The general goal of classical mechanics is to determine what happens to a given set of objects in a given physical situation. In order to figure this out, we need to know what makes the objects move the way they do. There are two main ways of going about this task. The first one, which you are undoubtedly familiar with, involves Newton's laws. This is the subject of the present chapter. The second one, which is more advanced, is the Lagrangian method. This is the subject of Chapter 6. It should be noted that each of these methods is perfectly sufficient for solving any problem, and they both produce the same information in the end. But they are based on vastly different principles. We'll talk more about this in Chapter 6.

Newton's laws

In 1687 Newton published his three laws in his Principia Mathematica. These laws are fairly intuitive, although I suppose it's questionable to attach the adjective “intuitive” to a set of statements that weren't written down until a mere 300 years ago. At any rate, the laws may be stated as follows.

• First law: A body moves with constant velocity (which may be zero) unless acted on by a force.

• Second law: The time rate of change of the momentum of a body equals the force acting on the body.

• Third law: For every force on one body, there is an equal and opposite force on another body.

Physics involves a great deal of problem solving. Whether you are doing cutting-edge research or reading a book on a well-known subject, you are going to need to solve some problems. In the latter case (the presently relevant one, given what is in your hand right now), it is fairly safe to say that the true test of understanding something is the ability to solve problems on it. Reading about a topic is often a necessary step in the learning process, but it is by no means a sufficient one. The more important step is spending as much time as possible solving problems (which is inevitably an active task) beyond the time you spend reading (which is generally a more passive task). I have therefore included a very large number of problems/exercises in this book.

However, if I'm going to throw all these problems at you, I should at least give you some general strategies for solving them. These strategies are the subject of the present chapter. They are things you should always keep in the back of your mind when tackling a problem. Of course, they are generally not sufficient by themselves; you won't get too far without understanding the physical concepts behind the subject at hand. But when you add these strategies to your physical understanding, they can make your life a lot easier.

Maxwell's Equations as presented in Chapters 1–4 apply to electric and magnetic fields in matter as well as in free space. However, when you're dealing with fields inside matter, remember the following points:

• The enclosed charge in the integral form of Gauss's law for electric fields (and current density in the differential form) includes ALL charge – bound as well as free.

• The enclosed current in the integral form of the Ampere–Maxwell law (and volume current density in the differential form) includes ALL currents – bound and polarization as well as free.

Since the bound charge may be difficult to determine, in this Appendix you'll find versions of the differential and integral forms of Gauss's law for electric fields that depend only on the free charge. Likewise, you'll find versions of the differential and integral form of the Ampere–Maxwell law that depend only on the free current.

What about Gauss's law for magnetic fields and Faraday's law? Since those laws don't directly involve electric charge or current, there's no need to derive more “matter friendly” versions of them.

Newton's laws hold only in inertial frames of reference. However, there are many noninertial (that is, accelerating) frames that we might reasonably want to consider, such as elevators, merry-go-rounds, and so on. Is there any possible way to modify Newton's laws so that they hold in noninertial frames, or do we have to give up entirely on F = m a? It turns out that we can in fact hold on to our good friend F = m a, provided that we introduce some new “fictitious” forces. These are forces that a person in the accelerating frame thinks exist. If she applies F = m a while including these new forces, then she will get the correct answer for the acceleration a, as measured with respect to her frame.

To be quantitative about all this, we'll have to spend some time figuring out how the coordinates (and their derivatives) in an accelerating frame relate to those in an inertial frame. But before diving into that, let's look at a simple example that demonstrates the basic idea of fictitious forces.

Example (A train): Imagine that you are standing on a train that is accelerating to the right with acceleration a. If you are to remain at the same spot on the train, then there must be a friction force between the floor and your feet with magnitude Ff = ma, pointing to the right.

Some years ago professional colleagues suggested that a new edition of An Introduction to Astronomical Photometry could be useful and timely. The decision to act upon this did not come, however, until the warm and conducive summer of 2003, in the stimulative environment of north-west Anatolia, once home to great forefathers of astronomy, such as Anaxagoras and Hipparchos. The former set up his school at the surely appropriately named Lampsakos, just a few miles from where the present authors are working: the latter hailed originally from what is now the Iznik district of neighbouring Bythinia. Eudoxus too, after learning his observational astronomy in Heliopolis, moved back to Mysia to found the institute at Cyzicus (today's Kapu Dagh), while Aristotle's thoughts on the heavens must have also been developing around the time of his sojourn in the Troad, after the death of Plato. In such surroundings it is difficult to resist thinking about the brightness of the stars.

But that was just the beginning. It quickly became clear that the proposed task could not be lightly undertaken. There were at least three main questions to clarify: (1) what branches of modern astronomy can be suitably associated with photometry; (2) what level of explanation can be set against the intention of an introduction; and (3) who could become involved with what aspect of the subject? An approximate size and scope were originally based on the model of the first edition.

Introductory background

‘Starspots’ are not a new notion. There was a time when starspots were offered as a general explanation of stellar variability. From that extreme, in the nineteenth century, attention to the hypothesis had dwindled away almost completely after the development of stellar spectroscopy, until, in a well-known pair of papers dealing with the cool binaries AR Lac and YY Gem in the 1950s, the photometrist Gerald Kron revived it. As events have turned out, there is now a good deal of evidence to support Kron's conjecture, for certain groups of cool variable.

The two stars that Kron referred to are good examples of somewhat different but related categories of ‘active cool star’ (see Figures 10.1 and 10.2) – stars of spectral type generally later than mid-F, i.e. associated with convective outer regions, and usually having a relatively rapid rotational speed. In the cases of AR Lac and YY Gem, both close binary systems, this rapid rotation is a consequence of tidally induced synchronism between rotation and orbital revolution. The M type dwarf components of YY Gem are typical flare stars, characterized by Balmer line emissions that on occasions become very strong, reminiscent of solar flares, but with much greater relative intensity. AR Lac also shows ‘chromospheric’ emission lines, but its particular configuration, G5 and subgiant K0 stars in a ∽2 day period binary, places it as a standard RS CVn type binary.

In this chapter we proceed to more general effects in close binary light curves, with a wider sample of data sets. Although through most of the twentieth century close pairs formed a distinct subset of double star research, engendering its own data, purposes, methods and outcomes, in present times this somewhat artificial separation, mentioned at the beginning of the book, is being bridged. The time is in sight when photometric data can be more easily joined with astrometry for fuller analysis and information retrieval from close binary stars. To proceed with this, we need first to consider the overall geometry.

Coordinate transformation

Ambiguities are possible in bringing data on close binary systems into the conventional framework used for double stars, since, for example, the ‘longitude of periastron’ used for the radial velocities of spectroscopic binaries is normally measured inward from the plane of the sky and not the line of nodes in the equator, as in the standard 3-dimensional specification for astrometric binaries. If, as is usual for close systems, we use the local plane of the sky as the reference, the nodal angle Ω should still refer the line of intersection of orbital and sky planes to the equatorial coordinate system. This angle has no effect on radial velocities or photometric effects, though it remains a basic parameter of a binary system.

Introductory background

The eclipsing binaries and spotted stars discussed in the previous three chapters still represent only about a quarter of all variables. The largest class of variable stars are those having some inherent physical variation in luminosity, as distinct from an effect of geometry. They are often referred to as pulsating, sometimes vibrating, stars: words suggesting the physical cause of the variation. Although, as noted before, all stars would vary over a sufficiently long timescale, an appreciable intrinsic variation of luminosity accessible to human inspection implies a very short period against general stellar time frames, giving perspective to such terminology. There are examples whose light pattern repeats in measurably the same form for many cycles, with a periodicity of comparable constancy to that of eclipsing variables. Others show varying degrees of chaotic behaviour. In ‘irregular’ cases, the light level wanders up and down with no pattern or predictability. But many show quasi-periodic variations of a ‘semiregular’ nature. Some examples of the different light curves are shown in Figure 11.1.

Spectroscopy shows that the variations in apparent magnitude are linked with changes of radius. By studying the Doppler shifts of absorption lines in the regular cepheid type variables (prototype δ Cephei), it was deduced that the star oscillates inward and outward in the same period as the brightness cycle, the star being faintest not far from, but somewhat before, the time when it is smallest.

Variable stars and periodic effects

The times of observed features, usually minima or maxima, on the light curves of variable stars (single or binary), can often be accurately determined. Many classes of variable star have a repetitive, cyclic behaviour regarding such features and time, which is understood in terms of their basic physical properties. Thus, timing data are applied to measuring rotation, pulsation, or orbital periods (P) of the different kinds of variable. It follows from angular momentum considerations that any change in the velocity field and/or mass distribution within or nearby a star, or binary system component, will cause a change in rotational or orbital motion. There will then result shifts of corresponding minima (or maxima) times that have relevance to physical changes of the object in question. The ‘O – C’ method is commonly used in measuring shifts in the observed times of photometric features.

The O – C method

Defining terms: The O – C – ‘observed minus calculated’ datum (regarded as a single item) – indicates any time difference between the epoch of an observed phenomenon, presumed periodic, and its prediction, based on an ephemeris formula. Eclipses are a typical example. In this case, O – C analysis often allows identification of the possible cause of period variation. These may arise from orbital eccentricity effects, mass loss from the system or exchange between the components, magnetic effects, the presence of a third body or other reasons.

The normal unit of time for this context is heliocentric Julian days.

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.

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.

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.