In the previous chapter we have dealt with models of the stellar structure under conditions of thermal and hydrostatic equilibrium. But in order to accomplish our first task toward understanding the process of stellar evolution – the investigation of equilibrium configurations – we must test the equilibrium configurations for stability. The difference between stable and unstable equilibrium is illustrated in Figure 6.1 by two balls: one on top of a dome and the other at the bottom of a bowl. Obviously, the former is in an unstable equilibrium state, while the latter is in a stable one. The way to prove (or test) this statement is also obvious and it is generally applicable; it involves a small perturbation of the equilibrium state. Imagine the ball to be slightly perturbed from its position, resulting in a slight imbalance of the forces acting on it. In the first case, this would cause the ball to slide down, running away from its original position. In the second case, on the other hand, the perturbation will lead to small oscillations around the equilibrium position, which friction will eventually dampen, the ball thus returning to its original point. The small imbalance led to the restoration of equilibrium by opposing the tendency of the perturbation. Thus a stable equilibrium may be maintained indefinitely, while an unstable one must end in a runaway, for random small perturbations are always to be expected in realistic physical systems.

As a star consists of a mixture of ions, electrons and photons, the physics of stellar interiors must deal with (a) the properties of gaseous systems, (b) radiation and (c) the interaction between gas and radiation. The latter may take many different forms: absorption, resulting in excitation or ionization; emission, resulting in de-excitation or recombination; and scattering. In order not to stray too far from our main theme, we shall only consider processes and properties that are simple enough to understand without requiring an extended physical background, and yet sufficient for providing some insight into the general behaviour of stars. The full-scale processes are incorporated in calculations of stellar structure and evolution, performed on powerful computers by means of extended numerical codes that include enormous amounts of information. These, however, should be regarded as computational laboratories, meant to reproduce, or simulate, rather than explain, the behaviour of stars. Our purpose is to outline the basic principles of stellar evolution and we are therefore entitled to some simplification. Eddington defends this right quite forcefully:

Observational evidence of mass loss

It is an acknowledged fact that stars lose mass. In addition to the outflow of photons, there usually is an outflow of material particles. But unlike the flow of radiation, which is supplied by energy generation in the interior, the flow of mass is not replenished. As a result, the stellar mass decreases at a rate that is usually measured in solar masses per year and denoted by Ṁ, where the negative sign is omitted. Shedding of mass may take two forms: a sudden ejection of a mass shell, usually following an explosion, or a continuous flow, usually referred to as a wind. We shall deal with explosive mass ejection in Chapter 10, and devote the present discussion to stellar winds.

Indirect evidence for mass loss was brought in the previous chapter and theoretical indication for its probable occurrence was mentioned in Chapter 5. There is, however, direct observational evidence for continuous rapid expansion of the outer layers of stars beyond the stellar photosphere that marks the outer edge, and into the interstellar medium. The most common is exhibited by a characteristic shape of spectral lines, known as P-Cygni lines, named after the star P Cygni – one of the brightest in our Galaxy, discovered in 1600 as a new star (see upcoming Chapters 10 and 11) – where they are prominent. A P-Cygni line profile consists of a blue-shifted absorption component and a red-shifted emission component.

What is a star?

A star can be defined as a body that satisfies two conditions: (a) it is bound by self-gravity; (b) it radiates energy supplied by an internal source. From the first condition it follows that the shape of such a body must be spherical, for gravity is a spherically symmetric force field. Or, it might be spheroidal, if axisymmetric forces are also present. The source of radiation is usually nuclear energy released by fusion reactions that take place in stellar interiors, and sometimes gravitational potential energy released in contraction or collapse. By this definition, a planet, for example, is not a star, in spite of its stellar appearance, because it shines (mostly) by reflection of solar radiation. Nor can a comet be considered a star, although in early Chinese and Japanese records comets belonged with the ‘guest stars’ – those stars that appeared suddenly in the sky where none had previously been observed. Comets, like planets, shine by reflection of solar radiation and, moreover, their masses are too small for self-gravity to be of importance.

A direct implication of the definition is that stars must evolve: as they release energy produced internally, changes necessarily occur in their structure or composition, or both. This is precisely the meaning of evolution. From the above definition we may also infer that the death of a star can occur in two ways: violation of the first condition – self-gravity – meaning breakup of the star and scattering of its material into interstellar space, or violation of the second condition – internally supplied radiation of energy – that could result from exhaustion of the nuclear fuel.

For over ten years I have been teaching an introductory course in astrophysics for undergraduate students in their second or third year of physics or planetary sciences studies. In each of these classes, I have witnessed the growing interest and enthusiasm building up from the beginning of the course toward its end.

It is not surprising that astrophysics is considered interesting; the field is continually gaining in popularity and acclaim due to the development of very sophisticated telescopes and to the frequent space missions, which seem to bring the universe closer and make it more accessible. But students of physics have an additional reason of their own for this interest. The first years of undergraduate studies create the impression that physics is made up of several distinct disciplines, which appear to have little in common: mechanics, electromagnetism, thermodynamics and atomic physics, each dealing with a separate class of phenomena.

Astrophysics – in its narrowest sense, as the physics of stars – presents a unique opportunity for teachers to demonstrate and for students to discover that complex structures and processes do occur in Nature, for the understanding of which all the different branches of physics must be invoked and combined. Therefore, a course devoted to the physics of stars should perhaps be compulsory, rather than elective, during the second or third year of physics undergraduate studies. The present book may serve as a guide or textbook for such a course.

Stars of the types considered in this chapter differ from those discussed so far, inasmuch as, for various reasons, they do not (or cannot) appear on the H–R diagram. As before, we shall rely on stellar evolution calculations to describe them. Whenever possible, we shall confront the results and predictions of the theory with observations, either directly or based on statistical considerations. We shall find that, as we approach the frontiers of modern astrophysics, theory and observation go more closely hand in hand.

What is a supernova?

We should start by making acquaintance with the astronomical concept of a supernova, as we did with main-sequence stars, red giants and white dwarfs in Chapter 1. Stars undergoing a tremendous explosion (sudden brightening), during which their luminosity becomes comparable to that of an entire galaxy (some 1011 stars!), are called supernovae. Historically, nova was the name used for an apparently new star; eventually it turned out to be a misnomer, novae being (faint) stars that brighten suddenly by many orders of magnitude. So are supernovae, but on a much larger scale. Not until the 1930s were supernovae recognized as a separate class of objects within novae in general. They were so called by Fritz Zwicky, after Edwin Hubble had estimated the distance to the Andromeda galaxy (with the aid of Cepheids) and had thus been able to appreciate the unequalled luminosity of the nova discovered in that galaxy in 1885, amounting to about one sixth of the luminosity of the galaxy itself.

We have learned a star to be a radiating gaseous sphere, made predominantly of hydrogen and helium. Radiation may be regarded as a photon gas, each ‘particle’ carrying a quantum of energy hν, proportional to the frequency ν of the associated electromagnetic wave, and a momentum hν/c, where h is Planck's constant and c is the speed of light. This mixture of gases that makes up a star is governed by frequent collisions between its particles, ions, electrons and photons alike. This is how Sir Arthur Eddington describes The Inside of a Star:

This chapter differs from previous ones by being descriptive rather than analytical. An account will be given of the evolution of stars as it emerges from full-scale numerical calculations – solutions of the set of equations (2.54), with accurate equations of state, opacity coefficients and nuclear reaction rates. Such numerical studies of stellar evolution date back to the early 1960s, when the first computer codes for this task were developed. The first to program the evolution of stellar models on an electronic computer were Brian Haselgrove and Hoyle in 1956. They adopted a method of direct numerical integration of the equations and fitting to outer boundary conditions. A much better suited numerical procedure for the two-boundary value nature of the stellar structure equations (essentially a relaxation method) was soon proposed by Louis Henyey; it is often referred to as the Henyey method and it has been adopted by most stellar-evolution codes to this day. Among the numerous calculations performed by many astrophysicists all over the world since the early 1960s, the lion's share belongs to Icko Iben Jr. The detailed results of such computations cannot always be anticipated on the basis of fundamental principles, and simple, intuitive explanations cannot always be offered. We must accept the fact that, being highly nonlinear, the evolution equations may be expected to have quite complicated solutions.

As the complete solutions of the evolution equations provide, in particular, the observable surface properties of stars, we shall focus in this chapter, more than we have previously done, on the comparison of theoretical results with observations.