Active regions

The concentrations of strong magnetic field occur in characteristic time-dependent configurations. A prominent configuration is the bipolar active region, which is formed during the emergence of strong magnetic flux. Around the time of its maximum development, a large active region comprises sunspots, pores, and faculae arranged in plages and enhanced network (Fig. 1.2). In this book, the term active region is used to indicate the complete area within a single, smooth contour that just includes all its constituents. We prefer the term active region over the older term center of activity. Classical descriptions of active regions are found in Kiepenheuer (1953, his Section 4.12), De Jager (1959), and Sheeley (1981).

The present chapter is restricted to simple bipolar active regions. Throughout this book we use the term bipolar active region for regions consisting of only two fairly distinct areas of opposite polarity, in contrast with complex active regions in which at least one of the polarities is distributed over two or more areas. Bipolar active regions are building blocks in complex active regions and nests (Section 6.2.1).

At maximum development, directly after all magnetic flux has emerged, bipolar active regions range over some 4 orders of magnitude in magnetic flux, size, and lifetime; see Table 5.1. The division between small active regions and ephemeral active regions is based on an arbitrary historical choice: there is no qualitative difference between the two groups of bipolar features other than gradual trends.

1897 Discovery of electron

1900 α, β and γ radioactivity

1905 Photon identified as quantum of electromagnetic field

1911 Discovery of atomic nucleus

1912 Discovery of cosmic rays

Invention of cloud chamber

1913 Bohr model of atom

1919 Discovery of proton

1923 de Broglie wave–particle duality

1925 Introduction of electron spin

1926 Wave mechanics

1927 Uncertainty Principle

1928 Dirac wave equation

1930 Neutrino hypothesis

1931 Operation of first cyclotron and of Van der Graaff accelerator

1932 Discovery of positron

Discovery of neutron

1933 Discovery of electromagnetic showers

1934 Theory of beta decay

Discovery of Čerenkov effect

1935 Yukawa theory of nuclear forces

1936 Breit–Wigner resonance formula

1937 First evidence for mesotron (= muon)

1939 Observation of mesotron (= muon) decay

1940 Spin-statistics theorem

1945 Phase stability in accelerators (synchrotron principle)

Introduction

Until recently, only surface measurements of the solar rotation rate were available. Since the mid-1980s, with the advent of helioseismology, much has been learned about the internal rotation of the Sun through the inversion of p-mode frequency splittings. As was noted in Section 1.2.2, it now appears that the observed surface pattern of differential rotation with latitude prevails throughout most of the solar convection zone, with equatorial regions moving faster than higher latitudes. In contrast, the underlying radiative core appears to rotate nearly uniformly down to r ≈ 0.1−0.2R⊙, at a rate that is intermediate between the polar and equatorial rates of the photosphere. Within the central region r ≲ 0.2R⊙, some measurements suggest that the angular velocity increases with depth, implying rotation at a rate between 2 and 4 times that of the surface; other measurements strongly suggest, however, that the solar inner core rotates rigidly down to the center.

The problem presented by the observed solar differential rotation is one of long standing and many efforts have been made to formulate a plausible flow pattern that reproduces the large-scale motions in the solar atmosphere. Following Lebedinski's (1941) pioneering work, many theories have been proposed to explain how the equatorial acceleration originated and is maintained in the solar convection zone.

The present situation looks very good for the inflationary cosmology. As we hope that we have demonstrated, the picture of structure formation based on the inflationary cosmology is very complete and elegant, bringing together quantum theory for the origin of the perturbations, general relativity for their evolution, and a variety of astrophysical processes to make the final link to observations.

Inflation-based structure formation models appear an excellent paradigm within which to understand the inhomogeneous Universe in which we live, even though, as we write, it is rather unclear precisely which type of model might best fit the data. For instance, should the Universe have critical density, and if so, is a component of hot dark matter necessary? Should it instead have lower density, with or without a cosmological constant? What is clear is the success of the basic paradigm, that everything originates from a Gaussian, adiabatic, density perturbation with spectral index close to 1.

For the future, we await improved observations to see if this picture can be sustained. Many types of observation will probe the cosmological modelling and, for inflationary cosmologists, none is more eagerly awaited than improved measurements of the microwave background anisotropies, which directly probe linear perturbation theory. The Microwave Anisotropy Probe and Planck satellites promise data of unprecedented quality when it comes to constraining inflationary and cosmological parameters, and before then, many ground- and balloon-based experiments should accurately probe at least the first acoustic peak; for them, any significant spatial curvature is easy prey, and other parameters may be pinned down too.

Classification

The weak interactions between quarks and leptons are those mediated by the socalled intermediate bosons W± and Z0, with a coupling which we generically label g. As indicated in Figure 7.1, there is a strong similarity between the Feynman diagrams for electromagnetic interactions mediated by photon exchange and weak interactions mediated by the intermediate bosons.

A familiar interaction is that between two straight wires hanging close together when electric currents – i.e. a flow of electrons and ions – pass through them. So, we can view the interaction of Figure 7.1(a) as that of two conserved electric currents jem, carried in this case by the electron e and the quark Q. Similarly, the interactions shown in Figures 7.1(b) and (c) can be viewed as between weak currents jweak. The difference is that these currents consist of a flow of conserved weak charge g rather than electric charge, e. In either case, these currents will contain the product of the (normalised) wavefunctions of the ‘in’ and ‘out’ particles that occur in the matrix element (2.16) for the interaction. Thus j ∝ ψ* ψ, where ψ* denotes an incoming and ψ an outgoing amplitude.

In Figure 7.1(b) one sees that the electric charges associated with the weak current actually change in the interaction, while in Figure 7.1(c) they do not. Rather inaccurately, these two are referred to as charged-current weak interactions (those in which the electric charge of each weak current changes) and neutral-current weak interactions (those in which the electric charges do not change).

Historical sketch of the study of stellar activity

It has long been known that emission components are present in the cores of the Ca II H and K resonance lines in the spectra of many stars of spectral type G and later. The discovery paper by Schwarzschild and Eberhard (1913) was followed by a stream of usually brief papers from which lists of stars with references were collected by Joy and Wilson (1949) and Bidelman (1954). In the early 1950s, a vigorous study of the Ca II H and K resonance lines in stellar spectra was started by Olin C. Wilson at the Mt. Wilson Observatory; see Section 9.3. This research was boosted once more when in 1977 the efficient Ca II HK photometer (Vaughan et al., 1978b) was installed at the Mt. Wilson 60-in. telescope. The productivity of the Ca II HK photometer was enhanced by letting researchers (including Ph.D. students) from other institutes use the instrument. The Catania Observatory has a long tradition of pursuing solar and stellar observations in parallel, also aimed at a better understanding of magnetic activity (Godoli, 1967).

From the early investigations onward, the researchers were aware that the mechanisms causing the stellar Ca II H and K emission are probably similar to those of the solar chromosphere.

The central premise of modern cosmology is that, at least on large scales, the Universe is homogeneous and isotropic. This is borne out by a variety of observations, most spectacularly the nearly identical temperature of microwave background radiation coming from different parts of the sky. Despite the belief in homogeneity on large scales, it is all too apparent that in nearby regions the Universe is highly inhomogeneous, with material clumped into stars, galaxies, and galaxy clusters. It is believed that these irregularities have grown over time, through gravitational attraction, from a distribution that was more homogeneous in the past.

It is convenient then to break up the dynamics of the Universe into two parts. The largescale behaviour of the Universe can be described by assuming a homogeneous and isotropic background. On this background, we can superimpose the short-scale irregularities. For much of the evolution of the Universe, these irregularities can be considered to be small perturbations on the evolution of the background Universe, and can be tackled using linear perturbation theory; we discuss this extensively, starting in Chapter 4. It is also possible to continue beyond the realm of linear perturbation theory, via a range of analytic and numerical techniques, which we discuss only briefly, in Chapter 11. In this chapter and the next, we concern ourselves solely with the evolution of the background, isotropic Universe. This usually is called the Robertson Walker Universe, often with Friedmann and occasionally with Lemaitre also named.

When I wrote my first book – Theory of Rotating Stars (Princeton: Princeton University Press, 1978) – I was not aware of the fact that the 1970s were a period of transition and that major unexpected developments would take place in the field of stellar rotation during the 1980s.

In the mid-1970s, we had no direct information about the internal rotation of the Sun. Little was known about the rotation of main-sequence stars of spectral type G and later, although it was already well established that the surface rotation rate of these stars decayed as the inverse square root of their age. We certainly had much more information about axial rotation in the upper-main-sequence stars, but the actual distribution of specific angular momentum within these stars was still largely unknown. On the theoretical side, important progress in the study of rotating stars had been made by direct numerical integration of the partial differential equations of stellar structure. However, because there was no clear expectation for the actual rotation law in an early-type star, the angular momentum distribution always had to be specified in an ad hoc manner. The presence of large-scale meridional currents in a stellar radiative zone was also a serious problem: All solutions presented to date had unwanted mathematical singularities at the boundaries, and the back reaction of these currents on the rotational motion had never been properly taken into account.

The magnetic field that corresponds to conspicuous features in the solar atmosphere is found to be confined to relatively small magnetic concentrations of high field strength in the photosphere. Between such concentrations, the magnetic field is very much weaker. The strong-field concentrations are found at the edges of convective cells: convective flows and magnetic field tend to exclude each other. These time-dependent patterns indicate that the magnetic structure observed in the solar atmosphere is shaped by the interplay between magnetic field, convection, and large-scale flows.

The construction of a comprehensive model for the main phenomena of solar magnetic activity from basic physical principles is beyond our reach. The complex, nonlinear interaction between turbulent convection and magnetic field calls for a numerical analysis, but to bring out the main observed features would require simulations of the entire convection zone and its boundary layers. The intricacy of convective and magnetic features indicates that extremely fine grids in space and in time would be required. Such an ambitious program is far beyond the power of present supercomputers. Hence, we must gain insight into the solar magnetic structure and activity first by studying the observational features, and subsequently by trying to interpret and model these with the theoretical and numerical means at hand.

Our approach is to map the domain of solar magnetic activity by a mosaic of models. Some of these models are well contained; others are based on ad hoc assumptions.

Historical development

The study of stellar rotation began at the turn of the seventeenth century, when sunspots were observed for the first time through a refracting telescope. Measurements of the westward motion of these spots across the solar disk were originally made by Johannes Fabricius, Galileo Galilei, Thomas Harriot, and Christopher Scheiner. The first public announcement of an observation came from Fabricius (1587–c. 1617), a 24-year old native of East Friesland, Germany. His pamphlet, De maculis in Sole observatis et apparente earum cum Sole conversione, bore the date of dedication June 13, 1611 and appeared in the Narratio in the fall of that year. Fabricius perceived that the changes in the motions of the spots across the solar disk might be the result of foreshortening, with the spots being situated on the surface of the rotating Sun. Unfortunately, from fear of adverse criticism, Fabricius expressed himself very timidly. His views opposed those of Scheiner, who suggested that the sunspots might be small planets revolving around an immaculate, nonrotating Sun. Galileo made public his own observations in Istoria e Dimostrazioni intorno alle Macchie Solari e loro Accidenti. In these three letters, written in 1612 and published in the following year, he presented a powerful case that sunspots must be dark markings on the surface of a rotating Sun. Foreshortening, he argued, caused these spots to appear to broaden and accelerate as they moved from the eastern side toward the disk center.

Classical and quantum pictures of interactions

Classically, interaction at a distance, e.g. in electromagnetism, is described in terms of a potential or field due to one charged particle acting on another, the field permeating the whole of space around the source charge. A conceptually more appealing idea, perhaps, is that of an exchange interaction, where the object exchanged carries momentum from one charge to the other, the rate of exchange of momentum providing the force. A frequently quoted everyday example is of two ice-skaters travelling side by side, initially moving parallel, who start to diverge as they repel one another by throwing a ball back and forth.

In quantum theory, action at a distance is indeed viewed in terms of an exchange interaction, the exchange being of a specific quantum (a boson) associated with the particular type of interaction. Since the quantum carries momentum and energy, say ΔE, the conservation laws can only be satisfied if the process takes place within a timescale Δt limited by the Uncertainty Principle, i.e. ΔE Δt ≃ ħ. Such transient quanta are said to be virtual.

The quantum concept of an electromagnetic interaction is of the continual exchange (emission and absorption) of virtual photons by the charges, which is no more or less fictitious than the classical concept of an all-pervading field surrounding a source charge. Neither the field nor the virtual quanta are directly observable; it is the force that is the measured quantity.

Introduction

Consider a single star that rotates about a fixed direction in space, with some assigned angular velocity. As we know, the star then assumes the shape of an oblate figure. However, we are at once faced with the following questions. What is the geometrical shape of the free boundary? What is the form of the surfaces upon which the physical variables (such as pressure, density, …) remain a constant? To sum up, what is the actual stratification of a rotating star, and how does it depend on the angular velocity distribution? For rotating stars, we have no a priori knowledge of this stratification, which is itself an unknown that must be derived from the basic equations of the problem. This is in sharp contrast to the case of a nonrotating star, for which a spherical stratification can be assumed ab initio.

In principle, by making use of the equations derived in Section 2.2, one should be able to calculate at every instant the angular momentum distribution and the stratification in a rotating star. Obviously, this is an impossible task at the present level of knowledge of the subject, even were the initial conditions known. Until very recently, the standard procedure was to calculate in an approximate manner an equilibrium structure that corresponds to some prescribed rotation law, ruling out those configurations that are dynamically or thermally unstable with respect to axisymmetric disturbances (see Sections 3.4.2 and 3.5).

Introduction

In Section 3.3.1 we noted that the conditions of mechanical and radiative equilibrium are, in general, incompatible in a rotating barotrope. This paradox can be solved in two different ways: Either one makes allowance for a slight departure from barotropy and chooses the angular velocity Ω = Ω(ϖ, z) so that strict radiative equilibrium prevails at every point or one makes allowance for large-scale motions in meridian planes passing through the rotation axis. The first alternative is mainly of academic interest because there is no reason to expect rotating stars to select zero-circulation configurations. Moreover, these baroclinic models are thermally unstable with respect to axisymmetric motions, as well as dynamically unstable with respect to nonaxisymmetric motions (see Sections 3.4 and 3.5). Hence, the slightest disturbance will generate three-dimensional motions and, as a result, a large-scale meridional circulation will commence. The second alternative was independently suggested by Vogt (1925) and Eddington (1925), who pointed out that the breakdown of strict radiative equilibrium in a barotrope tends to set up slight rises in temperature and pressure over some areas of any given level surface and slight falls over other areas. The ensuing pressure gradient between the poles and the equator thereby causes a flow of matter. In fact, it is the small departures from spherical symmetry in a rotating star that lead to unequal heating along the polar and equatorial radii, which in turn causes large-scale currents in meridian planes.