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).

In this chapter we generalize the results of Chapters 3 and 4 to the case of quasi-longitudinal whistler-mode propagation. As in Section 2.1 we consider whistler-mode propagation as quasi-longitudinal if either ⃒θ⃒ ≪ 1 (see inequalities (2.14) and (2.17)) in the plasma with arbitrary electron density (inequalities (2.10) and (2.12) are not necessarily valid), or inequalities (2.9) and (2.10) (or (2.12)) are valid simultaneously provided the whistlermode wave normal angle θ is not close to the resonance cone angle θR0 in & cold plasma (see equation (2.13)). First we consider the case ⃒θ⃒ ≪ 1 when whistler-mode waves propagate almost parallel to the magnetic field.

As was shown in Chapter 1, when we impose no restrictions on the electron density we should use the general relativistic expressions for the elements of the plasma dielectric tensor in the form (1.73). Assuming that the waves propagate through plasma with the electron distribution function (1.76) and imposing conditions (1.77) we write these expressions in a much simpler form (1.78). Also, we assume that the electron temperature is so low that it can only slightly perturb the corresponding whistler-mode dispersion equation in a cold plasma, i.e.

where N is the whistler-mode refractive index in a hot plasma, and N0 the whistler-mode refractive index in a cold plasma defined by (2.15).

Approximations

In view of the applications of our theory to the conditions of the Earth's magnetosphere the following assumptions are made:

1. The plasma is assumed to be homogeneous in the sense that its actual inhomogeneity does not influence its dispersion characteristics, instability or damping at any particular point, although in general these characteristics can change from one point to another. For low-amplitude waves this assumption is valid when the wavelength is well below the characteristic scale length of plasma inhomogeneity, a condition which is satisfied for whistlermode waves propagating in most areas of the magnetosphere (except in the lower ionosphere). For finite amplitude waves the condition for plasma homogeneity depends on wave amplitude, but the discussion of these effects is beyond the scope of the book (see e.g. Karpman, 1974).

2. The plasma is assumed to be collisionless in the sense that we neglect the contribution of Coulomb collisions between charged particles as well as collisions between charged and neutral particles leading to charge exchange. More rigorously this assumption can be written as: where qα is the particle's charge (index α indicates the type of particle: α = e for electrons, α = p for protons; 〈r12〉 is the average distance between particles; Tα is the particles' temperature in energy units.

The physical meaning of (1.1) is obvious: the average energy of interaction between charged particles is well below their average kinetic energy.

As was shown in Chapter 5, the quasi-electrostatic approximation becomes invalid when θ approaches the resonance cone angle θR defined by (2.23) or (2.24) (or θR0 defined by (2.13) in the case of a dense plasma, the contribution of ions being neglected). However, at θ equal to or close to θR we can use another approximation based on the following assumptions:

(1) The wave refractive index N is assumed to be so large that only the contribution of the terms proportional to the highest powers of N in the dispersion equation is to be taken into account.

(2) The plasma temperature is assumed to be so low that inequalities (1.77) are valid and the elements of εij can be written in the form (1.78), εijt and εijr being the perturbations of εij0.

The first assumption seems to be a straightforward one in a sufficiently low-temperature plasma, as in a cold plasma limit N20 ∞ and when θ → θR. However, the second assumption does not seem to be an obvious one since εijt ∼ N2 and εijt → ∞ as N2 → ∞. Thus N should be large enough to satisfy the first assumption but small enough to satisfy the second one. The validity of these assumptions can be checked by the actual value of N obtained from the solution.

In contrast to Chapter 2, in this chapter we restrict ourselves to considering the parallel whistler-mode propagation, but impose no restrictions on magnetospheric electron temperature except the condition (1.2) of weakly relativistic approximation. We begin with some general simplifications of the dispersion equation (1.42) with the elements of the plasma dielectric tensor defined by (1.73) in the limiting case θ → 0. In this limit we can assume that ⃒λα⃒ ≡ ⃒k⊥υ⊥/Ωα ⃒ ≪ 1 and expand the Bessel functions in (1.74) using the following formula (Abramovitz & Stegun, 1964).

Remembering (3.1) and keeping only zero-order terms with respect to λα we obtain the following expressions for the non-zero elements of the tensor Πij(n,α) defined by (1.74):

From (1.73) and (3.2) it follows that:

In view of (3.3) and remembering our assumption that θ = 0 we can simplify the dispersion equation (1.42) to

Parallel propagation (weakly relativistic approximation)

This equation can be further simplified if we take into account the contribution of electrons only (which is justified for whistler-mode waves) and present ∈11 = ∈22 as ∈11 = 1 + ∈+ + ∈_ and ∈12 = –∈21 as ∈12 = i(∈+ – ∈_), where ∈+ and ∈_ are the contributions of the electron currents corresponding to n = 1 and n = – 1 respectively in the term ∈11 in (1.73). In this case equation (3.4) can be written as

In order to satisfy (3.5) at least one of three factors in the left-hand side of this equation must be equal to zero. The equation ∈33 = 0 is the dispersion equation for electrostatic Langmuir waves, propagating along the magnetic field (see equation (1.13)).

The linear theory of whistler-mode propagation, growth and damping considered so far has been based on the assumption that waves with different frequencies and wave numbers do not interact with each other (superposition principle) and that the waves do not cause any systematic change in the background particle (electron) distribution function f0. A self-consistent analysis of both these processes creates, in general, a very complicated problem even for modern computers (see e.g. Nunn, 1990). However, in many practically important cases we can develop an approximate analytical theory which takes into account some of these processes and neglects others. This theory, known as the non-linear theory, has been developed during the last 30 years and its results are summarized in numerous monographs and review papers, such as those by Kadomtsev (1965), Vedenov (1968), Sagdeev & Galeev (1969), Tsytovich (1972), Karpman (1974), Akhiezer et al. (1975), Hasegawa (1975), Vedenov & Ryutov (1975), Galeev & Sagdeev (1979), Bespalov & Trakhtengertz (1986), Zaslavsky & Sagdeev (1988), Petviashvili & Pokhotelov (1991) and many others. I have no intention of amending this long list of references by one more contribution. Instead I will restrict myself to illustrating the methods of non-linear theory by two particularly simple examples: the quasi-linear theory of whistler-mode waves (Section 8.1) and the non-linear theory of monochromatic whistler-mode waves (Section 8.2).

Quasi-linear theory

(a) Basic equations

A theory which assumes that the wave amplitude is so small that the superposition principle remains valid, but large enough to provide a non-negligible change of the background electron distribution function ƒo under the influence of the waves, is known as the quasi-linear theory (Akhiezer et al., 1975).