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.

Introduction

Quasars are unique among objects of the universe in the observable span of their continuous spectra. In some cases, the same quasar can be seen with existing instruments at wavelengths from X-rays to radio, including everything in between. The quasar continuous spectrum is deceptive. Order-of-magnitude agreement over all wavelengths, from tens of centimeters to fractions of an angstrom, covering a range of >1011 in frequency, can be obtained by fitting a single power law spectrum, of form fν∞να, where a is ˜ – 1. It is tempting in the face of such a result to attribute all parts of the spectrum to related mechanisms. As has become very clear from more careful examination of spectra, that is not valid. Different components of the continuous spectra are produced by drastically different mechanisms, and there are sometimes no physical relations among these mechanisms. It is nevertheless assumed that all of these mechanisms are basically set in motion by a single underlying engine, such as gravitational accretion, but the radiation which comes out represents many ways of transforming gravitational to radiative energy. The greatest success of the intensive observational effort has been to show the exceptional similarities among spectroscopic properties for quasars covering a factor approaching 107 in luminosity. This is the single key fact to be explained by theoretical models of quasars. Whatever processes control the radiation must be capable of scaling over this range of energy release without fundamentally changing character.

Perturbations in the orbit of a single particle

In plasma physics it is traditional to use a mixture of the collective-medium approach and the single-particle approach in treating various processes. These two approaches can complement each other in providing physical insight. A relevant example is in the treatment of Landau damping. In the collective-medium approach this is treated by allowing the frequency to have an imaginary part which is determined by the anti-hermitian part of the response tensor (§2.5). In the single-particle approach, one calculates Cerenkov emission by a single particle, relates absorption to emission using the Einstein coefficients (or Kirchhoff's law) and hence finds that Landau damping is the absorption process corresponding to Cerenkov emission by thermal electrons (§6.3). Although it is possible in principle to use the collective-medium approach to treat spontaneous, e.g. Cerenkov, emission it is cumbersome to do so and it is difficult to build up a physical understanding using this approach. Thus spontaneous emission is treated using a single-particle approach. This approach may be extended to calculate the response tensors (§5.5) which are the basis of the collective-medium approach.

In this Chapter we discuss several aspects of the single-particle approach applied to particle-wave interactions. The object is to identify the physical processes involved in reactive and kinetic instabilities, and in their saturation.

If we look out from any point in an isotropic, homogeneous Universe it must appear to expand in the same way. This expansion, mentioned in Section 21, has the form H(t)x with H = Ṙ/R when the coordinate system is chosen to coincide with a point of zero systematic motion.

To see this pretend to be a ‘fundamental observer’ at point O, moving with the average flow. Looking out at time t to an arbitrary point P along the position vector x = OP, you see the velocity v(x, t) of P relative to you. Another fundamental observer is moving with the flow at O′ and his velocity relative to you is v(s, t) where s = OO′. (The reader may find it helpful to draw a diagram.) He measures the velocity of the same point P, located at x′ = x - s in his coordinate system, and finds it to be v′(x′, t) = v′(x - s, t) = v(x, t) - v(s, t). Now, since the Universe is homogeneous and isotropic, v′ must be the same function of x′ and t that v is of x and t. Therefore, v(x - s, t) = v(x, t) - v(s, t). By inspection the solution of this functional equation is that the velocity is a linear function of position, so it has the form v = f(t)x = Ẉ.

Any process, such as mass segregation, mass loss, or core–halo instability which redistributes the density of a gravitating system will also produce orbit segregation. Orbit segregation is caused primarily by changes in the mean gravitational field. It affects orbits according to their eccentricity.

In a globular cluster, for example, the timescale for mass segregation to redistribute density falls between the dynamical crossing timescale and the relaxation timescale for stars of average mass. Therefore orbits of average or light stars undergo secular changes governed by the slowly changing mean field. (We are isolating changes in the mean field which, averaged over orbits, lead to orbit segregation, so we now ignore close encounters and dynamical friction for the test stars.) Suppose, for clarity, we compare the extremes of eccentricity: circular and radial orbits. Let the cluster's density increase toward the center, as usual. If the cluster were in stationary equilibrium, stars in circular orbits would just go around and around, and those in radial orbits would just go in and out through the center, in equilibrium with the mean field. Stars with intermediate eccentricities, but constant angular momentum, would generally follow open orbits with constant amplitude.

Now suppose part of the cluster begins to contract slowly compared with the timescale for free fall.

Having introduced the basic descriptions of gravitational many-body physics, it is time to attend to some astronomical applications. Many basic processes still remain to be explored, but they are best introduced in their astrophysical contexts.

Occasionally in Part 1 I have mentioned that infinite homogeneous systems – indeed all homogeneous gravitational systems – are anomalies, idealizations which do not exist in nature. However convenient they may be for mathematical analyses they are too unstable to represent anything we see, except as a first approximation. Newton recognized this and described it qualitatively in his letter to Bentley. Jeans (see Section 15) formulated it quantitatively for a static universe. Important complications, and a new richness of results, occur in the expanding Universe. This is a fundamental problem, for it begins to describe how matter is distributed around us on the largest scales.

Some basic aspects of many-body theory remain to be developed in future sections where they will be connected more closely to their astronomical applications. Here I give several extensions which can be worked out as problems by the reader, or can be used to enter the literature. Other suggestions for practice problems are sprinkled lightly throughout the text.

The point mass approximation

So far we have usually supposed that gravitating bodies, from dust grains to galaxies, can be treated as point masses. This is clearly an idealization. Geometric collisions make this approximation fail; so does tidal disruption. A third reason for failure is angular momentum transfer from the orbit to the spin of an object. Consider two solid ellipsoids, each of mass m and semimajor axes a, b where a > b. Show that, as they pass by one another at average distance r and velocity v, each acquires an angular momentum of order G(mae)2/r2v, where e2 = 1 - (b/a)2. How close do they have to pass to transfer an appreciable fraction of their orbital energy into rotational energy? Is this a plausible process for stars? For galaxies? What happens if the masses are not approximated as solid bodies? What residual internal circulation would result? Is there a net vorticity? How does the transferred energy change the size of the galaxy? (See Harrison, MNRAS, 154, 167, 1971.)