In constructing models of evolving stars in volume 1 of this monograph, it has been assumed that particle diffusion can be neglected. However, the current abundance of Li at the solar surface is much smaller than predicted by models neglecting diffusion and this suggests that, during the gravitationally contracting phase which preceeds the main sequence, diffusion carries Li from the convective envelope into higher temperature regions below the base of the envelope where it can be destroyed. The abundance of Fe observed at the solar surface is smaller than the interior Fe abundance indicated by comparison between neutrino fluxes observed from the Sun and fluxes predicted by solar models, suggesting that Fe has diffused out of the convective envelope during the main sequence phase into regions below the base of the convective envelope. The fact that many low luminosity white dwarfs exhibit monoelemental surface abundances is a dramatic demonstration that, in regions where the gravitational acceleration is orders of magnitude larger than near the surface of the Sun, gravitationally induced diffusion is a first order effect. Thus, there is ample motivation for studying the physics of diffusion.

A description of the physics of particle diffusion can be given on many levels of sophistication. The description adopted in this chapter is based on an analysis of Boltzmann transport equations constructed on the assumptions that there exists an equilibrium distribution function for every species of particle and that, when a system is not in equilibrium, the time rate of change of each distribution function can be determined as the consequence of binary interactions between all particles.

History of s-process nucleosynthesis and outline

In the middle of the twentieth century, Paul W. Merrill (1952) detected the beta-unstable element technetium in the spectra of red supergiants known as S stars, which have long been recognized as being AGB stars. 99Tc is the longest lived isotope of technetium, and its half life of 211 000 yr is several orders of magnitude shorter than the lifetime of a core helium-burning precursor of an AGB star. When observed mass-loss rates are taken into account, the lifetime of a star in the AGB phase is roughly an order of magnitude larger than the half life of 99Tc. It is therefore incontrovertible that the 99Tc observed has been formed in AGB stars. On the other hand, given the temperatures and densities in AGB models, it is clear that an element of atomic charge as large as that of Tc cannot be a consequence of reactions between charged particles and it has become accepted that neutron capture must be involved.

The earliest theoretical studies of neutron-capture nucleosynthesis were carried out by A. G.W. Cameron (1955) and by E. Margaret Burbidge, Geoffrey R. Burbidge, William A. Fowler, and Fred Hoyle (1957) and, over the years, numerous studies have been devoted to examining how the distribution of abundances in an initial set of heavy elements is altered when this set is placed in a bath of neutrons at various assumed number densities for various exposure times.

Chapters 5 to 7 considered passive sensors, detecting naturally occurring radiation. In this chapter and the next we shall discuss active sensors, which emit radiation and analyse the signal that is returned by the Earth’s surface or atmosphere. We have already identified three possible classifications of remote sensing systems, distinguishing between passive and active and between imaging and non-imaging, as well as classifying them according to the wavelength of radiation employed. We can also classify active systems according to the use that is made of the returned signal. If we are principally concerned with the time delay between transmission and reception of the signal we shall call the method a ranging technique, whereas if we are also (or mainly) interested in the strength of the returned signal we shall call it a scattering technique. The distinction between the two cannot be made entirely rigorous, but it provides a useful way of thinking about active remote sensing systems. It is clear that ranging systems are simpler both to visualise and, because of their less stringent technical demands, to construct, and we shall therefore consider them first. In Chapter 9 we shall discuss the scattering techniques.

Laser profiling

Laser profiling (or laser altimetry) is the simplest application of the LiDAR (Light Detection and Ranging) technique. Conceptually it is extremely straightforward (Baltsavias 1999, Flood 2001). A short pulse of ‘light’ (visible or near-infrared radiation) is emitted towards the Earth’s surface by the instrument, and its ‘echo’ is detected some time later. By measuring the time delay and knowing the speed of propagation of the pulse, the range (distance) from the instrument to the surface can be determined. By transmitting a continuous stream of pulses, a profile of the range can be built up, and if the position of the platform as a function of time is accurately known the surface profile may then be deduced.

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.

In this chapter we complete our survey of the principal types of remote sensing instrument by discussing those active systems that make direct use of the backscattered power. Optical (LiDAR) systems are used for sounding clouds, aerosols and other atmospheric constituents, for characterising surface albedo, and for measuring wind speeds. These are discussed briefly in Section 9.1. However, the bulk of this chapter is concerned with microwave (radar) systems. (‘Radar’ is an acronym, originally standing for ‘radio detection and ranging’. The functions that can be performed by microwave scattering systems now extend far beyond detection and ranging, but the term continues to be in very general use.)

In Section 9.2 the ground-work established in Chapter 3 is extended to a derivation of the radar equation, which shows how the power detected by a radar system is related to the usual measure of backscattering ability, the differential backscattering cross-section σ0. The remainder of the chapter discusses the main types of system that employ this relationship. The first and simplest is the microwave scatterometer (Section 9.3), which measures σ0, usually only for a single region of the surface but often for a range of incidence angles. As described here, this is not an imaging system, although the distinction between microwave scatterometers and imaging radars is not a precise one.

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.

In Chapters 5 and 6 we considered passive remote sensing systems in which the diffraction resolution limit λ/D, while important, was not usually a critical parameter of the operation. In this chapter we consider our last major class of passive remote sensing system, the passive microwave radiometer. This is a device that measures thermally generated radiation in the microwave (usually 5–100 GHz) region. (Frequencies much below 1 GHz are unsuitable because of the large signal contributed by the Galaxy, as well as the difficulty of achieving adequate spatial resolution.) As we discussed in Section 2.6, the long ‘tail’ to the Planck distribution at relatively low frequencies means that measurable amounts of radiation are emitted even in this range of frequencies.

Because microwave wavelengths are so much greater than those of visible or even of thermal infrared radiation, the resolution limit plays a much more important role, and we shall need to give more attention to the factors that determine it. More detailed technical treatments of antenna theory are given by Ulaby, Moore and Fung (1981) and by Sharkov (2003), amongst others. Much of the technology and nomenclature of passive microwave radiometry was originally developed in the field of radio astronomy, and further details can also be found in works on that subject.

Aerial photography, as we remarked in Chapter 1, represents the earliest modern form of remote sensing system. Despite the fact that many newer remote sensing techniques have emerged since the first aerial photograph was taken over 100 years ago, aerial photography still finds many important applications, and there are many books that discuss it in more detail than will be possible in this chapter. The interested reader is referred, for example, to Berlin and Avery (2003) and a detailed treatment of photogrammetry and stereogrammetry is give by Kraus (2007). Aerial photography is familiar and well understood, and is a good point from which to begin our discussion of types of imaging system. In particular, it provides a convenient opportunity to introduce some of the imaging concepts that will be useful in discussing some less familiar systems in later chapters.

Photography responds to the visible and near infrared parts of the electromagnetic spectrum. It is, in the context of remote sensing, a passive technique, in that it detects existing radiation (reflected sun- and skylight), and an imaging technique in that it forms a two-dimensional representation of the radiance of the target area. In this chapter we shall consider the construction, function and performance of photographic film, especially its use in obtaining quantitative information about the geometry of objects. Although film-based aerial photography is still dominant, digital photography is beginning (at the time of writing) to rival it, so the chapter includes a brief comparison of the two methods. The chapter then discusses the effects of atmospheric propagation, and concludes by describing the characteristics of some real instruments and giving a brief account of the applications of the technique.

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.