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.

Almost all of the baryonic Universe is fluid, and the study of how these fluids move is central to astrophysics. This book originated in a 24-lecture course entitled ‘Astrophysical Fluids’ given by one of us (JEP) in Part III of the Mathematical Tripos at the University of Cambridge, comparable in level to a graduate course in the USA. The course was intended as a preparation for research, and the book reflects this. Preparing the lecture course and especially its booklist made it plain that there was a need to bring these ideas together in one place.

The book provides a brief coverage of basic concepts, but does assume some familiarity with undergraduate-level fluid dynamics, electromagnetic theory and thermodynamics. Our aim is to give a flavour of the fundamental fluid dynamical processes and concepts which an astrophysical theorist ought to know. To keep the book to a manageable size, we have had to be selective. In particular, we omit all discussion of dissipative fluid processes such as viscosity and magnetic diffusivity.

As well as covering a range of fluid dynamical concepts, we introduce some mathematical ideas and techniques. None of these is particularly deep or abstract, but some of the implementations do require some moderately heavy but straightforward algebra. Thus the reader will benefit from some familiarity with undergraduate-level mathematical methods, as well as some facility in mathematical manipulation. This takes practice and care, but more than anything it requires the ability to spot a mistake before proceeding too far.

This chapter is about the essential procedures for setting up and using an astronomical photometric system, and processing its data. The first two sections deal with two basic calibration experiments. There is then a section dealing with observational particulars in the application of a CCD camera. A brief introduction to variable star photometry, including an overview of the timing of photometric phenomena, follows. The treatment and interpretation of data is concentrated on.

The standard stars experiment

The purpose of this is to calibrate a given local photometric system to a standard or reference system, based on detailed comparisons of published magnitude and colour values of standard stars, with corresponding measurements made with local equipment. The experiment is sometimes related to the terms absolute, or all-sky photometry. To do it well normally requires very good, i.e. transparent and stable, sky conditions, but these nights are not so common at most observing locations. They are sometimes described as ‘photometric nights’, though certain kinds of high quality, differential photometry have been carried out (notably with multi-channel or areal photometers) in nights of lesser quality, even where light clouding is present.

The choice and finding of particular standard stars is related to observing experience and particulars of the task. Specialist programmes are underway that continue to produce improved and more extended lists of standards in various photometric systems. But there are also certain well-accepted primaries (Section 3.6), which are normally bright stars and easy to find.

In Chapter 1 we emphasized that one of the major differences between astrophysical flows and the typical flows encountered in the terrestrial or laboratory context is that astrophysical fluids are compressible. This means that pressure information takes a finite time to propagate through the fluid. Because this time is often comparable to flow timescales, this gives compressible flows a fundamentally different character. In such flows the sound speed plays a role similar in some respects to that of the speed of light in the theory of relativity. In particular, sound travel times express physical causality. Pressure changes cannot propagate upstream in a supersonic flow. Subtle differences from the causal structure of relativity arise because, unlike the speed of light, the sound speed is variable and depends on the local properties of the fluid.

It is important to remember that all flows are compressible at some level. While the incompressible approximation is extremely useful in studying most terrestrial flows, intuition based on it is often a misleading guide in the astrophysical context. Moreover the elaborate mathematical apparatus assembled to study incompressible flows has limited applicability to astrophysical flows. For example, in incompressible fluids the pressure is formally disconnected from the other fluid variables, and appears only in the equation of motion, and only through its gradient (this is a mathematical expression of the assumption that it can adjust instantaneously at each point).

Optical photometry

Astronomical photometry is about the measurement of the brightness of radiating objects in the sky. We will deal mainly with optical photometry, which centres around a region of the electromagnetic spectrum to which the human eye (Figure 2.1) is sensitive. Indeed, photometric science, as it concerns stars, has developed out of a history of effort, the greatest proportion of which, over time at least, has amounted to direct visual scanning and comparison of the brightness of stellar images. In this context, brightness derives from an integrated product of the eye's response and the energy distribution as it arrives from the celestial source to reach the observer. Still today there is a large amount of monitoring of the many known variable stars carried out (largely by amateurs) in this way.

With the passage of time, however, there has been a general trend towards more objective methods of measurement. The use of photometers with a non-human detector element has become increasingly widespread, though the term optical remains to denote the relevant spectral range (Figure 2.2), which significantly coincides with an important atmospheric ‘window’ through which external radiation can easily pass. This is presumably connected with biological evolution: in fact, the maximum sensitivity of the human eye is at a wavelength close to the maximum in the energy versus wavelength distribution of the Sun's output (∽5000 Å).