Motivation

The aim of this book is to describe the connection between the physics and evolution of relatively nearby luminous hot stars and more distant starburst phenomena occurring all the way to cosmological distances. There have been recent significant advances in our knowledge concerning hot stars and their contribution to highly energetic star formation episodes. The Hubble Space Telescope (HST) and Spitzer in particular have provided many new insights into these areas. Recent observations of the near infrared regions have also greatly aided our view of star formation processes. This book is aimed at those extragalactic astronomers who would be interested in hot star astrophysics and those stellar astrophysicists concerned with galaxy evolution. It would be of use in graduate astrophysics courses at a level suitable for advanced students. This monograph will be one of the first of its kind in spanning the connection from the astrophysics of hot stars to that of newly forming galaxies.

Observed properties

The Hertzsprung–Russell diagram

What are stars? They are fully gaseous, ionized, gravitationally bound entities, emitting large amounts of radiation over many wavelengths. In normal stars, the gas is in hydrostatic pressure equilibrium under the ideal gas law equation of state. It is held in balance against the inward forces of gravity by the outward pressure of radiation generated from nuclear reactions in the stellar interior (or from gravitational contraction adjustments).

In this chapter we discuss the evolution of massive stars (≥ 8M⊙) from their Main Sequence through their chemically evolved supergiant phases, to their ultimate demise as supernovae. The birth of massive stars is deferred until Chapter 7, and gamma ray bursts are discussed in Chapter 11.

The physical scale of stellar masses results from gravity and stellar structure considerations, whilst the ability for stars to maintain their hydrostatic equilibrium over millions or billions of years is as a consequence of the properties of their atomic nuclei. For non-degenerate stars the energy radiated is extracted from either their gravitational or nuclear reservoir. Gravitational contraction maintains mechanical equilibrium over a Kelvin–Helmholtz timescale, tKH, whilst the nuclear energy source can sustain the stellar luminosity during much longer times. Schematically, there exist four ranges of stellar mass, those that undergo (i) no nuclear burning (brown dwarfs, ≤ 0.08M⊙); (ii) hydrogen burning (0.08–0.5 M⊙); (iii) hydrogen and helium burning (0.5 –8 M⊙); (iv) beyond helium burning (≥ 8M⊙). Of these, solely the final mass range is discussed here, since low and intermediate mass stars are widely discussed in the literature (e.g. Iben 1967). A modern review of massive stellar evolution is provided by Maeder & Meynet (2000).

Nucleosynthesis

Arnould & Takahashi (1999) provide a modern review of nuclear astrophysics.

Starspots are analogues of sunspots that appear as dark patterns in a stellar atmosphere and modulate the radiative output over the visible hemisphere of the star. Early detections of starspots, beginning with the work of Kron (1947), are discussed in Section 2.8. Intensity patterns of various types have been detected on many stars, not all of which are analogous to sunspots; for instance, optical aperture synthesis has revealed convection cells on Betelgeuse. Here we shall restrict attention to those patterns that are most like sunspots, although in almost all cases the dark areas are substantially larger than spots on the Sun, for otherwise they would not have been detectable. A good example of such a large starspot has already been shown in Figure 1.8. We infer that starspots share with sunspots a magnetic origin.

The motivation to study starspots comes from many areas of investigation, including the study of magnetoconvection and the study of stellar activity and patterns of emergence of magnetic flux. Starspots and other surface intensity patterns provide the most accurate means of determining stellar rotation periods and also allow the detection of surface differential rotation, a key ingredient in understanding stellar dynamos. The possible effects of starspots often have to be considered in interpreting data related to stellar pulsations or searches for extrasolar planets.

Methods for detecting and mapping starspots on stellar surfaces have advanced rapidly over the past two decades.

Viewed as a star, the Sun exhibits very mild variability. Its total luminosity fluctuates by only 0.1%, following the sunspot cycle. Although most of the energy radiated is in the visible and infrared ranges, corresponding to the peak in the Planck spectrum for the photosphere, the temperature rises through the chromosphere, and then undergoes an abrupt transition to reach millions of degrees in the corona. Since the structure of the solar atmosphere is dominated by its magnetic field, it is not surprising that the Sun's ultraviolet, extreme ultraviolet and X-ray emission vary much more drastically as a result of solar activity. This activity also gives rise to flares and coronal mass ejections, generating energetic particles and enhanced magnetic fields that are carried outwards into interplanetary space by the solar wind. These then impinge upon the Earth, producing aurorae and magnetic storms.

In this chapter we describe the solar irradiance variations and the influence of solar activity on the heliosphere, which gives rise to ‘space weather’. We also explain how the incidence of geomagnetic storms, which have been studied since the time of Gauss, provides a proxy measure of the Sun's open magnetic flux, which is responsible for deflecting galactic cosmic rays. Finally, we consider the controversial topic of the influence of solar variability on terrestrial climate (Friis-Christensen et al. 2000; Haigh 2003, 2007); here we emphasize that this influence, though it is clearly present, remains small compared with the recent global warming caused by anthropogenic greenhouse gases.

After working on research topics related to seismic tomography for a quarter of a century, I decided it was time to write down all I know about the topic – but not in a grand unifying tome that covers everything from first principles to numerical applications. First of all, I have little patience for mathematical niceties; second, and more importantly, I wrote this book for the practitioners of the craft of seismic tomography. Those who go out into the field to collect data usually have no time for proofs of convergence or existence. The intended reader of this book is therefore an observational seismologist or helioseismologist who is not interested in lengthy derivations nor in the subtleties that fascinate the theoreticians, but who wants to understand the assumptions behind algorithms, even if these are mathematically intricate, and develop an understanding of the conditions for their validity, which forms the basis of that priceless commodity: scientific intuition. The level is such that it could be used for a one-semester course at upper undergraduate or beginning graduate level, perhaps following up on an introductory course based on Shearer or Stein and Wysession. Despite covering a wide range of topics, I have tried to keep it short (hence the title), while not economizing on references that may provide more detail if needed.

In this chapter we will take a brief look at several promising new developments. None of the new methods described here are as yet widely employed, and questions of viability remain for some of them. Some exciting new instruments are still in a stage of testing or exist only on the drawing board. The emphasis will be on showing the connection with the material presented in this book, rather than on a full development.

Beyond Born

Though the Born approximation improves on the ray-theoretical approximation, it also binds our hands considerably, since the amplitude of the perturbed wave δu must be much smaller than that of the zero-order wavefield in order to justify the neglect of higher-order scattering (the repeated scattering of scattered waves). Especially at higher frequency the limitations of Born theory discussed in Section 7.5 may become prohibitive. Iterative methods that start with low frequencies and slowly increase the frequency content as the model becomes more complicated may offer some relief, if ray theory can be used to model the first-order perturbation, as is done in methods like PWI. Meier et al. formulated a waveform inversion algorithm that combines 3D Born inversion with this strategy. It becomes potentially very powerful when combined with a 3D adjoint algorithm (see next section).

Encouraging progress has also been made to model the scattering of surface waves in a more complete manner.

Early in the 1970s, seismologists realized that a full three-dimensional (3D) interpretation was needed to satisfy variations in the observed seismic travel times. The starting point of modern seismic tomography (from σεισµός = quake and τόµος = slice) is probably the 1974 AGU presentation by MIT's Keiti Aki (Aki et al.) in which arrival times of P-waves were for the first time formally interpreted in terms of an ‘image’ as opposed to a simple one-dimensional graph of seismic velocity versus depth. That Aki's co-authors came from NORSAR – the Norwegian array to monitor nuclear test ban treaties – was caused by a quirk of history: Aki had originally planned a sabbatical in Chile, but when a military coup d'état brought the Allende government down in 1973 he changed plans and accepted an invitation from his former MIT student Eystein Husebye for a short sabbatical at NORSAR, which was equipped with a state-of-the-art digital seismic network and computing facilities. Even so, in the twenty-first century it is easy to underestimate the difficulties faced by early tomographers, who had to invert matrices of size 256 × 256 using a CPU with 512 Kbyte of memory. The collaboration between Aki, Husebye and Christoffersson was continued in 1975 at Lincoln Labs in Massachussets (Aki et al.).

The name that was later given to the new imaging technique is more than an accidental reference to medical tomography, because the earliest radiologic tomograms also attempted to get a scan of the body that focuses on a plane of interest, albeit using X-rays rather than seismic waves.

Historically, travel times were measured from seismograms recorded on smoked or photographic paper. An example is shown in Figure 6.1. This is a seismogram, dated July 26, 1963, from the World Wide Standardized Seismograph Network (WWSSN), the state-of-the-art at the time. Arrival times were picked from such recordings by measuring the distance to the nearest minute mark, visible as small deflections at regular intervals. One major shortcoming of such photographic recordings is that the trace becomes hard to read when the amplitude is large and the light source moves quickly, giving only a short exposure of the photographic film. Short period recordings, an example of which is shown in Figure 6.2 are to be preferred for the picking of the P-wave arrival time, but may be less suitable to correctly identify the later arrivals. Though the network is now obsolete, scanned images of seismograms for a growing number of historical earthquakes are available in the public domain. These images can be digitized with suitable vectorising software, such as Teseo (Pintore et al.,).

Modern, digital instrumentation has greatly changed the practice in seismographic stations around the world. Figure 6.3 shows an example from a modern digital seismographic station. Digitized seismograms are much easier to manage, archive and analyse.

In this chapter we consider only digitized signals. Formally, a seismogram s(t) is digitized by convolving it with a Dirac comb ∑i δ(t − i Δ t), resulting in N-tuple of values, e.g. (s1, s2, … sN).

As we have seen in Chapter 2, the energy density of a seismic wave decreases as the wave propagates because of geometrical spreading: the available energy spreads over a larger wavefront-surface, and therefore amplitudes should decrease to avoid an increase in the total energy of the system. Individual wave packets also lose energy because of scattering: part of the energy is redirected by refraction, or converted to a different kind of wave. Examples of this are P–S conversion at the Moho, or scattering at random heterogeneities.

In both cases, however, the total mechanical energy in the vibrating system (the Earth) remains the same. We know this cannot be true. At regular intervals, vibrational energy is added to the Earth, through release of potential energy (strain energy) in earthquakes, through the detonation of large explosions or simply when a train passes by. But even after a large earthquake, the activity on a short period seismograph returns to normal after a few hours. It may take days on a low frequency seismograph, but there, too, the energy eventually damps away. The mechanical energy of seismic waves is converted to other forms of energy, mostly heat. Such processes are inelastic, commonly referred to as ‘intrinsic attenuation’.

In this chapter we take a closer look at the factors that determine the amplitude of a body wave.

Geometrical spreading

In Section 2.7 we saw that the energy of a body wave in a spherically symmetric Earth is proportional to ∂2T/∂Δ2 = ∂p/∂Δ.

Since the Earth is a mechanical system with a well-defined boundary, the formal solution to the elastodynamic equations (2.3) with the boundary condition that the surface is stress-free yields a discrete spectrum of eigenfrequencies, just as is the case for a finite-length string in a violin. The reason that we have not taken this viewpoint earlier is that it is highly impractical for seismic tomography in the frequency band that is of most interest: above 10 mHz the number of eigenfrequencies in a small frequency band becomes very large. In addition, the spectral peaks are widened by the effects of attenuation and lateral heterogeneity and the discrete spectrum becomes, for all practical purposes, a continuous one because the peaks overlap. But below 10 mHz and even at higher frequency for some high Q modes it becomes feasible to measure individual eigenfrequencies (Figure 9.1).

The theoretical study of terrestrial eigenfrequencies started with the historical work of Love. Interest in the field really grew only after the first observations of normal modes, following the Chile earthquake of 1960. Major contributions to the development of the theory are by Pekeris et al., Backus and Gilbert, Dahlen, Gilbert, Woodhouse and Dahlen, Jordan and Park; and of the interpretation by Backus, Gilbert and Dziewonski, Jordan, Masters et al., Woodhouse and Girnius, Woodhouse and Dziewonski, Woodhouse and Wong.

Because of the stronger heterogeneity near the surface of the Earth, surface waves are even more prone to the effects of scattering than the teleseismic P- and S- waves. Only at rather low frequencies is it safe to assume that scattering can be ignored, but this is of course also the frequency band where the approximations of ray theory become questionable. Detailed studies on the validity of asymptotic approximations by Park, Kennett and Nolet and Clévédé et al. show that the approximations of ‘ray theory’ for surface waves can be problematic even for rather smooth Earth models. Lateral heterogeneity poses even stronger problems for the inversion of group velocity, which is theoretically an interference phenomenon between neighbouring frequencies, as expressed by the differentiation with respect to frequency in (10.4). When the Earth is laterally heterogeneous, waves travelling different paths may equally well interfere and significantly perturb the time of arrival of the maximum energy at a particular frequency, robbing the group arrival time of its conventional interpretation.

Phase velocity measurements also display significant oscillations due to interference of multipathed arrivals. When using ray theory (Chapter 10) there is no other solution but to average over many observations (Pedersen,). First-order scattering should at least be able to model some of the multipathed energy, and this has led to efforts to extend finite-frequency theory to surface wave observations.

In this chapter we introduce some basic concepts of the mathematics of wave propagation for acoustic and elastic waves. We shall often use heuristic or intuitive arguments rather than formal proofs, since our primary aim is to provide the necessary minimum background to readers not familiar with the fundamentals of continuum mechanics. Readers eager to educate themselves more extensively on the topics that we touch upon only briefly should consult the advanced seismology textbooks by Aki and Richards, Kennett or Dahlen and Tromp. Ĉervený and Chapman have written more specialized books on ray theory and its extensions.

We limit ourselves to wave propagation in isotropic media. This means that the elastic properties of the medium do not depend on its orientation in space: to shear a cube in the x-direction requires the same force as in the y- or z-directions. Even if the real Earth is locally anisotropic in regions of fine layering or of crystal alignment because of solid state flow, the background model – the model with respect to which heterogeneity is defined – is usually defined to be isotropic (and spherically symmetric). As we shall see in Chapter 16, anomalies with respect to the background model can be anisotropic. In the Sun, the magnetic field introduces anisotropy, but here too the background model is assumed to be isotropic (a gas).