The Structure and Evolution of Stars

The following sections are included:

• Dedication
• Foreword
• Preface
• Acknowledgements
• Contents

“A star is basically a pretty simple structure” announced Fred Hoyle at a colloquium in the Cambridge Observatory’s Library in 1954. From the audience Professor Redman retorted, “You’d look pretty simple, Fred, at ten parsecs.” This was at the dawn of the modern age of stellar astrophysics. The basic physics had been identified and the equations assembled. It remained to solve them in progressively more detail and so model the stars in all their glory, a process that continues apace. Today’s instruments and observing techniques mean that stars do not look quite so simple at ten parsecs any more, our understanding of the details of the physical processes has improved enormously and we have identified some important complications, particularly in the fluid dynamics of convection and mixing. Nevertheless stars, in their interiors, are indeed pretty simple structures. Not far below their visible surfaces, temperatures are so hot that the complexities of chemistry are not a burden. Even in the atmospheres much of the chemistry appears to be relatively simple and biology remains out of reach. If we understand physics we can understand stars. One of the beauties of stars is that they touch upon every aspect of physics from the quantum mechanics of electron degeneracy and nuclear fusion, the general relativity of the neutron stars and black holes, thermodynamics and electrodynamics, fluid dynamics and magnetohydrodynamics through to solid state crystalline structures of cold white dwarfs. In the coming chapters the necessary physics is described, with some mathematical detail, to the extent that the reader can grasp the essentials and understand the details to the level required to comprehend the current literature and to proceed to push forward the frontiers of our understanding of stars…

Four differential equations are required to model the structure of a star. These are actually partial differential equations in space, or mass, and time but we shall often treat a star as quasi-static and solve the equations of stellar structure in space at a given time and then consider changes, including in composition, with time as evolution from one such quasi-static model to another. Two equations describe the physical structure of the star, its distribution of mass and pressure. These form the body of this chapter. The other two equations for the transport and generation of energy determine the luminosity and temperature through the star. We consider them in Chapters 4 and 6.

In most stars, the equation of state is not everywhere sufficiently simple that we may write pressure P = P(ρ) as a function of density ρ only. Generally we need two state variables, such as density and temperature T, and a description of the composition, such as the set of mass fractions X_i of nuclide i. An equation of state might then be written in the form P = P(ρ, T, {X_i}). The first two equations of stellar structure (2.2) and (2.10) are no longer closed and we shall need to include two more, the equation of heat transport and the equation of energy generation that form the subjects of Chapters 4 and 6…

We now turn to the transport of energy within a star. Typically the energy flow is from the stellar interior, where nuclear fusion is taking place, towards the stellar surface where it can be radiated into space. We define the local luminosity L_r at radius r from the centre of the star to be the total energy flux outwards through the sphere of radius 4πr². This luminosity depends on the temperature gradient dT/dr through the star. Three major processes contribute,

• radiation — the diffusion of photons,
• conduction — the diffusion of particles and
• convection — bulk transport within the fluid.

From the surface of the star, the photosphere, photons escape to infinity so that their mean free path can no longer be considered short. Neither can the radiation field be expected to be in local thermodynamic equilibrium (LTE), nor is it isotropic because photons moving inwards are still absorbed or scattered after travelling only a short distance. However particles typically do have short enough mean free paths to remain in LTE to a somewhat greater height in the atmosphere. In this chapter we consider the radiation transport through the atmosphere more carefully and use this to derive more correct boundary conditions for the stellar interior.

One further equation, for energy generation, is required to close the set that describe stellar structure. The last physical ingredient required to make models of stellar structure and evolution is the source of energy that ultimately causes them to shine. Up until the start of the twentieth century the source of the Sun’s energy remained a puzzle. Early in the nineteenth century geologists had established that the Earth must be at least ten million if not a hundred million years old. It was easy to calculate that the energy liberated by chemical burning is insufficient. For example, consider the combustion of one mole of hydrogen to water. This requires half a mole of oxygen, so a total mass of Δm = 10g and liberates ΔQ = 286 kJ. Combustion of the entire mass of the Sun at a rate to maintain its current luminosity would take only (M_⊙/Δm)/(ΔQ/L_⊙) ≈ 4800 yr. In 1941 Julius Mayer proposed, as an alternative source, the gravitational energy released by the impact of meteors on the surface of the Sun but ruled this out himself based on his estimate of the rate they actually fall into the Sun. Hermann von Helmholtz took this idea a step further suggesting that the Sun itself is gradually contracting and releasing energy. He estimated that this self gravity could keep the Sun alight for some twenty million years. Towards the end of the nineteenth century Lord Kelvin refined Helmholtz’ theory by asserting that the Sun had contracted from a very large gas cloud. During this contraction its core was heated to several million degrees and this slowed the contraction to its current state similar to that proposed by Helmholtz. With energy from gravitational contraction they calculated that the Sun could last some one hundred million years in a similar state. However the theory was already under threat because geologists were by then claiming an age of two thousand million years for the Earth and so at least that for the Sun. Our current understanding was born in the early twentieth century when Albert Einstein’s theory of relativity hinted that matter could be converted to energy. In 1920, Arthur Stanley Eddington proposed that the fusion of hydrogen to helium was just the energy source that the Sun needs. However it was not until quantum mechanics and statistical physics were combined that we could develop a plausible theory of how fusion can proceed. We shall spend much of this chapter discussing nuclear fusion and its wide-ranging consequences but begin more generally by combining all energy sources and sinks into a single equation…

We have assembled the necessary physics to make detailed models of stars. To do so we solve the corresponding set of nonlinear partial differential equations gathered together in this chapter. We also discuss numerical techniques in common use. In Chapter 8 we describe typical results of such calculations in detail while in the final sections of this chapter we look at analytical approximations that can give us significant insight into both the structure and evolution of stars without the need of detailed solutions…

We have described the physical ingredients required to model a star and discussed how these are combined to create models of its structure and evolution. In this chapter we describe full models of the evolution of stars from their births to their deaths either in violent supernovae or gently cooling as white dwarfs. We take as our starting point not our Sun but a star of five times its mass because, with this, we can discuss most of the interesting phases of evolution. Subsequently we shall describe the differences in the evolution of both less and more massive stars.

In our Galaxy only a minority of stars are as isolated from other stars as is our own Sun. In our case the Sun rotates slowly and the angular momentum of the solar system lies in the orbits of the planets. For many systems the angular momentum is shared between two or more objects of sufficient mass to become stars. William Herschel (1803) introduced the name binary star for a gravitationally bound system of two stars. For the majority of such systems the physics is simply Newton’s gravity and the description of their motions is a generalised form of Kepler’s laws for planetary motion. Thus the scientific study of binary stars is in some respects older than that of the evolution of single stars. However if the two stars in a binary system are close enough their individual evolution can influence that of their companion. In the extreme case, material at the surface of one star can end up more attracted to its companion than itself giving birth to a number of interesting exotic objects, particularly when the companion is a dense degenerate object, a white dwarf, a neutron star of even a black hole. In this chapter we explore the physics governing binary star orbital evolution and interaction and look in more detail at some examples of exotic binary stars…

The following sections are included:

• Bibliography
• Index