A very important concept in physics is the symmetry or invariance of the equations describing a physical system under an operation – which might be, for example, a translation or rotation in space. Intimately connected with such invariance properties are conservation laws – in the above cases, conservation of linear and angular momentum. Such conservation laws and the invariance principles and symmetries underlying them are the very backbone of particle physics. However, one must remember that their credibility rests entirely on experimental verification. A conservation law can be assumed to be absolute if there is no observational evidence to the contrary, but this assumption has to be accompanied by a limit set on possible violations by experiment.

The transformations to be considered can be either continuous or discrete. A translation or rotation in space is an example of a continuous transformation, while spatial reflection through the origin of coordinates (the parity operation) is a discrete transformation. The associated conservation laws are additive and multiplicative, respectively.

Translation and rotation operators

In an isolated physical system, free of any external forces, the total energy must be invariant under translations of the whole system in space. Since there are no external forces, the rate of change of momentum is zero and the momentum is constant. So invariance of the energy of a system under space translations corresponds to conservation of linear momentum. Similarly, invariance of the energy of a system under spatial rotations corresponds to conservation of angular momentum.

Introduction

An inspection of Figure 1.6 shows that the mean projected equatorial velocity of main-sequence stars increases slowly with spectral type, reaching a maximum of about 200 km s−1 in the late B-type stars. Thence, the mean velocity 〈ν sin i〉 decreases slowly for later spectral types until about F0, where it starts dropping precipitously through the F-star region. As is well known, this rapid transition to very small rotational velocities occurs at approximately the spectral type where subphotospheric convection zones become suddenly much deeper on the main sequence. Accordingly, because Sunlike stars are most likely to develop episodic mass ejections and magnetically channeled stellar winds, it is generally thought that these stars are losing mass – and, hence, angular momentum – as they slowly evolve on the main sequence. Postponing to Chapter 7 the study of these low-mass stars (M ≲ 1.5M⊙), in this chapter we shall consider stars more massive than the Sun (M ≲ 1.5M⊙) that are in radiative equilibrium in their surface layers.

In Chapter 4 we have already discussed the large-scale meridional currents and concomitant differential rotation in the radiative envelope of an early-type star, when the departures from spherical symmetry are not too large. Admittedly, the aim of that chapter was to develop a clear understanding of the many hydrodynamical phenomena that arise in a rotating star. In the following sections of this chapter we shall instead examine a selection of practical topics dealing with rotation, meridional circulation, and turbulence in the early-type stars.

Gravitational collapse

Overview

The bulk of this book has concerned the linear regime, and it is fair to say that, ultimately, our main understanding of the parameters describing the Universe as a whole will come via observations of that sort, especially those of microwave background anisotropies. However, we would not exist were it not for structures having entered the nonlinear regime, ultimately collapsing to form gravitationally bound objects, and so, it is interesting to ask about this regime. In fact, at present, one of the most powerful constraints on the spectrum comes from this regime, being the present abundance of galaxy clusters.

The nonlinear regime has been investigated using a variety of techniques. The most popular of all is numerical simulations, particularly simulations of the gravitational force alone, known as N-body simulations, though recently many codes have been developed that also include a range of hydrodynamical gas processes and sometimes even radiative processes such as photoionization. We discuss some of these in this chapter, but not in any depth because this is primarily a book about analytical and semianalytical techniques.

Among this latter class are theories designed to predict the number of collapsed objects of a given mass. We discuss two such theories: the Press–Schechter (1974) theory and the theory of peaks (Bardeen et al. 1986). These are semianalytic (meaning that there usually exist integrals that must be performed numerically to deal with realistic spectra, but that the basic structure of the theory is analytic), though it is fair to say that they are only used with confidence because they have been compared and calibrated using N-body simulations.

In the preceding chapter we were on firm ground. The calculation of the vacuum fluctuation uses basic field theory, and is unlikely ever to be invalidated. Now we go to the more fluid topic of inflation model building. The subject has seen a renaissance in recent years, with an increasing emphasis on the particle-theory basis. After sketching the theoretical framework, we give a rapid survey of currently favoured models, corresponding to a snapshot of the present situation. More detail, with an extensive bibliography, is provided in a review by Lyth and Riotto (1999). The models give different predictions for the spectral index, which means that Microwave Anisotropy Probe and Planck satellite observations will be able to reject most of them. We end by discussing two special classes of model, not covered in the rest of the chapter; these are models invoking non-Einstein gravity, and models leading to an open Universe.

Overview

Although inflation most likely begins when V is at the Planck scale, we consider only the era that begins when cosmologically interesting scales start to leave the horizon. By that time, V¼ is at least 2 orders of magnitude below the Planck scale, from Eq. (7.108). Any memory of what happened earlier has been wiped out, except insofar as it sets the initial conditions.

At its most minimal, a model of inflation simply gives the form of the effective potential V during inflation (along with the kinetic term if it is nontrivial, and any modification of Einstein gravity).

Introduction

The main body of the book has been concerned with the effects of axial rotation upon the structure and evolution of single stars. As was pointed out in Section 1.4, further challenging problems arise from the study of double stars whose components are close enough to raise tides on the surface of each other. Indeed, tidal interaction in a detached close binary will continually change the spin and orbital parameters of the system (such as the orbital eccentricity e, mean orbital angular velocity Ω0, inclination ω, and rotational angular velocity Ω of each component). Unless there are sizeable stellar winds emanating from the binary components, the total angular momentum will be conserved during these exchange processes. However, as a result of tidal dissipation of energy in the outer layers of the components, the total kinetic energy of a close binary system will decrease monotonically. Ultimately, this will lead to either a collision or an asymptotic approach toward a state of minimum kinetic energy. Such an equilibrium state is characterized by circularity (e = 0), coplanarity (ω = 0), and corotation (Ω = Ω0); that is to say, the orbital motion is circular, the rotation axes are perpendicular to the orbital plane, and the rotations are perfectly synchronized with the orbital revolution.

After our discussion of the observed radiative emissions from solar and stellar outer atmospheres, we now turn to the mechanisms heating the outer-atmospheric domains to a temperature that even in apparently quiescent conditions is up to 3 orders of magnitude higher than that of the photosphere. Over the years, a multitude of mechanisms has been proposed to transport energy from the stellar interior into the outer atmosphere and to dissipate it there (see the compilations by Narain and Ulmschneider, 1990, 1996, and the summary in Table 10.1). It has become clear that the real question is no longer how the atmosphere can be heated, but which mechanisms dominate under specific conditions. The problem of outer-atmospheric heating can be separated into two parts: (a) what is the source of the energy, and (b) how is it transported and dissipated? Although at first sight the second part seems to be separable into the problems of transport and dissipation, this is not always the case, because some mechanisms intricately link these two processes through cascades or critical self-regulation, as we mention below.

Our discussion concentrates on coronal heating, with a substantial bias to current-based mechanisms. Space limitations do not allow an in-depth discussion of the many distinct wave-heating processes. Chromospheric heating follows similar principles as coronal heating, but the added complexity of radiative transfer and nonforce-free fields complicate studies in this area. In contrast, the larger viscosity and resistivity of the chromosphere make it easier to dissipate both currents and waves.

In this chapter, we put together a set of observations presently available to us (i.e., in 1999), which can be interpreted using the linear and quasi-linear approaches that we have described. At present, no single type of observation is dominant in providing constraints on models of structure formation; instead, the best results come from compiling as wide a set of data as possible, covering a range of scales from our present observable Universe down to the scales of galaxies.

No doubt, the observational details will be superseded quickly, but the general approaches to using them are well established. We also give this discussion here as a post hoc motivation for the models that we considered earlier, both the inflationary aspects and the structure formation scenarios.

A detailed comparison of models with observations requires numerical investigation, to probe the nonlinear regime where many of the observations of galaxy correlations and velocities are made. However, we have seen that there remain a very significant number of undetermined parameters on which the formation of structure depends; we might consider three inflationary parameters, δH, n, and r, and several cosmological parameters such as h, Ω0, ∧, Ωb, a possible admixture of hot dark matter ΩHDM, and the redshift of reionization Zion. In all the models that we discuss, we need cold dark matter (CDM); inflation-based models do not appear to work without it. It always has whatever density is required to make the total add up correctly.

This book is the first comprehensive review and synthesis of our understanding of the origin, evolution, and effects of magnetic fields in stars that, like the Sun, have convective envelopes immediately below their photospheres. The resulting magnetic activity includes a variety of phenomena that include starspots, nonradiatively heated outer atmospheres, activity cycles, the deceleration of rotation rates, and – in close binaries – even stellar cannibalism. Our aim is to relate the magnetohydrodynamic processes in the various domains of stellar atmospheres to processes in the interior. We do so by exploiting the complementarity of solar studies, with their wealth of observational detail, and stellar studies, which allow us to study the evolutionary history of activity and the dependence of activity on fundamental parameters such as stellar mass, age, and chemical composition. We focus on observational studies and their immediate interpretation, in which results from theoretical studies and numerical simulations are included. We do not dwell on instrumentation and details in the data analysis, although we do try to bring out the scope and limitations of key observational methods.

This book is intended for astrophysicists who are seeking an introduction to the physics of magnetic activity of the Sun and of other cool stars, and for students at the graduate level.

This book

The study of the early Universe came into its own as a research field during the 1980s. Though there had been occasional forays during the seventies and even before that, it was during the 1980s that a wide range of topics, united by the adoption of modern particle physics ideas in a cosmological context, were investigated in detail. This era of study culminated with the publication in 1990 of the classic book The Early Universe by Kolb and Turner, in which the authors described ideas across the whole range of what had become known as particle cosmology or particle astrophysics, including such topics as topological defects, inflationary cosmology, dark matter, axions, and even quantum cosmology.

Although all these topics matured during the 1980s, if we look back at the papers of that era, we are struck by the rarity with which any detailed comparison with observations could be made. In that regard, particle cosmology in the nineties and onward has become a very different subject from what it was during the eighties because, for the first time, there are observations of a quality that seriously constrains some of the possible physics of the early Universe. Those observations are of structure in the Universe, and a starring role among them is played by the first detection of microwave background anisotropies by the Cosmic Background Explorer (COBE) satellite, announced in 1992.