The Physics of Stars

A. C. Phillips

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118723272
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Astronomy & Astrophysics

The Physics of Stars, Second Edition, is a concise introduction to the properties of stellar interiors and consequently the structure and evolution of stars. Strongly emphasising the basic physics, simple and uncomplicated theoretical models are used to illustrate clearly the connections between fundamental physics and stellar properties. This text does not intend to be encyclopaedic, rather it tends to focus on the most interesting and important aspects of stellar structure, evolution and nucleosynthesis. In the Second Edition, a new chapter on Helioseismology has been added, along with a list of physical constants and extra student problems. There is also new material on the Hertztsprung-Russell diagram, as well as a general updating of the entire text. It includes numerous problems at the end of each chapter aimed at both testing and extending student's knowledge.

1

Basic concepts in astrophysics

The aim of this book is to explore the properties of stellar interiors and hence understand the structure and evolution of stars. This exercise is largely based on the application of thermal and nuclear physics to matter and radiation at high temperatures and pressures. However, before developing and applying this physics it is useful to consider the subject as a whole using elementary physics. In this brief and rapid overview we shall introduce some concepts which are fundamental to stellar evolution, fix the order of magnitude of some important astrophysical quantities and identify the basic observational information on stars. Many of the topics mentioned are covered in more detail later in the book and in the references listed at the end of the book. We begin by considering the processes which produced the raw material used in the construction of the first stars.

1.1 BIG BANG NUCLEOSYNTHESIS

To a first approximation matter in the universe consists of hydrogen and helium, with a smidgen of heavier elements such as carbon, oxygen and iron. It is now recognized that the bulk of this helium was produced by nuclear reactions which occurred during the first few minutes of the universe, a process called primordial or big bang nucleosynthesis. We shall begin this introductory chapter by giving a very brief outline of big bang nucleosynthesis so that the reader is aware of the origin and nature of the raw material used in the construction of the first stars.

A brief history of the universe

In order to understand the history of the universe it is necessary to account for two important facts regarding the present universe: firstly the universe is expanding in such a way that if we extrapolate back in time it appears that the universe had infinite density some 10 to 20 billion years ago. Secondly the whole of space is filled with a thermal radiation at a temperature of about 3 K, the cosmic microwave background radiation discovered by Penzias and Wilson in 1965. These facts are consistent with the idea that the universe began with a sudden decompression, a big bang.

The big bang is not a local phenomenon with matter being expelled in all directions from a point in space. The big bang happened simultaneously everywhere in space. Everywhere was a point at the time of the big bang if the universe is closed, i.e. a finite volume of space with no boundary. But if the universe is open, the big bang occurred all over a space of infinite extent. According to the standard model of the big bang, the universe developed along the following lines:

• Nanoseconds after the big bang the universe was filled with a gas of fundamental particles: quarks and antiquarks, leptons and antileptons, neutrinos and antineutrinos, and gluons and photons. When the temperature fell below 10¹⁴ K, the quarks, antiquarks and gluons disappeared, annihilating and transforming into less massive particles. Fortunately, because the number of quarks slightly exceeded the number of antiquarks, a few quarks were left behind to form the protons and neutrons present in today’s universe. The heavier leptons and antileptons were also annihilated as the temperature fell.
• In the interval between a millisecond to a second after the big bang the universe consisted of a gas of neutrons and protons, electrons and positrons, neutrinos and antineutrinos, and photons. As the temperature fell, the density of the universe became too low for the neutrinos to interact effectively with matter; this occurred when the temperature was about 10¹⁰ K. These non-interacting, decoupled neutrinos now form a universal gas which, because of the expansion of space, has cooled to a temperature of about 2 K. As yet it has not been possible to detect this universal background of neutrinos. Soon after the decoupling of the neutrinos, the annihilation of electron–positron pairs removed all of the positrons and most of the electrons.
• After 100 seconds, neutrons combined with protons to form light nuclei, ultimately leading to a universe in which approximately 75% of the mass consists of hydrogen and 25% is helium. We shall explain later how these percentages were determined by the ratio of neutrons to protons in the universe when the neutrinos decoupled.
• After 300 000 years the temperature fell to 4000 K, low enough for the formation of stable atoms. Hydrogen and helium nuclei combined with electrons to form neutral hydrogen and helium atoms. As a result, the photons in the universe ceased to interact strongly with matter; in other words, the universe became transparent to electromagnetic radiation. This radiation, freed from interaction with matter at a temperature near 4000 K, has now cooled to a temperature of about 3 K because of the expansion of the space. It is the cosmic microwave background radiation which was first detected by Penzias and Wilson. This radiation is slightly warmer than the as yet undetected neutrino background at 2 K because, unlike neutrinos, photons were warmed by the heat generated by electron–positron annihilation in the early universe.
• The universe continued to expand and cool until it reached its present lumpy condition with most of the matter assembled in stars, galaxies and clusters of galaxies.

This history of the universe is summarized in Table 1.1.

TABLE 1.1 A history of the universe according to the big bang. As the universe cooled quarks produced protons and neutrons, protons and neutrons formed helium and other light nuclei, and then nuclei and electrons combined to form neutral atoms. This led to today’s universe in which matter is assembled in stars and galaxies with a thermal universal background of photons and neutrinos at temperatures of about 3 and 2 K, respectively.

Cosmic time	Temperature	Temperature
t ≈ 10^−4 s	kT ≈ 10^2 MeV	Quarks form neutrons and protons
t ≈ 1 s	kT ≈ 1 MeV	Neutrinos decouple
t ≈ 4 s	kT ≈ 0.5 MeV	Electron–positron annihilation
t ≈ 3 min	kT ≈ 0.1 MeV	Helium and other light nuclei formed
t ≈ 3 × 10^5 years	kT ≈ 0.3 MeV	Atoms formed and photons decouple

The synthesis of helium

We shall now focus on the processes which led to the formation of helium and other light atomic nuclei. To understand these processes we shall follow what happened to the gas of neutrons and protons as the universe expanded and cooled from around 10¹⁰ to 10⁹ K. At temperatures above 10¹⁰ K, any deuteron formed from a neutron–proton collision was quickly disrupted by a collision because the thermal energies involved often exceeded the 2.2 MeV binding energy of the deuteron. The only nuclei existing at these temperatures were single protons and neutrons.

In normal circumstances a neutron beta decays with a mean life of about 15 minutes to a proton, an electron and an antineutrino,

However, at high temperature and density, neutrons can be transformed to protons, and protons can be transformed to neutrons in collisions involving thermal neutrinos, antineutrinos, electrons and positrons. In particular, neutrons and protons in the early universe were continually transformed into one another by the reactions:

(1.1)

Because neutrons are more massive than protons, more energy had to be borrowed from the gas to make a neutron than a proton. Hence the neutrons were outnumbered by the protons. Indeed, the ratio of neutrons to protons at equilibrium at temperature T is given by a Boltzmann factor:

(1.2)

where Δm is the neutron–proton mass difference, 1.3 MeV/c².

The Boltzmann factor in Eq. (1.2) implies that the neutron/proton ratio decreased rapidly as the expanding universe cooled. But as the temperature and density decreased the neutrino reactions (1.1) became less frequent, and neutrons and protons were transformed into one another at a slower rate. Eventually, the reaction rates became too slow to maintain thermodynamic equilibrium. The neutrino reactions fizzled out, and the numbers of neutrons and protons ceased to change rapidly. Calculations...