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).

This paper presents a discussion of the characteristic observables of stellar explosions and compares the observed signatures such as light curve and abundance distribution with the respective values predicted in nucleosynthesis model calculations. Both the predicted energy generation as well as the abundance distribution in the ejecta depends critically on the precise knowledge of the reaction rates and decay processes involved in the nucleosynthesis reaction sequences. The important reactions and their influence on the production of the observed abundances will be discussed. The nucleosynthesis scenarios presented here are all based on explosive events at high temperature and density conditions. Many of the nuclear reactions involve unstable isotopes and are not well understood yet. To reduce the experimental uncertainties several radioactive beam experiments will be discussed which will help to come to a better understanding of the correlated nucleosynthesis.

Introduction

Historically, the field of nuclear astrophysics has been concerned with the interpretation of the observed elemental and isotopic abundance distribution (Anders & Grevesse 1989) and with the formulation and description of the originating nucleosynthesis processes (Burbidge et al. 1957; Wagoner 1973; Fowler 1984). Each of these nucleosynthesis processes can be characterized by a specific signature in luminosity and/or in the resulting abundance distribution.

The Mexican School on Nuclear Astrophysics was held in the Hotel Castillo Santa Cecilia, Guanajuato, México, from August 13 to August 20, 1997. The goal of the school was to gather together researchers and graduate students working on related problems in astrophysics – to present areas of current research, to discuss some important open problems, and to establish and strengthen links between researchers. The school consisted of eight courses and material presented in these forms the basis of this book.

Non–stop interaction between the participants, through both formal and informal discussions, gave the school a relaxed and productive atmosphere. It provided the opportunity for researchers from a wide range of backgrounds to share their interests in and different perspectives of the latest developments in astrophysics.

The productivity of the meeting reflected the strong interest of the Mexican and Latin American scientific communities in the subjects covered, Indeed, a second school is planned for 1999.

Professor David Schramm very sadly died not long after the conference, in December 1997. His lectures at the School were fascinating. He will be sorely missed by us and the rest of the astrophysics community.

A brief review of key concepts in multifrequency observational astronomy is presented. The basic physical scales in astronomy as well as the concept of stellar evolution are also introduced. As examples of the application of multifrequency astronomy, recent results related to the observational search for black holes in binary systems in our Galaxy and in the centers of other galaxies is described. Finally, the recently discovered microquasars are discussed. These are galactic sources that mimic in a smaller scale the remarkable relativistic phenomena observed in distant quasars.

Introduction

There have been many outstanding observational and theoretical discoveries made in astronomy during the twentieth century. However, in the future this ending century will most probably be remembered not by these achievements, but by being the time when astronomers started observing the Cosmos with a variety of techniques and in particular when we started to use all the “windows” in the electromagnetic spectrum.

During our century we started to investigate systematically the Universe using:

• The whole electromagnetic spectrum. At the beginning of the century, practically all the data was coming from the visible photons (that is, those detected by the human eye) only.

• Cosmic rays. These charged particles hit the Earth's atmosphere and can be detected by the air showers they produce. The origin of the most energetic cosmic rays (1019 ergs or more) remains a mystery.

• […]

Introduction, formalism, and cosmological bound

In these four lectures I will present a brief and rather elementary description of the physics of massive neutrinos as it emerges from studies involving nuclear physics, particle physics, astrophysics and cosmology. The lectures are meant for physicists who are not experts in this field, which I believe covers most of the participants in this School, and many potential readers elsewhere. I hope that such readers can find here enough information that they will be able to understand and appreciate the connection between the hunt for neutrino mass and mixing described here, and their own field of expertize.

Throughout I will use original references sparingly. Instead I refer to several monographs, written and published during the last decade [Boehm & Vogel (1992), Kayser, Gibrat–Debu k Perrier (1989), Winter (1991), Mohapatra & Pal (1991), Kim & Pevsner (1993), Klapdor–Kleingrothaus & Staudt (1995)] where an interested reader can find references to the original papers. When appropriate I will also refer to review papers on various aspects of the neutrino mass or related topics. For the experimental data, including the list of the most recent original experimental papers, the best source is the Review of Particle Physics, periodically updated, with the latest printed version in PDG (1996). The update of this very useful publication is available even between printed editions on the World–Wide Web at http://pdg.lbl.gov/.

The mechanism for a core–collapse or type II supernova is a fundamental unresolved problem in astrophysics. Although there is general agreement on the outlines of the mechanism, a detailed model that includes microphysics self–consistently and leads to robust explosions having the observational characteristics of type II supernovae does not exist. Within the past five years supernova modeling has moved from earlier one–dimensional hydrodynamical simulations with approximate microphysics to multi–dimensional hydrodynamics on the one hand, and to much more detailed microphysics on the other. These simulations suggest that large–scale and rapid convective effects are common in the core during the first hundreds of milliseconds after core collapse, and may play a role in the mechanism. However, the most recent simulations indicate that the proper treatment of neutrinos is probably even more important than convective effects in producing successful explosions. In this series of lectures I will give a general overview of the core–collapse problem, and will discuss the role of convection and neutrino transport in the resolution of this problem.

Introduction

A type II supernova is one of the most spectacular events in nature, and is a likely source of the heavy elements that are produced in the rapid neutron capture or r–process. Considerable progress has been made over the past two decades in understanding the mechanisms responsible for such events.