Introduction

This chapter deals with the dynamics of electrically conducting fluids, usually called plasmas. The emphasis is on concepts that are of direct relevance to astrophysics. The basic ideas that are covered here will be used in several chapters of Vols. II and III.

The Mean Field and Collisions in Plasma

Several astrophysical systems are made of fully ionised gases, usually called plasmas, in which electromagnetic interactions between the constituents play a vital role. In this chapter, we treat fully ionised plasma as having two components: electrons of charge -e and positive ions of charge Ze. We begin by discussing several assumptions and approximations that will be inherent in our description.

The effective use of statistical methods in the study of neutral gases relies on the fact that the interaction between constituent particles are of short range and random in nature. This assumes that the gas is sufficiently rarefied and can be treated as ideal. To treat a plasma as a fluid, it is necessary to impose a corresponding condition that, however, has some important conceptual differences from that of neutral gases.

The condition for a plasma to be treated as ideal requires that the random kinetic energy of the particles be large compared with the electrostatic potential energy between two particles, that is, kBT ≫ e2/r ∼ e2nfrac13;, where T is the temperature, n is the number density of particles, and r ≃ n-⅓ is the mean interparticle distance.

Introduction

This chapter begins with a rapid overview of concepts from special relativity and develops the four-vector notation. Several aspects of electrodynamics are then introduced using special relativity and four-vector notation. Finally, this formalism is used to discuss some aspects of principles of optics that are relevant to astronomy. The concept of distribution functions, developed in Section 3.6, will be used extensively in several later chapters. This chapter will also be needed in the development of radiative processes (Chaps. 4 and 6), general relativity (Chap. 11), and in the study of cosmology in Vol. III.

The Principles of Special Relativity

The description of physical processes requires the specification of spatial and temporal coordinates of events that may be combined into a single entity characterised by the four numbers xi = (t, x). Throughout this chapter, the Latin indices a, b, …, i, j, etc., run over 0, 1, 2, and 3, with the 0 index denoting the ‘fourth’ dimension and 1, 2, and 3 denoting the standard space dimensions. The actual values of xi, attributed to any given event P, will depend on the specific coordinate system which is used. We pay special attention to a subset of all possible coordinate systems called inertial coordinate systems. Such coordinate systems are defined by the property that a material particle, far removed from all external influences, will move with uniform velocity in such systems.

Introduction

This chapter develops several basic ideas of dynamics, emphasizing general principles that are useful in classifying the behaviour of dynamical systems. The reader is assumed to be familiar with elementary concepts of classical mechanics. Concepts developed here will be needed in the study of special relativity (Chap. 3), statistical mechanics (Chap. 5), general relativity (Chap. 11), Sun and solar system (Vol. II), binary stars (Vol. II), and galactic dynamics (Vol. III).

Time Evolution of Dynamical Systems

Many systems encountered in nature can be described by a finite set of N real variables [q1(t), q2(t), …, qi(t), …, qN(t)] t h a t evolve in time. For example, in the study of two stars, moving under the influence of their mutual gravitational force, we are interested in the positions of the stars as functions of time. The position of each star can be described by three coordinates (in three-dimensional space) so that the full system can be described by a total of six functions of time. The quantities qi(t) (with i = 1,2, …, N) are called dynamical variables; obviously, we are free to choose any other set of N independent, single-valued functions of qi as dynamical variables to describe the system, with the particular choice often dictated by mathematical convenience. The central problem of dynamics is related to determining the time dependence of qi(t) and studying the general characteristics of motion.

Introduction

This chapter introduces several concepts from nuclear physics that are used in different areas of astrophysics, especially in the study of stellar evolution. The ideas developed here will be used in the theory of stellar structure and evolution (Vol. II) and in cosmology (Vol. III).

Nuclear Structure

The nuclei of atoms contain protons and neutrons that differ – most importantly – in their electrical charge; the proton is positively charged whereas the neutron is neutral. They are both fermions with spin half and their masses are almost equal (mpc2 = 938.3 MeV for a proton and mnc2 = 939.6 MeV foraneutron). Protons, being positively charged, will repel each other and a collection of protons will fly apart because of this repulsion. They are kept together along with neutrons in a nucleus because of the existence of a stronger, attractive, nuclear force. As far as nuclear force is concerned, protons and neutrons behave in very similar manner. We may say that the nuclear force provides the necessary binding energy to keep the atomic nucleus together.

A fundamental many-body theory describing the nuclear force that acts between the nucleons is not yet fully developed. Hence it is not possible to study aspects involving nuclear forces from first principles. In what follows we rely extensively on phenomenological facts and experimental results.

Different atomic nuclei are characterised by different values of the binding energy per nucleon, Ēb.

During the past decade or so, theoretical astrophysics has emerged as one of the most active research areas in physics. This advance has also been reflected in the greater interdisciplinary nature of research that is being carried out in this area in the recent years. As a result, those who are learning theoretical astrophysics with the aim of making a research career in this subject need to assimilate considerable amount of concepts and techniques, in different areas of astrophysics, in a short period of time. Every area of theoretical astrophysics, of course, has excellent textbooks that allow the reader to master that particular area in a well-defined way. Most of these textbooks, however, are written in a traditional style, focussing on one area of astrophysics (say stellar evolution, galactic dynamics, radiative processes, cosmology etc.) Because different authors have different perspectives regarding their subject matter it is not very easy for a student to understand the key unifying principles behind several different astrophysical phenomena by studying a plethora of separate textbooks, as they do not link up together as a series of core books in theoretical astrophysics covering everything which a student would need. A few books, which do cover the whole of astrophysics, deal with the subject at a rather elementary (“first course”) level.

Introduction

We use the concepts developed in some of the previous chapters (especially Chap. 5) to study the physical processes in fluids in this chapter. The emphasis is on aspects of fluid mechanics that are of relevance in astrophysics. This chapter will be needed in Chaps. 9 and 10 and in several sections of Vols. II and III.

Molecular Collisions and Evolution of the Distribution Function

At the microscopic level, a fluid can be thought of as a collection of molecules. Ignoring the internal structure of the molecules, we can specify the state of any molecule by giving its position x and momentum p. Let dN = f(x, p, t) d3xd3p denote the number of molecules in a phase volume d3xd3p at time t. We are interested in the form and evolution of this distribution function.

The distribution function changes because of two kinds of physical processes. Macroscopic force fields, such as the gravitational field, can exert forces on the molecules and influence their motion. Such a force, Fsm(x, t) ≡ -ΔUsm (x, t), will vary smoothly over the microscopic scales and can be derived from a suitable potential Usm. A particular molecule will also experience the force Fcoll that is due to collision with another molecule whenever it is close to another molecule. At a fundamental level, collisions arise because of two molecules interacting by means of the intermolecular force.

Introduction

This chapter discusses physical systems involving large number of particles, conventionally called statistical mechanics. Because these concepts are used in several later chapters, a complete and pedagogical discussion is presented here. We begin with the equilibrium statistical mechanics of classical systems and derive macroscopic thermodynamics from statistical mechanics. In the second half of the chapter we deal with quantum statistical mechanics, including the physics of Fermi gas. This chapter depends on the concepts developed in the previous three chapters and will be needed for Chaps. 6 (radiative processes), 8 (neutral fluids), 9 (plasma physics) as well as in the study of stellar evolution and stellar remnants (Vol. II) and the thermal history of the universe (Vol. III).

Operational Basis of Statistical Mechanics

The dynamical evolution of any system can be studied most conveniently by use of the concept of phase space developed in Chap. 2, Section 2.2. For a system of N-point particles the phase space will be 6N dimensional. Given the initial state of the system as a point in the phase space, the dynamical evolution of the system traces out a one-dimensional curve in the 6N-dimensional phase space, starting from the given point. If the equations of motion for the system are solved exactly (with the given initial conditions), then this curve in the phase space can be determined exactly.

For any realistic N-particle system, this task is impossible even with the best computers available today.

Introduction

The subject of astrophysics involves the application of the laws of physics to large macroscopic systems in order to understand their behaviour and predict new phenomena. This approach is similar in spirit to the application of the laws of physics in the study of, say, condensed-matter phenomena, except for the following three significant differences:

1. (1) We have far less control over the external conditions and parameters in astrophysics than in, say, condensed-matter physics. It is not possible to study systems under controlled conditions so that certain physical processes dominate the behaviour. Identifying the causes of various observed phenomena in astrophysics will require far greater reliance on statistical arguments than in laboratory physics.

2. (2) The astrophysical systems of interest span a wide range of parameter space and require inputs from several different branches of physics. Typically, the densities can vary from 10-25 gm cm-3 (interstellar medium) to 1015 gm cm-3 (neutron stars); temperatures from 2.7 K (microwave background radiation) to 109 K (accreting x-ray sources) or even to 1015 K (early universe); radiation from wavelengths of meters (radio waves) to fractions of angstroms (hard gamma rays); typical speeds of particles can go up to 0.99c (relativistic jets). Clearly we require inputs from quantum-mechanical and relativistic regimes as well as from more familiar classical physics.

3. (3) […]

This book is in a sense a sequel to my previous book Nonlinear Magnetohydrodynamics, which contained a chapter on magnetic reconnection. Judging from many discussions it appeared that it was this chapter that was particularly appreciated. The plan to write a full monograph on this topic actually took a concrete shape during a stay at the National Institute for Fusion Science at Nagoya, where I found the time to work out the basic conception of the book. It became clear that resistive theory, to which most of the previous work was restricted, including that chapter of my previous book, covers only a particular aspect of this multifaceted subject and not even the most interesting one, in view of the various applications, both in fusion plasma devices and in astrophysical plasmas, where collisionless effects tend to dominate over resistivity.

While resistive reconnection theory had reached a certain level of maturity and completion about a decade ago (few theories are really complete before becoming obsolete), the understanding of collisionless reconnection processes has shown a rapid development during the past five years or so. The book therefore consists of two main parts, chapters 3–5 deal with resistive theory, while chapters 6–8 give an overview of the present understanding of collisionless reconnection processes. I mainly emphasize the reconnection mechanisms, which operate under the different plasma conditions, to explain the apparent paradox that formally very weak effects in Ohm's law account for the rapid dynamic time-scales suggested by the observations.

Since the early 1950s, when magnetohydrodynamics – MHD in short – became an established theory and along with it the concept of a “frozenin” magnetic field within an electrically conducting fluid, the problem of how magnetic field energy could be released in such a fluid has been generally acknowledged. In the early days the major impetus came from solar physics. Estimates readily showed that the energies associated with eruptive processes, notably flares, can only be stored in the coronal magnetic field, all other energy sources being by far too weak. On the other hand the high temperature in the corona, which makes the coronal plasma a particularly good electrical conductor, appeared to preclude any fast magnetic change involving diffusion. For a coronal electron temperature Te ～ 106 K the magnetic diffusivity is η ～ 104 cm2/s, hence field diffusion in a region of diameter L ～ 104 km as typically involved in a flare would require a time-scale τη = L2/η ～ 1014 s, whereas the observed flash phase of a flare takes less than ～ 103 s.

It had, however, soon been realized that the discrepancy is not quite as bad as this. Contrary to magnetic diffusion in a solid conductor, a fluid is stirred into motion by the change of the magnetic field. As it carries along the frozen-in field, it may generate steep field gradients typically located in sheet-like structures, and hence lead to much shorter diffusion times.