Introductory remarks

We now begin the study of plasmas, which are gases in which the constituent particles are charged. In a gas made up of neutral particles, two particles are assumed to interact only when they collide, i.e. are physically very close. Between collisions, the neutral particles move along straight lines. In contrast, the particles in a plasma always interact with each other through long-range electromagnetic interactions and the trajectories of individual particles can be quite complicated. Before investigating how collections of charged particles behave, it is worthwhile developing some ideas about the motions of individual charged particles in electromagnetic fields. This topic is referred as the plasma orbit theory and often turns out to be very useful in handling problems involving plasmas. While developing the theory of neutral gases in Chapters 2–3, it was not necessary to pay much attention to motions of individual neutral particles, as these motions are quite simple. After discussing motions of individual plasma particles in this chapter, we shall, however, follow a course of development roughly similar to that which we followed for neutral fluids. In the next chapter, we shall begin developing theoretical techniques for treating plasmas as collections of charged particles and eventually we shall end up with continuum models in Chapters 14–16.

As soon as we start discussing electromagnetic quantities, we face the vexing question of choosing units.

Thermodynamic properties of a perfect gas

Most of the problems studied in Chapters 4 and 5 did not involve considerations of compressibility, the only exception being §4.4 where the static equilibrium of compressible fluids was considered. We now wish to study the dynamics of compressible fluids. We saw in §4.7 that the irrotational flow of an incompressible fluid around an object gives rise to the Laplace equation, which is an elliptic partial differential equation. The similar problem of high-speed flow of a compressible fluid around an object can give rise to a hyperbolic partial differential equation, provided the flow speed is larger than the sound speed. In other words, the mathematical character of the equations governing high-speed compressible flows can be quite different from that of the equations governing incompressible flows (although we begin from the same hydrodynamic equations!), and consequently the solutions can also be of profoundly different nature. We give here only a brief introduction to gas dynamics, which is the branch of hydrodynamics dealing with compressible flows. Even the subject of supersonic flows past solid objects just mentioned above, which leads to a two-dimensional problem, is not treated in this elementary introduction. See Landau and Lifshitz (1987, Chapter XII) or Liepmann and Roshko (1957, Chapter 8) for a discussion of this subject. We restrict ourselves to a discussion of one-dimensional gas dynamics problems only.

The mathematical analysis of compressible fluids becomes more manageable if we assume the fluid to behave as a perfect gas.

The fundamental equations

We discussed in the previous chapter how the one-fluid or the MHD model of the plasma can be developed starting from microscopic considerations. It was not possible to give as thorough or as systematic a presentation of the subject as we did in Chapter 3, where the hydrodynamic model for neutral fluids was developed from the microscopic theory. We have not rigorously established the conditions under which the one-fluid model of a plasma holds. We saw in Chapter 3 that frequent collisions make a neutral gas behave like a continuous fluid. Collisions certainly help in establishing fluidlike behaviour. It was, however, mentioned in §11.7 that a strong magnetic field in a plasma can also keep charged particles confined within local regions for sufficient time, thereby giving rise to fluidlike behaviour even in the absence of collisions.

Between the microscopic model based on distribution functions and the macroscopic one-fluid model, there exists the intermediate twofluid model of the plasma discussed in Chapters 11–13. This was referred to in Table 1.1 as the 2½ level. When we consider phenomena in which electrons and ions respond differently (such as the propagation of electromagnetic waves through a plasma), the two-fluid model has to be applied rather than the MHD model. The MHD model is applicable only when charge separation is negligible. The condition for it is that the length scales should be larger than the Debye length and the time scales larger than the inverse of plasma frequency.

Introduction

In this chapter, we shall mainly study what happens when a plasma in thermodynamic equilibrium is slightly disturbed. If the effect of collisions can be neglected in studying the evolution of the disturbance, then we call it a collisionless process. In stellar dynamics, one often studies systems with relaxation times larger than the age of the Universe so that collisions have never been important in the system. Here, however, we shall consider plasmas which are close to thermodynamic equilibrium presumably as a result of collisions. But collisions happen to be unimportant in the particular processes we are going to look at.

If a plasma is close to thermodynamic equilibrium, then hydrodynamic models may be applicable as we pointed out in §11.5. We shall mostly use the two-fluid model which was developed in the previous chapter. Only §12.4 will provide an example of how calculations can be done with the help of the Vlasov equation. We shall see that the Vlasov equation can give rise to a conclusion different from what we get by using the two-fluid model due to rather subtle reasons.

We shall mainly consider waves of high frequencies (so that collisions are unlikely to occur during a period of the wave), as such waves are the most important examples of collisionless processes. For low-frequency processes (with frequency ≪ the plasma frequency ωp defined in §12.2), we shall see in later chapters that charge separation can be neglected and the plasma may be regarded as a single fluid.

Fluids and plasmas in the astrophysical context

When a beginning student takes a brief look at an elementary textbook on fluid mechanics and at an elementary textbook on plasma physics, he or she probably forms the impression that these two subjects are very different from each other. Let us begin with some comments why we have decided to treat these two subjects together in this volume and why astrophysics students should learn about them.

We know that all substances are ultimately made up of atoms and molecules. Ordinary fluids like air or water are made up of molecules which are electrically neutral. By heating a gas to very high temperatures or by passing an electric discharge through it, we can break up a large number of molecules into positively charged ions and negatively charged electrons. Such a collection of ions and electrons is called a plasma, provided it satisfies certain conditions which we shall discuss later. Hence a plasma is nothing but a special kind of fluid in which the constituent particles are electrically charged.

When we watch a river flow, we normally do not think of interacting water molecules. Rather we perceive the river water as a continuous substance flowing smoothly as a result of the macroscopic forces acting on it. Engineers and meteorologists almost always deal with fluid flows which can be adequately studied by modelling the fluid as a continuum governed by a set of macroscopic equations.

This lecture series provides an overview of modern physical cosmology with an emphasis on nuclear arguments and their role in the larger framework. In particular, the current situation on the age of the universe and the Hubble constant are reviewed and shown now to be in reasonable agreement once realistic systematic uncertainties are included in the estimates. Big bang nucleosynthesis is mentioned as one of the pillars of the big bang along with the microwave background radiation. It is shown that the big bang nucleosynthesis constraints on the cosmological baryon density, when compared with dynamical and gravitational lensing arguments, demonstrate that the bulk of the baryons are dark and also that the bulk of the matter in the universe is non–baryonic. The recent extragalactic deuterium observations as well as the other light element abundances are examined in detail. Comparison of nucleosynthesis baryonic density arguments with other baryon density arguments is made.

Introduction

Modern physical cosmology has entered a “golden period” where a multitude of observations and experiments are guiding and constraining the theory in a heretofore unimagined manner. Many of these constraints involve nuclear physics arguments, so the interface with nuclear astrophysics is extemely active. This review opens with a discussion of the three pillar of the big bang: the Hubble expansion, the cosmic microwave background, and big bang nucleosynthesis (BBN).

This review concentrates on nucleosynthesis processes in general and their applications to massive stars and supernovae. A brief initial introduction is given to the physics in astrophysical plasmas which governs composition changes. We present the basic equations for thermonuclear reaction rates and nuclear reaction networks. The required nuclear physics input for reaction rates is discussed, i.e. cross sections for nuclear reactions, photodisintegrations, electron and positron captures, neutrino captures, inelastic neutrino scattering, and beta–decay half–lives. We examine especially the present state of uncertainties in predicting thermonuclear reaction rates, while the status of experiments is discussed by others in this volume (see M. Wiescher). It follows a brief review of hydrostatic burning stages in stellar evolution before discussing the fate of massive stars, i.e. the nucleosynthesis in type II supernova explosions (SNe II). Except for SNe la, which are explained by exploding white dwarfs in binary stellar systems (which will not be discussed here), all other supernova types seem to be linked to the gravitational collapse of massive stars (M>8M⊙) at the end of their hydrostatic evolution. SN1987A, the first type II supernova for which the progenitor star was known, is used as an example for nucleosynthesis calculations. Finally, we discuss the production of heavy elements in the r–process up to Th and U and its possible connection to supernovae.

This chapter is a review of the background and status of several current problems of interest concerning cosmic rays of very high energy and related signals of photons and neutrinos.

Introduction

The steeply falling spectrum of cosmic rays extends over many orders of magnitude with only three notable features:

1. (a) The flattened portion below 10 GeV that varies in inverse correlation with solar activity,

2. (b) The “knee” of the spectrum between 1015 and 1016 eV, and

3. (c) the “ankle” around 1019 eV.

4. For my discussion here I will divide the spectrum into three energy regions that are related to the two high–energy features, the knee and the ankle: I: E < 1014 eV, II: 1014 < E < 1018 eV and III: > 1018 eV.

In Region I (VHE) there are detailed measurements of primary cosmic rays made from detectors carried in balloons and on spacecraft. These observations, and related theoretical work on space plasma physics, form the basis of what might be called the standard model of origin of cosmic rays. Cosmic rays are accelerated by the first order Fermi mechanism at strong shocks driven by supernova remnants (SNR) in the disk of the galaxy. The ionized, accelerated nuclei then diffuse in the turbulent, magnetized plasma of the interstellar medium, eventually escaping into intergalactic space at a rate that depends on their energy.

The structure of neutron stars is discussed with a view to explore (1) the extent to which stringent constraints may be placed on the equation of state of dense matter by a comparison of calculations with the available data on some basic neutron star properties; and (2) some astrophysical consequences of the possible presence of strangeness, in the form of baryons, notably the Λ and Σ−, or as a Bose condensate, such as a K− condensate, or in the form of strange quarks.

Introduction

Almost every physical aspect of a neutron star tends to the extreme when compared to similar traits of other commonly observed objects in the universe. Stable matter containing A ∼ 1057 baryons and with a mass in the range of (1 − 2) M⊙ {M⊙ ≅ 2 × 1033 g) confined to a sphere of radius R ∼ 10 km (recall that R⊙ = 6.96 × 105 km) represents one of the densest forms of matter in the observable universe. Depending on the equation of state (EOS) of matter at the core of a neutron star, the central density could reach as high as (5 − 10)p0, where p0 ≅ 2.65 × 1014 g cm−3 (corresponding to a number density of n0 ≅ 0.16 fm−3) is the central mass density of heavy laboratory nuclei (compare this to P⊙= 1.4 g cm−3).