In Chapter 4 we considered the stability of a static fluid configuration against convective instabilities, or buoyancy. We found that for stability the specific entropy must increase upwards. In this chapter we again consider the stability of a static atmosphere, but with the complication of an added magnetic field. We shall find that the magnetic field can act either to stabilize or to de-stabilize the fluid. It is possible to derive a variational principle for perturbations of a fluid containing a magnetic field, just as we did for a non-magnetic fluid in Chapter 4 (see Problem 4.7.4). Of course, the expressions we would derive in doing so contain all the information required to decide stability. But we found in Chapter 4 that we had to manipulate carefully the expressions we derived in the variational principle to extract a useful stability criterion – the Schwarzschild criterion. Adding a magnetic field makes the expressions in the variational principle much more complex, simply because the geometry of the magnetic field and its interaction with the fluid add more degrees of freedom, and there is no simple stability criterion. Accordingly we adopt a simpler, less comprehensive, approach here.

There are some guiding concepts with which a theoretical astrophysicist should be familiar, and we illustrate these here. We discuss the two distinct, but often confused, modes of instability – the buoyancy instability and the Parker instability. As before we keep try to keep the situation simple, in order to bring out the physics of the situation without obscuring it in mathematical detail.

The book which follows has grown out of my experiences in carrying out and teaching optical astronomy. Much of the practical side of this started for me when I was working with Professor M. Kitamura at what is now the National Astronomical Observatory of Japan, Mitaka, Tokyo, in the mid seventies. Having already learned something of the theoretical side of photometric data analysis and interpretation from Professor Z. Kopal in the Astronomy Department of the University of Manchester, when I later returned to that department and was asked to help with its teaching programme I started the notes which have ultimately formed at least part of the present text. I then had the pleasure of continuing with observing at the Kottamia Observatory, beneath the beautiful desert skies of Egypt, in the days of Professor A. Asaad, together with a number of good students, many of whom have since gone on to help found or join university departments of their own in different lands of the world.

In recent years – particularly since moving to Carter Observatory – another dimension has been added to my experience through my encounters with that special feature of the astronomical world: the active amateur! In previous centuries many creative scientists were, in some sense, amateurs, but in the twentieth century the tide, for fundamental research at least, has been very much in the direction of government, or other large organization, supported professionals, no doubt with very persuasive reasons.

In the next few chapters we consider what happens if we perturb a stationary fluid configuration. The unperturbed configuration we have in mind is a body of fluid at rest in a stationary gravitational potential well. This potential might result from the self-gravity of the fluid itself, as for a star, or it might be produced by some external agency. An example of the latter case is the potential well produced by the dark matter component of a cluster of galaxies. The intracluster medium sits in this potential, without significantly contributing to it.

Studying perturbations in this way is important for a number of reasons. We can often use a linear analysis, and thus make things mathematically tractable. Working out when perturbations grow or not often provides us with a good idea of how a system will react, even to finite (non-infinitesimal) perturbations. In particular, we may be able to decide if the system is likely to react with drastic changes (instability), or settle down again to a state rather like its original one (stability). A system's reaction to perturbations also tells us a lot about its structure. Just as geophysicists learn about the Earth's interior by studying how it reacts to perturbations such as earthquakes, astronomers can use a similar technique (asteroseismology) to study the interior of stars.

Models of stars

To be specific we shall mainly consider perturbations to models of stars, although the results we find are generally applicable.

Scope of the subject

This book is aimed at laying groundwork for the purposes and methods of astronomical photometry. This is a large subject with a large range of connections. In the historical aspect, for example, we retain contact with the earliest known systematic cataloguer of the sky, at least in Western sources, i.e. Hipparchos of Nicea (∽160–127 BCE): the ‘father of astronomy’, for his magnitude arrangements are still in use, though admittedly in a much refined form. A special interest attaches to this very long time baseline, and a worthy challenge exists in getting a clearer view of early records and procedures.

Photometry has points of contact with, or merges into, other fields of observational astronomy, though different words are used to demarcate particular specialities. Radio-, infrared-, X-ray-astronomy, and so on, often concern measurement and comparison procedures that parallel the historically well-known optical domain. Spectrophotometry, as another instance, extends and particularizes information about the detailed distribution of radiated energy with wavelength, involving studies and techniques for a higher spectral resolution than would apply to photometry in general. Astrometry and stellar photometry form limiting cases of the photometry of extended objects. Since stars are, for the most part, below instrumental resolution, a sharp separation is made between positional and radiative flux data. But this distinction seems artificial on close examination.

In the previous chapter we showed that if a star is stable it oscillates about its equilibrium configuration when perturbed. To do this we looked at the global properties of the whole star. We showed that the star acts like an organ pipe in that it oscillates in a distinct set of modes with a distinct set of frequencies. But to investigate the details of the oscillation modes of the star, we need to look at the details of how the oscillations propagate through the star. In line with our philosophy expressed previously, we shall simplify matters by only considering flat ‘stars’, or equivalently we can think of the analysis as applying to a plane-parallel atmosphere, whose vertical thickness is small compared with the star's radius.

To understand the physics of the oscillations, we need to ask what the restoring forces are. That is, if we perturb the fluid, what tries to push it back to where it was? We have so far come across two types of restoring force in non-magnetic media, and we can expect both to operate in a star. They are pressure and buoyancy.

1. (i) Pressure. If we compress a fluid element, we increase its pressure, and this increase produces a restoring force. The resulting oscillations are sound waves, with local speed. Oscillation modes in which pressure is the main restoring force are called p-modes.

2. […]

Overview of basic instrumentation

Figure 5.1 schematizes the essential optical photometric system in its observatory setting. At the heart of the system is the detector. The excitation of electrons responding to incident photons, whose acquired energy then allows them to be registered in some way, forms the basis for practical flux measurement in more or less all photometric systems. With this there has been a continued trend towards more reliably linear, efficient and informative detectors. The photocathode (photomultiplier) tube, a single channel, spatially non-resolving device, has been the simplest type of linear detector in general use. However, the advent of efficient areal detectors, bulk data-handling facilities and software improvements that can efficiently control observing instruments, as well as process raw data into manageable form, have revolutionized ground-based astronomical photometry in the last decade or so. This was already evident from the widening new horizons of the previous chapter's review, and will be reinforced as we progress. But our more immediate purpose is to draw out the practical groundwork on which useful data relies. Overviewing this allows us to see how common elements in photon flux measurement have developed, sometimes along new and separate paths, but with generally parallel aims, often involving the same or similar principles.

Referring to the scheme of Figure 5.1, the optical arrangements in front (foreoptics) are designed to direct suitably selected fluxes from astronomical sources to the detector element. After electrons have been photo-energized, their activity is to be registered by appropriate electronics.

Light curve analysis: general outline

In this chapter we approach some classic problems of photometric analysis. We start with the light curves of eclipsing binary stars. These reduce, in their simplest form, to regular patterns of variation which can be understood by reference to relatively simple models of stars in a simple geometrical arrangement. First estimates for key parameters can often be directly made from inspection of the salient features of a light curve. This is a useful preliminary to more detailed analysis. The main issue underlying this and subsequent chapters, however, concerns the setting out of a comprehensive procedure for parameter value estimation. This represents a subsection of the field of optimization analysis, or the optimal curve-fitting problem.

In the version of this problem that faces us, we are given a set of N discrete observations lo (ti), i = 1, …, N, in a data space, which have a probabilistic relationship to an underlying physical variation, dependent in a single-valued way on time t. This real variation, whatever its form, is approximated by a fitting function, lc(aj, t), say, which is, formally, some function of the independent variable t, and a set of n parameters aj, j = 1, …, n. In general, we regard a subset m of these parameters as determinable from the data.

The object is to transfer the lo(ti) information from data space to the aj information in parameter space.

The subject of this book is how the matter of the visible Universe moves. Almost all of this matter is in gaseous form, and each gram contains of order 10 particles (atoms, ions, protons, electrons, etc.), all moving independently except for interactions such as collisions. At first sight it might seem an impossible task to describe the evolution of such a complicated system. However, in many cases we can avoid most of this inherent complexity by approximating the matter as a fluid. A fluid is an idealized continuous medium with certain macroscopic properties such as density, pressure and velocity. This concept applies equally to gases and liquids, and we shall take the term fluid to refer to both in this book. The structure of matter at the atomic or molecular level is important only in fixing relations between macroscopic fluid properties such as density and pressure, and in specifying others such as viscosity and conductivity.

Describing a medium as a fluid is possible if we can define physical quantities such as density ρ(r, t) or velocity u(r, t) at a particular place with position vector r at time t. For a meaningful definition of a ‘fluid velocity’ we must average over a large number of such particles. In other words, fluid dynamical quantities are well defined only on a scale l such that l is not only much greater than a typical interparticle distance, but also, more restrictively, much greater than a typical particle mean free path, λmfp.