Progenitors of PN play an important role in the chemical enrichment of the Galaxy because He and significant amounts of C and N are produced by nucleosynthesis during the MS, RG, and AGB stages of their evolution. These elements are brought to the surface of the star by convection events called dredge ups (Iben, 1991). The first dredge up occurs in all stars on the RGB following the exhaustion of H in the core. Convection extending into the interior brings material processed in the CNO cycle to the surface. The second dredge-up occurs only in higher mass (≥3 M⊙) stars during the early AGB phase and results in an increase in the surface abundance of 4 He and 14 N at the expense of 12 C and 16 O. The third dredge-up occurs on the AGB after each He-shell flash, resulting in He, C, and s-process elements being brought to the surface (Renzini and Voli, 1981). This repeated enhancement of C abundance in the stellar surface may in some stars bring the C/O ratio to be greater than unity. Since the less abundant of these two elements is completely tied up in the CO molecule, the photospheric spectrum of the star will change from O-based (e.g., TiO and VO) to C-based (e.g., C2 and CN), and a carbon star is created (see Section 10.2). Stellar winds during the AGB eject these elements from the stellar surface and therefore enrich the interstellar medium.

In Chapter 11 we discussed that the central stars of PN originate from the electrondegenerate carbon-oxygen core of AGB stars. After the thin H envelope has been depleted by nuclear burning and mass loss, CSPN will change from deriving their energies from the CNO process to gravitational contraction. This is accompanied by a drop in luminosity by at least an order of magnitude. He burning is unimportant (except during a thermal pulse) and also dies away eventually. The star now enters the “cooling track” with a gradual decline in both luminosity and temperature. These blue and faint stars are referred to as white dwarfs (WDs).

WDs were discovered as faint stars that are unusually dense that perturb the orbits of their normal companions through gravitational interaction. Most of the early WD discoveries are nearby single stars, which are found in surveys of stars with large proper motions (Luyten, 1979). More recent discoveries have resulted from surveys of faint blue stars, such as the quasar survey of Palomar-Green (Green et al., 1986). The PG survey has led to the discovery of hot, H-deficient stars known as PG 1159 stars (see Section 7.3). These are the hottest WDs and therefore can be considered as possible transition objects between the PN and WD stages.

For very low-mass stars, it is possible that mass loss or binary mass transfer on the RGB removes sufficient mass from the envelope to stop nuclear evolution before the He flash.

Since molecules were not expected to survive in the hostile high temperature, ionized environment of PN, the first detection of CO in the PN NGC 7027 by Mufson et al. (1975) therefore came as a complete surprise. The CO profile in NGC 7027 is almost 40 km s–1 wide, suggesting that the molecular gas is in an expanding envelope (Fig. 5.1). The amount of the molecular gas inferred from the CO line strength is over 1 and is much higher than the ionized mass usually associated with PN.

The CO profile of NGC 7027 resembles the CO profiles observed in the AGB stars, for example, IRC+10216 (Solomon et al, 1971). By the mid-1970s, CO emission had been detected from many AGB stars (see Section 10.4.3). The amounts of mass observed in the circumstellar envelopes of AGB stars are found to be similar to that observed in NGC 7027. The most likely explanation of the origin of molecular gas in PN is that they are the remnants of the circumstellar envelopes of AGB stars (Kwok, 1982).

The presence of molecules implies that there is more material in PN than what is suggested by the optical images. Vibrational and rotational states of molecules can be excited either collisionally or radiatively, and the observations of these transitions allow new ways to probe the physical structure of PN.

Paradoxically, observations of PN in external galaxies offer several advantages over the study of galactic PN. Because of the heavy obscuration of the center of our Galaxy, our catalog of galactic PN is highly incomplete. However, PN in nearby galaxies, for example, M31, can be optically identified to within a few parsecs from the nucleus. PN were first detected in M31 by Baade (1955). Five PN southwest of M31's center were detected in [OIII] with a 1,500-å filter and the Palomar 5-m telescope. By using a narrower filter (23 å), Ford and Jacoby (1978) were able to go much fainter and detected 315 PN in the bulge of M31. Later surveys have cataloged PN in M32, NGC 205, NGC 185, and NGC 147.

A survey of PN in an external galaxy can in fact provide a much better picture of the spatial distribution of PN. As discussed in Chapter 16, the distances to galactic PN are highly uncertain. Since the PN in an external galaxy can be safely taken to be at the same distance, the identification of a large number of PN in a galaxy of known distance can lead to an accurate luminosity function.

CSPN are cores of AGB stars and are therefore very luminous objects. If the nebula is ionization bounded, the central star's radiation is reprocessed into line radiation. For example, in the nebular model shown in Fig. 4.3, ∼ 3% of the stellar luminosity (or ∼300 L⊙) comes out in the [OIII] line.

The evolutionary stage between the end of the AGB and PN phases has long been a missing link in our understanding of single-star evolution. As we discussed in Chapter 10, the AGB is terminated by the depletion of the H envelope by mass loss, and this occurs before the onset of carbon detonation. When mass loss reduces the mass of the H envelope (Me) below a certain value (Me ∼ 10–3M⊙ for a core mass [Mc] of 0.60 M⊙, Schönberner 1983), the star will begin to evolve toward the blue side of the H-R diagram. The effective temperature of the star will increase as the remaining H envelope continues to diminish by H-shell burning. This phase will last until the central star is hot enough (T* ∼ 30,000 K) to ionize the circumstellar nebula. The emergence of recombination lines of H, He, and forbidden lines of metals will make the nebula easily observable in the visible, signaling the beginning of the PN phase. A sketch of the evolutionary tracks of proto-planetary nebulae (PPN) in the H-R diagram is shown in Fig. 14.1.

PPN can be defined as the stage of evolution in which their central stars have stopped the large-scale mass loss on the AGB, but have not evolved to be hot enough to emit a sufficient quantity of Lyman continuum photons to ionize the surrounding remnants of the AGB envelope.

The study of evolution of the central stars of PN is motivated by the observed distribution of PN in the H-R diagram. Using Shklovsky distances (Section 16.1.1) and H Zanstra temperatures (Section 7.1.1), O'Dell (1963) was the first to construct a luminosity-temperature diagram for the central stars of PN. Similar results were obtained by Harman and Seaton (1964) and Seaton (1966) using Hell Zanstra temperatures. They found that both low(T* = 30,000 K) and high temperature (105 K) central stars had low luminosities (102 L⊙) and that the intermediate temperature (50,000 K) stars have high luminosities (104 L⊙). Apparently, the distribution of PN in the H-R diagram had the shape of an upside-down horse shoe, which was interpreted as an evolutionary sequence and named the Harman-Seaton sequence.

Harman and Seaton (1964) suggested that PN were formed at the end of the horizontal branch, followed by a rapid increase in temperature and luminosity of the central star. Such rapid evolution in 104 yr (the dynamical age of PN) posed a great challenge to theorists. Early efforts have concentrated on remnant stellar cores without nuclear burning, and undergoing gravitational and thermal adjustments while contracting toward the WD stages. Although artificial models with specific initial conditions could be made to mimic the Harman-Seaton sequence, the results are far from satisfactory (cf. Salpeter, 1968; Shaviv, 1978).

We begin the discussions on the physics of PN with the classical static model of PN. PN are assumed to be made up of two components: a central star and a surrounding gaseous nebula. If the star is hot enough, much of its energy will be emitted in the ultraviolet (UV). These UV photons will be able to ionize the atoms in the nebula. The electrons ejected in the ionization process provide a pool of kinetic energy for the collisional excitation of the heavy atoms (carbon, nitrogen, oxygen, etc.). Spontaneous emissions from the various excited states of different atoms and ions are responsible for the rich emission-line spectrum seen in the visible.

The first excited state of hydrogen (H) is at 10.2 eV above the ground state, corresponding to an excitation temperature (E/k) of ∼105 K. This is much higher than the typical kinetic temperatures of ∼104 K found in PN. Even for electrons with energies high enough to overcome this energy gap, the low densities in PN imply that the excitation or ionization rates by electron collisions are much slower than the spontaneous emission rate (see Section 3.4), and the collisionally excited electron will remain at an excited state for a very short time. As the result, the population of an excited state of H is determined not by collisions from below, but by the recombination between free protons and electrons and the subsequent electron cascades via spontaneous emissions.

The traditional way of comparing stellar evolutionary models and observations is through the Herztsprung-Russell (H-R) diagram. However, we should remember the H-R diagram has its origin in color-magnitude diagrams of stellar clusters, where all stars are at the same distance. For galactic and globular clusters, the stars have relatively small bolometric corrections, and therefore the luminosities and temperatures can be easily derived from apparent magnitudes and colors. The plotting of PN on the H-R diagram turns out to be a much more difficult task. In order for luminosities to be obtained, accurate distances and total fluxes are needed. CSPN have very high temperatures, and the conversion from visual magnitudes to bolometric magnitudes requires a large and uncertain correction factor. A large part of the stellar flux is transferred to the nebula, so the total emitted fluxes from the CSPN have to be inferred from the nebular spectrum. The accounting of the UV flux would be incomplete if the nebula is not ionization bounded. Furthermore, PN emit a large fraction of their total fluxes in the far infrared (see Section 6.2), and observations in the optical region alone are insufficient to account for all the emitted fluxes.

In spite of numerous efforts, the determination of distances to galactic PN remains uncertain to a large factor. Since many of the CSPN are not directly observable, their temperatures have to be inferred from the nebular properties.

By the early 1970s, the field of PN had achieved a high degree of success. The nebular spectrum in the visible was reasonably well understood and PN had served well as a laboratory for atomic physics. Laboratory or theoretically derived atomic parameters such as recombination rates, collisional excitation rates, and spontaneous decay rates had been used to interpret the observed strengths of the line fluxes. The accounting of processes not observable in the terrestrial environment (e.g. the 2γ radiation, forbidden lines, etc.) is a particularly noteworthy accomplishment. The model of PN (which we refer to as the classical model), consisting of a nebular gas shell of fixed mass photoionized by a hot central star, seemed to be adequate in explaining the nebular spectrum. The combination of sophisticated observations (in particular spectroscopy) with theoretical calculations has made physics of gaseous nebulae one of the most successful examples of modern astrophysics.

Although astronomers were justifiably elated by the success of PN research, a number of problems were lurking under the surface. Here we summarize several examples of problems with the classical model that were starting to be recognized in the early 1970s.

1. • The nebular mass problem: in the classical model in which the PN is made up of a uniform-density shell of a fixed mass, the ionized masses of PN should be well determined by the measurement of the Hβ flux or the radio continuum flux (see Section 4.5). However, in cases in which the distances were reasonably well known, the actual derived masses were found to spread over several orders of magnitude, in contradiction to the traditional assumption of a fixed-mass nebula.

2. […]

Like most fundamental concepts in physics, magnetic reconnection owes its appeal to its ability to unify a wide range of phenomena within a single universal principle. Virtually all plasmas, whether in the laboratory, the solar system, or the most distant reaches of the universe, generate magnetic fields. The existence of these fields in the presence of plasma flows inevitably leads to the process of magnetic reconnection. As we shall discuss in more detail later on, reconnection is essentially a topological restructuring of a magnetic field caused by a change in the connectivity of its field lines. This change allows the release of stored magnetic energy, which in many situations is the dominant source of free energy in a plasma. Of course, many other processes besides reconnection occur in plasmas, but reconnection is probably the most important one for explaining large–scale, dynamic releases of magnetic energy.

Figures 1.1–1.4 illustrate the rich variety of plasma environments where reconnection occurs or is thought to occur. The evidence of reconnection in laboratory fusion machines such as the tokamak [Fig. 1.1 (a)] and the reversed–field pinch [Fig. 1.1 (c)] is so strong that there is no longer any controversy about whether reconnection occurs, but only controversy about the way in which it occurs (§9.1). However, as one considers environments which are further away from the Earth, the evidence for reconnection becomes more circumstantial. Most researchers who study the terrestrial aurorae [Fig. 1.2(a)] believe that they are directly or indirectly the result of reconnection in the Earth's magnetosphere, but the evidence for similar phenomena in other planetary magnetospheres [Fig. 1.2(b)] is much smaller (§10.6).

When solving partial differential equations, either analytically or numerically, the form, value, and number of the boundary conditions is of crucial importance. Indeed, often much physics is incorporated in the boundary conditions and, in the setting up of a numerical experiment with nonstandard boundary conditions, it is often the implementation of the boundary conditions that causes the most trouble. Petschek's mechanism, in which the boundary conditions at large distances are implicit, has been generalised in two distinct ways by adopting different boundary conditions to give regimes of almost-uniform reconnection (§5.1) and non-uniform reconnection (§5.2). Whereas Petschek's mechanism may be described as being almost-uniform and potential (§4.3), the first of these new families is in general nonpotential and the second is nonuniform. Also, surprisingly late in the day, a theory of linear reconnection was developed, which occurs when the reconnection rate is extremely slow (§5.3).

Almost-Uniform Non-Potential Reconnection

Vasyliunas (1975) clarified the physics of Petschek's mechanism by pointing out that the inflow region has the character of a diffuse fast-mode expansion, in which the pressure and field strength continuously decrease and the flow converges as the magnetic field is carried in. (This characterization of the inflow does not mean that a standing fast-mode wave is present in the inflow, since such a standing wave is not possible in a sub-fast flow.) A fast-mode disturbance has the plasma and magnetic pressure increasing or decreasing together, while a slow-mode disturbance has the plasma pressure changing in the opposite sense to the magnetic pressure.

We introduced briefly in Section 1.3.2 the idea of a current sheet as a narrow region across which the magnetic field changes rapidly. In this chapter we consider in detail the formation of such sheets in a medium where the magnetic field is frozen to the plasma (§1.4), and then in Chapter 4 we describe how they diffuse through the plasma.

There are several ways in which current sheets may form. One is by the collapse of an X-type neutral point (§2.1). Such a formation in two dimensions through a series of static potential field states may be described by complex variable theory, in which the sheet is treated as a branch cut in the complex plane (§2.2). Other techniques are required for three-dimensional axisymmetric fields (§2.2.5), force-free fields (§2.3.1) or more general magnetostatic fields (§2.3.2). The concept of magnetic relaxation as developed by Moffatt is described in Section 2.4, and a self-consistent theory for slow time–dependent formation is discussed in Section 2.5. Finally, two other ways of forming current sheets are described, namely by shearing a field with separatrices (§2.6) and by braiding (§2.7).

X-Point Collapse

As we shall discuss in detail in Section 7.1, an X-type neutral point in a magnetic configuration tends to be locally unstable, provided the sources of the field are free to move (Dungey, 1953).

Magnetic reconnection is a fundamental process with a rich variety of aspects and applications in astrophysical, space, and laboratory plasmas. It is one that has fascinated us both for much of our research careers, so that we have from time to time felt drawn to return to it after working on other topics and to ponder it anew or view it from a different angle.

Indeed, it was reconnection that brought us together in the first place, since one of us (TGF) went to work as a postdoc with the other in 1980 on the subject of reconnection in solar flares. Our initial meeting in Edinburgh at the start of this collaboration was rather amusing, since a friend had misleadingly described Eric Priest as a tall old man with ginger hair, and so the inaccuracy of this description did not exactly help us to find each other in the crowded airport!

At present the whole field of reconnection is a huge, vibrant one that is developing along many different lines, as can be seen by the fact that a recent science citation search produced a listing of 1,069 published articles written on this subject in only the past three years. We are therefore well aware of the impossibility of comprehensively covering the whole field and apologise in advance to those who may be disappointed that we have not found space to discuss their work on reconnection.

This book is devoted almost entirely to magnetohydrodynamic theories of reconnection and does not review the extensive literature on collisionless processes. Such processes are critical for the production of energetic particles by reconnection, but they constitute an enormous topic on which a whole text could easily be written. However, since the production of fast charged particles is one of the main consequences of reconnection in the cosmos, we feel it is important to devote some space to a brief discussion of this subject.

As we shall see, many particle acceleration theories rely, either implicitly or explicitly, on MHD concepts to supply information about the large-scale distribution of magnetic and electric fields. In the simplest theories (§13.1), the energetic particles are directly accelerated by an electric field whose behaviour is assumed to be known. In this case MHD models are often used to justify the particular choices made for the fields. However, even in more complex theories, MHD concepts such as turbulence (§13.2) and shock waves (§13.3) are often invoked. MHD theory is complemented by kinetic theory, which provides a great deal of additional information about the local plasma behaviour.

Energetic particles are common throughout the universe and reveal at once that it is not as a whole in thermodynamic equilibrium. They show up, for example, as cosmic rays incident on our atmosphere; they produce synchrotron emission from distant radio galaxies; and they are detected in the solar wind upstream of shocks produced by the Earth's magnetosphere and by coronal mass ejections.

The theory of reconnection in two dimensions is now fairly well understood and is highly developed, and, as we have seen, the type of reconnection that is produced depends very much on the reconnection rate, the configuration, the boundary conditions, and the parameter values. Many questions do remain, however, such as: what are the properties of turbulent or impulsive bursty reconnection; why does the diffusion region in Petschek reconnection lengthen when the resistivity is uniform; what is the effect of outflow boundary conditions on fast reconnection; how does reconnection occur in a collisionless plasma; and how do the different terms in the energy equation such as radiation and conduction affect reconnection?

The theory of three-dimensional reconnection is much less developed. We have only just started a voyage of discovery that will last many years, but some important directions have already been indicated. Many features are quite different in three dimensions. For example, we discuss here the definition of reconnection (§8.1), the structure of null points (§8.2), the nature of the bifurcations (§8.3), the global magnetic topology (§8.4), and the nature of the reconnection itself (§§8.6, 8.7).

In this chapter we introduce several new concepts. At null points, magnetic reconnection can take place by spine reconnection, fan reconnection, or separator reconnection (§8.6). Regions where magnetic field lines touch a boundary and are concave towards the interior of the volume are referred to as bald patches (§8.4.1). When no null points or bald patches are present, the mapping of field lines from one boundary to another is continuous, so they all have the same topology and there are no separatrices.