The solar wind is the extension of the solar corona to very large heliocentric distances. As we shall see later in this chapter the solar wind exists because of the huge pressure difference between the hot plasma at the base of the corona and the interstellar medium.

The existence of a continuous solar wind was first suggested by Ludwig Biermann based on his studies of the acceleration of plasma structures in comet tails. The detailed mathematical theory of the solar wind was put forward by Eugene Parker. The solar wind was first sporadically detected by the Soviet space probes Lunik 2 and 3, but the first continuous observation of the solar wind was made with the Mariner 2 spacecraft.

In this chapter we shall describe the “classic” theory of the solar wind, which is based on the fluid approximation of coronal and interplanetary plasmas.

Hydrostatic Equilibrium — It Does Not Work

The simplest theoretical description of the solar corona is based on the assumption of a spherically symmetric, steady-state hydrostatic corona. Single-fluid equations can be applied since the gas is assumed to be a fully ionized, quasineutral proton—electron plasma. The effects of magnetic field and heat conduction are neglected in this simple approximation.

Maxwell's velocity distribution function together with Clausius's mean free path concept were the key elements that made it possible to develop connections between the motions of microscopic molecules (molecules are defined in a very broad sense here, referring to any neutral or charged particle composing the gas) and the macroscopic properties of gases (observable by the methods of classical physics).

In this chapter we introduce some fundamental concepts of kinetic theory. For more details we refer the reader to books on classical kinetic theory (cf. Gombosi 1994).

Collisions

First, we introduce several statistical quantities such as the mean free path, collision frequency, collision rate, collision cross section, and differential cross section. We shall also see that these quantities are closely related to each other and to other fundamental molecular quantities. We will use very simple physical models to emphasize the basic concepts behind these new statistical quantities.

Mean Free Path

The free path is the distance traveled by a molecule between two successive collisions. The mean free path is the average distance between two successive collisions of a single molecule.

Consider a single molecule with velocity v. Assume that this particle suffers a collision with another molecule at a distance of s = 0. Let Ppath(s) denote the probability that this molecule survives a distance s without suffering a second collision.

Understanding the fundamental properties of the energetic particle population in the heliosphere is very important for two reasons:

1. These particles represent considerable hazard for both humans and radiation-sensitive systems in space, because they can penetrate through large amounts of shielding material.

2. They carry information about the large-scale properties of the heliosphere and the galaxy.

High-energy cosmic ray particles carry a large amount of kinetic energy. The deposition of this energy can cause permanent effects in the material through which the cosmic ray particle passes. In the case of biological materials or miniature electronic circuits, these effects can be very serious. In order to provide adequate shielding for radiation-sensitive systems, we need to know the basic properties of the high-energy particle radiation, including its elemental composition, energy spectrum, and temporal variations.

A significant portion of our present knowledge about the global structure of the heliosphere comes from energetic particle observations. These particles travel through space at velocities considerably higher than the characteristic velocities of the local plasma population. Because the propagation of the energetic particles is greatly affected by various physical properties of the medium, energetic particles sample regions of the heliosphere and the galaxy that are currently not accessible to spacecraft.

Nature produces many different kinds of waves and oscillations. In general, these periodic phenomena are very different from each other. However, certain physical and mathematical properties are common to small-amplitude waves almost irrespective of their nature. For instance, all small-amplitude waves can be characterized by dispersion relationhips and they transport physical quantities with the group velocity of the wave.

In this chapter we will examine some fundamental proprties of small-amplitude waves in neutral and conducting fluids. We will see that although neutral fluids exhibit a relatively small number of fundamental wave phenomena, conducting fluids (especially when they are magnetized) are a very fertile medium for the generation of a huge variety of plasma waves.

Here we shall concentrate on small-amplitude waves, when the wave equations can be linearized. This does not mean that nonlinear phenomena are unimportant — they are just too complicated for this introductory text. Also, we will limit our discussions to single species gases (or in the case of plasmas, to single ion plasmas). The results can be generalized to multispecies plasmas, when needed.

Linearized Fluid Equations

First of all, let us consider the linearized version of the ideal MHD equations. We choose to use the MHD equations, because mathematically the Euler equations represent a subset of these equations (one just has to set B to zero everywhere at all times). Let us assume that we have a solution of the full equation set, that is, ρm0, u0, p0, and B0 represent a steady-state solution of Eqs. (4.89).

In this chapter we will briefly consider some of the basic theoretical tools used in describing the transport of superthermal particles. By superthermal particles we mean a very small fraction of the total particle population with energies far exceeding the average thermal energy. These superthermal particles contribute negligibly to the particle density and bulk velocity (due to their very small number compared to the total number of particles), but in some cases they may represent a significant contribution to the pressure and heat flow.

We will consider the basic transport equations describing two kinds of superthermal particles: energetic solar particles and photoelectrons. Since our goal is to provide an introduction to the theoretical tools of space physics, we will constrain our derivations to the most fundamental processes. More sophisticated treatments can be found in the literature.

Transport of Energetic Particles

As in most cases, we start from the Boltzmann equation describing the evolution of the particle distribution function. The main difference this time is that because superthermal particles can be relativistic, we need to derive a transport equation that is valid for relativistic particles as well. To achieve this we use the form of the Boltzmann equation given by Eq. (2.36), where the variables of the distribution function are time, location, and full (inertial) velocity.

In view of the complexity of the various kinetic models discussed near the beginning of the previous chapter and of the vast array of cross sections and other atomic data required for their implementation, there is a great need for a more basic theory. Such a theory will have a restricted region of validity but can be used with confidence for the interpretation of plasma spectroscopic experiments performed under appropriate conditions. This basic theory is provided by equilibrium statistical mechanics or thermodynamics. It, together with the various conditions for its validity, is the subject of the present chapter. Before introducing this new topic, a second role for the more basic theory should be emphasized, namely its service in testing kinetic models under conditions for which thermodynamic equilibrium should hold.

The practical goal of any of the kinetic models and of the local thermodynamic equilibrium (LTE) relations introduced here is the calculation of populations of the various states of atoms or ions and of the free electron density for specified temperature(s), pressure or mass density, and assumed chemical composition. As already discussed in the preceding chapter, one deals with local, and usually also transient, equilibria, because strict thermodynamic equilibrium would either require an unbounded, spatially and temporally homogeneous plasma or a plasma enclosed in an ideal blackbody hohlraum (cavity).

In the preceding chapters, the basic radiative and collisional processes governing local radiative properties of a plasma were introduced; and their quantitative evaluation or acquisition was discussed to help the reader in the selection of data needed for analysis or prediction of a spectrum. We also learned about various kinetic or thermodynamic models designed for comprehensive descriptions of level populations.

There are two reasons for this seemingly all-encompassing body of theory and basic data to be insufficient, nevertheless, for both analysis and predictions. First, one generally cannot measure local values of the plasma emission, but must infer them from some distance and averaged over the various contributing volume elements. Second, and even more fundamentally, radiative processes also influence level populations so that the emission or absorption in one location depends on the radiation flux coming from the rest of the plasma.

Therefore a self-consistent treatment of radiation transport and level populations is necessary, requiring in general a nonlocal and nonlinear theory. Such theory has been developed over many years, mainly by astronomers and astrophysicists. Much progress has been made after the two basic treatises (Chandrasekhar 1950 and Sobolev 1963) were written, mostly by computational methods (see, e.g., Athay 1972, Kalkofen 1984, Crivellari, Hubeny and Hummer 1991), but also through more or less analytic models (see, e.g., Thomas and Athay 1961, Jefferies 1968, Ivanov 1973).

Depending on various plasma conditions, such as size, composition, densities and temperatures, the electron density and related quantities can be measured using a number of techniques. Of these, Langmuir probes (Tonks and Langmuir 1929, Hutchinson 1987) provided the first means to infer local values of the electron density, mostly at relatively low densities. Much more recently, Thomson scattering of laser light has become a method of choice for localized electron density measurements (Kunze 1968, Evans and Katzenstein 1969, DeSilva and Goldenbaum 1970, Sheffield 1975) over a range of about 1011 to 1021 cm−3. Then there are the usually inherently most accurate interferometric techniques, mostly laser-based as well (Jahoda and Sawyer 1971, Hauer and Baldis 1988), which span a similar range, recently extended to Ne ≈ 3 × 1021 cm−3 (DaSilva et al. 1995), but provide localized values of the electron density only indirectly. Using two or more wavelengths, one can, however, separate the free-electron contribution (1.37) to the refractive index from any boundstate contribution (2.103). In single-species, partially ionized plasmas, this boundstate contribution is a direct measure of the neutral atom density. Interferometric local values of the density can, in principle, be determined by methods analogous to those discussed for emission or absorption measurements in section 8.5.

In a fully ionized plasma, in which all bound electrons have been removed from their original atoms or molecules, there is besides the continuous spectrum no line emission or absorption, except possibly for features related to plasma resonances or waves (Bekefi 1966, Stix 1992, Swanson 1989). These often nonthermal features are usually at such low frequencies that they do not obscure or interfere with atomic radiation, the subject of principal interest here. Since atoms and incompletely stripped ions possess a continuous spectrum, besides the discrete spectrum providing the pairs of states involved in line radiation, continuous emission and absorption spectra underly and accompany the line spectra discussed in the preceding chapters. These processes are not only important as background to line emission, but also because continuum intensities can provide relatively direct measures of electron density and temperature (see sections 10.2 and 11.4, respectively).

With the usual convention for the energies of bound states as being negative relative to those of zero kinetic energy electrons at large distances from the nucleus of any isolated atom or ion, one might infer that all positive energy states belong to the continuous spectrum. In practice, this is an oversimplification, because there are states corresponding to the excitation of two or more bound electrons which are almost discrete.

Observable intensities of spectral lines and continua depend just as strongly on the appropriate level populations as on the transition probabilities and related quantities discussed in chapters 2 and 3, and on the radiative transfer processes to be discussed in chapter 8. The level population and transfer problems are not really separable. One must therefore strive for internal consistency, following the example of astrophysicists (see, e.g., Mihalas 1978). Nevertheless, it is reasonable to discuss first the various population and depopulation processes as if these interconnections were not very important. Often only processes affecting discrete states that are not very close to the corresponding continuum limits need to be evaluated explicitly. This important simplification arises because the rates of collisional processes controlling the relative populations of highly excited states tend to be very large, thus facilitating the establishment of so-called partial local thermodynamic equilibrium (PLTE, see sections 7.1 and 7.6). However, the lower excited level populations often need explicit evaluation, especially because these levels tend to give rise to the strongest lines. Note also that electron collisional rates usually, but not always, dominate because of their high velocities and of their energy-dependent cross sections that are similar to those for ions.

The quantitative description of atomic radiation reviewed in the previous chapter requires knowledge of energy levels and of level populations, which will be the principal subjects of chapters 6 and 7, and of oscillator strengths or of the closely related line strengths. Their calculation and measurement are the main topics of the present chapter. As to energy levels and wavelengths, a large body of high quality empirical data is available (Moore 1949-1958, Martin, Zalubas and Hagen 1978, Cowan 1981, Kelly 1987a and b, Bashkin and Stoner 1975, 1978, 1981), and atomic structure calculations (Cowan 1981, Sobel'man 1992) are usually of an accuracy that is sufficient for most plasma spectroscopy applications. Our knowledge of oscillator strengths, and therefore also of transition probabilities for spontaneous transitions and line strengths, has also greatly improved since the predecessor of this monograph (Griem 1964) was written. These advances are the result of improved experiments and computations and of critical evaluations of data (Wiese, Smith and Glennon 1966, Wiese, Smith and Miles 1969, Fuhr and Wiese 1995, Martin, Fuhr and Wiese 1988, Fuhr, Martin and Wiese 1988). Especially complete and accurate data are now available for atoms and ions of carbon, nitrogen and oxygen (Wiese, Fuhr and Deters 1995).

Although most of the electromagnetic radiation from many natural and laboratory plasmas is atomic in origin and therefore subject to quantum effects, it remains useful to introduce some of the basic radiative processes via classical theory. Other important foundations of plasma spectroscopy are atomic physics and plasma physics, especially the statistical mechanics of ionized gases. Generally, the basic theory is well established in these parent disciplines. However, the large variety of processes contributing to emission or absorption spectra often requires more or less drastic simplifications in their theoretical description or computer modeling. Critical experiments are playing an essential role in checking the reliability of various models and in delineating the region of their applicability.

Plasma spectroscopy, although being a highly specialized subfield, is at the same time a very interdisciplinary science. It not only owes its origins largely to astronomy, but also returns to astronomy and astrophysics methods of analysis of spectra and a multitude of basic data, which have both been subjected to experimental scrutiny. The state of stellar plasmas is significantly influenced by radiation, and the latter is more or less controlled by radiative energy transfer. Internally consistent treatments of the states of matter and radiation first developed by astronomers are now also becoming important for the description of plasma experiments.

In addition to the most general applications of spectroscopic methods to density and temperature measurements, which were discussed in the two preceding chapters, there are numerous special applications. Some of these will be discussed in this concluding chapter, without any prejudice against any well-established or recently developed methods which are omitted. If there is any common thread, it is in the strong role played by atomic collision theory and by the physics of atoms and ions in electric and magnetic fields. As in the other chapters, no attempt will be made to describe the often very sophisticated instrumentation or other experimental details, which the interested reader should be able to find with the help of the references in the original papers.

The first two special applications to be discussed, namely charge exchange recombination and beam emission spectroscopy, obviate the essential difficulty in emission or absorption spectroscopy in obtaining spatial resolution along the line of sight. This is accomplished by injecting heating or diagnostic neutral beams into magnetically confined plasmas either to preferentially populate some excited states of plasma ions by charge exchange recombination, as discussed in section 6.5, or by having the atoms in the beam ionized and excited by the plasma electrons, as discussed in sections 6.2 and 6.3, or even by protons and other plasma ions (Mandl et al. 1993), see section 6.5.