Because of the large Coulomb barrier between an alpha particle and another, heavier nucleus, temperatures at which helium-burning reactions occur in stars at interesting rates are considerably larger than temperatures at which hydrogen-burning reactions occur. Under appropriate conditions, a substantial fraction of interacting alpha particle, heavy nucleus pairs have relative kinetic energies such that the compound nucleus formed during a collision has an excitation energy close to the energy of a discrete level in the compound nucleus. For a reaction in which the compound nucleus decays with the emission of a gamma ray or a nucleon to form a stable nucleus, the cross section can become very large and the reaction is said to be a resonant one. In Section 16.1, the formation and decay of a compound nucleus is examined and the Breit–Wigner form for a single level resonant cross section is derived heuristically. In Section 16.2, the set of processes whereby three 4He nuclei combine to form 12C through resonant states of intermediate compound nuclei is examined. It is conventional to call the set of processes the triple-alpha process.

After the exhaustion of hydrogen over the inner 13% or so of its mass, a low mass star develops an electron-degenerate helium core and evolves upward in the HR diagram along a red giant branch burning hydrogen quiescently in a shell. The core grows in mass until, when it reaches a mass of about 0.5 M⊙, it has become sufficiently hot for the triple alpha process to terminate further core growth.

The discipline of stellar structure asks: given the opacity and the energy-generation rate as functions of composition, density, and temperature, and given the composition as a function of mass, what is the model structure in the static approximation? The discipline of stellar evolution asks: how, due to a combination of nuclear transformations and mixing processes, does the distribution of composition variables in a model star change with time, and how does the structure respond to these changes and to the loss of energy in the form of photons from the surface and neutrinos from the interior by the conversion of gravitational potential energy into heat and work and by the conversion of heat and work into gravitational potential energy. For a wide variety of situations, it is possible to explore evolution in the quasistatic approximation, which follows when bulk acceleration in an equation relating pressure-gradient and gravitational forces is neglected and the contribution to the internal energy of the kinetic energy of bulk motions is neglected. Nevertheless, meaningful estimates of bulk velocities follow as a consequence of changes in gravothermal characteristics required by the conservation of energy.

In order to reveal the full character of the quasistatic approximation, structure equations are derived in Section 8.1 without assuming spherical symmetry or placing restrictions on the acceleration. By invoking the conservation of mass, linear momentum, and energy, it is shown how the work done by gravity is translated by pressure-gradient forces into a primary component of the local gravothermal energy-generation rate ϵgrav and how, in regions where particles are being created and destroyed, another component of ϵgrav depends on the rates of creation and destruction of particles. An important theorem is derived which shows that, although the local rate at which gravity does work differs from the local rate at which pressure-gradient forces do work, the global rate at which gravity does work is identical with the global rate at which pressure-gradient forces do work.

By the end of the third decade of the twentieth century, it had become clear that, in nuclear beta-decay events, beta particles are emitted in a continuous energy spectrum rather than with a unique energy equal to the change in energy of the emitting nucleus (although the maximum energy of the beta particle is equal to the change in nuclear energy), that the change in the electrical charge of the nucleus is exactly equal in absolute value but of opposite sign to the charge of the emitted beta particle, and that beta decay events often involve a unit change in the spin of the nucleus. In 1930, Wolfgang Pauli began communicating to other physicists the idea that, in order to account for these facts, another, previously unknown, “penetrating” particle must also be emitted in beta decay events with the properties: mass much smaller than the electron mass, no electrical charge, and an intrinsic spin equal to that of the electron. According to M. Mladjenović (1998), for over three years Pauli considered the idea too speculative to publish, and his first formal account appeared in the proceedings of an international conference on physics held in London in 1934. Enrico Fermi dubbed the hypothetical particle the “neutrino” (little neutral one) and formulated a mathematical theory of beta interactions involving neutrinos (Fermi, 1934) which has guided experimental and theoretical work on the weak interaction up to the present time.

In this chapter,models ofmass 1, 5, and 25 M☉ and of population I composition (Z = 0.015 and Y = 0.275) are evolved through all phases of hydrogen burning up to the point when helium is ignited in a hydrogen-exhausted core. The model masses have been chosen with the aim of representing three broad classes of stars. Models of mass less than 0.5 M☉ have been excluded from consideration not only because they evolve on a time scale much longer than a Hubble time, but, because they remain completely convective throughout their nuclear burning lives, they can be described adequately by a sequence of polytropes, as discussed in Section 5.6, without invoking the elaboration of an evolutionary calculation. The 1 M☉ model is representative of a class of stars which evolve in less than a Hubble time into red giants with an electron-degenerate helium core and ignite helium in a semiexplosive fashion. These stars, of mass extending to ˜2.25 M☉, eventually become AGB stars with a carbon-oxygen core and, after ejecting a nebular shell, evolve into CO white dwarfs of mass ˜0.55 M☉ The 5 M☉ model is representative of stars in the approximate mass range 2.25 <M/M☉ <10.5 which ignite helium under non electron-degenerate conditions, but during and following the quiescent helium-burning phase evolve in a fashion similar to the evolution of lower mass stars during and after the quiescent helium-burning phase.

The evolution of a 1 M⊙ population I model star of initial composition (Z, Y) = (0.015, 0.275), begun in Volume 1 (Section 11.1) and carried there to the ignition of helium on the red giant branch, is continued in this chapter through four distinct helium- and hydrogen-burning phases. In Section 17.1, evolution is followed from the off-center ignition of helium at the tip of the red giant branch through a series of helium shell flashes which lift electron degeneracy in shells successively closer to the center of the hydrogenexhausted core.

Once helium burning reaches the center, shell flashes of this sort no longer occur. As described in Section 17.2, the model metamorphoses into a horizontal branch star, converting helium quiescently into carbon and oxygen at the base of a convective core which grows in mass, while hydrogen burning continues to convert hydrogen quiescently into helium in a shell outside of the core. Once helium is exhausted at the center, the model continues to burn helium quiescently in a shell. The helium exhausted core contracts until electrons in the core become degenerate, converting the core into a hot white dwarf composed of carbon and oxygen. The envelope of the model expands to giant dimensions, the strength of the hydrogen-burning shell at the base of the envelope declines significantly, and the surface luminosity is provided primarily by a helium-burning shell which increases in strength as the model climbs upward in the HR diagram.

This chapter describes the evolution of a 25 M⊙ population I model during core and shell helium-burning phases and during core and early shell carbon-burning phases. The initial model, described in Volume 1 (Section 11.3), is burning hydrogen in a shell and has just begun to burn helium in central regions. In Section 20.1 of this chapter, the evolution of central and surface characteristics of the model during the bulk of its quiescent nuclear burning lifetime is compared with the evolution of the same characteristics in 1 M⊙ and 5 M⊙ models during quiescent nuclear burning phases up to the TPAGB phase. The location in the HR diagram of a theoretical pulsational instability strip is compared with the location of a band defined by where core helium burning takes place on a long time scale in models of different mass. The fact that the strip and the band have slopes of opposite sign and intersect makes it possible to understand why there exists a peak in the distribution of Cepheids in an aggregate of stars of the same composition, but of different masses and ages. The peak occurs at the intersection of the strip and the band.

In Section 20.2, the evolution of internal structure and composition characteristics of the 25 M⊙ model during the core helium-burning phase is described in some detail, with particular attention paid to the neutron-capture nucleosynthesis in the convective core occasioned by the activation of the 22Ne(α, n)25Mg neutron source.

The characteristics of the Sun and of other bright stars for which distances can be estimated constitute the major observational foundation of the disciplines of stellar structure and stellar evolution. The Sun's basic global characteristics, which provide natural units for cataloguing the global characteristics of other stars, are descibed in Section 2.1. Properties of some bright stars in familiar constellations and of some nearby stars for which masses have been estimated are described in Section 2.2, with several of the more familiar stars being shown in the Hertzsprung–Russell (HR) diagram, where a measure of intrinsic brightness (luminosity) is plotted against surface temperature (color). It is evident that nearby and/or visually bright stars form distinctive sequences in the HR diagram.

Mass and luminosity estimates for stars in relatively wide binary systems as well as for those in close, but detached systems are presented in Section 2.3. Comparing these estimates in a mass-luminosity (ML) diagram, one may infer that, for systems in which proximity does not imply mass transfer between components or significant tidal interaction, the relationship between mass and luminosity for either component is not greatly affected by the presence of a companion. Comparing the locations of individual stars in the HR and ML diagrams, one can infer that stars probably evolve between sequences in the HR diagram.

In Section 2.4, the evolution of the interior and global characteristics of theoretical stellar models is sketched and, in Section 2.5, the theoretical results are employed to interpret the significance of the different branches defined in the HR diagram by nearby and/or visually bright stars and to identify the evolutionary status of familiar stars in the night sky.

Considerable insight into the interior characteristics of main sequence stars can be obtained by means of back of the envelope estimates. Such estimates are vital, for, while being engrossed in the construction of numerical solutions of the rigorous equations of stellar structure, one can lose sight of the basic physics underlying the equations.

In Section 3.1, the balance between the pressure gradient force and the gravitational force is used in conjunction with the equation of state for a perfect gas to estimate interior temperatures in homogeneous main sequence stars, emphasizing that the thermal energy of a particle in the deep interior of the star is comparable to the gravitational potential energy of a particle near the surface. In Section 3.2, the effects on the equation of state of electrostatic forces, electron degeneracy, and radiation pressure are examined, with the conclusion that the first two effects become important in stars less massive than the Sun and the third effect becomes important in stars considerably more massive than the Sun. In Section 3.3, theorems relating the binding energy of a star with the kinetic energy of material particles in the star and to the overall gravitational binding energy of the star are constructed.

In Section 3.4, three modes of energy transport – radiation, convection, and conduction – are explored. In the discussion of radiative flow, emphasis is placed on the microscopic physical processes involved in estimating the radiative opacity and a theorem relating stellar luminosity to stellar mass, mean molecular weight, and mean interior opacity is constructed.

In the last four decades of the twentieth century, the detection of neutrinos from the Sun became a reality. Using a detection scheme beginning with a chloroethylene-filled tank in the Homestake mine in South Dakota, Raymond Davis and his collaborators (Davis, Harmer, & Hoffman, 1968) established upper limits on the fluxes of neutrinos made in the Sun by the reactions 8B→8Be* + e+ + ve and 7Be + e−→ 7Li + ve and reaching the Earth as electron-flavor neutrinos. The experiment relied on the reaction 37Cl(ve, e−)37 A, which has a threshold (at 0.814 MeV) approximately twice as large as the 0.43 MeV maximum energy of the neutrino emitted in the pp reaction, slightly smaller than the 0.861 MeV energy of the neutrino emitted in the 7Be + e−→ 7Li +ve reaction, and much smaller than the maximum energy of the neutrino emitted in the 8B(e+ve)8Be* reaction.

The limits established by Davis et al. were an order of magnitude smaller than fluxes which had been predicted on the basis of solar models that incorporated the then best guesses as to the appropriate input physics. Among other consequences, the discrepancy, commonly referred to as the solar neutrino problem, led to a re-examination of available data on relevant nuclear cross sections, revisions and new measurements of these cross sections, and to a refinement over time in the solar models.

The bright stars in the familiar constellations of the Milky Way have intrigued mankind for millennia. Over the past several centuries we have obtained by observations a quantitative understanding of the intrinsic global and surface characteristics of these stars, and over the past century we have learned something about their internal structure and the manner in which they change with time. An awareness that one kind of star can transform into another kind of star and an appreciation of how this transformation is achieved have been accomplishments of the last half of the twentieth century. One of the objectives of this monograph is to describe some of the transformations and to understand how they come about.

The microscopic and macroscopic physics that enters into the construction of the equations of stellar structure and evolution is described in many other monographs and texts. For highly personal reasons, this physics is nonetheless developed here in some detail. My undergraduate and graduate training was in physics, but I did not fully appreciate the beauty of physics until, just prior to my second year of college teaching, during an enforced sedentary period occasioned by a collision between myself on a bicycle and an automobile, I discovered the book Frontiers of Astronomy by Fred Hoyle and became entranced with the idea that the evolution of stars could be understood by applying the principles of physics. During my next two years of teaching, I embarked on a self study course heavily influenced by the vivid discription of physical processes in stars by Arthur S. Eddington in his book The Internal Constitution of the Stars and by the straightforward description of how to construct solutions to the equations of stellar structure by Martin Schwarzschild in his book The Structure and Evolution of the Stars. These books taught me that stars provide a context for understanding physics on many different levels.

Because faster moving particles at higher temperatures transfer energy to more slowly moving particles at lower temperatures, the very existence of a temperature gradient implies a flow of energy in the direction in which the temperature decreases. In the stellar interior, because of their small mass and consequent high velocities, free electrons are the dominant contributors to this mode of thermal energy transfer, which is called thermal or heat conduction.

Thermal conduction does not play a significant role in transporting heat in stars during most of the main sequence phase. However, towards the end of the main sequence phase, as detailed in Chapter 11 of Volume 1 (Section 11.1), low mass stars develop hydrogenexhausted cores in which electrons become increasingly degenerate, and evolve into red giants with fully electron-degenerate helium cores. Under electron-degenerate conditions, only those electrons with energies within about kT of the Fermi energy ϵF participate in transporting heat, but their average cross section for scattering from ions and other electrons is reduced by a factor of the order of (kT/ϵF)2 relative to their average cross section under non-degenerate conditions. Hence, conduction becomes very effective in slowing the rate at which temperatures increase in the electron-degenerate cores and prevents low mass red giants from igniting helium until the degenerate core has grown to almost one-half of a solar mass. As described in Chapters 17 and 18 of this volume, electron conduction plays a similar role in both low and intermediate mass stars after they have exhausted helium at their centers and become asymptotic giant branch stars with electron-degenerate carbon–oxygen or oxygen–neon cores.