Preliminary remarks

One of the most obvious features of the plasma state is the rich variety of wave motions which plasmas can support. Waves of a particular kind are said to be in a particular wave mode. The idea of a wave mode is familiar from other contexts. For example in a compressible gas there are sound waves and, if there is a gravitational field present, there are also internal gravity waves. These waves have specific dispersion relations, which relate the frequency ω to the wave vector k. For sound waves and internal gravity waves the dispersion relations are ω = kcs and ω = (gk)½ respectively, where cs is the sound speed and g is the gravitational acceleration. One could cite numerous examples of wave modes in other media, e.g. spin waves in ferromagnetic media and seismic waves in the solid Earth, each of these is characterized by its dispersion relation and other properties which determine the nature of the wave motion. The wave modes of a plasma depend on the plasma properties and these are described in terms of various plasma parameters. Of particular importance are the natural frequencies of the plasma: the electron plasma frequency ωp, the electron cyclotron frequency Ωe, the corresponding ion frequencies ωpi and Ωi, and various collision frequencies (vei, vee, vii) between electrons (e) and ions (i).

The bump-in-tail instability

Many microinstabilities have both reactive and kinetic forms. From a mathematical viewpoint one treats the reactive form by ignoring the imaginary part of the dielectric tensor and solving a real dispersion equation to find complex solutions, and one treats the kinetic form by assuming real frequencies to a first approximation and then including weak (negative) damping. To be more specific, in the weak-beam instability (§3.3) thermal motions are neglected and the correction to the real part of the dielectric tensor due to the presence of trie beam leads to a cubic equation (3.11) for the frequency shift of the Langmuif waves; this cubic equation has a real solution and a pair of complex conjugate solutions in the regime of interest. The kinetic version of this instability is known as the bump-in-tail instability. It is treated by first finding the imaginary contribution of the beam to the dielectric tensor and using this to evaluate the imaginary part of the frequency shift.

It is apparent from the foregoing discussion that the reactive and kinetic versions should be limiting cases of a single instability. This may be shown by finding both the real and imaginary parts of the frequency shift simultaneously. The real and imaginary parts of the frequency can be found as a complex solution for ω as a function of real k to the complex dispersion equation KL(ω, k) = 0, where both real and imaginary parts of KL are retained.

Resonant scattering

The scattering rate due to Coulomb interactions between a fast particle (speed v) and thermal particles decreases with increasing v as v−3. Thus one might expect that fast particles are scattered very ineffectively. The reverse is the case in the low density (low β) plasmas in the magnetosphere, and in space and astrophysical plasmas generally. Fast particles are scattered very efficiently due to resonant interaction with low frequency waves, called resonant scattering.

The evidence which led to the initial development of the theory of resonant scattering came from the properties of the trapped particles in the magnetosphere. By the early 1960's it was clear that for the stability of the distributions of trapped magnetospheric particles (in the terrestrial ‘radiation’ or ‘van Allen’ belts) to be consistent with the observations of precipitation of these particles, both the fast electrons and the fast ions must be scattered very efficiently. The development of the theory of resonant scattering led to a satisfactory qualitative and semi-quantitative explanation for these magnetospheric observations. Resonant scattering also offered ways of resolving serious difficulties connected with the scattering and acceleration of fast particles in astrophysical plasmas. The most obvious of these concerns the confinement of galactic cosmic rays (§13.4). Another serious difficulty was with the acceleration of fast particles: early theories for the acceleration required (either implicitly or explicitly) very efficient scattering.

Introduction

In the chapters that follow, nearly a quarter century of intensive research and substantial progress in understanding the quasars will be summarized. From the perspective of an astronomer, it would be satisfying to report that this progress was primarily attributable to the cleverness and diligence of astronomers in attacking the problem. To be honest, however, most of the progress should be credited to the engineers and physicists who have developed the tools that allow our wide ranging probes into the mysteries of quasars. Trying to understand quasars forced astronomers into realizing that observations must extend over the broadest possible spectral coverage, that we must learn how to use X-ray, ultraviolet, optical, infrared and radio astronomy. It is now possible for an individual astronomer to have access to telescopes that access all of these spectral regions, and I am convinced that the definition of a ‘good’ observer in the next few decades will weigh heavily on the ability to be comfortable with all of these techniques. To realize how much this has changed the science of astronomy, one need only recall that the analogously important talents in 1960 were the ability to work efficiently at night, to withstand cold temperatures, and to develop photographic emulsions without accidently turning on the lights. At that time, the technically sophisticated astronomer was one who could use a photomultiplier tube.

Dielectric tensors

The inclusion of a magnetic field leads to a considerable increase in the richness and variety of the wave motions which can exist in a plasma. It also leads to a qualitative change in the orbits of the particles, which become spirals about the magnetic field lines. This affects the nature of particle-wave interactions. Not surprisingly, the generalization from unmagnetized to magnetized plasmas involves a marked increase in algebraic complexity of the relevant formulas. However the basic principles do not change.

In this Chapter the generalization (to the magnetized case) of the calculations of the dielectric tensor (§10.1) and of the quasilinear equations for wave-particle interactions (§10.5) are presented, and the properties of important classes of waves are discussed. The case of cold plasma wave modes is treated in a formal way in §10.2, the magnetoionic wave modes are discussed in §10.3, and low frequency wave modes are treated in §10.4. The waves discussed in detail in this chapter can nearly all be regarded either as magnetized versions of the waves in an unmagnetized plasma or as collisionless analogs of the MHD waves. There are other wave modes which are intrinsic to collisionless magnetized plasmas; some of these are discussed in Chapter 12.

For an unmagnetized plasma there are three equivalent methods for calculating the response tensors: the cold-plasma method (§2.1), generalized as discussed following (2.25), the Vlasov approach (§2.2) and the forward-scattering method (§5.5).

The discovery of quasars nearly a quarter of a century ago made a new science out of astronomy. There were two factors in this invigorating revolution. One was the conceptual shock of learning that some very important sources of energy exist in the universe that are not related to the nuclear fusion processes in stars. The other was the fact that the discovery was made with a new technology, in this case radio astronomy. For the theorist, there was suddenly an open season for wide ranging and creative speculations on cosmological processes, energy generation, and radiation physics. For the technologist, there was proof that opening new observational spectral windows could reveal extraordinary and totally unanticipated things. Radio astronomy was quickly followed by ultraviolet, infrared and X-ray astronomy.

That first phase of the theoretical and technical regeneration of astronomy is now complete. Astronomers are more open to heretical theoretical suggestions and unconventional observational techniques. Exceptional telescope facilities are at our disposal worldwide and in space. Two decades of effort have not answered all of the fundamental questions about quasars, nor have they led to the discovery of objects any more puzzling. We still have the problems, but we now have the tools, and so can get on with the work of learning what, where, when, and why are the quasars.

My initiation into the subject began during a few spare hours left over from another project while observing with the 36-inch telescope at McDonald Observatory, in the fall of 1967.

Introduction

It should be clear from the discussions in the preceding chapters that an overwhelming amount of information is now available for describing quasar properties. Observationally, the study of quasars has been a great success. Also, it should be no surprise that, as the data have accumulated, it becomes more difficult to produce models that can explain everything. As might be expected for the most energetic objects in the universe, quasars are complex. This should not be a source of discouragement. It is not necessary to understand all details of the solar surface to know why the Sun shines. It is not necessary to understand all sedimentary rocks to know why continents drift. It is not necessary to memorize the taxonomy of all living creatures to realize why evolution occurs. When we are after the fundamental understanding of why something happens, all of the details are not required. In the study of quasars, we are still struggling to the point of knowing which details can be safely ignored, and it is for guidance in this regard that existing models are most useful.

The single most significant observational datum about quasars is that their spectra are so extraordinarily similar, even over ranges of 107 in luminosity, for objects separated by more than ten billion light years in the universe.

This book is intended as an introduction to the theory of plasma instabilities. It is directed at graduate students, advanced undergraduate students with some background knowledge of plasma physics, and to researchers seeking to become more familiar with the field.

In most applications of plasma physics, plasma instabilities of various kinds play important roles. Some laboratory examples include instabilities limiting inertial or magnetic confinement of fusion plasmas, instabilities which produce enhanced radiation and anomalous transport coefficients in current-carrying plasmas, and instabilities which provide coherent sources of radiations in gyrotrons and free electron masers. Some examples from space plasmas include instabilities which produce nonthermal wave and particle distributions in the magnetosphere and the interplanetary medium and nonthermal radiation from the planets and the solar corona, instabilities involved in nonlinear propagation effects in the ionosphere, and instabilities leading to scattering and acceleration of fast particles in astrophysical plasmas. The richness and variety of the plasma instabilities and the diversity of their applications preclude any thorough treatment in a single book. In this book I have attempted to be thorough only in the coverage of the qualitative kinds of plasma instability which are possible. The main emphasis is on instabilities at moderate to high frequencies, that is frequencies from about the ion gyrofrequency to above all the natural frequencies.