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.

The magnetosphere is the cosmic plasma laboratory nearest to the Earth, which is therefore accessible to detailed ground and in-situ observations. It is, loosely speaking, a magnetic cavity generated by the interaction of the solar wind with the Earth's dipole field, which shields the Earth from direct bombardment by high-energy particles. This shield is, however, rather leaky, allowing solar-wind plasma to penetrate into the magnetosphere, which gives rise to a variety of different phenomena, the most spectacular being the aurora. The leakiness is mainly due to large-scale reconnection processes occurring at the front and in the tail of the magnetosphere. These processes form the main topic of this chapter.

The magnetosphere has a complex onion-like structure consisting of various plasma layers of distinctly different properties separated by rather sharp boundary surfaces. In section 8.1 we give a brief overview of the main features and outline the mechanisms leading to this layered structure. For a more detailed introduction to magnetospheric physics see, e.g., Baumjohann & Treumann (1996).

Reconnection is believed to be the main mechanism responsible for the magnetic processes observed in the magnetosphere, commonly called geomagnetic activity. The basic model of magnetospheric reconnection and plasma convection has been proposed by Dungey (1961) and this is considered in section 8.2. Reconnection of the dipole field (which is essentially oriented northward) with a southward component of the interplanetary field opens the magnetic cavity. The reconnected field lines are swept along by the solar wind to the nightside, until the increasing magnetic tension leads to a second reconnection process in the tail, reclosing the dipole field lines, which then contract back toward the Earth.

It thus appears that a long-standing riddle has now been solved. Fast quasi-Alfvénic magnetic reconnection may occur under rather general conditions with a rate rather independent of the particular reconnection physics, both in high- and low-β plasmas. Ironically, the case of stationary resistive MHD, which has been regarded as the most natural framework of reconnection theory, does not allow fast merging. The pecularities of resistive reconnection have been the origin of the long controversy dividing the community into two camps, the adherents of the Petschek model and those of the Sweet-Parker model emphasizing current sheets. Actually, physical conditions for a stationary high-Lundquist number MHD model to apply are rarely satisfied, neither in nature nor in laboratory plasmas. Either the plasma is strongly resistive, which often implies relatively low S or, at large S-value, collisionless effects are more important than resistivity. In addition, the plasma behavior is usually highly nonstationary, sometimes fully turbulent, and such a system allows fast reconnection even in resistive MHD. It is true that the usual quasi-stationary 2D models for collisionless reconnection are also highly idealized, far from real plasma conditions. Real plasmas which tend to exhibit a whole maze of fluctuations, but these do not seem to control the reconnection rate.

A concept which had also polarized the community, the distinction between driven and spontaneous reconnection, has to a good deal lost its significance. It is true that in many cases an external driving agent can be identified.

In an electrically conducting magnetized plasma even slow motions do not, in general, preserve a smooth magnetic field, but give rise to sheetlike tangential field discontinuities, called current sheets. These are the natural loci of magnetic reconnection. In this chapter we consider the properties of current sheets and reconnection via current sheets in the traditional framework of resistive magnetohydrodynamics (MHD). This restriction is justified, since macroscopic current sheets mostly occur, when the reconnection process R in Ohm's law (2.1) is dissipative. Collisionless reconnection processes, which are treated in chapters 6 and 7, usually give rise to microcurrent sheets.

The chapter starts with a brief introduction to MHD theory, discussing the basic equations, magnetostatic equilibria, and linear MHD waves. In low-β plasmas it is often convenient to eliminate the fast compressional MHD wave by using a reduced set of equations. In section 3.2 we first consider the conditions under which a current sheet arises and how it is formed by rapid thinning, which continues until finite resistivity leads to a stationary sheet configuration. Then the structure of a resistive current sheet, called a Sweet–Parker sheet (Sweet, 1958; Parker, 1963), is analyzed. While the global properties of such sheets follow from the basic conservation laws, the detailed structure requires a more specific analysis. Section 3.3 deals with the role of current sheets as centers of reconnection in a magnetic configuration. Syrovatskii (1971) has developed a simple and elegant theory of current sheets which captures many features of the fully dynamic resistive theory, in particular the complicated structure of the sheet edges, the so-called Y-points.