In this chapter we give a brief introduction to various types of non-linear effects that can occur in a plasma. Almost all of the basic equations in plasma physics have non-linear terms. For example, these include the (E + v × B) · ∇vf term in the Vlasov equation (5.2.15), the U · ∇U term in the convective derivative (5.4.26), the ρmU term in the MHD mass continuity equation (6.1.32), and the J × B term in the MHD momentum equation (6.1.33). All of these terms represent potential sources of non-linear effects. There are many more. In our analysis of waves in the previous chapters, we always assumed that the wave amplitude was small, so that the governing equations can be linearized. This assumption provides a remarkably accurate description of many types of small amplitude waves. However, if the wave amplitude becomes large, as always occurs for an instability, the linearization assumption breaks down. Non-linear effects must then be taken into account. There are many such non-linear effects, more than we can possibly discuss in this introductory textbook. In order to limit the scope of the discussion, we will concentrate on two quite different types of non-linear analyses that have a wide range of applications. These are (1) quasi-linear theory, and (2) time-stationary electrostatic potentials. For a more comprehensive discussion of non-linear effects in plasmas, the reader is referred to one of the specialized books on non-linear effects, such as Kadomtsev [1965], Sagdeev and Galeev [1969], and Davidson [1972].

In this chapter we investigate the propagation of small amplitude waves in a hot unmagnetized plasma. Because of the shortcomings of the moment equations, the approach used is to solve the Vlasov equation directly using a linearization procedure similar to that used in the analysis of cold plasmas. Although both electromagnetic and electrostatic solutions exist, the discussion in this chapter is limited to solutions that are purely electrostatic, i.e., the electric field is derivable from the gradient of a potential, E = −∇Φ. Electromagnetic solutions are discussed in the next chapter.

From Faraday's law it is easily verified that electrostatic waves have no magnetic component. This greatly simplifies the Vlasov equation by eliminating the v × B force. For electrostatic waves, it is usually easier to solve for the potential rather than for the electric field. Therefore, in the following analysis, the electric field is replaced by E = −∇Φ, and the potential is calculated from Poisson's equation, ∇2Φ = −ρq/∊0.

The Vlasov approach

In an initial attempt to analyze the problem, we assume that normal modes of the form exp(−iωt) exist and represent them by using Fourier transforms, following the same basic procedure used in Chapter 4. This is the approach used by Vlasov [1945], who first considered this problem. As we will see, the Vlasov approach encounters a mathematical difficulty that can only be resolved by reformulating the problem in terms of Laplace transforms.

In this chapter we discuss the propagation of small amplitude waves in a hot magnetized plasma. Just as for a cold plasma, the presence of a static zero-order magnetic field in a hot plasma leads to a wide variety of new phenomena. Because the zero-order motions of the particles in a magnetized plasma consist of circular orbits around the magnetic field, some type of resonance can be expected when the wave frequency is equal to the cyclotron frequency. In a cold plasma, this resonance is the same for all particles of a given charge-to-mass ratio, and gives rise to the well-defined cyclotron resonances described in Chapter 4. In a hot plasma, the frequency “felt” by a particle is Doppler shifted by the thermal motion of the particle along the static magnetic field. For a given parallel velocity, resonance occurs when the frequency in the guiding center frame of reference of the particle is at the cyclotron frequency, i.e., ω′ = ω − k∥ν∥ = ωc. Because of the thermal spread in the particle velocities, the resonance is no longer sharp, as it was in a cold plasma, but is now broadened by the thermal motion. The resonant interaction also produces damping, called cyclotron damping, in a manner somewhat analogous to Landau damping. If the cyclotron radius of the particle is a significant fraction of the wavelength, the phase shift introduced by the periodic cyclotron motion of the particles back and forth along the perpendicular component of the wave vector produces a phase modulation at the cyclotron frequency.

A complete mathematical model of a plasma requires three basic elements: first, the motion of all particles must be determined for some assumed electric and magnetic field configuration; second, the current and charge densities must be computed from the particle trajectories; and third, the electric and magnetic fields must be self-consistently determined from the currents and charges, taking into account both internal and external sources. To be self-consistent, the electric and magnetic fields obtained from the last step must correspond to the fields used in the first step. It is this self-consistency requirement that makes the analysis of a plasma difficult.

To develop an understanding of the processes occurring in a plasma, a useful first step is to forget about the self-consistency requirement and concentrate on the motion of a single particle in a specified field configuration. This approach can be useful in a variety of situations. If the external fields are very strong and the plasma is sufficiently tenuous, the internally generated fields are sometimes small and can be safely ignored. This situation arises, for example, in radiation belts at high energies and in various electronic devices such as vacuum tubes and traveling wave amplifiers. In other situations the self-consistent electric and magnetic fields may be known from direct measurement. In this case, it is often useful to follow the motion of individual tracer particles in the known electric and magnetic fields in order to gain insight into the physical processes involved, such as particle transport and energization.

A plasma is an ionized gas consisting of positively and negatively charged particles with approximately equal charge densities. Plasmas can be produced by heating an ordinary gas to such a high temperature that the random kinetic energy of the molecules exceeds the ionization energy. Collisions then strip some of the electrons from the atoms, forming a mixture of electrons and ions. Because the ionization process starts at a fairly well-defined temperature, usually a few thousand K, a plasma is often referred to as the “fourth” state of matter. Plasmas can also be produced by exposing an ordinary gas to energetic photons, such as ultraviolet light or X-rays. The steady-state ionization density depends on a balance between ionization and recombination. In order to maintain a high degree of ionization, either the ionization source must be very strong, or the plasma must be very tenuous so that the recombination rate is low.

The definition of a plasma requires that any deviation from charge neutrality must be very small. For simplicity, unless stated otherwise, we will assume that the ions are singly charged. The charge neutrality condition is then equivalent to requiring that the electron and ion number densities be approximately the same. In the absence of a loss mechanism, the overall charge neutrality assumption is usually satisfied because all ionization processes produce equal amounts of positive and negative charge. However, deviations from local charge neutrality can occur.

This textbook is intended for a full year introductory course in plasma physics at the senior undergraduate or first-year graduate level. It is based on lecture notes from courses taught by the authors for more than three decades in the Department of Physics and Astronomy at the University of Iowa and the Department of Applied Physics at Columbia University. During these years, plasma physics has grown increasingly interdisciplinary, and there is a growing realization that diverse applications in laboratory, space, and astrophysical plasmas can be viewed from a common perspective. Since the students who take a course in plasma physics often have a wide range of interests, typically involving some combination of laboratory, space, and astrophysical plasmas, a special effort has been made to discuss applications from these areas of research. The emphasis of the book is on physical principles, less so on mathematical sophistication. An effort has been made to show all relevant steps in the derivations, and to match the level of presentation to the knowledge of students at the advanced undergraduate and early graduate level. The main requirements for students taking this course are that they have taken an advanced undergraduate course in electricity and magnetism and that they are knowledgeable about using the basic principles of vector calculus, i.e., gradient, divergence and curl, and the various identities involving these vector operators. Although extensive use is made of complex variables, no special background is required in this subject beyond what is covered in an advanced calculus course.

Abstract

Observations over the last decade have shown that neutron stars receive a large kick velocity (of order a few hundred to a thousand km s-1) at birth. The physical origin of the kicks and the related supernova asymmetry is one of the central unsolved mysteries of supernova research. We review the physics of different kick mechanisms, including hydrodynamically driven, neutrino — magnetic field driven, and electromagnetically driven kicks. The viabilities of the different kick mechanisms are directly related to the other key parameters characterizing nascent neutron stars, such as the initial magnetic field and the initial spin. Recent observational constraints on kick mechanisms are also discussed.

Evidence for neutron star kicks and supernova asymmetry

It has long been recognized that neutron stars (NSs) have space velocities much greater than their progenitors'. A natural explanation for such high velocities is that supernova (SN) explosions are asymmetric, and provide kicks to the nascent NSs. Evidence for NS kicks and NS asymmetry has recently become much stronger. The observations that support (or even require) NS kicks fall into three categories:

Abstract

HETE-2 has confirmed the connection between GRBs and Type Ic supernovae. Thus we now know that the progenitors of long GRBs are massive stars. HETE-2 has also provided strong evidence that the properties of X-Ray Flashes (XRFs) and GRBs form a continuum, and therefore that these two types of bursts are the same phenomenon. We show that both the structured jet and the uniform jet models can explain the observed properties of GRBs reasonably well. However, if one tries to account for the properties of both XRFs and GRBs in a unified picture, the uniform jet model works reasonably well while the structured jet model fails utterly. The uniform jet model of XRFs and GRBs implies that most GRBs have very small jet opening angles (∼ half a degree). This suggests that magnetic fields play a crucial role in GRB jets. The model also implies that the energy radiated in gamma rays is ∼100 times smaller than has been thought. Most importantly, the model implies that there are ∼ 104–105 more bursts with very small jet opening angles for every such burst we see. Thus the rate of GRBs could be comparable to the rate of Type Ic core collapse supernovae. Accurate, rapid localizations of many XRFs, leading to identification of their X-ray and optical afterglows and the determination of their redshifts, will be required in order to confirm or rule out these profound implications.

Abstract

We present the V light curve and optical/infrared spectra of the Type Ic SN 1997B. We show that (1) this SN displayed lines of the He I series; (2)the expansion velocities were higher than those of SNe with traces of H or large He masses in their envelopes (like SN 1993J); the light curve of SN 1997B decayed slower than that of SN 1993J. The smaller mass to kinetic energy ratio and shallower light curve of SN 1997B are inconsistent with it being a He stripped version of some of the best studied Type Ib or II-transition SNe. We infer that Type Ib/c and II-transition SN progenitors come, at least, with two different types of inner structure.

Introduction

A few years ago the presence of He in the atmospheres of Type Ic SNe, the nature of their progenitors, and the relation between Type Ib and Type Ic SNe was subject of debate. On the one hand, empirical evidence and theoretical interpretation supported the view that SNe of Type Ib and Ic are different enough to insure that their progenitors result from different paths of stellar evolution. If so, Type Ic SNe originated in bare C+O cores and were expected not to display He I lines in their spectra. On the other hand, it was stressed that Type Ib and Ic SNe could originate in similar stars evolving as interacting binaries.

Abstract

There are many interesting topics at the intersection of physics and astrophysics we call Supernova Theory. A small subset of them include the origin of pulsar kicks, gravitational radiation signatures of core bounce, and the possible roles of neutrinos and rotation in the mechanism of explosion. In this brief communication we summarize various recent ideas and calculations that bear on these themes.

What is the mechanism of pulsar kicks?

Radio pulsars are observed to have large proper motions that average ∼400–500 km s-1 (Lyne & Lorimer 1994) and whose velocity distribution might be bimodal (Fryer, Burrows, and Benz 1998; Arzomanian, Chernoff, & Cordes 2002). If bimodal, the slow peak would have a mean speed near ∼100 km s-1 and the fast peak would have a mean speed near 500–600 km s-1. A bimodal distribution implies different populations and different mechanisms, but what these populations could be remains highly speculative.

Many arguments suggest that pulsars are given “kicks” at birth (Lai 2000; Lai, Chernoff, and Cordes 2001), and are not accelerated over periods of years or centuries. The best explanation is that these kicks are imparted during the supernova explosion itself. We think that this view is compelling. The two suggested modes of acceleration and impulse are via net neutrino anisotropy during the neutrino emission phase (which lasts seconds) and anisotropic mass motions and aspherical explosion which impart momentum to the residual core.

Abstract

In the popular progenitor scenario, Type Ia supernova are the result of a white dwarf exploding in a binary system. The presence of a nearby companion star could cause a substantial asymmetry in the supernova ejecta — according to the models of Marietta et. al. (2000), the companion carves out an hole in the ejecta. The opening angle of the hole is as large as 40°. Such an asymmetry would leave signatures in the supernova flux and polarization spectra. We explore this possibility using a three-dimensional Monte-Carlo LTE radiative transfer code which includes gamma ray transport and a temperature correction procedure. We calculate synthetic spectra and polarization levels from multiple lines of sight to see how an ejecta hole model compares to observations.

Introduction

While some Type Ia supernovae (SNe Ia) are known to be aspherical, the exact nature of the asymmetry is unknown. The direct evidence of the asphericity is the detection of non-zero intrinsic polarization in, for example, SN 1999by [4] and SN 2001el [17]. In both cases, the polarization level was rather low (∼0.7% for SN 1999by, ∼0.4% for SN 2001el), which indicates a mild asymmetry along the line of sight. In addition, the polarization angle was fairly constant across the majority of line features, indicating that the bulk of the ejecta obeyed a near axial symmetry. The exact shape of the supernova ejecta is an important question, as it must be closely tied to the explosion processes and progenitor systems of SNe Ia.

Abstract

Current massive single star evolution models with rotation, especially when magnetic fields are included, appear to get close in reproducing the spin rates of young neutron stars. This, however, excludes them as progenitors of gamma-ray bursts within the collapsar model. Close binary evolution models with rotation, on the other hand, suggest that the mass receiving star is spun-up appreciably and may retain enough angular momentum in its core until collapse, while the mass donor is spun-down to produce core rotation rates below those of single stars.

Introduction

The evolution of a single star can be strongly influenced by its rotation (e.g., Heger & Langer 2000; Meynet & Maeder 2000), and evolutionary models of rotating stars are now available for many masses and metallicities. While the treatment of the rotational processes in these models is not yet in a final stage (e.g., magnetic dynamo processes are just about to be included; Heger et al. 2003), they provide first ideas of what rotation can really do to a star.

Effects of rotation, as important as they are in single stars, can be much stronger in the components of close binary systems: Estimates of the angular momentum gain of the accreting star in mass transferring binaries show that critical rotation may be reached quickly (Packet 1981; Langer et al. 2000). Therefore, we need binary evolution models which include a detailed treatment of rotation in the stellar interior, as in recent single star models.

Abstract

We study a thermonuclear explosion of a carbon-oxygen white dwarf (WD) using a three-dimensional hydrodynamic model with a simplified mechanism for nuclear reactions and energy release. The explosion begins as a deflagration with the flame front highly distorted by the Rayleigh-Taylor instability. Turbulent combustion and convective flows produce an inhomogeneous mixture of burned and unburned materials that extends from the center to about 0.8 of the radius of the expanding WD. At this stage, a detonation is ignited and propagates through the layers of unburned material with the velocity about 12,000 km/s, which is comparable to the expansion velocities induced in outer layers of the WD by the subsonic burning. During the period of detonation propagation, the density of the expanding unreacted material ahead of the shock can decrease by an order of magnitude compared to its value before the detonation started. Because the detonation burns material to different products at different densities, it can create a large-scale asymmetry in composition if it starts far from the WD center. In contrast to the 3-D deflagration model, the 3-D delayed-detonation model of SN Ia explosions does not leave carbon, oxygen, and intermediate-mass elements in central parts of a WD. This removes the key disagreement between simulations and observations, and confirms that the delayed detonation is currently the most promising mechanism for SN Ia explosions.

Introduction

Type Ia supernovae (SNe Ia) [1–10] result from the most powerful thermonuclear explosions in the Universe.

Abstract

Stellar abundance observations are providing important clues about the relationship between supernovae (SNe) and the rapid neutron capture process (i.e., the r-process). Although the site for the r-process is still not identified, events in and around SNe have long been suspected. Abundances of heavy neutron-capture elements in a number of stars suggest a robust r-process operating over billions of years, constraining astrophysical and nuclear conditions in supernova models. Variations in lighter n-capture element abundances — observed only very recently in any stars — could be explained as a signature of certain supernova models, or might require multiple r-process sites with different mass ranges or frequencies of SNe. Recent observations of elemental abundance scatter in the early Galaxy are consistent with earlier suggestions of a restricted range of SNe responsible for the r-process.

Introduction

The elements heavier than iron are synthesized in neutron processes, either in the (s)-low or (r)-apid process. In the s-process the timescale for neutron capture (τn) is much longer than the electron (beta)-decay (τβ) timescale. For the r-process, however, τn << τβ with many neutrons captured in a very short time period. As a result, neutron captures proceed into very neutron-rich regions far from the stable nuclei, where very little experimental nuclear data is available. This element synthesis is intimately connected to the late stages of stellar evolution, with the s-process occurring in the thermally pulsing helium shells of asymptotic giant branch (AGB) stars of low- and intermediate-mass (M ∼ 0.8–8 M⊙) (see, e.g., the review by Busso, Gallino, & Wasserburg 1999).