Introduction

Chapter 5 described the one-dimensional characteristics of the solar activity cycle in terms of variations of the scalar function N(t), where N is a number representing the number of sunspots (or sunspot groups or faculae or any other indicator) and t is the time. The butterfly diagram, shown again in Figure 8.1, provides a two-dimensional characterization of the cycle in terms of a function N(λ, t), where λ is the latitude, a representation which yields additional information of obvious importance to the heuristic models discussed in Chapter 6. (Strictly speaking, the existence of active longitudes emphasizes that N is a function of three variables, N(λ, φ, t), where φ is the Carrington longitude.)

It has been known for ∼ 130 years that the wings of the butterfly diagram overlap to some extent. The overlap is marginal between some cycles, e.g. 18 and 19, but in other cases, e.g. 19 and 20, it extends over ∼ 2 years. Sunspot minimum, therefore, is a point on the one-dimensional plot determined partly by the decay of the old cycle and partly by the rise of the new.

In the one-dimensional approach, all activity lying between successive minima is associated with that particular cycle but, in the two-dimensional approach of the butterfly diagram, active regions of each cycle are distinguished by two properties: latitude and orientation of the magnetic axis (see§ 2.10). Spots of the new cycle should appear at higher latitudes (20° – 40°) and exhibit a reversed magnetic orientation compared with those of the old cycle for a given hemisphere.

Introduction

The measurement and interpretation of the travel times of earthquake signals have been used for many years to study the structure of the Earth's interior. In addition to the classical method, observations of free oscillations have been used since the great Chilean earthquake of 1960, and this branch of seismology has been recently applied to the Sun with considerable success. For a given solar model, the eigenfunctions and eigenfrequencies of the radial velocity perturbations may be calculated by standard methods and compared to the observed frequencies. Improved models of the internal structure of the Sun may be obtained by bringing as many calculated frequencies as possible into agreement with observed frequencies. Such calculations have, for example, forced the rejection of ‘low-Z’ models of the solar interior which, it had been hoped, might resolve the solar neutrino problem (Christensen-Dalsgaard and Gough 1982).

Of particular interest for studies of cyclic phenomena is the rotational modulation of the eigenmode frequencies. Because the eigenfunctions of the many non-radial p-modes exhibit different depth and latitude dependence, the measurement of frequency splittings in the intermediate orders of the waves has led to inferences regarding the internal rotation rate as a function of depth and latitude (Duvall et al. 1986, Brown and Morrow 1987, Morrow 1988, Brown et al 1989, Libbrecht 1989). Surprisingly, these results at first suggested that, within certain error bars, the angular velocity is independent of radial distance across the convection zone; i.e. that the differential rotation (with latitude) at the base of the convection zone is very similar to that at the surface.

Introduction

It is now time to summarize and conclude but, regrettably, it is not possible to follow the King's advice, for the end of this story is not yet in sight. As in any good detective story, both clues and red herrings have been scattered liberally throughout the foregoing chapters. Unlike the novelist, however, we cannot be certain which is which and so cannot now discard the red herrings, marshal the significant clues, and reveal the solution. At best, we can indicate what appear to be the important themes which must form part of the solution and suggest the crucial areas of investigation which may lead to a satisfactory resolution.

Two important questions currently remain without clear and unequivocal answers:

1. (i) Do both cyclic and non-cyclic stellar activity arise as a result of dynamo action and, if so, of what kind (or kinds)?

2. (ii) Is the aperiodicity of solar cyclic activity evidence of a chaotic system, and, if so, can the existence of a strange attractor be demonstrated and its dimension determined?

Until firm answers can be provided for these questions, the problem of cyclic stellar activity remains without a satisfactory solution.

The input from stellar cycles

Because of the more extensive data-base regarding the solar cycle, the discussions in this volume are weighted heavily in that direction, but this in no way implies a greater importance to solar over stellar studies. Although still in their infancy, studies of stellar activity have already made a significant contribution to our understanding of the problem of cyclic activity.

The discovery of sunspots

Although naked-eye observations of sunspots have been recorded sporadically since the first Chinese observations several centuries before the birth of Christ, the year 1611, when sunspots were observed for the first time through the telescope, marks the beginning of the science of astrophysics. Four men share the honour of this discovery: Johann Goldsmid in Holland (1587-1616), Galileo Galilei in Italy (1564-1642), Christopher Scheiner in Germany (1575-1650), and Thomas Harriot in England (1560-1621). It is uncertain which of this international quartet made the first observations, but priority of publication belongs to Goldsmid, or Fabricius, as he is known by his Latinized name. Although his equipment was probably inferior to that of Galileo or of Scheiner, Fabricius made observations of sunspots and used them to infer that the Sun must rotate but did not carry this work beyond these initial observations.

When Scheiner, a Jesuit priest teaching mathematics at the University of Ingolstadt, first observed the spots, he suspected some defect in his telescope. He soon became convinced of their actual existence but failed to persuade his ecclesiastical superiors, who refused to allow him to publish his discovery. This indignity was later shared by the French astronomer, Messier, who in 1780 was similarly prevented from announcing his observation of the return of Halley's comet in that year. Regrettably, such instances of scientific censorship are not uncommon and, in Scheiner's case, played a major role in the controversy that led to the denouncement of Galileo to the Italian inquisition.

Introduction

Although not the most spectacular phenomenon of the solar cycle and undetectable in stellar cycles, the large-scale magnetic field patterns on the Sun play an important role in solar cyclic activity and in the attempts to understand the solar cycle discussed in Chapter 6. In relaxation models such as the Babcock model (see§ 6.2), the reversal of the polar fields by the poleward drift of large-scale fields is crucial, and in the Leighton-Sheeley flux-transport model the large-scale fields arise solely from the decay of active-region fields.

However, magnetic flux emerges at the surface of the Sun in the form of magnetic bipoles whose dimensions range across a wide spectrum, from the largest active regions with dimensions up to 100 000 km, down to the small intra-network elements of order 500 km. Stenflo (1992) has noted that the total flux emerging in the smaller elements exceeds that emerging as large active regions by several orders of magnitude, and there is no a priori reason why regions from the large-scale end of this spectrum should be the only\ contributors to the large-scale field patterns and thus to the reversal of the polar fields.

In this chapter we describe recent studies of the evolution of the large-scale field patterns at the beginnings of Cycles 20, 21, and 22. In Chapter 10 we look at the polar fields near the maximum of Cycle 22.

The solar-stellar connection

Hale's conviction that solar physics is an essential component of astrophysics was shared by some, but unfortunately not by all, astrophysicists. Nevertheless, a further important initiative in this spirit was taken at Mount Wilson in 1966, when, using the 100-inch Hooker telescope, Olin Wilson and his colleagues began a long-term study of 91 cool dwarf stars (Wilson 1978). In 1978 this project (the ‘HK Project’) was transferred to the Mount Wilson 60-inch telescope, which was dedicated entirely to the continuation and extension of this work and has become identified with a particular methodology known as the solar-stellar connection.

Although the Sun permits the detailed two-dimensional study of its activity phenomena, it exhibits only a single set of stellar parameters, since its mass, size, composition, and state of evolution are necessarily fixed at this point of time. On the other hand, stars, as observed from earth, are essentially one-dimensional objects, but they offer a wide range of physical parameters which permit a more thorough testing of theories and conjectures regarding common phenomena than is possible for the Sun alone. The solar-stellar connection aims to bring these two lines of investigation together in order to further our understanding of the properties of the Sun and other late-type stars.

Even before the solar-stellar connection methodology became recognized as such, there were many examples of the successful application of solar results to stellar investigations.

Introduction

The standard concept of the sunspot cycle is of an 11-year variation in the number of sunspots present on the Sun, N(t), at time t. The data from which N(t) must be determined are the daily values of the Zurich sunspot number Rz (defined in §2.5), but, because only half the Sun is visible at any one time, Rz is a measure of the number of spots on the visible hemisphere, and it is not possible, even in principle, to determine N(t) at any instant t. For this reason N(t) must be derived from a time average of Rz over a longer period, such as a Carrington rotation or a calendar year. The variable t is therefore discrete rather than continuous, and the function N(t) is strictly a sequence, Ni, which represents twice the mean value of Rz during the ith time interval.

The sunspot number Rz is not the only scalar quantity that exhibits cyclic variations with an 11-year period. Other such quantities include total sunspot area, active region counts, flare counts, the strength of Call emission, the 10 cm radio flux, the incidence of aurorae, the flux of cosmic rays as measured through certain indicators, and even the widths of terrestrial tree rings. Each of these quantities exhibits a slightly different pattern of variation, and the investigation of the various time series can provide different insights into the nature of the cycle. Such a variety of indicators must, however, raise the question as to what is the fundamental physical variable which generates these variations in secondary phenomena which we call the solar cycle.

Introduction

Chapter 1 discussed the relationship of the solar cycle to the terrestrial environment and offered the hope that a greater understanding of solar and stellar cycles might lead to improved predictions of the parameters of a given solar cycle and, consequently, more reliable forecasts of solar-induced geospheric phenomena.

The more important terrestrial consequences arise during and just after sunspot maximum, when the EUV-UV and total solar irradiance are also at a maximum, along with the probability of occurrence of large solar flares, particle bursts, and solar cosmic rays. These enhanced solar emissions disrupt communications, shorten the orbital lifetimes of satellites in low Earth orbit, cause failure in solid-state components in satellites, and generally introduce major or minor disruptions to our environment (see §1.3). The likelihood of the occurrence of disruptive events follows closely the intensity of the activity cycle, so that accurate forecasting of solar activity on time scales of weeks, months, and years is of considerable importance to those agencies which plan and operate space missions.

Apart from space missions, other possible terrestrial effects of the influence of solar activity are discussed in Chapter 1 and elsewhere in this book. Perhaps the most significant for us and our descendants are the small variations of the solar output which accompany the activity cycle. Although the measured variations are only a few tenths of a per cent, it is generally accepted that, because of the non-linear nature of the interaction of solar radiation with the geosphere (which is poorly understood), the effect of these small fluctuations on our environment may be considerably amplified.

Introduction

Recent work in the theory of non-linear dynamical systems has centred on the concept of chaos, a term that applies to a great variety of situations and configurations. This relatively new subject is fascinating in its own right, and the rapidly growing body of knowledge surrounding it has uncovered a number of characteristics shared by all chaotic systems. The concept has not only achieved an extensive currency throughout the mathematical and scientific communities but has also captured the interest of many in the nonmathematical world, the latter largely due to an excellent popular discussion of the history and basic ideas by James Gleick (1988)

If the dynamics of the solar magnetic field are due to magnetoconvective dynamo action within the Sun, then the activity cycle is governed by the non-linear equations of magnetohydrodynamics, discussed in Chapter 11. A number of investigators have suggested that solar and stellar activity cycles are chaotic phenomena and have begun to explore the implications of cyclic systems which are chaotic.

If stellar activity cycles are indeed examples of chaotic systems, then they will share in the universal characteristics of such systems. In order to discuss the implications for cyclic activity, a brief outline of the relevant concepts in the theory of chaos is called for.

Introduction

The recognition that magnetic fields are an essential component not only of solar and stellar activity but also of the structure of galaxies, quasars, and pulsars has focussed considerable theoretical interest on the origin and maintenance of cosmic magnetic fields. Since the length scales associated with many cosmic magnetic fields are very large, the ohmic decay times (see §4.1 and below) are long, and there is no difficulty in explaining the continued existence of primordial or fossil fields, such as the megagauss fields found in magnetic A-type stars; but the changes observed to occur in many cosmic magnetic fields, over periods which may be short compared with the decay time, entail an interaction between the plasma motions and the existing fields which may also maintain these fields against ohmic decay. This has become known as dynamo action, and, in order to understand evolutionary changes occurring in the solar magnetic cycle, it is necessary to probe further into the underlying theory.

Parker (1970) drew attention to the curious asymmetry throughout the universe between electric and magnetic charge on the one hand, and the corresponding fields on the other.

Basic data

The historical studies traced in the previous chapter provided an introduction to our knowledge of the structure of the Sun and of cyclic activity. We now offer a brief summary of the general state of our knowledge of the physical properties of the Sun and of solar-type stars, together with some basic theory relevant to an understanding of cyclic phenomena. The interested reader who desires further information is referred to the more general accounts listed in the references (e.g. Mihalas 1978, Foukal 1990, Stix 1989, Zirin 1989).

Stars are generally classified according to their luminosity and surface temperature, a classification scheme which has been codified as the Hertzprung-Russell (H-R) diagram. In this diagram the absolute magnitude (or logarithm of the total luminosity) is plotted against the logarithm of the surface temperature. In the Harvard classification scheme the categories O, B, A, F, G, K, M, R, and S represent decreasing surface temperatures and increasingly complex spectra, and the Sun (of type G2) sits squarely in the middle. It is about 4.5 x 109 years old, less than half the age of the oldest stars in our galaxy, and in these, as in many other respects, it is a very ‘ordinary’ star.

The central aim of this book is the development of the results and techniques needed to determine when it is possible to extend a space-time through an “apparent singularity” (meaning, a boundary-point associated with some sort of incompleteness in the space-time). Having achieved this, we shall obtain a characterisation of a “genuine singularity” as a place where such an extension is not possible. Thus we are proceeding by elimination: rather than embarking on a direct study of genuine singularities, we study extensions in order to rule out all apparent singularities that are not genuine. It will turn out, roughly speaking, that the genuine singularities which then remain are associated either with some sort of topological obstruction to the construction of an extension, or with the unboundedness of the Riemann tensor when its size is measured in a suitable norm.

I had at one stage hoped that there would be a single simple criterion for when such an extension cannot be constructed, which would then lay down once and for all what a genuine singularity is. But it seems that this is not to be had: instead one has a variety of possible tools and concepts for constructing extensions, and when these fail one declares the space-time to be singular on pragmatic grounds. The main such tools are the use of Hölder and Sobolev norms of functions, used for measuring the extent to which the metric or the Riemann tensor is irregular.