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.

Introduction

In a conducting medium a typical current sheet tends to diffuse outward at a slow rate with a time-scale of τd = l2/η, where 2l is the width of the current sheet and η = (μσ)-1 is the magnetic diffusivity. During the process of magnetic diffusion, magnetic energy is converted ohmically into heat at the same slow rate. However, in practice, the magnitude of τd is often far too large to explain the time-scale of dynamical cosmic processes. Nevertheless, Furth et al. (1963) showed how the diffusion can drive three distinct resistive instabilities at a rate which is often fast enough to be physically significant. These instabilities occur when the sheet is wide enough that τd ≫ τA, where τA = l/υA is the time it takes to traverse the sheet at the Alfvén speed υA = B0(μρ0)-½. The instabilities occur on time-scales τd(τA/τd)λ, where, and they have the effect of creating in the sheet many small-scale magnetic loops. In other words, resistive instabilities produce current filaments in current sheets (or, indeed, in any sheared structure) subsequently, the filaments and associated magnetic loops diffuse away, releasing magnetic energy in the process.

The gravitational and rippling modes (§6.3) occur when the density or resistivity varies in the direction across the sheet. They create a small-scale structure in the sheet (Fig. 6.1) and so are relatively harmless as far as the large-scale global stability of the configuration is concerned, although they may produce a turbulent diffusivity.

Although the existence of the Earth's magnetic field has been known since ancient times, the fact that it does not extend indefinitely into space, but is confined to a cavity, is a discovery of the twentieth century. This cavity is created by the solar wind, which shapes the Earth's field into a comet-like structure as shown in Fig. 10.1. The region in which the Earth's field is confined is called the magnetosphere, a term coined by T. Gold in 1959 before the true shape of this region had become known (Gold, 1959). Upstream of the magnetosphere there is a detached bow shock, which occurs because the solar wind flows past the Earth at a speed approximately eight times greater than either the Alfvén or sound speeds. The bow shock is located about 15 RE sunward of the Earth, where one Earth-radius (1 RE) is about 6,370 km. In front of the bow shock is a region called the fore shock, which contains energetic particles and waves. In MHD terms, the bow shock and fore shock together constitute a fast-mode MHD shock, while the bow shock itself may be thought of as a subshock (Kennel, 1988), where the solar wind is compressed. The region downstream of the bow shock is called the magnetosheath, and it is here that the shocked solar wind is deflected around the magnetospheric cavity.

The concept of the magnetosphere was first introduced by S. Chapman and V. Ferraro (1931), who thought the Earth's magnetic field would be confined to a cavity whenever the Earth is impacted by a plasma cloud ejected from the Sun.

Reconnection is not difficult to achieve in a laboratory environment. When two simple dipole magnets are held near each other in air, two null points will generally be present, and when the magnets are moved relative to each other, their field lines easily reconnect. It is only when a conducting plasma is present in the vicinity of a null point that reconnection starts to become difficult and therefore interesting.

The principal application of reconnection theory in the laboratory has been in the development of magnetic containment devices for controlled thermonuclear fusion, but plasma experiments have also been designed specifically to study reconnection dynamics. Containment devices try to confine a sufficiently hot plasma inside a magnetic bottle for a period long enough to achieve a sustained nuclear reaction. Reconnection can both hinder and help in this regard. For example, in one device (known as the tokamak, §9.1.2) reconnection is involved in several different instabilities which degrade the confinement, but in another device (known as the spheromak, §9.1.3) reconnection is necessary to create the field configuration which actually confines the plasma.

Laboratory experiments specifically designed to study reconnection dynamics are motivated by a desire to understand reconnection as a general physical process, in the hope that this knowledge can be applied to both fusion, space, and astrophysical applications. However, as with numerical simulations, laboratory experiments cannot easily replicate the conditions that occur outside the Earth, primarily because of the problem of scale. Laboratory devices typically have dimensions of a metre or less, which is many orders of magnitude smaller than occurs in cosmical applications.

In this chapter we look at two theories for time-dependent reconnection that are not as well known as the tearing mode. The first of these is X-type collapse, which was first considered by Dungey (1953) and has been briefly described in Section 2.1. The second is the time-dependent, Petschek-type theory developed by Semenov et al. (1983a). Both theories provide new perspectives on reconnection because they describe behaviour which is not encompassed within the scope of either the steady-state or tearing-mode theories.

X-Type Collapse

Dungey's (1953) work on X-type collapse is the earliest analysis ever done on magnetic reconnection and predates both the tearing-mode (Furth et al., 1963) and the Sweet–Parker (1958) theories. Dungey considered what happens when a small, but uniform, current perturbation is imposed at a current-free X-line (i.e., an X-point in any intersecting plane). Before the current is imposed, the separatrices are at right angles to one another, but after the current is added the separatrices scissor, as shown in Figure 7.1. Assuming that the plasma pressure in a strongly magnetized plasma can be ignored, Dungey argued that the initial perturbation would grow with time and rapidly lead to the formation of a current sheet at the X-line. Cowling (1953) objected that the growth of the current density would violate Lenz's Law, but this was eventually resolved by Dungey (1958), who pointed out the role of the v × B term in the evolution of the plasma.

Much of the impetus for the development of reconnection theory began with R. G. Giovanelli's efforts to develop an electromagnetic theory of solar flares. His first paper on this subject had the title “A Theory of Chromospheric Flares” (Giovanelli, 1946), which is somewhat outdated because flares are now thought of as primarily a coronal phenomenon rather than a chromospheric one. The prodigious advances in X-ray and radio astronomy since Giovanelli's time have made it increasingly clear that what is seen in the chromosphere by ground-based optical telescopes is a response to coronal activity. During a flare, the heated plasma in the corona becomes so hot (T > 107K) that the emitted radiation lies outside the visible portion of the spectrum. Thus, the coronal flare is imperceptible to telescopes observing in the visible spectrum.

Despite the title of his paper, Giovanelli proposed what is essentially a coronal theory of flares. He argued that the chromospheric emissions seen during a flare are produced by a bombardment of energetic electrons accelerated along field lines by electric fields in the vicinity of coronal magnetic null points. This concept still prevails today. However, Giovanelli thought that the electric fields could be produced by induction as the magnetic fields in the photosphere shifted their position slowly over time. Nowadays, such a process is considered to be far too slow to produce a sufficiently strong and impulsive electric field.

After Dungey's (1953) pioneering work on reconnection, Sweet (1958a) and Parker (1957) reconsidered Giovanelli's neutral-point model as a reconnection process which taps the energy stored in the magnetic field associated with coronal currents.

Introduction

In most of the universe the magnetic Reynolds number (Rm, §1.2.2) is very much larger than unity and so the magnetic field is attached very effectively to the plasma. It is only in extremely thin regions where the magnetic gradients are typically a million times or more stronger than normal that the magnetic field can slip through the plasma and reconnect. Thus, for example, a field line initially joining a plasma element at A to one at B in Fig 4.1 may be carried towards another oppositely directed field line CD and a narrow region of very strong magnetic gradient (containing an X-type neutral point) may be formed between them. Then the field lines may diffuse, break, and reconnect, so that element A becomes linked instead to element C (Fig. 4.1).

There are several important effects of this local process:

1. (i) The global topology and connectivity of field lines change, affecting the paths of fast particles and heat conduction, since these are directed mainly along field lines;

2. (ii) Magnetic energy is converted to heat, kinetic energy, and fast particle energy;

3. (iii) Large electric currents and electric fields are created, as well as shock waves and filamentation, all of which may help to accelerate fast particles (Chapter 13).

As we discussed in Chapter 1, two questions that many of the early researchers tried to answer are: what is the nature of field-line breaking and reconnection when it takes place in a steady-state manner; and what is the rate at which it occurs - that is, what is the speed with which magnetic field lines can be carried in towards the reconnection site?

The application of reconnection theory to astrophysical systems is a relatively recent development in comparison with applications to the terrestrial magnetosphere and the solar corona. The extreme remoteness of objects outside our solar system presents an enormous challenge for plasma physicists, because there are few spatially resolved observations on stellar scales with which to constrain theory. However, advances in Doppler imaging and the development of high-resolution instruments such as the Hubble Space Telescope are beginning to provide some help. Astrophysical magnetism is a huge field which we can only touch upon briefly here, but for an in-depth account the reader is referred to the new monograph by Mestel (1999).

The two astrophysical topics to which reconnection theory has been extensively applied are stellar flares (Mullan, 1986) and accretion disks (Verbunt, 1982). The analysis of stellar flares relies heavily upon the assumption that they are basically similar to solar flares except more energetic (e.g., Gershberg, 1983; Poletto et al., 1988). Flare stars can release 104 to 106 times the amount of energy seen in a large solar flare, but only modest increases in magnetic field strengths and scale-sizes are required to account for this extra amount. The use of reconnection theory in accretion disks has a dual purpose. One is to explain flare-like outbursts generated within disks, and the other is to account for the viscosity needed to allow material in the disks to fall inwards.

Introduction

Critical discussions of membership of individual galaxies in the Local Group have recently been given by van den Bergh (1994a,b, 1999), by Grebel (1997), and by Mateo (1998). A galaxy distance ≲1.5 Mpc was used as a preliminary selection criterion for Local Group membership. A detailed discussion of additional selection criteria will be given in Chapter 16 of the present volume, which deals with the membership of individual galaxies located near the outer fringes of the Local Group. Table 2.1 presents a summary of the observational parameters for the 35 most probable members of the Local Group. This table lists the name and (where appropriate) one alias for each Local Group member, its DDO classification (mostly from van den Bergh 1966a), the J2000 coordinates of each galaxy, its observed heliocentric velocity, its integrated magnitude in the V band, and its reddening E(B – V). The derived parameters for each of these galaxies, basedon the discussion in the present volume, will be given in Table 19.1.

Incompleteness of the sample

The fact that IC 1613 (MV = -14.9) has been known (Dyer 1895) for about a century indicates that our Local Group sample (at least outside the zone of avoidance at low Galactic latitudes) is almost certainly complete for objects brighter than MV = -15. The fact that Irwin (1994) discovered only a single new Local Group member during a survey of ∼ 20,000 square degrees at high Galactic latitudes suggests that the search for Galactic satellites at high latitudes is probably complete to at least MV =-10.

Introduction

We mainly owe the “discovery” of the fact that the Milky Way system is a galaxy to the work of Shapley, Lindblad, and Oort. Shapley (1918a) noted that “we may say confidently that the plane of the Milky Way is also a symmetrical plane in the great system of globular clusters.” He (Shapley 1918b) also found that “The center [of this system of globular clusters], which lies in the region of the rich star clouds of Sagittarius near the boundary of Scorpio and Ophiuchus, has the coordinates R.A. = 17h 30m, Decl.= –30°.” Shapley concluded that “A consideration of the foregoing results leads naturally to the conclusion that globular clusters outline the extent and arrangement of the total [G]alactic organization.” Shapley's discovery may be thought of as the second Copernican revolution. Copernicus had shifted the center of the Universe from the Earth to the Sun, and Shapley took the even greater step of moving it from the solar system to the center of the Galaxy. Subsequently Lindblad (1927) and Oort (1927, 1928) were able to show that (a) the Galactic disk is in rapid differential rotation about a center that coincides with that of Shapley's globular cluster system and that (b) globular clusters and high-velocity stars rotate more slowly around the same center than does the Galactic disk. The notion that the Milky Way system is a spiral galaxy received its ultimate confirmation when Morgan, Whitford & Code (1953) were able to outline three Galactic spiral-arms in the vicinity of the Sun. A year later, the existence of global spiral-structure in the Milky Way was demonstrated by the 21-cm observations of Westerhout (1954) and Schmidt (1954).

Introduction

To the naked eye the Magellanic Clouds appear as detached portions of the Milky Way. They have probably been known to the inhabitants of the southern hemisphere for thousands of years. In the north the Large Cloudwas, like M31, already known to al-Sufi in the 10th Century. The Large Magellanic Cloud (see Table 6.1) is the largest and the brightest (external) galaxy in the sky. Because of this proximity (D ≈ 50 kpc), its stellar content can be studied in more detail than that of any other external galaxy. Observations that require a 5-m class telescope in M31 can be carried out with a 0.5-m telescope in the LMC. The Large Cloud belongs to the barred subtype of Hubble's irregular class. Its DDO classification is Ir III–IV [i.e., it has a morphology intermediate between that of giant (III) and subgiant (IV) galaxies]. Based on the presence of a faint streamer of nebulosity, which extends from α = 5h, δ = –73° to α = 3ḥ, δ = –55° (de Vaucouleurs 1954a,b, 1955), the Large Cloud xhas often been described as a late-type spiral. However (de Vaucouleurs & Freeman 1972), this spiral arm–like feature actually appears to be a faint streamer of Galactic foreground nebulosity. This feature in the constellation Mensa has also been observed in the infrared. From multicolor photometry and high-dispersion photometry of 38 stars in the vicinity of this filament Penprase et al. (1998) conclude that it produces a mean reddening E(B – V) ∼ 0.17 and is located at a distance of 230±30 pc.

Introduction

The galaxy M33 (= NGC598) is the third-brightest member of the Local Group. It is a late-type spiral of type Sc II–III. The large angular size of the Triangulum galaxy, and its intermediate inclination i = 56° (Zaritsky, Elston & Hill 1989), make it particularly suitable for studies of spiral structure and stellar content (see Figure 5.1). Only M31, the Magellanic Clouds, and the tidally disrupted Sagittarius dwarf have larger angular diameters than the Triangulum galaxy. The spiral nature of this galaxy was first hinted at by visual observations made by the Earl of Rosse (1850).

The modern era of exploration of M33 began with the independent discovery of variable stars in this object by Duncan (1922) and by Wolf (1923). Neither of these papers show any indication that these authors anticipated the revolutionary impact that the discovery of variable stars in nearby “nebulae” would soon have on Man's view of the Universe. In the words of Hubble (1926) “The nature of nebulae and their place in the scheme of the universe have been favorite subjects of controversy since the very dawn of telescopic observations.” Hubble writes that his investigation “followed naturally upon the partial resolution of Messier 33 into swarms of actual stars.” He concluded that “The data lead to a conception of the object as an isolated system of stars and nebulae, lying far outside the limits of the [G]alactic system.” In his paper Hubble was able to show that 35 of the variables in M33 were classical Cepheids, thus demonstrating, beyond reasonable doubt, that galaxies were “island universes, ” and ending the reat debate (Heatherington 1972, Hoskin 1976) on the nature of spiral nebulae.