynthesis, supernova cores, jets and/or magnetic
bubbles inside rotating stellar cores, and binary compact
remnant interactions. In secondary production scenarios, neutrons are rapidly captured onto seed nuclei
made by a previous generation of stars. These would
include explosive helium or explosive carbon burning inmassive stars, neutrino irradiation in the helium layer ofmassive stars, helium core flashes in low mass stars, intermediate and low mass stars as they pass through theasymptotic giant branch phase of evolution, accretion
disks around neutron stars, and classical novae. The detection of solar r-process abundances in the 56<Z<76
range in very metal-poor (thus presumably very old)
halo stars strongly suggests a primary r-process originsite that contributed its nucleosynthetic products to theinterstellar medium (e.g., Type II supernovae of Pop III
or II) prior to the formation of currently observed halostars (Sec. VIII). The present observational data for
Z,56, however, is not so conclusive and could permit asecondary origin site.
Much of the recent theoretical work has centered oncore collapse supernovae, especially within the region
where a neutrino-driven wind is blown from the surface
of a nascent neutron star following a delayed Type II or
Ib supernova explosion. As the gravitational binding energy of the neutron star is liberated during its coolingtime scale ( t KH;10 s), the enormous neutrino flux
drives the mass loss and provides unique physical conditions under which neutron-rich heavy element synthesis
can occur. The r process is primary in this scenario, andis not a purely neutron driven process. It begins withseed nuclei synthesized through a reassembladge of
nucleons from an alpha-rich freeze out from nuclear statistical equilibrium in which there is a large neutron excess. This process has been termed the ‘‘a process’’(Woosley and Hoffman, 1992; Witti et al., 1994). The
same name was originally used by B 2 FH to describe
what is presently referred to as ‘‘neon burning.’’ The
resulting distribution of seed nuclei is centered on thekrypton to zirconium region with A;100, not on ironpeak nuclei (as would be the case for a freeze out with asmall neutron excess). The larger neutron excess is dueto the intense neutrino flux from the protoneutron star,
which drives a large number of neutron liberating
charged current neutrino capture reactions ( n ,n) on
free nucleons. A freeze out with a large neutron excessbypasses the first closed neutron shell at N550, bridging
the waiting point there, and shortens the overall timescale needed for a solar abundance producing r-process
to operate. During the subsequent neutron capture
phase of this r-process scenario, the presence of theseheavier seed nuclei reduces the overall neutron to heavy
seed ratio required of a purely neutron-driven process.

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Wallerstein et al.: Synthesis of the elements

The total mass of r-process material ejected is typically;1025 M ( per supernova, and is set from first principle
calculations of the neutrino wind’s physical properties.
This value of the mass ejected per supernova is consistent with estimates of the Galactic Type II + Ib supernovae rate (1021 events per century) and the total massof r-processed material (.10 4 M ( ) inferred to presently exist in the .10 Gyr old Galaxy. That is, neither
too little material is ejected, nor too much.
The r process described above that may occur withina given mass element can be succinctly described bythree parameters: evolution of the entropy per baryon
S/N A k (which specifies the thermodynamics of the material), evolution of the electron fraction Y e (which
specifies the composition of the material and is related
in a simple algebraic way to the neutron excess), and the
dynamic timescale (t dyn) over which the material cools.
Larger entropies give smaller abundances of heavy seed
nuclei. This is due to the large density dependence of the
intrinsically weak three-body reaction sequencesa ( aa , g ) 12C and a ( a n, g ) 9 Be( a ,n) 12C. These two reaction chains are very important because they bridge theunstable mass gaps at A55 and A58. Physically, a highentropy means many photons per baryon. A significant
fraction of these photons can be on the high energy tailof the Bose-Einstein distribution, and therefore canmaintain, for example, a low abundance of 9 Be throughthe highly efficient photodisintegration reactions. The
neutron excess determines the overall availability ofneutrons in the material. It chiefly affects the path of thenuclear flow and hence the distribution of heavy seed
nuclei produced at the end of the freeze out from
nuclear statistical equilibrium. In general, lower valuesof Y e give smaller abundances of heavy seed nuclei. A
shorter expansion time scale also reduces the seed abundance by limiting the time over which the freeze out
operates. A smaller heavy seed abundance at the end of
the a process gives a larger neutron-to-seed ratio, which
enhances the production of heavier r-process nuclei. It is
important to emphasize that the entropy and neutron
excess of a given mass element are time-dependent
quantities. The total r-process yield is the sum over allejected mass elements, each of which follows a somewhat different nucleosynthetic pathway as a result of thethermodynamic history and initial neutron excess ofeach mass element. Hydrodynamics (especially in multiple dimensions) are therefore important in our understanding of neutron-rich heavy element synthesis (Burrows et al.; 1995, Bazan and Arnett, 1994a, 1994b).
C. Early model results, from conflict to clarity

Detailed r-process nucleosynthesis studies in spherically symmetric 20M ( stars have been studied by Woosley et al. (1994) and Takahashi et al. (1994). In both
works a proper amount of r-process material wasejected, a reasonable agreement with the solar r-process
isotopic distribution was achieved, and high entropies(S/N A k;400) were considered to be required to reproduce the solar abundance of platinum peak (A5195)
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

nuclei. Both works also shared the common drawback of
producing unacceptably large amounts of the N550
closed neutron shell nuclei 88Sr, 89Y, and 90Zr during
the a -process phase of their calculations. This wascaused by having too many neutrons present (Y e ;0.46)
during the early evolution of the neutrino-powered
wind. If the number of neutrons was decreased slightly(Y e ;0.48), the production factors for the N550 closed
neutron shell nuclei became smaller, and hence morepalatable (Hoffman et al., 1996). In this case, some of
the light p-process nuclei were coproduced, most notably the hitherto unproduced isotope 92Mo (Sec. XIV).
Several correlations have been noted between the production ratios of some light-p and r-process isotopes ascalculated by these studies and the production ratiosthat have been measured in the Allende and Touluca
meteorites. This is suggestive, but certainly not conclusive, evidence that such isotopes were produced in acommon environment. The isotope 64Zn is also produced, and appears to be a unique signature of the aprocess. This might be important for understanding theobserved zinc to iron [Zn/Fe] ratio in Galactic dwarf
stars and the QSO absorption line systems, which show
a solar [Zn/Fe] ratio for all metallicities larger than
0.001Z ( .
Each group shared the dubious distinction of claiming
significantly different entropy per baryon structures in
the neutrino-powered wind. This spurred a quest for amore physical understanding of the conditions extant inthese winds. Inspired by the early work of Duncan, Shapiro, and Wasserman (1986) analytic equations for theentropy per baryon, dynamic time scale, and mass-loss
rate were derived as functions of the neutron-star mass,
radius, and emergent neutrino luminosities and spectra(Qian and Woosley, 1996). An equation for the electron
fraction in terms of the charged current lepton capture
reactions on free nucleons was also derived. These analytical expressions, with the transparent physical insighttheir derivations provide, were verified to a high degreeof accuracy by resolved numerical simulations of steadystate neutrino winds. For typical neutron star properties,
neutrino luminosities and spectra, the derived entropy
per baryon fell short, by roughly a factor of 2, of thevalue apparently needed to produce the A5195 platinum peak r-process nuclei (Hoffman et al., 1997). Whena modest increase to the energy produced by the wind
was added at the base of the neutrino atmosphere, sufficient conditions for producing platinum peak nuclei
were achieved. Possible sources of energy capable of justifying this energy perturbation, which were not included in either the analytical equations or the numerical simulations, could include rotation or vibration of
the neutron star, general relativistic effects, and magnetic fields (Qian and Woosley, 1996).
It is important to stress that high entropy is not anecessary requirement for a successful neutrino-winddriven r process (Fig. 19; see also Hoffman et al., 1997).
A shorter dynamic time scale will limit the duration of
the precursor a process. This reduces the heavy seedabundances and increases the neutron to heavy seed ra-Wallerstein et al.: Synthesis of the elements

FIG. 19. The minimum entropy required to make platinum
peak nuclei (A;195) by neutron capture following an a process for a given electron fraction (Y e ) and expansion timescale (t exp). Filled circles connected by solid lines are the results of a numerical calculation for a constant entropy of
S53.33T 39 / r 5 . Filled squares (not connected by a line) are theresults of a numerical calculation for an exact adiabatic equation of state. Open circles connected by dotted lines are fromanalytic expressions for the minimum entropy. Figure from
Hoffman et al. (1997).

tio for the subsequent neutron capture phase. The sameneutron to seed ratio can be obtained at 21 the entropy if
the expansion time scale is about 8 times shorter. The
chief utility of such parametrized reaction network surveys could be to determine the suitability of detailed
core collapse simulations (or other astrophysical environments, e.g., Ruffert et al., 1997) as potential r-process
sites, provided the r process is envisioned as comingfrom a neutron-rich, a -rich freezeout from nuclear statistical equilibrium.
D. Twisting in the windAs with any evolving theory, our understanding of ther process in neutrino winds is far from complete, andnew twists to our current thinking about the time evolution of the key parameters that describe it continue topermeate the literature. At issue are uncertainties in the
nuclear equation of state, neutrino-nucleus interaction
cross sections, possible neutrino-flavor transformations,fallback of freshly synthesized material onto the compact remnant, and neutrino transport. Their continuing
resolution will play a central role in determining
whether the neutrino-driven wind will remain an attractive site for the r process.
Neutrino transport is probably the most serious, andperhaps the most challenging, of these issues. The transport of neutrinos directly impacts the nucleosynthesis
through the time evolution of Y e ; previous studies suggest an almost uncomfortable sensitivity of the nucleosynthesis to this parameter (Sec. XIV). Y e also couples
to the hydrodynamical question of how much mass fallsback onto the remnant. Even if all supernovae coreRev. Mod. Phys., Vol. 69, No. 4, October 1997

1049

collapse events can make a successful r process, much(or all) of it might not be injected into the interstellar
medium. Neutrino transport also affects the magnitudeof any asymmetrical energy deposition (kicks), which
may manifest itself in the observed pulsar proper motions (Burrows and Hayes, 1996). Studies to date havegenerally relied upon flux-limited multigroup neutrino
diffusion schemes. A less ambiguous, if more ambitious,
treatment would solve the Boltzmann equation for all
the neutrino species (Mezzacappa and Bruenn, 1993)
and couple it to a multidimensional hydrodynamics
solver. Such tools are just now coming to the forefrontof supernova research.
Neutrino-nucleus interactions during the early phasesof the core collapse are probably essential, and were notincluded in the r-process studies cited above. The electron fraction is set by a competition between chargedcurrent lepton (e 2 , e 1 , n e , and n ē ) capture processes onfree nucleons, the neutrino capture reactions being themost important. Neutral current spallation of a particles
is also important, but only in the limit of a high entropy
r process. Neutrino interactions with heavy nuclei do
not affect Y e (McLaughlin et al., 1996; Hoffman et al.,1997), however, they do exhibit interesting effects during the neutron-capture phase by spreading and damping the peaks in the abundance distribution. Haxton
et al. (1997) find that nuclei in the mass regions
124<A<126 and 183<A<187 show an inordinate sensitivity to neutrino postprocessing effects, which could
impose stringent bounds on the freeze-out radii and dynamic time scales governing the r process in future corecollapse models. Freeze-out from statistical equilibrium
has already been deemed necessary to account for otherunique isotopic signatures within the solar r-process
abundance pattern using b -decay properties and abundances of the r-process nuclei themselves (also see Kratz
et al. 1993).
The evolution of Y e might also be affected by neutrino flavor transformation near the surface of the nascent neutron star due to the Mikeheyev-SmirnovWolfenstein (MSW) effect (Fuller et al., 1987).
Transformation between n¯ m ( t ) and n¯ e can occur if thevacuum masses for these species satisfy m n¯ e .m n¯ m , t andDm 2 .1 eV 2 . Such transformations, however, are notexpected to drive Y e .0.5, and hence do not pose any
difficulty for the envisioned r-process origin site. In
stark contrast, n mt to n e conversions, in which the neutrino energy hierarchy of the n e is reversed, could havedamaging consequences (Qian and Fuller, 1995).

E. Concluding remarks

The search for an overarching r-process paradigm
that encompasses the measured solar r-process abundances, observed halo and disk star r-process abundances, calculated r-process nucleosynthesis, and observed chemical evolution of the r-process abundances
has not been without its sucesses, but it remains unfulfilled. The attempt to satisfy these demands with a natu-

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Wallerstein et al.: Synthesis of the elements

FIG. 20. Fractions of neutron-capture solar system element abundances attributable to the r process (F ( @ r # ) as a function of
atomic number. These abundance fractions have been determined by Sneden et al. (1996) and Cowan and Burris (1997) bycombining r- and s-process solar system isotopic fractions (Käppeler et al., 1989) with solar system elemental abundances (Anders
and Grevesse, 1989; revised abundance recommendations of Grevesse et al., 1996, for a few elements would not materially affectthese fractions).

ral astrophysical origin site encompassing these aspects
promises to be an exciting adventure into the next millennia.
XIII. OBSERVATIONS OF THE r PROCESS
A. Defining The r-process elements

In this section we will review the stellar spectroscopic
data that confirm the existence of r-process nucleosynthesis at the early epochs of our galaxy, and that show
the growth of the r-process element abundances withtime. Only analyses of cool stars will be considered, inorder to avoid the spectroscopic peculiarities of hot stars
that are more indicative of surface and outer envelope
anomalies rather than true element nucleosynthesis.
Stellar metallicity will of necessity be the stand-in forstellar age, with the most metal-poor stars being assumed to be the oldest stars. Much of the discussion will
center on r-process elements in old, very metal-poorstars of the Galactic halo, for r-process elements can bemost easily disentangled from other elements in suchobjects. Although we will concentrate on positive evidence for r-process nucleosynthesis, we will also comment on the relative contributions of r and s processes
as a function of stellar metallicity.
B 2 FH devoted nearly 20 pages to a detailed description of physical conditions for and expected yields fromthe r process. However, their discussion of direct proofof the existence of the r process in stars was limited to apossible interpretation of Type I SNe light curves as being due to the decay of 254Cf (which later turned out tobe incorrect). There is still no direct evidence of
r-process species in the spectra of SNs or their remnants
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

(Wallerstein et al., 1995). While B 2 FH could display
spectra of stars with very obvious enhanced s-processcompositions (their Plate 3), no such dramatic examples
of stars with enhanced r-process compositions were
known for a very long time after publication of B 2 FH.
The firm identification of r-process nucleosynthesis
products in stars was difficult, because r-process elements are not easy to observe in cool stars. To amplify
this point we first identify all elements whose abundances are sensitive to r-process synthesis, rememberingthat stellar spectroscopy is mostly limited to determination of elemental, and not isotopic abundances. In Fig.
20 we display the solar system elemental abundance
fractions attributable to the r process (F ( @ r # ) and the sprocess (F ( @ s # ) of all stable ‘‘neutron-capture’’ elements in the range 31<Z<83. We also show the solar
system abundance fraction of the important unstableelement Th (Z590), which is produced only in the rprocess (F ( @ r # 51.0) because s-process nucleosynthesis
cannot proceed past Z583. The values of F ( @ r # andF ( @ s # have been determined by Cowan and Burris(1997) in the manner described by Sneden et al. (1996);
see Fig. 20 caption for details of this procedure. In thisfigure, element symbols are printed near the F ( @ r # values, but only for those elements whose solar abundanceis due chiefly to the r process (arbitrarily defined here as
F ( @ r # .0.75) or the s process (F ( @ r # ,0.25). Therefore
for convenience we shall call the labeled elements Sr,
Zr, Ba, Ce, La, and Pb ‘‘s-process elements,’’ and theremaining labeled elements will be called the ‘‘r-processelements.’’ It must be acknowledged that the neutroncapture element abundances in a given star may be due
to a very different mix of r- and s-process syntheses thanthose that produced the solar system composition.

Wallerstein et al.: Synthesis of the elements

There are 24 r-process elements identified in Fig. 20,
but unfortunately only a handful of these elements areeasily observable in stars. Several r-process elements(As, Br, Te, I, Xe, Cs, Re) have never been detected in
the solar photospheric spectrum (e.g., Grevesse et al.,1996). Several other r-process elements (Ag, Lu, Os, Ir,
Pt, Au) display only one or two usable solar spectralfeatures, and then only in the crowded and difficult toobserve UV spectral region (l<3500 Å). Thus the ‘‘observable’’ r-process elements usually come down tothose in the range 62<Z<69, and of those elements
only Eu (Z563, F ( @ r # .0.97) has spectral featuresstrong enough to be detected in many very metal-poorstars.

B. Early r-process discoveries

Most stellar r-process surveys have concentrated on
Eu, and assessments of the r- and s-processes contributions to stellar compositions usually compare Eu abundances to those of Ba (Z556), a nearby s-process element (F ( @ s # .0.88) with strong spectral features. To
begin a brief history of r-process observations, Wallerstein et al. (1963) detected a single Eu II feature in theirspectrum of HD 122563, which is by far the brightest
very metal-poor star ([Fe/H];–2.7) in the sky. Their
analysis, and Pagel’s (1965) reanalysis of the same dataset, did not find an easy match to r-process predictions,for they noted that Eu was deficient with respect to Fe inthis star: [Eu/Fe];–0.4. However, the Ba abundance of
this star was found to be even lower: [Ba/Fe];–1.0, leading Pagel to note that the relative abundances of neutron capture elements in HD 122563 were ‘‘an almost
exact mirror-image’’ of the abundance enhancements of
CH stars. Therefore, since the neutron-capture element
enhancements of CH stars are due to the s process(much greater overabundances of Ba, La, and Ce thanEu), this statement then constitutes the first solid evidence for r-process contributions to the material of very
old, very metal-poor stars.
A pioneering survey of neutron-capture elements inmany metal-poor stars by Spite and Spite (1978) showeda steep decline in relative Ba abundances with decreasing metallicity (that is, [Ba/Fe]}[Fe/H]) for stars in themetallicity range –1.3 < [Fe/H]<–2.6, while finding littlechange in the relative Eu abundances ([Eu/Fe];0 at all
metallicities). Consideration of these abundance ratios,
along with those of lighter neutron-capture elements Sr,
Y, and Zr led Truran (1981) to claim that ‘‘the heavy
element abundance patterns characteristic of the mostiron-deficient stars . . . are therefore entirely compatible with their having an r-process origin.’’
Sneden and Parthasarathy (1983) investigated the HD
122563 spectrum over greater spectral range and withhigher signal-to-noise data, producing first detections ofseveral r-process elements (Gd, Dy, Er) in a very metalpoor star. The relative underabundance of all neutroncapture elements in this star was confirmed, but theRev. Mod. Phys., Vol. 69, No. 4, October 1997

1051

element-to-element abundance pattern could only bematched by a dominant r-process contribution to theelement distribution.
Soon it became apparent that some very metal-poorstars possess neutron-capture element overabundances,most notably HD 115444 (Griffin et al., 1982), and HD
110184 (Sneden and Pilachowski, 1985). The large overabundances of Eu in both of these stars signaled a strong
r-process contribution. Finally, a 20-star survey by Gilroy et al. (1988) demonstrated the pervasiveness of
r-process nucleosynthesis products in metal-poor stars,
and argued strongly for significant star-to-star scatter inthe overall levels of the neutron-capture element abundances relative to Fe for stars with [Fe/H]<–2. Gilroy
et al. suggested that significant contributions bys-process nucleosynthesis to the neutron-capture elements are only apparent in stars more metal-rich than
[Fe/H];–2. Results from other investigations ofneutron-capture elements from this era may be found inreviews such as Wheeler et al. (1989).

C. Recent r-process surveys

Emphasis in recent years has turned to large surveys
of stars over a wide range of metallicity. These surveys
try to elucidate abundance ratio trends that might beapparent even though the ratios for individual stars may
still have substantial uncertainties. Major boosts to thesestudies have come from new catalogs of low metallicity
stars. To cite two examples, Laird et al. (1988) have produced a large set of [Fe/H] values for nearby (mostly
low luminosity dwarf and subgiant) high proper motion
stars, and Beers et al. (1985, 1992) have an ongoing program to discover extremely low metallicity, high luminosity giant stars at large distances into the galactic halo.
These metallicity surveys are proving especially usefulin finding stars with [Fe/H]<–2; such stars surely were
the recipients of the first wave of galactic nucleosynthesis, and abundance ratios of many element groups withrespect to Fe show major departures from solar ratios insuch stars. In Fig. 21 we show average abundance ratiosof three neutron-capture element groups from some recent studies of low metallicity stars. Since the purpose ofthis figure is only to show rough abundance ratio trends,
several stipulations accompany this figure. First, the datachosen for inclusion here are representative (emphasizing large-sample abundance ratio surveys), and are notmeant to include all studies of low metallicity stars. Second, no attempt has been made to renormalize the data
sets of different studies to a common scale (see the individual cited papers for discussions of this point).
Third, some bright ‘‘standard’’ stars (e.g., HD 122563)
appear in more than one survey, but we have chosen toplot all the points from each survey. Fourth, we havemade these three-element averages in an attempt tominimize analysis uncertainties; in most studies often
only one or two lines of a given element have been analyzed, and there are difficulties in deriving accurate
abundances from the very strong spectral features of Sr

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Wallerstein et al.: Synthesis of the elements

FIG. 21. Mean relative abundance ratios of ‘‘light’’ s-processelements Sr, Y, and Zr (top panel), ‘‘heavy’’ s-process elements Ba, La, and Ce (middle panel), and r-process elements
Sm, Eu, and Dy (bottom panel), as functions of stellar metallicity [Fe/H]. The relative abundances are defined as@ A/B # [log10(N A /N B ) star – log 10(N A /N B ) ( for elements A
and B. In each panel the dotted horizontal lines represent thesolar abundance ratios of these elements: filled circles, McWilliam et al. (1995); open circles, Gilroy et al. (1988); open
squares, Ryan et al. (1996); plus signs, Ryan et al. (1992); filled
circles, Gratton and Sneden (1994); filled triangles, Magain(1989), Zhao and Magain (1990, 1991); crosses, Edvardsson
et al. (1993) in the top and middle panels, Woolf et al. (1995) inthe lower panel.

II (e.g., Gratton and Sneden, 1994) and Ba II (e.g., Magain, 1995, Sneden et al., 1996). In many stars only a Sror a Ba abundance has been determined, and thoseabundances must of necessity stand for the threeelement averages in such cases. Finally, we have eliminated some stars that are chemically evolved (e.g., CHstars) and thus are unrepresentative of the chemical mix
from which they formed.
Even with all the preceding caveats, several abundance trends are apparent in Fig. 21. If we arbitrarilylabel stars with [Fe/H]<–2.5 as ‘‘ultra-metal-poor’’(hereafter UMP) stars, at first glance the dominant
abundance signature for UMP stars is an extraordinarily
large star-to-star scatter in the overall level of neutroncapture elements with respect to Fe. This abundance
scatter, which, in the case of the Ba-La-Ce group, ismore than a factor of 100 at a given [Fe/H] and is wellRev. Mod. Phys., Vol. 69, No. 4, October 1997

beyond any possible observational and analytical uncertainties. The reality of the star-to-star scatter can bedemonstrated with simple comparative plots of the stellar spectra (e.g., Gilroy et al., 1988, McWilliam et al.,1995). Clearly, knowledge of a UMP star’s Fe metallicity
gives one absolutely no insight to its neutron-capture
element abundance levels. More importantly, this verylarge scatter means that the neutron-capture element
abundances in UMP stars are products of one or very
few prior nucleosynthesis event(s) that occurred in thevery early, poorly mixed galactic halo, an interpretation
favored by Truran (1981) from different observational
clues.
Beyond the obvious star-to-star scatter, the data displayed in the top panel of Fig. 21 show that the lights-process elements on average are nearly as abundantwith respect to Fe in UMP stars as they are in the sun.
However, there is a pronounced decrease in the BaLa-Ce group level in most UMP stars (see the middle
panel of Fig. 21). Like the observed star-to-star abundance scatter, the observed decline in the Ba-La-Cegroup with metallicity is far too large to blame on observational and analytical uncertainties. In contrast to theheavy s-process element decline, the r-process SmEu-Dy element group remains quite abundant in UMP
stars (see the bottom panel of Fig. 21), at least to metallicities as low as [Fe/H];–3. In many UMP stars,^ @ Ba,La,Ce# & / ^ @ Sm,Eu,Dy# & ; – 0.6, while for higher
metallicity halo stars the average abundance ratio ofthese two element groups is ;0. It is tempting to argue,as did Gilroy et al. (1988), that the Ba-La-Ce deficiency
argues against significant s-process contributions to galactic nucleosynthesis until the halo ISM metallicity attained [Fe/H];–2. Then, if one adopts the common assumption that significant s-process nucleosynthesis
occurs mainly during the He-burning phases of intermediate to low mass stars (M,10M ( ), the time scale forbuildup of galactic halo metallicity to [Fe/H];–2 musthave been no less than 10 7 years (the rough time scale
for the evolution of a 10M ( star).
How dominant is the r process in UMP star neutroncapture elements? The bottom panel of Fig. 21 clearlyshows the presence of a vigorous synthesis of the standard r-process Sm-Eu-Dy element group early in thehistory of the galaxy. But was it also responsible for the
low level of the element groups that are s-process products in solar system material? The nearly solar abundance ratios of the light s-process groups, Sr-Y-Zr, argues for significant contributions by the s process inmany UMP stars. Moreover, attempts to match a scaled
set of r-process abundance fractions to the detailed
abundance ratios among these elements have not met
with much success (e.g., Gilroy et al., 1988, Cowan et al.,1995).
Among the heavier s-process elements, Magain(1995) has estimated isotopic ratios of Ba from analysis
of a Ba II line profile in one UMP dwarf star (HD140283), concluding that no clear evidence exists for asignificantly nonsolar, enhanced r- to s-process ratio in
the total Ba abundance of this star. On the other hand,Wallerstein et al.: Synthesis of the elements

FIG. 22. Neutron capture element abundances in the UMP
star CS 22892-052 (filled circles with error bars), comparedwith scaled solar system abundance distributions.
The abundances are ‘‘absolute,’’ defined as loge(A)[log10(N A /N H )112.0 for any element A. Anders andGrevesse (1989) and Grevesse et al. (1996) are the sources forthe solar system e values, and the r- and s-process curves are
generated by multiplying these values by the appropriate
F ( @ r # and F ( @ s # fractions discussed earlier in this section.
Each solar loge curve has been shifted by a single vertical
additive constant to match the CS 22892-052 data points. The
r-process curve has been shifted to make the best average
match to the abundances in the range 56<Z<72. The total
solar system curve has been normalized to agree with the observed Eu abundance. The s-process solar system curve has
been shifted by an approximate amount, for it does not match
any set of observed abundance ratios.

McWilliam et al. (1995) use the subsolar Ba/Eu ratios ofnearly all UMP stars of their sample to assert that earlygalactic synthesis of Ba was carried out via the r process.
Moreover, serendipitous inclusion of the star CS 22892052 in the McWilliam et al. survey provides perhaps thestrongest case yet uncovered of r-process dominance inearly galactic nucleosynthesis.
CS 22892-052 is a UMP giant star with extraordinary overabundances of all neutron capture elements. It has the largest r-process/Fe abundance contrast of any known star: In the bottom panel of Fig. 21,
this star appears at [Fe/H];–3.1, ^[Sm,Eu,Dy/Fe]&;+1.5.
Sneden et al. (1996) have presented a detailed neutron
capture element analysis of CS 22892-052 using newhigh signal-to-noise spectra. In Fig. 22 we display theirderived abundances for elements in the range 56<Z<
90. Among the 17 elements detected in this star are several (Tb, Ho, Tm, Hf) never before found in cool star
spectra. Additionally, the abundances of elements previously detected in other metal-poor stars are based onlarger numbers of transitions (18 features in the case ofNd) in CS 22892-052, increasing confidence in the derived abundances. Clearly CS 22892-052 is one of thebest stellar ‘‘laboratories’’ for investigation of early galactic neutron capture nucleosynthesis. In Fig. 22 wecompare the observed CS 22892-052 abundances toscaled solar-system r process, s process, and total abunRev. Mod. Phys., Vol. 69, No. 4, October 1997

1053

dance distributions [see Sneden et al. (1996) for detailsof this procedure]. Neglecting for the moment the element Th (Z590), the data match the r-process curve
exceedingly well, and fail substantially when comparedto the other curves (the scaled total solar system abundance curve does indeed match the elements in therange 62<Z<72, but that is because these are r-processelements in the solar system). There is little reason to
postulate a significant s-process component to any ofthese elements in this star. Since the r process apparently cannot occur during quiescent stellar evolution,these neutron-capture species must have been created ina previous stellar generation, almost certainly during the
explosive death of a prior high mass star. Additionally,
the extreme overabundance of all these elements withrespect to Fe strongly suggests that they were created bya single prior supernova that was ‘‘local’’ to the ISM
from which CS 22892-052 was created. Finally, it seems
remarkable that a scaled solar system r-process abundance distribution should match that of an old, UMP
halo star so well (Truran, 1996). This seems to imply auniversality to r-process nucleosynthesis: the conditions
in supernovae that lead to production of r-process elements that subsequently survive to seed the ISM apparently are always quite similar. Perhaps there is only arelatively narrow mass range of supernovae that contribute to bulk production of r-process elements in the galaxy.
D. Thorium and the age of the halo and disk

Thorium (Z590) is synthesized only in the r process
and it decays with a half-life of 14 Gyr, and so a substantial fraction of any Th created at any galactic epochshould still survive today. For this reason much effort
has been devoted to the derivation of Th abundances(from the single available Th II spectral feature at 4019
Å) and their use in estimations of the age of the galaxy.
Butcher (1987) pioneered stellar Th cosmochronology,
estimating the age of the galactic disk by comparing Th
abundances to those of the stable neutron capture element Nd. His work has been further refined by Morell
et al. (1992). Unfortunately, interpretation of Th/Nd ratios for cosmochronology is clouded by the necessity toconsider only that part of the Nd abundance in a starthat is due to the r process (a bare majority in the sun:F ( @ r # 50.53). Therefore, Pagel (1989) suggested using
Th/Eu ratios instead, since Eu is an almost purer-process element (F ( @ r # 50.97). François et al. (1993)
carried out an extensive study of Th/Eu in low metallicity halo stars. They found approximately solar abundance ratios of these elements at all metallicities, butthey did not derive galactic ages from their data because
they felt that the production ratios for these two elements were insufficiently understood to make meaningful age estimates. Frustratingly, in stellar Th cosmochronology one cannot perform the classical parent/daughter
test to discover the original content of the parent (whichof course must be known to determine a radioactive age
for the star).

1054

Wallerstein et al.: Synthesis of the elements

CS 22892-052 and other UMP stars that are very richin r-process elements strengthen the use of Th in cosmochronometry (Wallerstein, 1994). The key is the determination of many stable r-process abundances (morethan just Eu), particularly those with atomic numbers as
close to Th as possible. Notice in Fig. 22 what little flexibility in the scaled solar system r-process distribution ispermitted from the 16 stable elements observed in CS
22892-052. This makes plausible the assumption used by
Sneden et al. (1996) that the implied, scaled Th abundance (corrected back to the start of the solar system)from the scaled r-process curve may be a good representation to the original Th in CS 22892-052. Sneden et al.
used the mismatch between predicted and observed Th
abundances in this star to suggest that the progenitor
supernova to this star exploded 15.263.7 Gyr ago.
Cowan et al. (1997) have considered uncertainties in all
aspects of Th observations, nuclear reaction rates, andgalactic chemical evolution to suggest that the ‘‘age’’ ofthe r-process elements in CS 22892-052 might be as old
as 17 Gyr, but ages much younger than about 12 Gyr are
unlikely.

ucts, then at least we must derive abundances of severalof the heaviest stable elements in UMP stars. This isnecessary so that we may predict with more confidence
the original Th abundances ejected by supernovae andnow residing in diminished numbers in stars with detected Th features. In practice, the elements Os, Ir, Pt,
Au, and Pb are candidates for observation, but all strongtransitions of these elements occur in the UV below theatmospheric cutoff. Therefore the high resolution spectrographs of the Hubble Space Telescope must be employed for this work, and Cowan et al. (1996) report detections of Os, Pt, and Pb in the moderately metal-poor
HD 126587 ([Fe/H];–1.7). The derived abundances forthese elements are consistent with a scaled, mixed r- ands-process solar abundance distribution of elements from
Z556 to Z582. In new HST observations, this samegroup has detected Pt and Os in HD 115444, a UMP starwith r-process overabundances. Preliminary analysis
suggests that abundances of these elements also agreewith those that would have been predicted from the contents of the neutron capture elements near Z560.
XIV. THE p PROCESS

E. Filling out the picture
A. The p processr-process observations have progressed from nonexistent at the writing of B 2 FH to the rich data sets thatare available today. However, many years of effort remain to complete our knowledge of these elements andhow they relate to galactic nucleosynthesis. Here wehighlight a few areas of special interest.
Light neutron capture element synthesis. The unsatisfactory understanding of the Sr-Y-Zr element abundances has been mentioned above. The large abundances of these elements even in stars with very
deficient heavier s-process elements (the Ba-La-Cegroup) may mean that the s process did produce thelightest neutron capture elements during the quiescentevolutionary stages of the high mass stars that eventually manufactured the r-process elements at the supernova stage. Alternatively, the r process may have atleast two components, perhaps produced in supernovaeof different mass. A ‘‘second’’ r process that would produce just the lighter neutron capture elements has beensuggested by Wasserburg et al. (1995), but definitive observational tests will be difficult. One possibility might
be to search for spectroscopic features of elements in theatomic number range 41<Z<55, none of which havebeen studied extensively in stars other than the sun.
Most elements in this range cannot be detected spectroscopically. One possibility is to search for Ag (Z547),
which has two strong lines in the near-UV solar spectrum. Even this one element filling in a very large neutron capture element gap would allow us to eliminate
some of the possible scenarios to explain the lighter neutron capture elements. At present we lack a definitive
observational signature to understand the abundances oflighter neutron capture elements in UMP stars.
Detection of the heaviest stable neutron capture elements. If we cannot observe radioactive Th decay prodRev. Mod. Phys., Vol. 69, No. 4, October 1997

One process for which B 2 FH anticipated much of themodern description is the p process. It creates the
proton-rich nuclides heavier than iron that are blockedfrom formation by either the r or s processes by stable
nuclides. This definition immediately suggests the majortask of any p-process description: how to find ways otherthan b decay to process the usually more abundant rand s-process nuclides into the usually less abundant p
nuclides. B 2 FH described two possible mechanisms bywhich p nuclides could be formed: proton radiative captures in a hot (T9;2–3) proton-rich environment, or
photon-induced n, p, and a -particle removal reactions,also in a hot environment. B 2 FH also noted some of themodern p-process problem nuclei. Finally, they anticipated Type I and Type II supernovae as possible sites
for the p process, both of which remain fashionable today, although for rather different reasons than thosesuggested by B 2 FH.
Following their qualitative description of the p process, B 2 FH developed some numerical estimates for nuclei in the A;100 region that suggested that (p, g ) reactions dominated over photonuclear processes for
p-nuclide synthesis in this mass region. They also notedthat the p process could proceed by successive proton
capture until equilibrium between (p, g ) and( g ,p)1( g ,n) reactions was achieved, which they estimated, from assuming a generalized b decay life-time
formula, to lie 2–3 nucleons to the proton-rich side ofstability. Finally they noted that proton capture rates did
not appear to be a limitation to the p process; captures
would proceed at a sufficiently rapid rate to drive the pprocess. However, those b -decay rates could limit the
number of protons that could be added during thep-process event.

Wallerstein et al.: Synthesis of the elements

The p process is now thought to take place at somelevel in a high temperature proton-rich environment,
both by proton captures on light seed nuclides and byphotonuclear processes on the heaviest abundant stable
nuclides. However, development of the a -rich freezeout model in Type II supernovae, discussed elsewhere inthis review, has suggested a new site for synthesis ofsome p nuclides. This would occur along with the r process in the neutrino wind resulting from supernova collapse. And other p nuclides are now thought to be synthesized by neutrino-induced nucleon removal fromnearby nuclides in the ‘‘n process.’’ Although the idea ofconcurrent r, n , and p processes is relatively new, somefeatures thereof are very appealing. They are discussed
in greater detail below.
However, it would be premature to declare the p process to be understood, partly because production of all
the p nuclides in the periodic table seems to require
several processes (although this may not be surprising),
and partly because there seem to be several explanations for production of some of those nuclides. A few ofthe p-process nuclides are so difficult to make by global
mechanisms that their synthesis seems to necessitatevery specific processes. In other cases, processes thatseem inevitable in supernovae overproduce p nuclides
that are also thought to be produced in other sites, creating an apparent abundance excess. Also, the predictions of the heavy nuclide production via photodissociation are based on cross sections no one of which has
been measured. Finally, it has been sufficiently difficult
to identify a site in which the conditions that p-processmodels require that any p-process scenario must be
greeted with caution.
The separate mechanisms that are suggested as contributing to p-process nucleosynthesis, as well as theirstrengths and weaknesses, are discussed below. Additional detail about the p process can be found in thereview article by Lambert (1992). In particular, he describes many of the details associated with determining
the isotopic abundances essential to pinpointing the nature of the p processes of nucleosynthesis. Specifically,studies of meteoritic samples have provided most ofthese details, since astronomical observation of isotopes
in even the lightest p-process nuclides is usually not possible.
The generally accepted list of p nuclides is given in
Table VI. Their most general characteristic is their rarity, with a few notable exceptions, with respect to theother isotopes of each element. Note that some of them
can also be made by r or s processes, but those processes
are not thought to contribute a large fraction of any
particular p nuclide’s total abundance.
B. Early p-process models

The basic features of p-process nucleosynthesis, andtherefore some framework for the conditions required
to synthesize the p nuclides, were first set forth by Ito(1961). He assumed a very high-temperature hydrogenrich environment, T 9 ;2.5–3, and (p, g ) and ( g ,n) reacRev. Mod. Phys., Vol. 69, No. 4, October 1997

1055

tions to produce significant abundances of all the p nuclides. Truran and Cameron (1972) studied theproperties of a potential site for p-process nucleoysnthesis, a shock heated hydrogen-rich environment, andfound that the lighter p nuclides could be produced byproton capture reactions, while the photodissociative reactions were required for synthesis of the heavier nuclides, in basic agreement with B 2 FH. They concluded
that temperatures of around T9;2.5 or more were required to synthesize the p nuclides throughout the periodic table.
Audouze and Truran (1975) included (p, g ), (n, g ),( a , g ), (p,n), ( a ,p), and ( a ,n) reactions and their inverses in their network to study the successes and failures of the early p-process description. They assumed adensity of 10 4 g cm 23 and temperatures of T9;2 that
decreased exponentially with an e-folding time of either
0.446 s, the adiabatic expansion time, or 1 s. Their results identified some deficiencies, e.g., underproductionof the mass 92–98 p nuclides. They also noted that thesite for the requisite temperature and density might notexist. Arnould (1976) considered hydrostatic burning
conditions that would produce simultaneous r and p
processes, but found that very long times, greater than
104 s, were required for appreciable p-nuclide production. These concerns were amplified by Woosley andHoward (1978), who noted that the energy required toheat an appropriately thick hydrogen-rich shell to T9;2
for 10 s exceeds the entire mechanical energy output of
the supernova. Woosley and Howard also found that theseed nuclei in the Audouze-Truran model would have to
be enhanced in abundance by a factor of 100 for supernovae to produce the required p-nuclide overproduction
factors. Thus, the early p-process description could havebeen correct only if a site that produces its requisite
conditions can be found, and that site is not readily identifiable with either supernovae or novae.
C. The g process

The g process appears to be capable of producing
nearly all the heavy (A.100) p nuclei. As noted above,it was first studied by Ito (1961) and advanced by Audouze and Truran (1975). The current description wasformulated two decades ago by Woosley and Howard(1978) who showed that, with a few notable exceptions,the predicted ratios of the abundances of all the heavy p
nuclides corresponded reasonably well with those observed. One of the exceptions, 146Sm, was subsequently
explained (Woosley and Howard, 1990) by a more careful handling of the nuclear physics details. The othertwo, 180Ta and 138La, are discussed below. Both solar
and s process enhanced seeds were tried. The latter, ofcourse, enhanced the p-nuclide abundances, but did
little redistribution thereof. The Woosley-Howard work
was updated in 1990 (Woosley and Howard, 1990).
The g process operates essentially as the B2FH p process, i.e., photons from a high-temperature (T9;2–3)
bath containing previously synthesized heavy nuclides
initiate successive ( g ,p), ( g ,n), and ( g , a ) reactions on1056

Wallerstein et al.: Synthesis of the elements

TABLE VI. The p-process nuclides.
Element

Z

A

N Aa

(N A /SN A )3100

Se
Kr
Sr
Nb
Mo

34
36
38
41
42

Ru

44

Pd
Cd

46
48

In
Sn

49
50

Te
Xe

52
54

Ba

56

La
Ce

57
58

Sm

62

0.55
0.15
0.13
0.0
0.38
0.24
0.10
0.035
0.014
0.020
0.014
7.931023
0.037
0.025
4.331023
5.731023
5.131023
4.831023
4.531023
4.131023
2.231023
2.831023
8.031023
0.0
2.231024
3.831024
3.531024
3.231024
2.531024
2.531026

0.88
0.34
0.56
0.0
14.84
9.25
5.52
1.88
1.02
1.25
0.89
4.3
0.97
0.66
0.09
0.12
0.11
0.11
0.10
0.09
0.19
0.25
3.10
0.0
0.06
0.10
0.14
0.13
0.16
0.01

1.731024
1.231024
1.731024
5.231024

0.13
0.02
0.01
0.15

Dy

66

Er
Yb
Hf
Ta

68
70
72
73

74
78
84
92
92
94
96
98
102
106
108
113
112
114
120
124
126
130
132
138
136
138
144
146
156
158
162
168
174
180

W
Os
Pt
Hg

74
76
78
80

180
184
190
196

Comments

b decays to 92Mo; (T 1/2 ) b53.5310 7 y
Also produced by s process

Also produced by r and s processes
Also produced by s processb decays to 138Ce; (T 1/2 ) b51.5310 11 y

a decays to 142Nd; (T 1/2 ) b51.033108 y

Actually 180Ta m ; (T 1/2 ) b51.231015 y
b 2 decays to 180W or e 2 captures to 180Hf
Also produced by s process
Also possibly produced by s processaAnders and Grevesse (1989).
Tuli (1995).

b

those nuclides to synthesize the heavy p nuclides. Typical processing times must be less than a second, as
longer times would, at these high temperatures, destroy
all the heavy nuclides by photonuclear processes. Blocking of the b 2 decays from the abundant nuclides on theneutron-rich side of stability to the p nuclides, the standard inhibition to p-nuclide production, is circumvented
by the photon-induced reactions that populate the p nuclides directly. For example, any sequence of photonuclear reactions in which 14 neutrons and 4 protons are
removed from 208Pb would produce the p nuclide 190Pt.
Initially the ( g ,n) reactions would dominate, but ( g ,p)and ( g , a ) reactions would become more probable as
the photodisintegrations proceeded. In some cases
proton-rich unstable nuclides are formed which b 1 decay back to the stable p nuclides after the g -process
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

conditions have subsided. Of course, the Q values and,for charged particles, Coulomb barriers will determine
the probability of each reaction from any intermediate
nucleus. In addition to photonuclear production, thelight p nuclides might also be formed by rapid proton
captures on lighter (and more abundant) seed nuclides,
in which the seed nuclei are driven to the proton drip
line. They decay back to stability after the high temperature environment has cooled.
A major issue for the g process appears to be whether
or not a site could be found for it that would yield appropriate overproduction factors for the p nuclides.
Rayet et al. (1990) extended the original WoosleyHoward work to an oxygen burning shell of a supernova. Their calculations involved (n, g ), (p, g ), ( a , g ),(n,p), (n, a ), and (p,a ) reactions and their inverses, asWallerstein et al.: Synthesis of the elements

well as reactions between several heavy nuclei. The peaktemperatures ranged from T 9 =2.2–3.2, and fell off exponentially with an e-folding time of either 0.446 s or 1 s,
and densities were 10 6 g cm 21 . Despite the differences
between the studies of Rayet et al. (1990) and of Woosley and Howard (1978), their results did not differ fromeach other in most cases by more than a factor of 2.
Prantzos et al. (1990) extended the study of Rayetet al. (1990) by using their reaction network in a modelof the thermodynamic conditions of a Type II supernova. The model used had been found (Nomoto andHashimoto, 1988) to give a good representation of SN
1987A. The temperatures involved ranged from T 9 of 2
to 3.2 which was found to be the range relevant to the pprocess, as very little processing occurred at lower temperatures, and the seeds were destroyed at higher temperatures. It was assumed that the seed nuclei were
those resulting from the s process. Finally Rayet et al.
(1995) considered p-nuclide production from Type II supernovae spanning a mass range from 13 to 25 M ( . The
results of these three studies were significant p-nuclide
overproduction factors, but a wide variation in theirrelative abundances. Most notably, the Mo and Ru p
nuclides were badly underproduced, as were a numberof others. However, it should be noted that the overproduction factors of the p nuclides lighter than Mo wereroughly the same as those for the more massive nuclides
excepting the usual p-nuclide problems. It is noteworthy
that the model of Prantzos et al. (1990) produced
roughly the correct overproduction factor of 180Ta, a nuclide that is a problem for all other g -process models.
A recent extension of the Woosley-Howard work by
Howard et al. (1991) has identified a plausible site forthe g process. They find the requisite thermodynamic
and seed conditions to occur when a carbon-oxygen
white dwarf explodes, either as Type Ia or a subclass ofa Type II supernova. The resulting high temperatures,
T9;2–3, produce the high-energy photons necessary to
photodissociate the heavy nuclear seeds to yield theheavy p nuclides, and the carbon-burning reactions of
the white dwarf produce enough protons to fuel the(p, g ) reactions necessary to synthesize the lighter p nuclides. Virtually all of the p nuclei heavier than 92Mo
have overproduction factors within a factor of 9 of eachother (excepting 180Ta, 146Sm, and 138La), and absolute
overproduction factors of order 10 4 , a number large
enough to suggest that the site should be taken seriously. The overproduction factors (ratios of averaged
stellar to solar system mass fractions) of order of the pnuclides from this model are shown in Fig. 23.
The exploding carbon-oxygen white dwarf model does
have some deficiencies, even aside from 180Ta and 138La.
Specifically, 152Gd is badly underproduced, and the light
p nuclides are overproduced relative to their more massive counterparts. In addition, the 74Se, 78Kr, and 84Sr
overproduction factors fall as much as a factor of 2
above those of the heavier nuclides. While this alone
might not constitute a fatal flaw, other processes also
produce those lighter p nuclides. The resulting total
overproduction would thus present a serious conflict
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1057

FIG. 23. Overproduction factors for p nuclides, relative to theaverage overproduction factor, as a function of atomic mass.
The values shown are an average of equally weighted contributions from temperatures in the range T9=2.4 to 3.2 in stepsof 0.1. From Howard et al. (1991).

with the observed abundances. Finally, as usual, the Mo
and Ru p nuclides are underproduced.
Essential inputs to any g -process description are theproperties of the very proton-rich lighter nuclides thatare produced primarily by rapid proton captures. In particular, the masses are crucial, as the high temperatureenvironment that characterizes the g process will certainly produce ( g ,p) and ( g , a ) reactions to competewith the (p, g ) reactions, and the probabilities of thephotonuclear processes depend critically on the reaction
Q values. Those nuclides with long half-lives will tend to
inhibit the flow of the g process and will build up in
abundance. Finally, the decay modes may be importantas, e.g., b 1 decay and b 1 -p decay will produce different
final nuclides from the g process. Thus significant effort
has gone into observing these properties of the very
proton-rich nuclides (Boyd, 1994; Winger et al., 1993;
Blank et al., 1995).
D. The rp process

Some nucleosynthesis of the lighter p nuclides is provided by the rp process. The modern formulation of therp process, proposed by Wallace and Woosley (1981)and described also in Sec. IV.H, describes the explosive
nucleosynthesis that would occur in a fairly hightemperature hydrogen-rich environment. Wallace andWoosley considered temperatures as high as T9;2, andfound that even at temperatures appreciably less than a
T 9 of 1 they could produce appreciable abundances of74
Se and 78Kr. Possible sites for such environments
could result from accretion from the periphery of a red
giant onto a companion white dwarf or neutron star.
The seed nuclei contained in the rp process environment are driven to the proton-drip line, from which they
decay after the high temperatures have subsided. Note

1058

Wallerstein et al.: Synthesis of the elements

that, at high temperatures, the rp and g processes are
qualitatively similar; high-energy photons photodissociate the heavy nuclei and proton radiative captures boost
the masses of the lighter nuclei.
Beyond iron, the rp process begins to populate someof the nuclides that are the progenitors of the p nuclides.
The details of the burning were worked out by van
Wormer et al. (1994), who assumed (constant) temperatures ranging from T 9 of 0.15–1.5, densities around 10 4 g
cm 23 , and processing times of up to 1000 s for the lowest temperatures. At the upper end of their temperature
range significant abundances of 74Se and 78Kr could be
produced. However, 69Br, which might be thought to beformed by 68Se(p, g ) 69Br, was subsequently found to be
proton unbound (Blank et al., 1995). Since the half-lifeof 68Se is 35.5 s, appreciably longer than the highest
temperature conditions can persist, the rp process will
be slowed appreciably at mass 68. Of course, some leakage past 68Se will occur from the decays that do occur,and this could be a fairly large fraction of those that end
up at 68Se if the high-temperature conditions last a significant fraction of the 68Se half-life.
The yields were reexamined by Herndl et al. (1995)
with reaction rates determined from the nuclear structure results of shell model calculations. The possibility
that two-proton captures could influence the results
were also studied (Schatz et al., 1997). It was found thatthe usual rp-process termination points could be circumvented, but only at rather extreme conditions(T 9 5151.8, density=106 g cm 23 ). In such environments, the first capture might also circumvent the termination points (Schatz et al., 1997) in particularly hot
dense environments, as the first might result in a sufficiently long-lived state that a second can occur before
the initial proton is reemitted. Finally, Thielemann et al.
(1994) studied the general features of the rp-process
path, most notably, that it tends to involve cycles superimposed on its general flow. Indeed the general featuresof the rp process were found to be considerably more
complex than those of, e.g., the r process, due to theinterplay of proton captures, b decays, possible photodisintegrations, and a particle induced reactions
throughout the rp-process path.
Radioactive beam studies have made the present description of the rp-process possible. It had been thoughtthat 64Ge would be the end of the rp process, as mass
formulae suggested that 65As, the nucleus formed by
64
Ge(p, g ) 65As, would be unstable to proton decay.
This, however, was found (Winger et al., 1993) not to be
the case, allowing the rp process to proceed up to 68Se.
The 68Se roadblock would also be circumvented if therp process occurred in a Thorne-Żytkow object (Biehle,1991), a hypothetical star in its red giant phase that has aneutron star at its center. Such a composite star would
be sufficiently convective that material might undergo
multiple processing episodes at the high temperature region near the surface of the neutron star. Thus the proton rich nuclides synthesized in each processing interval
would undergo b 1 decay to provide seeds for the next
processing sequence. Such objects might be expected toRev. Mod. Phys., Vol. 69, No. 4, October 1997

make copious amounts of the light p-process nuclides(Cannon, 1993), although, as shown by van Wormer
et al. (1994), T 9 must be less than about 2 or ( g , a )reactions will terminate the nucleosynthesis at aboutmass 56. In addition, recent studies (Chevalier, 1996;
Fryer et al., 1996) have cast some doubt on the stabilityof a dense gas just above the surface of a neutron star.
Most troublesome, though, is the absence of a definitive
discovery of a Thorne-Żytkow object.
E. The n process

In 1990, the n process was proposed by Woosley et al.
(1990), after having first been suggested by Domogatskii
et al. (1977), and was shown to be capable of producing
some of the lowest abundance p nuclides. The process isthought to occur in the neutrino wind generated by supernova collapse. The nuclides synthesized clearly depend on the shell in which the n process occurs. For
example, 11B would be expected to be made in a shell inwhich the dominant constituent was 12C, and 19F would
be made in a shell with 20Ne, both by processes in whicha neutrino would excite the target nucleus via theneutral-current interaction (which therefore allows interactions with all neutrino flavors) to an excited statefrom which the nucleus could emit either a proton or aneutron.
However, the n process could make two of the rarest
stable nuclides in the periodic table: 138La and 180Ta.
180
Ta would be made by the 181Ta( n , n 8 n) 180Ta (neutral
current) reaction, which appears to produce an abundance consistent with that observed. Similarly, the139
La( n , n 8 n) 138La reaction produces 138La. However, itappears that a comparable amount of 138La would also
be produced by the 138Ba( n ,e 2 ) 138La (charged current)
reaction; these two reactions appear capable of synthesizing roughly the observed 138La abundance.
Note that the requirement of heavy seeds for production of 138La and 180Ta is not in conflict with the photodissociation to a particles and neutrons that occurs during a -rich freeze out, discussed in Sec. XII and briefly
below. The n process occurs outside the bubble in which
the a -rich freeze out occurs, so it can operate on its
requisite previously synthesized nuclei. Indeed, the nprocess seems to provide a natural mechanism for synthesis of 138La and 180Ta, whose synthesis has evaded
description for several decades.
While the n process appears very promising for synthesizing some of the more difficult to explain p-process
nuclides, its results are somewhat in question due to uncertainties about the neutrino spectrum resulting from a
Type II supernova (Myra and Burrows, 1990) and about
the neutrino cross sections. Thus its full acceptance requires that the details of its nucleosynthesis be worked
into a realistic supernova model.
F. Recent developments

In the past several years the p process, along with ther and n processes, has undergone a revolution that mightWallerstein et al.: Synthesis of the elements

solve some of the most troublesome problems of nucleosynthesis. The process that has emerged is a -rich freeze
out in neutrino-driven winds occuring after core bounce
in a supernova (Woosley and Hoffman, 1992); thisshows promise of providing a site for both the r and nprocesses in addition to a site for synthesis of some of
the most challenging p nuclides. This process occurs
about 1 s after a supernova collapse undergoes corebounce in the high-entropy high-temperature (initially
T9;10) bubble near the core.
In this process, the photons in the high temperatureenvironment dissociate all the preexisting nuclei down
to essentially a particles and neutrons. A critical parameter in characterizing the environment is the electron
fraction Y e 5p/(n1p), where n and p are the neutron
and proton densities. As the environment cools, nuclear
statistical equilibrium dictates the abundances createduntil the nuclear reactions initiated with charged particles freeze out. Then the remaining neutrons synthesize all the r-process nuclides, provided that Y e is appropriately chosen. This environment has also been shownto create the light p nuclides through 92Mo (Hoffman
et al., 1996), both in the correct relative abundances andin absolute magnitude, again if Y e is chosen appropriately. Some of these are actually synthesized fromnearby r-process products through charged current interactions with the electron neutrinos. For example(Fuller and Meyer, 1995), neutrino capture will dominate over antineutrino capture, so that processes such as
92
Zr( n ,e 2 ) 92Nb and 92Nb( n ,e 2 ) 92Mo become possible,and could make a great deal of 92Mo. Since such processes can proceed through stable nuclei, they do allow
production of p nuclides from r- and s-process seeds.
However, the extent to which such processing can occur
is limited, as too many neutrino-initiated interactions
would smear out and shift the well-established r-process
abundance peaks resulting from the neutron closed
shells.
The a -rich freeze-out model does have some deficiencies that have persisted since its inception. Those associated with the r process are discussed in Sec. XII. Fromthe perspective of the p process; however, although thisprocess does make 92Mo with enough abundance to provide all that observed in nature, it is unable to produce
abundances even close to those of the next several p
nuclides. In addition, the results of the model calculations exhibit an uncomfortable sensitivity to Y e . Only
future work will tell if these problems can be solved, andthe a -rich freeze-out model can provide the descriptionof the nucleosynthesis of at least part of the p nuclides.
G. Summary

Although the various subprocesses by which the synthesis of the p nuclides is explained seem promising,some of them are not yet fully developed. Furthermore,
the overview study of these processes that is required todemonstrate that together they produce the correct
overproduction factors of the p nuclides has certainly
not been done. Most notably, the promising a -richRev. Mod. Phys., Vol. 69, No. 4, October 1997

1059

freeze-out model produces appropriate abundances ofthe p nuclides up through 92Mo. But the exploding
white dwarf model also produces those same nuclides,
excepting 92Mo, at such abundances, as to possibly create an excess of those nuclides. And no model has yet
shown how to produce 94Mo, 96Ru, and 98Ru in adequate quantities. Thus, although the present p-processdescriptions have advanced greatly beyond that of
B2FH, serious questions remain as to the viability of ourpresent descriptions of that process.
XV. THE e PROCESS AND THE IRON-GROUP NUCLEI
A. Energetics and equilibria

In even a cursory glance at the abundance distributionof nuclides in the solar system, the iron peak leaps out at
the eye. It stretches from nuclear mass number A'40 toA'60, comprising isotopes from Ca to Zn. The peak is
at 56Fe, which is the 10th most abundant species in thesolar system. 56Fe makes up ;0.1% of the solar system’s
mass. This is even more impressive when one realizes
that 56Fe is ;6% of the solar system’s mass in species
other than H and He and ;20% of the mass in everything other than H, He, C, N, and O! The other irongroup isotopes are less abundant than 56Fe, but theircontribution to the mass of the solar system is still significant: ;5% of the contribution of 56Fe. Clearly something special about 56Fe and its iron-group neighbors
makes them so abundant.
That something special is their strong nuclear binding
energy: the iron-group nuclides are the most tightly
bound isotopes in nature. The binding energy B(Z,N)
of a nucleus with Z protons and N neutrons is
B ~ Z,N ! 5Zm p c 2 1Nm n c 2 2M ~ Z,N ! c 2 ,(28)

where m p and m n are the masses of the proton and neutron, respectively, and M(Z,N) is the mass of thenucleus (Z,N). Figure 24 illustrates the strong bindingof the iron-group isotopes. Among isotopes with equal
numbers of neutrons and protons, the iron-group nuclide 56Ni has the greatest binding energy per nucleon.
As defined in Eq. (28), the larger the binding energy, thesmaller the nuclear mass relative to the total mass of theconstituent nucleons, and the more tightly bound thenucleus. The binding even increases for somewhat moreneutron-rich isotopes. For isotopes with 0.86<Z/
N<0.88, Fig. 24 shows that the dominant iron-group nuclide 56Fe is the most tightly bound isotope. And among
all nuclei, the iron-group isotope 62Ni has the largest
binding energy per nucleon (8.794 MeV) according to
the mass compilation of Tuli (1995).
The strong binding energy per nucleon of iron-groupnuclei means that these isotopes will tend to dominate
the nuclear abundances if the system reaches nuclear
equilibrium. Hoyle (1946) recognized this fact, andB 2FH elaborated on it. In the intervening years, theprogress in understanding the synthesis of iron-groupnuclei has been in a broadening of the picture of possible statistical equilibria and how nature realizes these1060

Wallerstein et al.: Synthesis of the elementswith Z as Z 2 /A 1/3. For nuclei with A&32, the symmetry
term is sufficiently important relative to the Coulomb
energy that Z5N nuclides can be the most bound isobar. Nuclear pairing favors even-Z–even-N nuclei over
odd-Z–odd-N nuclei, which explains why 30P, a Z5N
nuclide, is less bound than 30Si. By contrast, the Coulomb term begins to dominate for nuclei with Z.16,
and the Z5N nuclide is no longer the most bound isobar. Nevertheless, for A536 and A540 the Z5N nuclide is still stable. In the case of A536, although 36Cl is
more tightly bound than 36Ar, the total mass of thenucleons in 36Cl is greater than that in 36Ar. As a consequence, the mass of 36Ar is in fact less than that of36
Cl. This means that the decay 36Ar → 36Cl does notoccur. For A540, 40Ca is more bound than 40K. Above
A540, however, the Coulomb energy is so strong that it
dominates the single-particle effects that allowed lower
mass Z5N nuclides to be stable. The consequence isthat there is no stable Z5N nuclide having A.40, andthe iron-group elements are the first not to have stable
Z5N isotopes. Because nature has chosen to synthesize
the bulk of the iron-group isotopes in matter with equal
numbers of neutrons and protons, these isotopes generally had radioactive progenitors. This has enormous implications for astronomy, but it was not known to B 2 FH!

FIG. 24. The binding energy per nucleon in MeV for nuclides
with Z5N (top panel) and with 0.86<Z/N<0.88 (bottom
panel). 56Ni has the strongest binding energy per nucleon for
Z5N nuclei while 56Fe has the strongest binding for nuclei
with Z/N'0.87. 60Ni is also tightly bound, but not as much as
56
Fe. No isotope of vanadium (Z523) or chromium (Z524)
has 0.86<Z/N<0.88. 56Fe could be made as itself in equilibrium if the ratio of neutrons to protons in the nucleosynthetic
environment were around 0.87. This is B 2 FH’s e process. In
fact, nature seems to have chosen to assemble most of the solar
system’s iron-group nuclei in matter that had equal numbers ofneutrons and protons. In this case, 56Ni was made and later
decayed to 56Fe. The masses needed for this figure are from
Tuli (1995).

equilibria. Summarizing this progress is the dominanttheme of this review. Before proceeding to this, however, a few further comments on the nuclear physics of
the iron-group nuclei are in order.
Figure 25 shows the binding energy per nucleon for
the A=28, 32, 36,40, 44, and 56 isobars (isobars are nuclides with the same mass number A). What is clear isthat for A.32, the Z5N nuclide is not the most tightly
bound isobar. This is understandable in terms of theliquid-drop model of the nucleus (von Weizsäcker, 1935;
for a modern descendant of this model, see Möller et al.,1995). In this model, the nuclear mass is the sum of (1)the masses of the constituent neutrons and protons, (2) avolume term, (3) a surface term, (4) a symmetry term,and (5) a Coulomb term. Refinements account for the
single-particle effects of nuclear pairing and the nuclear
shells. It is decisive that the symmetry term falls off with
A as @ (A/2)2Z # 2 /A while the Coulomb term grows
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

B. Statistical equilibrium

The idea of statistical equilibrium has played a specialrole in the history of stellar nucleosynthesis theory.
Hoyle recognized in 1946 that the dramatic peak in theabundances of the iron-group nuclei called for synthesis
under conditions of temperature and density such thatstatistical equilibrium was attained between nuclei andfree neutrons and protons, and he showed that evolvedstars reached appropriate thermal conditions (Hoyle,1946). This achievement launched the paradigm of element formation in stars and led eleven years later to thestill-celebrated review by B 2 FH. The name that B 2 FH
bestowed on this mode of element synthesis was the ‘‘eprocess’’ (for ‘‘equilibrium’’). B 2 FH had a rather specific definition in mind for the e process, one using thenuclear properties of iron; but modern usage of the term
often is looser than the original in simply focusing on
statistical equilibrium. This review will distinguish between B 2 FH’s e process, which ultimately was not thecorrect explanation for the origin of the bulk of the solar
system’s iron isotopic abundances, and the more general
notion of nuclear statistical equilibrium (NSE). That distinction facilitates better understanding of modern ideasof iron-group elements synthesis and of the historicaldevelopment of those ideas.
The essential statistical notions have not changed
from those contained in Hoyle (1946) and B 2 FH. The
years since 1957 have, however, seen a broadening ofthese ideas. It is now becoming clear that a hierarchy ofpossible statistical equilibria exist and that differing
natural systems can achieve these equilibria to high accuracy. This perspective may not be familiar to mostWallerstein et al.: Synthesis of the elements

1061

FIG. 25. The binding energy per nucleon of the isobars with A=24, 32, 36, 40, 44, and 56. Only for A&32 is the Z5N nuclide themost bound isobar. For A.32 the Z5N nuclide is not the most bound isobar. This is due to the fact that the Coulomb energy inthese nuclei is sufficiently strong that it favors fewer protons and more neutrons. Nevertheless, the Z5N isobars of A536 andA540 are still stable. For nuclei with A.40, however, the Coulomb energy is so strong that the Z5N nuclide is no longer stable.
Nature has chosen to assemble most of the iron-group nuclei present in the solar system in matter with equal numbers of neutronsand protons. As a consequence of that choice, most of these nuclei were originally synthesized as radioactive Z5N progenitors.
The masses needed for this figure are from Tuli (1995).

readers of this review, and it may require some effort todevelop. Nevertheless, the insight garnered is well worth
the intellectual investment.
The entropy of a system provides the most natural
framework for understanding statistical equilibrium.
The entropy S is properly a function of the energy E,
the volume V, and the numbers $ N i % of each species i in
the system. S is defined as
S ~ E,V, $ N i % ! 5klnG,(29)

where k is Boltzmann’s constant and G is the number ofavailable allowed macroscopic states of the system. A
macroscopic state is a particular arrangement of thevarious species i, and an allowed macroscopic state is
one that has energy E. For nucleosynthetic systems, forexample, a macroscopic state would be a particular setof abundances of nuclear species, but arranged in such a
way that the total energy is E.
The value of the concept of the entropy is that it gives
a clear notion of the evolution of a system. If the system
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

is thermally isolated, and the volume is left constant, thenatural evolution is towards maximum entropy. If any
macroscopic states with energy E were not available tothe system at time t 1 , the system evolves in such a way as
to make those states available at a later time t 2 . The
system will continue to evolve until all allowed macroscopic states are available. Once this condition is attained, the entropy is a maximum, and there is no further evolution. The system has reached complete
statistical equilibrium.
Systems rarely reach complete statistical equilibrium.
The problem is that some key interaction time scale ortime scales may exceed the duration of the natural
event. This means that a subset of allowed macroscopic
states will be closed off to the system. Nevertheless, if allother interaction time scales are sufficiently short, allallowed macroscopic states excluding those in the closed
off subset will be available. That system maximizes theentropy as best it can, but its search is ‘‘constrained’’ tothe available subset of states. This leads to a hierarchy of1062

Wallerstein et al.: Synthesis of the elements

possible statistical equilibria in which systems havemaximum entropy subject to different numbers of constraints.
Stellar nucleosynthesis systems present an example ofsuch a constrained system. Conditions in stars are rarely,
if ever, sufficiently extreme to allow nucleons to convert
into other types of subatomic particles such as hyperons
or free quarks. The time scale for the nucleons to comeinto equilibrium with these other particles is too long.
This means that some macroscopic states with energy E,
but with hyperons present in equilibrium with the nucleons, are not available to the system. The system is thus
constrained to a smaller number of macroscopic states
than the number in the full ensemble. This constraint
may be incorporated by writing(i A i Y i 51,

nuclearf512(i A i Y iand to introduce a Lagrange multiplier l. The problemof finding the entropy maximum subject to conservationof the number of nucleons thus becomes that of finding
the abundances Y i such that d(s2lf)50. From Eqs.
(31), (32), and (33), one finds that l5 m n /kT,m n 5 m p 1 m e , and m i 5Z i m p 1(A i 2Z i ) m n , where m n
and m p are the chemical potentials of neutrons and protons, respectively. With the appropriate ideal gas expressions, the following equations result:Y ~ Z i ,A i ! 5G ~ Z i ,A i !@ z ~ 3 ! A i 21 p ~ 12A i ! /22 ~ 3A i 25 ! /2#
2 3 ~ A i 21 ! /2 12A i
3A 3/2f
i ~ kT/m N c !

and

Tds52

(i m i dY i ,(31)

where T is the temperature, m i and Y i are the chemical
potential and the abundance per nucleon of species i,
respectively, and the sum runs over all species present.
Because photons always and neutrinos generally havezero chemical potentials in nucleosynthetic environments, Eq. (31) becomes
nuclearTds52(i m i dY i 2 m e dY e ,(32)

where now the sum runs only over nuclear species andwhere m e and Y e respectively refer to the chemical potential of electrons and the abundance per nucleon of
net electrons (i.e., electrons in excess of positrons). By
charge neutrality,
Y e5(i Z i Y i ,(33)

where Z i is the nuclear charge of species i.
The maximum of the entropy per nucleon s (the condition ds50) subject to the constraint in Eq. (29) is themost general statistical equilibrium for a nucleosynthetic
system. If no constraint is imposed on Y e , this statisticalequilibrium includes weak equilibrium between neutrons and protons. To find the entropy maximum subject
to the constraint in Eq. (29), it is convenient to define afunction f such that
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

Z

A 2Z i

exp@ B ~ Z i ,A i ! /kT #

(35)

S

D

(36)

3Y p i Y n i(30)

where A i is the mass number of species i, Y i the abundance per nucleon of species i, and the sum runs only
over nuclear species.
Statistical equilibrium is an entropy extremum, so it isfound by considering changes in the entropy. In thethermally isolated system, infinitesimal, quasistaticchanges in the entropy per nucleon s are given by(34)

Y p /Y n 5

r N A Y e 2 p \ 2 3/2
exp~ Q 0 /kT ! .
2m e kT

In these equations G(Z,A) is the nuclear partition function, z (3) is the Riemann zeta function of argument
3 @ z (3).1.202], m N is the nucleon mass, f is the photon
to nucleon ratio, N A is Avagadro’s number, m e is themass of the electron, r is the density, andQ 0 5m p c 2 1m e c 2 2m n c 2 . The photon to nucleon ratio is
given byf5

1
2 z ~ 3 !~ kT ! 3.
p 2 ~ \c ! 3
rNA

(37)

With Eqs. (36), (35), (34), and (29), it is possible to determine the abundances given T and r .
It may seem paradoxical that definite expressions exist
for the abundances of the nuclear species in statisticalequilibrium. Statistical equilibrium is the condition of
maximum entropy and thus of maximum randomness:
all allowed macroscopic states, that is, all sets of abundances yielding total energy E and satisfying Eq. (31),are available to the system, and all are equally likely.
How then are definite abundances [Eq. (34)] possible?
The answer is that the statistical equilibrium abundancesin Eqs. (34) and (35) actually represent an average overthe ensemble of allowed macroscopic states. The vast
majority of the allowed macroscopic states have abundances $ Y i % very similar to those in Eqs. (34) and (35).
Because of the huge number of allowed macroscopic
states in the ensemble, the deviation of the abundancesof any macroscopic state from the average is unlikely to
be large, and Eq. (34) accurately gives the abundance inequilibrium at any instant in time. This point is not
merely pedantic—it is crucial for understanding how order develops during nucleosynthesis.
As Hoyle (1946) and B 2 FH recognized, the timescales present in nucleosynthetic environments may not
be sufficiently long to allow weak equilibrium. In thiscase Y e is slowly varying and may be taken to be instantaneously constant; thus, a new constraint is imposed on

Wallerstein et al.: Synthesis of the elementsallowed macroscopic states. This constraint vastly reduces the number of allowed macroscopic states in theensemble from the number when that constraint is absent. The statistical equilibrium in which Y e is fixed istherefore much more ordered than that for which weakequilibrium holds.
When the entropy is maximized subject to the constraints in Eqs. (29) and (32) the abundances again are
given by Eq. (34). Now, however, Eq. (35) is replaced by
the requirement that the abundances also satisfy Eq.
(32). The abundances for specified T, r , and Y e may
now be found from Eqs. (34), (32), and (28). This particular statistical equilibrium at constant Y e is what mostresearchers mean when they refer to ‘‘nuclear statisticalequilibrium,’’ or NSE.
An even more ordered equilibrium in the hierarchy ofstatistical equilibria is quasiequilibrium (QSE). Thereexists a whole sequence of possible QSE’s, but the least
restricted arises from the additional possible constraintthat Y h , the number of heavy nuclei (those nuclei with
A>12), is instantaneously fixed. This equilibrium is infact realized in nature because the reactions assembling
heavy nuclei can be much slower than the other reactions occurring in the system (except those changing
Y e ). Y h is calculated as
nuclearY h5( Yi .

(38)

i,A i >12

This QSE is even more ordered than NSE because of
the further reduction in the number of allowed macroscopic states in the ensemble. Meyer, Krishnan, andClayton (1997) have found the abundances in this QSE
by maximizing the entropy subject to the constraints ofconstant nucleon number, Y e , and Y h , that is, Eqs. (29),(32), and (37). The result is that the abundance of eachheavy nucleus (A>12) is given by
A2Z NSEY ~ Z,A ! 5e m h /kT R Z
Y~ Z,A ! ,p Rn

R p 5Y p /Y NSEp

R n 5Y n /Y NSE
n

(39)

and
measure thewhereoverabundances of free neutrons and protons in the
QSE relative to NSE and where Y NSE(Z,A) is the NSE
abundance of nucleus (Z,A). Meyer et al. (1997) wereable to interpret m h as the chemical potential of heavy
nuclei in the system. The abundances in QSE for specified T, r , Y e , and Y h may now be found from Eqs. (38),(37), (32), and (29). It is also necessary in solving for
QSE to solve for the NSE abundances via Eq. (34).
More restricted QSE’s arise from the possibility thatthe total number of nuclei in different ranges of atomic
number or mass number might be slowly varying. For
example, it may be that not only are the total number ofheavy nuclei only slowly changing with time, but also thenumber Y h1 of nuclei with 10<Z<20 and the number
Y h2 with 21<Z<30 might separately be nearly constantin time. In this case, there arises an additional constraint
on the equilibrium, and the system is even further ordered. The abundances are given by equations similar toEq. (28), but now with chemical potentials m 1 and m 2
appropriate for the separate equilibrium clusters.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1063

FIG. 26. The hierarchy of possible statistical equilibria in nucleosynthetic systems. The statistical equilibria are conditionsof maximum entropy subject to certain constraints. The
smaller the number of constraints present, the more disordered the system is and the higher up the hierarchy the equilibrium sits. All possible statistical equilibria in nucleosynthetic
systems have the constraint of constant nucleon number andthus lie below the dotted line. Since one or a few key timescales control the evolution of the system in many cases, nature is able to approximate these equilibria to high accuracy.

Figure 26 summarizes the hierarchy of statistical equilibria possible in nucleosynthesis. At the very top, theonly constraint is that the energy of the system is E.
There is no constraint on the number of nucleons in thesystem. As argued above, such an equilibrium is impossible in any nucleosynthetic environment; therefore,
only statistical equilibria lying below the dotted line in
Fig. 26 are possibly relevant to nucleosynthesis. The
highest relevant statistical equilibrium is NSE with weakequilibrium. Adding constraints leads to a descent of thehierarchy and an increase in order in the system. On theother hand, temporal evolution may be such as to ascend the hierarchy by removing constraints.
A system in a dynamical environment is unlikely to
precisely achieve any of the statistical equilibria shownin Fig. 26. This depends on the interplay of many
nuclear reactions, all occurring on different time scales.
Nevertheless, many systems do nearly manage to maximize the entropy subject to one or a few key time scales.
This means that the concept of a hierarchy of statisticalequilibria will be helpful in understanding these systems.
Such is the case with the iron-group nuclides.
C. A brief history of the ideas of iron-group element
synthesis

For a valuable chronicle of this major intellectual step,
the reader is referred to Clayton (1996).
B 2 FH ascribed synthesis of the iron-group isotopes tothe e process, by which they meant the ‘‘equilibrium’’
process. They envisioned NSE occurring nearY e 526/5650.464 in order that 56Fe be made as itself incorrect proportion to other Fe isotopes. In fact, they
fixed the temperature and the ratio Y p /Y n of free protons to neutrons in their calculations. Their argument
for fixed Y p /Y n was that once the system falls out of1064

Wallerstein et al.: Synthesis of the elements

weak equilibrium, the weak reactions will be slow andwill change Y p /Y n very slowly throughout the rest of
the expansion and cooling of the material. This is actually a bit of a shaky assumption. It is true that the slowness of the weak reactions will keep Y e nearly constant,
but there is no guarantee, for example, that free neutrons would not capture onto nuclei more quickly thanfree protons. This would certainly alter Y p /Y n . B 2 FH
recognized this difficult point and called for detailed astrophysical calculations to get Y p /Y n correct throughout
a stellar explosion. Much of the progress since 1957 hascome in precisely this quest.
B 2 FH obtained their best fit to the solar abundancesof iron-group nuclei in the mass range 50<A<62 for atemperature T 9 53.78 and a ratio Y p /Y n '300. With theassumption of weak equilibrium Eq. (36), this implies adensity of r '1.53105 g/cm 3 . Their results included
some estimates for freeze-out corrections.
It is interesting to note that the conditions B 2 FH
found best fit the solar system’s abundances of Fe-groupnuclei, corresponding at T 9 53.78 to an entropy perbaryon s/k'50, which is fairly large. By comparison,modern supernova calculations show that Fe-group nuclei are produced in Type Ia supernovae at entropies
s/k&1 and in Type II supernovae at entropies s/k'10.
A fundamental obstacle to the synthesis of 56Fe instellar evolution was that, in a sequence of burning
phases, the synthesis of 28Si would precede the e process. Direct conversion of 28Si nuclei into 56Fe is difficult
because (1) the Coulomb barriers for interactions of 28Si
with each other are too large and (2) Y e would have todrop from ;0.5 for 28Si to ;0.464 for 56Fe. The correct
answer would turn out to be what we now call siliconburning.
First, network calculations showed that 28Si burned
into 56Ni (Truran, Arnett, and Cameron, 1967). This isbecause Y e changes little during the burning of silicon
and 56Ni is the most bound nucleus with Y e near 0.5 (seeFig. 24). The 56Ni subsequently decays into 56Fe since
the latter is a more tightly bound nucleus, and a profoundly important question became whether that decay
occurred during presupernova evolution or after ejection of the matter. A second crucial point is the way
silicon burning actually proceeds. Because the reaction
28
Si1 28Si→ 56Ni does not occur directly, some 28Si must
break down into alpha particles which then capture onto
the remaining 28Si. The effect is to decrease the numberof heavy nuclei in accordance with the dictates of NSE.
The crucial point recognized by Bodansky et al. (1968) isthat, as the 28Si nuclei are stripped down by ( g , a ), additional reactions such as 20Ne( g , a ) 16O increase theabundance of alphas. This leads to an increase in therate of ( a , g ) reactions which impede the 28Si disintegration flow and slow the decrease in the number of heavy
nuclei. At the same time, reverse reactions between
A528 and A558 come into a steady-state balance.
Thus, while most reactions occur quickly and drive thesystem toward equilibrium, the equilibrium achieved is
not NSE because the abundance of heavy nuclei is notthat demanded by NSE. The equilibrium achieved isRev. Mod. Phys., Vol. 69, No. 4, October 1997

rather QSE, the third of the possible nucleosynthesis
statistical equilibria in the hierarchy in Fig. 26. Only forlong times can Y h reach its NSE value and allow thesystem to ascend the hierarchy further. The agreement
between the solar system Fe-group abundances and results of QSE and network calculations showed that QSE
silicon burning was important in the synthesis of thebulk of the solar system’s 56Fe. It also offered the hopeof testing this theory through observation of gamma rays
emitted by specific nuclei (Clayton et al., 1969). The possibility of confirming nucleosynthesis in this way was notforeseen by B 2 FH.
Woosley et al. (1973) provided further refinements in
ideas about equilibrium synthesis of Fe-group nuclei.
Most importantly, they considered higher entropies insilicon (and oxygen) burning, such as those that might
be established by dissipation of a supernova shock wave
in the Si shell. The interesting result was that, unlike incalculations of lower-entropy expansions of material
from high temperature and density, the final asymptotic
mass fraction of free alphas was not negligibly small.
Woosley et al. (1973) termed this the ‘‘alpha-rich freeze
out.’’ The astronomical relevance is that synthesis of56
Ni in Type II supernovae appears to occur via thisprocess. Also of significance is the fact that an alpha-richfreeze out naturally produces the radioactive species
44
Ti in sufficiently high abundance to account for natural 44Ca.
The second interesting point was that Woosley et al.
(1973) studied in detail the approach both to QSE andto NSE. The initial state of the system in these alpharich freeze-out scenarios, the 28Si-rich shell of a massive
star, is a highly ordered state because nearly all nucleons
are locked into 28Si nuclei. As a shock wave passes
through this zone during the supernova explosion, thematerial is heated. Normal silicon burning occurs assome 28Si nuclei are stripped down into alphas whichthen are captured by other Si nuclei. Eventually small
clusters of neighboring isotopes come into equilibrium
under exchange of n’s, p’s, and a ’s. This type of statistical equilibrium is low in the hierarchy of statisticalequilibria, but it is more disordered than the originalstate. In the subsequent evolution, the different equilibrium clusters begin to merge, first to the A528 to A542
region, later up to the A558. This reduction in time in
the number of constraints on the system enable the equilibria to climb up the hierarchy of statistical equilibria,with the system becoming more disordered. Given sufficient time, the clusters merge into a single large equilibrium cluster containing almost all of the heavy isotopes,
but the abundance of heavy nuclei still differs from thatdesired by NSE. In many cases in Type II supernova
explosions, this is the farthest up the statistical equilibrium hierarchy (and the most disordered) the systemgets during explosive silicon burning because of theshort hydrodynamical timescales. As the system expands
and cools, the evolution reverses in the sense that larger
clusters break up into smaller ones, the system descends
the hierarchy of statistical equilibria, and order reap-

Wallerstein et al.: Synthesis of the elements

pears. For a recent discussion of the evolution of QSEclusters in silicon burning, see Hix and Thielemann(1996).
Natural systems do reach the higher ranks of the statistical equilibrium hierarchy. The cores of massive stars
can evolve nearly all the way to the top (NSE with weakequilibrium) because of the relatively long time scales
and high densities in presupernova evolution. Interestingly, 56Fe can there dominate the abundance, just as
B 2FH envisioned. Were this material (and only this material) simply ejected, 56Fe would indeed be made by thee process! In fact, calculations show that little of thismatter actually gets ejected during the explosion. Mostof it remains in the compact remnant. That little which isejected is heated to high temperatures and undergoes analpha-rich freeze out (Thielemann et al., 1990; Woosleyand Hoffman, 1991, Woosley and Hoffman, 1992). Silicon burning in the Si and O shells makes the bulk of the
56
Fe, via 56Ni, ejected by the massive star.
The nucleosynthesis in Type Ia supernovae (that is,
deflagration or detonation of carbon-oxygen whitedwarf stars) also can ascend to near the top of the statistical equilibrium hierarchy. The bulk of the matter inthe explosion can certainly achieve an NSE (e.g., Thielemann et al., 1986; Woosley and Weaver, 1994) having
Y e 50.5. Thus there is copious production of 56Ni. In the
cores, the density may even get high enough that thematerial approaches weak equilibrium. In such a case,electron captures occur and Y e drops so that moreneutron-rich species (including 56Fe!) may be produced
directly. It is here that nature may produce and eveneject 56Fe by B 2 FH’s e process! In certain rare TypeIa’s, the central density may even be high enough thatthe approach to weak equilibrium drives Y e low enoughto produce rare iron-group isotopes like 48Ca, 50Ti, and54
Cr (Woosley et al., 1995; Meyer et al., 1996; Woosley,1996).
While B 2 FH correctly surmised that the bulk of the
Fe-group isotopes in the solar system were made inequilibrium processes, they also found that the s process
had to be responsible for the synthesis of some of theiron-group nuclides. Work since 1957 has confirmed thisto be the case (e.g., Peters et al., 1972; Lamb et al., 1977;
Baraffe et al., 1992). In this process, seed iron-group nuclei from previous generations of stars are exposed to aflux of neutrons. This shifts the abundances over to themore neutron-rich of the iron-group isotopes. The weaks process most likely occurs in the helium burning coresof massive stars where ( a ,n) reactions on 13C and 22Ne
can generate a sufficient supply of neutrons. For moredetails, the reader should turn to Secs. IX and X of thisreview.

D. Significance for astrophysics

With the key ideas of iron-group nucleosynthesis
more or less clear since the early 1970’s, the recent work
has centered on obtaining improved yield estimates forthese nuclei and on exploring their significance for astroRev. Mod. Phys., Vol. 69, No. 4, October 1997

1065

physics. This subsection provides a very brief discussionof these aspects of synthesis of the iron-group elements.
Advances in computer technology have fostered great
increases in the sophistication of astrophysical irongroup yield estimates. The yields are now computed in
postprocessing calculations using detailed nuclear networks and hydrodynamics from realistic stellar models.
This is true for both Type Ia (e.g., Thielemann et al.,
1986; Woosley and Weaver, 1994) and Type II (e.g.,Thielemann et al., 1996; Woosley and Weaver, 1995) supernova models (see also Arnett, 1996). Further advances will come as the remaining uncertainties in themodels are dealt with realistically. In Type Ia models,
key uncertainties affecting the Fe-group yields include
the correct treatment of the nuclear flame speed andacceleration (e.g., Woosley and Weaver, 1994; Niemeyer
and Woosley, 1997; also Arnett, 1996) and the rates forelectron capture on the Fe-group isotopes (e.g., Fuller
et al., 1980, 1982a, 1982b, 1985; Aufderheide et al.,1994). Stellar spectroscopists hope to clarify the Ia
mechanism through observations of time-dependent
atomic lines. In Type II models, key uncertainties include the location of the mass cut that separates matterejected from that that stays in the remnant (e.g., Weaver
and Woosley, 1993), the role of convection near the nascent neutron star (e.g., Herant et al., 1992), and the correct treatment of the explosion shock (e.g., Aufderheide
et al., 1991).
Astronomical observations will play a crucial role indetermining the improvements needed in theoreticalyield estimates. Both gamma line astronomy and light
curves rely on the key fact that so many of the irongroup species had radioactive progenitors. Observationsof supernova light curves, powered in large measure by
the decay of the radioactive iron-group species 56Ni and56
Co, 57Co, and 44Ti, provide important estimates of theyields of these isotopes (e.g., Woosley et al., 1989; Suntzeff et al., 1992), replacing the 254Cf hypothesis of B 2 FH.
Long standing theoretical expectations were confirmed,
with surprises, by observations of g rays from the decayof these radioactive species in the debris of individual
supernovae: 56Co (e.g., Leising and Share, 1990) and57
Co (Kurfess et al., 1992) from SN 1987A; 44Ti (Dupraz
et al., 1996; The et al., 1996) from Cas A. While the observations of the Ni isotopes are in fairly good agreement with theoretical predictions (agreement within;50%), the differences present valuable challenges thatwill ultimately lead to improved supernova models. Of
further astronomical interest, 56Co and 44Sc also eject
positrons which may be the dominant source of the 511
keV g ray line and positronium continuum flux from the
Galactic plane (e.g., Lingenfelter and Ramaty, 1989).
Astronomical observations of iron-group isotopes
may also provide information on the Galaxy’s history.
For example, stellar observations of iron and oxygen instars of varying age (e.g., Peterson et al., 1990, Gratton
and Sneden, 1991) provide valuable constraints on thegrowth of elemental abundances during the Galaxy’s
history. Detailed models of Galactic chemical evolution(e.g., Timmes et al., 1995) must confront these con-

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straints. Among the pieces of information such a confrontation will yield is the relative contribution of Type I
and Type II supernovae to the solar system’s budget of
56
Fe. The current models seem to suggest they contributed roughly equally (e.g., Timmes et al., 1995). Other
valuable information on the Galaxy’s history comesfrom gamma ray astronomy. Gamma ray observations of
the relatively long-lived iron-group radioactivity 60Fe
provide useful information on the Galactic supernova
rate over the last million years (Leising and Share 1994).
Another area of astronomy on which the iron-groupisotopes may have a great impact is in solar system studies. Strong evidence exists that the iron-group radioactivities 53Mn and 60Fe were alive in small part at thetime of formation of solar system solid bodies (Harperand Wiesmann, 1992; Shukolyukov and Lugmair, 1993).
This information constrains the last nucleosynthesis
events contributing to the solar system’s supply of isotopes (e.g., Harper, 1996) and the time scales of solidbody formation in the early solar system (e.g., Shukolyukov and Lugmair, 1993). Correlated anomalies of theneutron-rich iron-group species 48Ca, 50Ti, and 54Cr in
calcium-aluminum-rich inclusions in meteorites (e.g.,
Loss and Lugmair, 1990; Volkening and Papanastassiou,
1990) strongly suggest these isotopes retain somememory of their site of nucleosynthetic origin, while endemic anomalies in 50Ti in all classes of carbonaceous
meteorites (Niemeyer, 1988) hint at an inhomogeneous
distribution of this isotope in the interstellar dust thatcomprised the original building material of the solar system. Especially noteworthy are the huge 44Ca excesses
found in graphite and silicon carbide X grains in primitive meteorites (e.g., Hoppe et al., 1996). The excesses
undoubtedly come from in situ decay of 44Ti (half life of
47 years) and provide proof that these grains condensed
in outflowing supernova debris. In 1957, B 2 FH could
only have dreamed of having such tangible evidence ofelement formation in stars!
XVI. CARBON STARS: WHERE THEORY MEETSOBSERVATIONS
A. Prologue

To this observer the most remarkable aspect of theclassic tale ‘‘Synthesis of the Elements in Stars’’ told by
Burbidge, Burbidge, Fowler, and Hoyle (1957, here
B 2FH—of course) is, perhaps, their confidence in theoretical ideas as revealed by the almost complete lack of
appeal to observations of the compositions of the stars
to whom element synthesis was attributed. In large part,the omission of observational data on the compositionsof stellar atmospheres reflected, however, the simple
fact that in 1957 the available quantitative data weresparse in the extreme. Fortunately, key clues to the
nuclear astrophysics of various kinds of carbon stars
were obtainable from qualitative inspection of stellar
spectra: for example, the presence of 13C-containing
molecules suggested exposure of material to a hydrogenburning region; furthermore, the presence of technetium
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

in some cool carbon stars implied, as in the S-type
oxygen-rich stars, recent or even continuing operation of
the s process. Curiously, B 2 FH were extremely cautious
in their sifting of spectroscopic clues for evidence of helium burning. In contrast to several lines of evidence
cited in support of hydrogen burning in carbon andother stars, B 2 FH described the evidence for helium
burning as ‘‘less direct’’ and among a few snippets ofsuch evidence did not cite carbon stars.
In 1957, origins of carbon stars were shrouded in mystery. Bidelman (1956) opened the review ‘‘The Carbon
Stars—An Astrophysical Enigma’’ with the declarationthat the stars have ‘‘posed a vexing astrophysical problem’’ since Father Secchi’s identification of the cool carbon stars in the previous century. Forty years on, problems remain but for some classes of carbon stars
observation and theory have converged on an evolutionary scenario in which nuclear astrophysics occurring in astellar interior and spectroscopy of an atmosphere are
related by a complete model of a stellar interior, envelope, and atmosphere. In this brief review, I sketch theconvergence of theory and observation, as well as thelack of convergence that remains for some classes ofcarbon stars. I do not discuss at all the contributions of
the carbon stars to the chemical evolution of the Galaxy,as Gustafsson and Ryde (1997) give a comprehensive
review of that topic.
B. Carbon stars—An observer’s view
1. What is a carbon-rich star?

An observer’s working definition is that ‘‘carbon star’’is certainly an appropriate label when the ratio of carbon to oxygen by number of atoms exceeds unity, here
C/O>1.
Thanks to the preeminent role of the CO molecule incontrolling the partial pressures of carbon and oxygen at
low temperatures, spectra of oxygen (C/O<1) and carbon (C/O>1) rich cool stars are readily distinguishable ateven low spectral resolution: optical spectra of oxygenstars are dominated by metal oxide (e.g., TiO, YO, ZrO)
bands, and spectra of carbon stars by bands of carbon
containing molecules, principally C 2 and CN. (For C/O<1, the presence of oxides other than TiO—i.e., YO andZrO—implies a C/O ratio close to unity. Such molecularfeatures are a defining mark of the S-type stars.)
At higher temperatures when the influence of the COmolecule is weak or unimportant, mildly carbon-richstars may go unnoticed by mere inspection of low resolution spectra. I have in mind here the case of the classical barium giants discovered by Bidelman and Keenan(1951). These G- and K-type giants with enhanced CH
and C 2 bands are presumably carbon enriched relative
to normal giants of the same type. CO still has an influence on the partial pressures of carbon and oxygen but adetailed analysis is needed to show if C/O exceeds unity.
Such analyses show that Ba II stars of roughly solar metallicity have C/O<1 but C/O is higher than in normal
giants (Tomkin and Lambert, 1979; Smith, 1984; Barbuy

Wallerstein et al.: Synthesis of the elements

et al., 1992). For the CH stars discovered by Keenan(1942) which are metal-poor halo counterparts of the Ba
II stars, Vanture (1992a, 1992b) found C/O to run fromjust in excess of 1 to 8. Clearly, the CH stars deserve theappellation ‘‘carbon star’’ but, in a strict sense, the Ba II
stars do not. Since it is now evident that the two groups
have a common evolutionary origin, I consider Ba II
stars here with stars that do satisfy the criterion C/O>1.
For the several classes of carbon stars alluded to in
the preceding paragraphs, it appears that they possess aroughly normal hydrogen abundance, which implies anormal helium abundance. Certainly, hydrogen is seenin the spectra either as the Balmer lines in absorption oremission or as lines of a H-containing molecule such as
CH. Bidelman (1956) included several categories of veryhydrogen-poor stars with strong spectral features of carbon among the carbon stars. B 2 FH also refer to thesestars: ‘‘Some rare specimens among the carbon stars
have weak or absent CH bands and hydrogen lines.’’ For
these stars, there is the possiblity that carbon is enrichedin the sense that the carbon mass fraction is many times
the low value it has in normal stars and even in the
carbon stars of normal hydrogen abundance. A few ofthe hydrogen deficient stars are cool enough to be surethat C/O>1 but others are too warm for the C/O ratio to
be inferred except by a rather detailed analysis. For lackof space I do not consider hydrogen deficient stars suchas the R Coronae Borealis variables, the hydrogen deficient cool carbon stars, and the extreme helium hotstars.
2. What makes a carbon-rich star?

At the present time, the interstellar medium is oxygen
rich. Models of galactic chemical evolution (e.g.,
Timmes et al., 1995) predict that the interstellar medium
has always been oxygen rich thanks to a greater production of oxygen relative to carbon. Oxygen (here, 16O) is
synthesized by massive stars. Carbon (here, 12C) is synthesized not only by massive stars but also by lower
mass stars. Prantzos et al. (1996) summarize theoreticalevidence on the stellar sites for the synthesis of all stable
isotopes of carbon, nitrogen, and oxygen. On the assumption that the surface abundances of carbon andheavier elements for an unevolved or main sequencestar are identical to the abundances in the parental interstellar cloud, all stars began life with oxygen-rich atmospheres. For the sun and stars of approximately solar
metallicity, the C/O ratio is about 0.6. Analyses of main
sequence stars show that the C/O ratio in metal-poor([Fe/H]<21) stars is about 60% lower than the solar
value (Tomkin et al., 1995). Models of galactic chemical
evolution can account for this drop in the C/O ratio. In
short, carbon stars are not born but are transformed
oxygen-rich stars.
The galactic experience as revealed by local disk andhalo stars is possibly not universal as to location and ageof a stellar system. Since considerable production of carbon but not of oxygen is provided by low mass stars, theC/O ratio depends on the history of star formation and,Rev. Mod. Phys., Vol. 69, No. 4, October 19971067

in particular, on the relative numbers of high and low
mass stars. If formation of high (M.10M ( ) mass stars
is consistently suppressed relative to low mass stars in aparticular environment, the C/O ratio will evolve tohigher values than the present value and may well exceed unity. No such regions have been identified, as yet.
Certainly, oxygen-rich main sequence stars are considered the norm. Then, the question ‘‘What makes acarbon-rich star?’’ becomes ‘‘How is an oxygen-rich startransformed to a carbon-rich star?’’ Existence of carbonrich stars might be accounted for in three broad ways: (i)the stars formed from an interstellar cloud locally enriched in carbon (relative to oxygen); (ii) nucleosynthesis followed by internal processes (diffusion, mixing,
mass loss) transformed the star from oxygen-rich tocarbon-rich; and (iii) carbon was transferred to the starafter its birth to increase the surface C/O ratio.
Possibility (i) seems unlikely but data on the composition of dark clouds, the sites of ongoing or prospective
star formation, are too sparse for its exclusion on observational grounds. Limited data on the carbon abundanceof H II regions in conjunction with ample measurementsof their oxygen abundance suggests that the C/O ratio is
close to the solar value and never exceeds unity. Since
low mass stars eject lots of carbon at low speeds into theinterstellar medium, it may be possible for pockets
within clouds to be made carbon rich and to have eluded
observers.
Possibility (ii) provides opportunities for transforming
evolved stars with C/O<1 to stars with C/O>1 but opportunities for transforming main sequence stars seem limited. Among the chemically peculiar A and B-type main
sequence stars is one class—the SrCrEu stars—withmembers that are carbon-rich by the measure C/O>1(Roby and Lambert, 1990). These stars for which elemental segregation driven by the combination of gravitational settling and radiation pressure is a likely explanation will not be discussed here, as they are notgenerally considered under the rubric of ‘‘carbon stars.’’
Surface abundances of carbon and oxygen will be affected by internal mixing only if it extends into regionsthat are experiencing or have experienced nuclear reactions. Since hydrogen-burning reactions at the interiortemperatures of main sequence stars reduce the C/O ratio, mixing cannot result in a carbon-rich star. Mixing
occurs in massive main sequence stars where it is probably driven by rapid rotation and revealed by the presence of carbon-poor but nitrogen-rich stars (Lyubimkov,
1993; Gies and Lambert, 1992). Hydrogen burning is followed by helium burning with a product that has C/O>1.
Wolf-Rayet stars of the WC types are carbon stars by
the definition C/O>1 but are not considered here. In thecase of the WC stars, severe mass loss exposes the
carbon-rich layers that were at a prior time deep in thestar and exposed to helium burning.
For evolved stars, possibility (ii) offers a richer varietyof possibilities. In particular, low (M,10M ( ) mass stars
evolved beyond helium-core burning experience heliumshell flashes (thermal pulses) which, in some calculations, are predicted to be followed by extension of the1068

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giant’s deep convective envelope into the top of the(now quiescent) helium shell. This extension or third
dredge-up occurs repeatedly and brings carbon ands-process products into the photosphere. (I. Iben, Jr.,
discusses this phase of AGB evolution in Sec. III.) Thereis no doubt that the third dredge up in AGB stars is a
primary process for converting oxygen-rich to carbonrich giants. Questions remain about, for example, theefficacy of the third dredge up in low mass stars, but it
may be claimed that theory and observation have converged for carbon-rich AGB stars (N-type stars—see below), as well as for the carbon-rich stars produced bymass transfer across a binary system (see below). Thereremain speculative possibilities for internal mixing leading to a carbon-rich star. Invocation of the helium core
flash in low mass stars is one such possibility. Severe
mass loss is another possibility. Such speculations are
aired in connection with carbon-rich stars for which
theory and observation have yet to converge, as in thecase of the R stars (see below).
Mass transfer [possibility (iii)] is the accepted mechanism for the formation of several classes of main unevolved and evolved carbon stars (barium dwarfs, CH
subgiants, barium and CH giants). In this case, a star
evolves to become a cool carbon star (N-type AGB star,see below) and transfers mass to its less evolved companion, which is converted to a carbon-rich star. The
companion remains as a white dwarf orbiting the newcarbon star. Mass transfer from an AGB star to a companion occurs only in a binary of wide separation. Binaries of closer separation experience mass transfer before
the more massive star has become a large AGB star.
Perhaps, in certain cases, the two stars develop a common envelope, coalesce, and the single star that results
has experienced such extensive mixing that carbon fromthe core of the previously more evolved star appears at
the surface.
3. The principal types of carbon-rich stars

In the following sections, I highlight some of the properties of several classes of galactic carbon stars. In general, the highlights are restricted to observations that are
particularly pertinent to establishing the evolutionary
state of the stars, i.e., luminosity, and aspects of thechemical compositions.
a. R-type carbon stars

The Harvard system of spectral classification introduced types R and N for cool stars with strong bands of
CN and C 2 . This separation remains useful. The R stars,
which are systematically warmer than the N stars, are, as
judged by temperature and luminosity, K giants with enhanced bands of CN and C 2 . The N stars are cooler andmore luminous. A further broad division is helpful: theN stars may be divided into those that do and do notshow strong spectral features attributable to the isotope
13
C: bands of 13CN and 12C 13C or even 13C 2 are often
distinguishable at low to moderate spectral resolution
from their parent bands of the more abundant 12C isoRev. Mod. Phys., Vol. 69, No. 4, October 1997

tope. Stars with strong 13C bands are termed J-type. A
small fraction of N-type stars are of the J-type. All or
almost all of the R-type stars qualify for the designationof J-type. Thus observers challenge theorists to provide
an evolutionary scenario for three types of cool carbon
giant stars: N, N(J), and R. In broad terms, observations
and theory have met and reached agreement about the
N stars: asymptotic giant branch (AGB) stars in which
the third dredge up (Iben, this review) has converted an
oxygen-rich to a carbon-rich envelope. By contrast, theR stars are unexplained by theory despite more extensive data on their chemical compositions than are available for the N stars.
A sample of 11 early R stars was analyzed by Dominy(1984) from high-resolution optical and infrared spectra.
His abundance analysis provided the following resultsthat an evolutionary scenario must explain:(1) Metallicity. The stars are of approximately solar
metallicity: [Fe/H] ranges from +0.2 to 20.6.
(2) Lithium. Lithium was not detected. It appears that
R and normal K giants are not distinguishable by theirlithium abundances.
(3) C, N, and O. The R stars are carbon stars in thesense that C/O .1: Dominy gave C/O ratios in the range
1–3. The oxygen abundance is normal. Nitrogen wasmarkedly enhanced: [N/Fe]=0.7 where [N/Fe]=0.3 ischaracteristic of normal K giants. The 12C/ 13C ratios of
the R stars are in the range 4–15 for a mean value of763.
(4) s process. The s-process elements are not significantly enhanced: @ s/Fe]=0.2560.18. Certainly, thes-process abundances are not at the level that is characteristic of Ba II stars.
These results imply that carbon from a heliumburning region has been added to the envelope after
substantial exposure to hot protons that convert some of
the 12C to 13C and 14N. The stars are too low in luminosity for the addition to have occurred as part of a third
dredge-up in the R star itself. Dominy speculated thatthe helium core flash led to extensive mixing. Since alllow (M,2.3M ( ) mass stars experience a helium core
flash but R stars are rare among the K giants (i.e., no R
stars in the Bright Star Catalogue, which contains morethan a thousand K giants), a special condition must besought that ensures that, in just a few cases, the helium
core burning giant is a R rather than a normal K giant.
Dominy suggested a rapidly rotating core might induce
mixing and create a R star. That the helium core flash is
the agent creating R stars has yet to be demonstrated
theoretically. Theorists appear to consider the flash tobe benign (i.e., mixing does not result). Perhaps recent
startling advances in numerical hydrodynamics may be
applied to this problem.
In light of the success of the mass transfer hypothesis
at explaining the Ba II and CH stars, suspicions havesurely lingered that the R stars might also be binaries inwhich the nucleosynthesis occurred in a luminous highly
evolved companion star (now extinct) and transferred to
the less evolved R star. This idea has now been put effectively to rest by the radial velocity survey of R stars

Wallerstein et al.: Synthesis of the elementsof McClure (1997). The remarkable result of his longterm survey is that, in a sample of 22 R stars, ‘‘no evidence for binary motion has been detected in any ofthem’’ while 20% of normal K giants are spectroscopic
binaries. McClure suggests that R stars may result fromcoalescence of two stars with coalescence providing thetrigger for mixing at the helium core flash. One might
expect coalescence to result in a rapidly rotating R staras orbital angular momentum is transferred to the coalesced pair but, as McClure notes and Dominy’s spectra
clearly show, there is no evidence that the R stars are
more rapidly rotating than K giants.
b. N-type carbon stars

Molecular spectra so dominate the spectra of the Ntype stars that abundance analyses have been seriously
hampered. Even with the advent of spectrum synthesis
as an analytical tool, progress has been slow. In part, thisis due to the fact that the many molecular bands aremuch more richly represented in the stellar spectra thanin laboratory sources; unidentified lines are commonly a
nuisance. The principal results can best be set out in theformat used above for the R stars.
(1) Metallicity. Kinematics of N stars suggests solarlike metallicities. Spectroscopic estimates are few, uncertain, and based on few lines. Lambert et al. (1986)
found [Fe/H]=0 to within about 60.3 dex from a handfulof infrared lines of Na, Ca, and Fe in a sample of 30
bright carbon stars.
(2) Lithium. Lithium abundances of the majority of N
stars appear to be at the same level as in normal M
giants (Denn et al., 1991; Abia et al., 1993) which implies
that transformation of M giants to N stars by operationof the third dredge up has not substantially altered thelithium abundance. This result seems compatible withcalculations of the third dredge up. A few stars are so
remarkably enriched in lithium that production oflithium must have occurred. These stars are the J-typestars having a low 12C/ 13C ratio and little or no enhancement of s-process elements (Dominy, 1985). Abia andIsern (1996) suggest that lithium increases with decreasing 12C/ 13C ratio but this may not reflect a physical relation as the N and N(J) stars appear so very different ins-process abundances. One is tempted to identify the Rstars as progenitors of the N(J) stars. Then, the N(J)stars must manufacture lithium or lithium-rich R stars
have gone undetected. Unusual constraints are placedon lithium manufacture in this case: no prior or attendant s production or reduction of the C/O ratio.
(3) C, N, and O. Elemental abundance were given byLambert et al. (1986) from infrared spectra that contained CO, C 2 , CN, and other molecular lines. Three
results deserve brief mention. First, the oxygen abundances [O/H] were on average slightly subsolar ([O/H]'20.2). The oxygen abundances were in good agreement with those found for M giants (progenitors of N
stars?) and for planetary nebulae (descendants of N
stars?). Second, the C/O ratios show the carbon enrichment to be relatively small: the most carbon-rich star
had C/O;1.6 and about half of the sample had C/O<1.1.
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1069

These values do show that substantial amounts of carbon have been added to convert an M giant (C/O;0.3)
to a N star but there are planetary nebulae that are significantly more C rich (Zuckerman and Aller, 1986).
Third, the nitrogen abundances of the N stars were lessthan those derived for the M (and K) giants by about 0.6
dex. M giants presumably may evolve to N stars, but it isdifficult to imagine that nitrogen can be destroyed by an
AGB star. (Many M giants may lose their envelope as
O-rich AGB stars and not evolve to N stars.) The mean
nitrogen abundance of the N stars ([N/H]=20.360.2) isless than that from C-rich planetary nebulae for which[N/H]=0.1 60.3 from Zuckerman and Aller (1986).
Lambert et al. (1986) stress that their N abundance determinations are particularly susceptible to systematic
errors.
Isotopic abundances of carbon were derived in three
ways using infrared lines of two molecules; CN and CO.
The three results were in good agreement. The range
30, 12 C/ 13C<70 encompassed the larger part of thesample. The N(J) stars had, of course, lower values close
to the equilibrium value for the CN cycle. Lambert et al.
(1986) argued that the 12C/ 13C ratios of N stars resulted
from the addition of 12C but not 13C to the atmosphereof a M giant with its lower carbon abundance and lower
12
C/ 13C ratio. Recently, Ohnaka and Tsuji (1996) havemeasured 12C/ 13C ratios from near-infrared CN bands
obtaining lower ratios than those given by Lambert et al.
(1986). Although it does not seem critical to evolutionary interpretation of N stars, this discrepancy needs tobe examined.
A general point of great importance may be inserted
at this juncture. Two different observational approaches
now provide information on the isotopic carbon (andother) ratios. First, observations of molecular millimeterand radio line emission from circumstellar shells provide
independent data. Presently, the reliable data refer to
AGB stars whose photospheric spectra have not been
analyzed for the same ratios. Kahane et al. (1992) provide the 12C/ 13C ratio for shells of five carbon stars. The
16
results (>65, 32 110
27 , >30, 4463, and 3125 ) might beviewed as supporting the results given by Lambert et al.
(1986). Second, the isolation of SiC grains from carbonaceous chondrites also provides new data on carbon
stars, which are the most likely source of these surviving
circumstellar grains. The distribution of 12C/ 13C ratios
for SiC grains mimics well the stellar distribution foundby Lambert et al. (1986) ( see also Anders and Zinner,
1993).
Oxygen isotopic abundances in carbon stars were estimated by Harris and Lambert (1987) following earlier
determinations of the abundances for oxygen-rich stars
on and off the AGB (Harris and Lambert, 1984, 1987;
Harris et al., 1985, 1988). Kahane et al. (1992) provide
the ratio 16O/ 18O and 16O/ 17O for circumstellar shells of
five carbon stars. Analyses of oxide grains from primitive meteorites (Nittler et al., 1994) has also provided
these ratios which pertain presumably to circumstellar
shells of oxygen-rich (now deceased) AGB stars. Theoretical work on interpreting the measured oxygen isoto-1070

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pic ratios has been quite extensive. Forestini and Charbonnel (1997) indicate that their models of intermediate
mass AGB stars can explain the ratios except where the16
O/ 17O ratio is high; for example, they remark that themeasured isotopic carbon, nitrogen, and oxygen ratios
for nearby dust-shrouded carbon star IRC+10216 (CWLeo) are consistent with their predictions for an AGB
star of mass 4 to 5 M ( and solar metallicity except thatthe observed 16O/ 17O ratio is not well explained. Wasserburg et al. (1995) invoke deep mixing in red (H-shellburning) giants to explain the oxygen isotopic abundance in giants (before and on the the AGB) and in themeteoritic oxide grains.
(4) s process. The N stars are s-process enriched but
the N(J) are not. Quantitative assessments of thes-process abundances in N stars come from a pioneering
study by Utsumi (1967, 1970, 1985) based on photographic spectra in ‘‘windows’’ at 4750–4900 Å and 4400–
4500 Å between C 2 bands. Busso et al. compare theabundances of the s elements with those of Ba II andother s-process enriched stars and with predictions for
AGB stars experiencing the third dredge up. In short,the observations for the carbon stars are consistent withpredictions as to neutron exposure and mixing fraction(see V. V. Smith, Sec. XI, for further discussion). Moreover, the observations of N stars are consistent with the
hypothesis that Ba II and other stars result from transfer
of mass to the present star from an AGB s-process enriched star; the s-process enrichment of Ba II stars is lessthan that of the carbon stars (i.e., diluted following masstransfer) but of a similar pattern as regards light toheavy s elements. The large enhancement of s-process
species in the CH stars is discussed in Sec. XI.
A clear signature of active or recent s processing is
the presence of technetium, as noted by B 2 FH for carbon stars (Merrill, 1956). Little-Marenin’s (1989) reviewof technetium in stars reports observations of 15 N starsof which 12 showed this unstable element. One N(J) star
did not show technetium. The absence of technetiumfrom the N(J) is expected as these stars are not s-processenhanced. The absence of the technetium from 3 N stars
does not necessarily imply the stars are not active AGB
stars. Detection of the Tc I lines in spectra of cool carbon stars is difficult. There is also the possibility that
continuous decay of technetium may reduce the abundance below detectable limits.
No review of carbon stars, especially one discussing
the convergence of observation and theory, could beconsidered complete without mention of the carbon
stars in the Magellanic Clouds. Since the Clouds are at aknown distance, stars may be assigned absolute luminosities. In contrast, distances to carbon stars in our Galaxy and, therefore, luminosities are quite uncertain. The
luminosity distribution function for carbon stars in the
Clouds showed an absence of the predicted very luminous carbon stars and peaked at a luminosity too low tocorrespond to the intermediate-mass AGB stars forwhich the third dredge up had been predicted to operate. A seminal discussion of the luminosity function isthat by Iben (1981). Two points should be stressed. First,
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

the marked absence of luminous carbon AGB stars is
now attributed primarily to two effects: (i) severe massloss that may terminate AGB evolution before high luminosities are attained, and (ii) the onset of hot bottom
burning that reconverts the carbon-rich to an oxygenrich AGB star. Second, as emphasised by Iben (1981),
the ‘‘low’’ mean luminosity of carbon stars implies that
low mass AGB stars also experience the third dredge
up, an observation not predicted by then extant calculations. Iben wrote ‘‘a significant fraction of the carbon
that is freshly synthesised in an AGB star during thermal pulses is brought to the surface of every AGB star in
the Clouds, whatever the mass of the underlying carbonoxygen core.’’ Understanding mass loss and the third
dredge up in low mass AGB stars remain the key theoretical problems in AGB evolution.
Luminous AGB stars do exist in the Clouds, as emphasised by Wood et al. (1983). These are S stars (i.e.,enriched in s elements) and all are lithium rich (Smith
and Lambert, 1989, 1990; Plez et al., 1993; Smith et al.,1995). The luminosities of the lithium-rich stars and thelithium abundances are consistent with the predictions
for hot bottom burning (Sackmann and Boothroyd,1992). In this phase, a very luminous AGB star supports
hydrogen burning at the base of the convective envelope
where previously the hydrogen burning shell was below
the base of the convective envelope. Hot bottom burning converts carbon to nitrogen and reduces the C/O
ratio below unity. Lithium is produced by the chain3
He(4He,g) 7Be(e, n ) 7Li (Cameron and Fowler, 1971)
where the 3 He was produced in the main sequence star.
This cannot account for all the lithium-rich stars in the
Clouds as some have lower luminosities than predicted
for hot bottom burning. One is reminded of the lithiumrich galactic carbon stars.
c. Barium and related stars

To B 2 FH the Ba II stars were a laboratory in which totest predicted abundance patterns for the s process. Certainly, then and now, the spectra of Ba II K giants are
more tractable than the spectra of the N stars with similar s-process enrichments. Early attempts to account for
the Ba II and CH giants, which are metal-poor C-richstars, focused on the helium core flash that occurs in low(M,2.3M ( ) mass stars when helium is ignited in theelectron degenerate core. This explosive event was seen
by some observers as the driver for the nucleosynthesis
and mixing needed to transform a star to a barium star.
An early call on the helium core flash as the deus ex
machina was made by Warner (1965) with, perhaps, thelast call by Tomkin and Lambert (1983). The luminosities of the barium stars seemed compatible with the idea
too; the mean luminosity appeared to be that of a clumpor horizontal branch (i.e., a post-flash helium core burning star).
Discovery of the CH subgiants by Bond (1974)
showed that stars with compositions of the Ba II giants
existed at luminosities too faint for helium core burning
giants. The Ba II phenomenon appeared to extend tothe main sequence. A speculation was advanced that theWallerstein et al.: Synthesis of the elements

helium core flash led to such extensive mixing that, withhydrogen replenished in the core, the postflash star assumed again the structure (and luminosity) of a main
sequence star (see also Luck and Bond, 1982).
All attachments to the helium core flash by observers
were shattered with the delightful discovery that all Ba
II and CH stars are long period spectroscopic binaries.
This result was first reported by McClure et al. (1980)—
reviews by McClure (1985, 1989) should be read! This
discovery was a reminder that binary stars must be considered to obtain a complete picture of stellar evolution
and a full understanding of peculiar stars. The discovery
led to the idea that mass transfer between components
of a binary star may create a low luminosity star with acomposition similar to that of an AGB star. The helium
core flash was cast aside by observers.
A prediction of the mass transfer hypothesis is thatmain sequence barium stars should exist. Happily, examples have been found by Tomkin et al. (1989), Edvardsson et al. (1993), and North et al. (1994).
For a discussion of the compositions of barium stars of
all kinds, the reader is referred to Lambert (1985, 1988),
Barbuy et al. (1992), Vanture (1992a, 1992b), with V. V.
Smith, Sec. XI of this review, providing remarks on the sprocess. A comparison of the compositions of bariumstars with those of active AGB stars including the Ntype carbon stars shows the great similarities required
under the mass transfer hypothesis. Mass transfer ismost probably effected by accretion of gas and dust
from the AGB star’s wind (Boffin and Jorissen, 1988).
The main sequence barium stars just mentioned are
all stars with spectral types of F and G. Cooler stars willshow strong molecular bands and may look at low spectral resolution like N-type (giant) stars. The first suchcase was G77-61 with a classical carbon star spectrum
but a high proper motion suggesting a main sequenceluminosity. It is a spectroscopic binary and mass transfer
is a likely explanation for it (Dearborn et al., 1986). Additional examples have now been found (Green et al.,
1991; Green et al., 1992; Warren et al., 1993). An abundance analysis of G77-61 (Gass et al., 1988) indicatesthat the dwarf carbon stars may not be simply cool close
relatives of the barium F and G main sequence stars.
Gass et al. (1988) conclude that G77-61 has low metallicity: [Fe/H]'25.6 but a normal hydrogen to helium
ratio. Relative to iron, carbon is substantially enriched([C/H]'21). No s-process element was measured. Gasset al. suppose the star was initially a very metal-poor starwith a composition reflecting contamination of primordial gas by debris from a very massive star. Perhaps,
there is a simpler alternative to consider: gas but notdust was accreted as part of the mass transfer processthat led to the dwarf carbon star. Gas in circumstellar
winds off AGB stars is highly depleted in those elementsthat form or adhere to grains. Iron and other elements
will be underabundant in the gas. If gas and not dust isaccreted by the companion in large amounts, the resulting star could be metal poor. Until other dwarf carbon
stars are analyzed, scepticism might be maintained
about their origin and their link to other main sequence
Rev. Mod. Phys., Vol. 69, No. 4, October 1997

1071

barium stars. This is an important area for future spectroscopic work because de Kool and Green (1995) suggest that ‘‘dwarf C stars may be the numerically dominant type of carbon star in the Galaxy.’’
C. Epilogue

In 1957, B 2 FH’s bold discussions of nuclear astrophysics and element synthesis were many years ahead ofan understanding of stellar evolution, which has come in
two main areas. First, enormous progress has been madein observational abundance studies and theoretical modeling of single AGB stars. In broad terms, observed
AGB stars and the theoretical models of these stars
have converged. It may be said that the N-type carbon
stars are understood. Models also show how hot bottom
burning converts the carbon star back to an S star. Second, the appreciation that Ba II and related stars arebinaries introduced the idea that mass transfer from an
AGB star suffices to account for these stars. Addition of
an innocent companion in a wide orbit provides the opportunity to transform that companion to a barium star.
Classical Ba II stars, CH giants, and chemically similarstars of lower and higher luminosity surely result fromaccretion of mass shed by an oxygen or carbon rich
AGB star. These expressions of convergence between
observation and theory should not hide vital topics requiring additional scrutiny. On the theoretical side,
present treatments of the third dredge up and of massloss by a wind need to be replaced by accurate ab initio
theories. On the observational side, quantitative spectroscopy of the cool carbon stars needs to be refined,
and the advent of large telescopes in the south should be
exploited through high-resolution spectroscopy of AGB
stars in the Magellanic Clouds.
Classes of stars for which observation and theory have
yet to converge particularly deserve further examination. In this review, we have mentioned just some ofthese recalcitrant classes. We recall the R [and N(J)]stars: do they result from an aberrant heliuim core flash
or coalescence of stars in a binary system? The dwarf C
stars deserve a mention, particularly if G77-61 with
[Fe/H]'25.6 is seen to be typical. Observers shouldtake note of this tip of the dwarf carbon star iceberg.
Other classes of carbon stars were not mentioned; forexample, the R CrB and related hydrogen deficientstars, stars that may be examples of ‘‘born again’’ supergiants experiencing a final helium shell flash (FG Sge?),and carbon stars with circumstellar shells of silicate
grains. Perhaps, at the next major anniversary of theclassic tale ‘‘Synthesis of the Elements in Stars,’’ theconvergence of observation and theory will have been
extended to all carbon stars. Perhaps!
XVII. CONCLUSIONS

Forty years after the publication of B 2 FH, we findthat new discoveries and developments in nucleosynthesis and stellar evolution (astronomical observations,laboratory measurements, and model calculations) have

1072

Wallerstein et al.: Synthesis of the elements

in large measure borne out and expanded the insights of
the original authors. For example, the development ofmass spectroscopy techniques have made possible theisotopic analysis of individual meteorite grains, while theexpansion of the observational spectrum to include
gamma rays and neutrinos have made possible the studyof such diverse processes as the distribution of 26Al in
our galaxy, neutrinos from the center of our sun, and theneutrinos and 56Fe gamma rays from the collapse andsubsequent decay of SN-1987a in the Large Magellenic
Cloud, some 55 kiloparsecs away. At the same time anumber of important open questions still remain, relating to areas ranging from the resolution of the solar neutrino problem to the determination of the important
14
N(p, g ) and 12C( a , g ) nuclear reaction rates both ofwhich are still uncertain to approximately a factor of 2
due to questions associated with the role of subthresholdresonances in the low-energy extrapolation of laboratory measurements.
The remaining uncertainties on the astronomy side
are substantial and, in some cases, difficult to overcome.
While the opacity calculations for stellar structure calculations have been greatly improved by the inclusion ofbound-bound transitions of iron and some other species(Rogers and Iglesias, 1992) the problems of convection
theory remain. In the deep interior of stars more massive than about 1.5M ( the core in which the CNO cycle
and later helium burning is taking place is convective. A
problem arises with the extent to which convection penetrates beyond the nuclear reaction region thereby
bringing in fresh fuel and extending the lifetime of thestar in that particular state. In stars cooler than about
6000 K at their surfaces an outer convection zone develops that homogenizes the layers in which it is active. The
depth to which this convection zone penetrates is uncertain and the temperature gradient in the superadiabatic
region is difficult to calculate.
The analysis of the chemical composition of stars has
improved vastly during the past 40 years due to the replacement of photographic plates by digital detectors
and the construction of new large telescopes such as
Keck I and II. Very soon we should see high resolution
spectra of stars in nearby systems at distances of 100
kiloparsecs. The analyses of stellar spectra have been
greatly improved by the routine use of model stellar atmospheres, but the models have their limitations. The
most important of these is the difficult problem of theboundary temperature for a star of given effective temperature, since the strengths of an absorption line is very
sensitive to the temperature gradient, not just the stellar
effective temperature. Two stars of the same effective
temperature may have different degrees of chromospheric heating and hence different backwarming of theouter layers of the atmosphere (Kurucz, 1996).
Finally, we must mention the ever-improving but
never good enough atomic data, especially the transition
probabilities, that are vital to the analysis of atomic absorption lines. The need is especially great for the rareearth and heavier species produced by the s, r, and p
processes. Similarly, for both oxide and carbon molRev. Mod. Phys., Vol. 69, No. 4, October 1997

ecules, we need transition probabilities accurate to better than 0.1 dex and dissociation potentials that are accurate to better than 0.1 eV to derive abundances ofvarious elements in cool stars.
Finally, we call attention to the complexity of theevents during a supernova explosion which is responsible for the production of almost all the species from Si
to Zn as well as the r-process elements. Both the great
progress and remaining uncertainties are fully discussed
in the monograph of Arnett (1996).
ACKNOWLEDGMENTS

This paper originated in the invitation by Michael
Wiescher to one of us (G.W.) to present the introductory talk at the ‘‘Nuclei in the Cosmos III’’ conference atNotre Dame University in June, 1996. The research of
the many authors has been supported by the National
Science Foundation, Department of Energy, The National Aeronautics and Space Administration, the Robert Welch Foundation, and the Kennilworth Fund of the
New York Community Trust. In addition, we want toexpress our appreciation to our home institutions forproviding us with the opportunity to have conducted research in nuclear astrophysics, stellar evolution, and observational astronomy over the (in some cases, many)
years. Finally, each individual author acknowledges assistance, comments, and criticism from a number of colleagues, the total list of whom is too long to include
here. Special acknowledgment is due Ms. Karen Fisher
who assembled the various manuscripts and translated avariety of Tex forms into the single form of this journal.
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