Introduction

In this chapter I shall consider that a galaxy consists of stars alone. To a first approximation this is true of a real galaxy apart from the possibility of hidden mass in the form of weakly interacting elementary particles; in what follows the gravitational field of any hidden mass is assumed to be added to that of the stars. For most of the life history of a galaxy the mass in the form of stars is much greater than the mass of the interstellar gas and the stars exert a much stronger gravitational influence on the gas than the reverse. Although there is a continual mass exchange between the stars and the gas, this occurs slowly compared to the time taken by individual stars to travel about the galaxy, certainly once the formation phase of the galaxy is completed. To a large extent in what follows I shall refer to the Galaxy, but it may be regarded as typical of other galaxies.

The first point to be made about the Galaxy, if it is considered as a system of stars, is that to a good approximation stars may be regarded as point masses. Except in the densest regions of galaxies, stellar separations very much smaller than 1016 m are rare, whilst only the very largest stars have radii greater than 1010 m.

Introduction

A study of the chemical composition of stars in our Galaxy indicates that this composition varies from star to star and that the variation is not random. In particular, there is some correlation between:

1. (a) Stellar chemical composition and stellar age, and

2. (b) Stellar chemical composition and stellar position or, more accurately, place of origin.

The correlations are in the sense that the oldest stars and the stars formed in the halo region of the Galaxy have a lower heavy element content than the youngest stars and those formed in the disk. It is not completely clear whether this is one correlation or two. To the accuracy to which stellar ages are known all halo stars could be older than all disk stars and their low heavy element content could just be a consequence of their time of formation. There is, however, also some evidence that halo stars of similar age have very different heavy element content, with those halo stars which were formed furthest from the galactic centre being most deficient in heavy elements. The present estimates of stellar age are not accurate enough to decide whether time of formation is the major factor determining stellar chemical composition or whether there have always been important variations of composition with position.

Introduction

This chapter contains a description of the properties of the Galaxy. It is concerned mainly with obervations, although as I have mentioned on page 18 many of the observations require a considerable amount of interpretation before they are very useful. The Galaxy is primarily a system of stars and I start this chapter by summarising some of the properties of stars of different types. As I shall explain later, there is some considerable uncertainty about the total mass of the Galaxy and about the masses of its individual components. In particular we shall learn that much of the mass of our own Galaxy and other galaxies may be invisible. Although this hidden matter might be very low luminosity stars or dead stellar remnants, there is a general belief that it is composed of weakly interacting elementary particles. At present I shall concentrate attention on the visible components. For them it may not be too far wrong to suppose that 95 per cent of the mass is stellar (including dead remnants) and about 5 per cent is in the form of interstellar gas and dust. In addition the Galaxy contains cosmic rays, very high energy charged particles, which contribute very little to the total mass but whose total energy is very important in discussions of the structure of the interstellar medium, as we shall see in Chapter 6.

The discovery of galaxies

In the medieval world picture the stars were regarded as points of light attached to a sphere, whose surface was a long way outside the Solar System but whose volume was thought to be very much smaller than the space which we now know the stars to occupy. Attempts had indeed been made to determine the distance to the stellar sphere based on the possibility that stars might appear to be in different directions when observed from two points on the Earth's surface (fig. 1). Although this method worked for the Sun and Moon and other objects in the Solar System, it failed for the stars, indicating that they were very distant. After the Scientific Revolution in the 16th and 17th centuries, culminating in Newton's explanation of the motion of the planets in terms of a universal law of gravitation, it was realised that the stars were probably also suns or equivalently that the Sun was but one star amongst many and that the fixed stars should, in fact, be moving through space and should be influenced by the same law of gravitation. This led to a renewed interest in trying to determine not only their positions but also their motions.

Initially it was thought that there was just one system of stars filling the Universe and this view persisted until the second decade of this century.

As was shown in Section 2.1, the general solution of the wave dispersion equation in a ‘cold’ plasma (see equation (2.4)) describes either wave propagation without damping or growth (N02 0), or the absence of the waves (N02 ≤ 0; N0 = 0 corresponds to plasma cutoffs). However, wave propagation in a plasma with non-zero temperature is accompanied, in general, by a change of amplitude of the waves, Aw, which is described by the increment of growth (γ > 0) or the decrement of damping (γ < 0) (Aw ∼ {\rm exp}(γt)). When ⃒γ⃒ ≪ {\rm min} (ω0, Ω − ω0) (which is satisfied in most cases of whistlermode propagation in the magnetosphere and will hereafter be assumed to be valid) then γ is described by equation (1.17). In this chapter, the latter equation will be applied to the analysis of whistler-mode growth (γ > 0) or damping (γ < 0) in different limiting cases of wave propagation considered in Chapters 3–6.

Parallel propagation (weakly relativistic approximation)

As was shown in Section 3.1, the general dispersion equation for parallel whistler-mode propagation in a weakly relativistic plasma, with the electron distribution function (1.76) (with j = 0), can be written in the form (3.10). This equation describes both wave propagation (see equation (1.16)) and the growth or damping (see equation (1.17)). The processes of parallel whistlermode propagation in a weakly relativistic plasma in different limiting cases were analysed in Sections 3.2 and 3.3.

Following Helliwell (1965) we can define whistlers as radio signals in the audio-frequency range that ‘whistle’. Usually a whistler begins at a high frequency and in the course of about one second drops in frequency to a lower limit of about 1 kHz, although the duration of the event may vary from a fraction of a second to two or three seconds. Occasionally this ‘lower’ branch of a whistler's dynamic spectrum is observed simultaneously with the ‘upper’ branch where the frequency of the signal increases with time, so that the whole dynamic spectrum appears to be of the ‘nose’ type. Typical dynamic spectra of such whistlers observed at Halley station in Antarctica (L = 4.3) are shown in Fig. I.

The energy source for a whistler is a lightning discharge where the waves are generated over a wide frequency range in a very short time. These waves propagate from their source in all directions. Part of their energy propagates in the Earth-ionosphere waveguide with a velocity close to the velocity of light and almost without frequency dispersion.

The general dispersion equation for whistler-mode propagation, instability or damping in a non-relativistic plasma with the electron distribution function in the form (1.90) has already been derived in Section 3.1 (see equation (3.20)). Assuming, as in Sections 1.2 and 3.2, that whistler-mode growth or damping does not influence wave propagation we can simplify equation (3.20) to:

where N, ω and Y hereafter in this chapter are assumed to be real, the argument of the Z function is ξ1 = ξ = (1 Y)/Nῶ∥, ῶ∥ = w∥/c (cf. similar assumptions in Section 3.2),

(cf. the definition of the Z function by equation (1.21)), and Ae = (j + 1) w⊥2/w∥2 (when deriving (4.1) we have generalized equation (3.20) for arbitrary integer j).

As follows from the analysis of Chapter 3, the non-relativistic approximation and, in particular, equation (4.1) is valid in a relatively dense plasma when ν ≫ 1 and N2 ≫ 1, in general. Hence, the second term ‘1’ in equation (4.1) will either be neglected altogether or taken into account when calculating the perturbation of N2 due to non-zero ν− 1 (cf. equation (3.34)).

Although equation (4.1) is much simpler than the corresponding weakly relativistic dispersion equation (cf. equation (3.10)), it still has no analytical solution in general.

As mentioned in the Introduction, interest in the theory of whistler-mode waves was stimulated mainly by the observations of these waves at groundbased stations and in the Earth's magnetosphere, their applications to the diagnostics of magnetospheric parameters and their role in the balance of the electron radiation belts. An overview (even a brief one) of theoretical models of all manifestations of whistler-mode waves in the magnetosphere is obviously beyond the scope of this book (it would require writing a separate monograph) and we will restrict ourselves to illustrating the application of the theoretical analysis developed in the previous chapters to interpreting only three particular phenomena. In Section 9.1 we consider the problem of the diagnostics of magnetospheric parameters with the help of whistlers generated by lightning discharges. Theoretical models of natural whistler-mode radio emissions observed in the vicinity of the magnetopause are discussed in Section 9.2. In Section 9.3 we apply one of the quasi-linear models described in Chapter 8 to the interpretation of mid-latitude hiss-type emissions observed in the inner magnetosphere. The approaches developed for these three illustrative examples can be extended with some modifications to several other related whistler-mode phenomena. These will be discussed in the appropriate sections.

Whistler diagnostics of magnetospheric parameters

Dynamic spectra of whistlers generated by lightning discharges are shown at the beginning of the book in Fig. I.

As mentioned in Section 1.1, for a really cold plasma (Tα → 0) the condition (1.1) is no longer valid and all the theory developed in Chapter 1 breaks down. Thus when speaking about cold plasma we will assume that its temperature is so low that the contributions of thermal and relativistic corrections to ∈ij (the terms ∈ijt and ∈ijr in (1.78)) to the process of wave propagation are small when compared with the contribution of ∈ij0, but at the same time this temperature is high enough for condition (1.1) to remain valid. This definition of a cold plasma obviously depends on the type of waves under consideration. The cold plasma approximation allows us to write the dispersion equation for various waves in a particularly simple form and it has been widely used for the analysis of waves (in particular, whistler-mode) in the magnetosphere. Some results of plasma wave theory based on this approximation will be recalled below.

Neglecting the contribution of the terms ∈ijt and ∈ijr in (1.78) we can assume ∈ij = ∈ij0 in the expressions for A, B and C defined by (1.43)–(1.45) and rewrite them as:

where index 0 indicates that the corresponding coefficients refer to a cold plasma approximation; S, R, L and P are the same as in (1.79).