Introduction

As we pointed out in §1.6, most of our knowledge about the astrophysical Universe is based on the electromagnetic radiation that reaches us from the sky. By analysing this radiation, we infer various characteristics of the astrophysical systems from which the radiation was emitted or through which the radiation passed. Hence an understanding of how radiation interacts with matter is very vital in the study of astrophysics. Such an interaction between matter and radiation can be studied at two levels: macroscopic and microscopic. At the macroscopic level, we introduce suitably defined emission and absorption coefficients, and then try to solve our basic equations assuming these coefficients to be given. This subject is known as radiative transfer. At the microscopic level, on the other hand, we try to calculate the emission and absorption coefficients from the fundamental physics of the atom. Much of this chapter is devoted to the macroscopic theory of radiative transfer. Only in §2.6, do we discuss how the absorption coefficient of matter can be calculated from microscopic physics. The emission coefficient directly follows from the absorption coefficient if the matter is in thermodynamic equilibrium, as we shall see in §2.2.4.

Theory of radiative transfer

Radiation field

Let us first consider how we can provide the mathematical description of radiation at a given point in space. It is particularly easy to give a mathematical description of blackbody radiation, which is homogeneous and isotropic inside a container.

Introduction

We have pointed out in §10.2 that in general relativity we have to deal with the curvature of spacetime and that tensors provide a natural mathematical language for describing such curvature. We now plan to give an introduction to tensor analysis and then an introduction to general relativity at a technical level. It will be useful for readers to be familiar with the qualitative concepts introduced in §10.2 before studying this chapter.

Since general relativity is a challenging subject, it is helpful to clearly distinguish the purely mathematical topics from the physical concepts of general relativity. So, when we develop tensor analysis in the next section, we shall develop it as a purely mathematical subject without bringing in general relativistic concerns at all. The two-dimensional metrics (10.7), (10.8) and (10.9) introduced in §10.2 will be used as illustrative examples repeatedly to clarify various points. When various formulae of tensor analysis are applied to metrics of dimensions higher than two, the algebra can be horrendous. It is, therefore, advisable to develop a familiarity with tensors by first applying the important results to two-dimensional surfaces.

After introducing the basics of tensor analysis in the next section, we shall start developing the basic concepts of general relativity from §12.3.

Particle physics, condensed matter physics and astrophysics are arguably the three major research frontiers of physics at the present time. It is generally thought that a physics student's training is not complete without an elementary knowledge of particle physics and condensed matter physics. Most physics departments around the world offer one-semester comprehensive courses on particle physics and condensed matter physics (sometimes known by its more traditional name ‘solid state physics’). All graduate students of physics and very often advanced undergraduate students also are required to take these courses. Very surprisingly, one-semester comprehensive courses on astrophysics at a similar level are not so frequently offered by many physics departments. If a physics department has general relativists on its faculty, often a one-semester course General Relativity and Cosmology would be offered, though this would normally not be a compulsory course for all students. It has thus happened that many students get trained for a professional career in physics without a proper knowledge of astrophysics, one of the most active research areas of modern physics.

Of late, many physics departments are waking up to the fact that this is a very undesirable situation. More and more physics departments around the world are now introducing one-semester comprehensive courses on astrophysics at the advanced undergraduate or beginning graduate level, similar to such courses covering particle physics and solid state physics.

Setting the time table

The present uniform expansion of the Universe suggests that there was an epoch in the past when the Universe was in a singular state with infinite density. Since most of the known laws of physics become inapplicable to such a singular state, we cannot extrapolate to earlier times before this epoch of singularity. We therefore concern ourselves only with what happened after this epoch of singularity, which is called the Big Bang. In the solutions discussed in §10.6 and §10.7, the time t was measured from the Big Bang.

The spacetime dynamics discussed in Chapter 10 sets the stage of the Universe. Now we shall look at the dramatis personae who were involved in the grand drama which unfolded and is still unfolding against this background stage of spacetime. How the temperature of the early Universe varied with time is given by (10.67) and (10.69). At times earlier than 1 s after the Big Bang, typical photons had energies somewhat larger than 1MeV. Since such photons are known to produce electron-positron pairs, the Universe at these early times must have been full of electrons and positrons which would have been approximately as abundant as photons. When photons had energies larger than 2GeV at still earlier times, they would have given rise to proton-antiproton pairs and neutron-antineutron pairs along with pairs of many other particles listed in elementary particle physics textbooks and their antiparticles.

The possibility of nuclear reactions in stars

We have seen in the previous chapter that many aspects of stellar structure can be understood without a detailed knowledge of stellar energy generation mechanisms. This is indeed fortunate because not much was known about energy generation mechanisms when Eddington was carrying out his pioneering investigations of stellar structure in the 1920s. Eddington (1920) correctly surmised that the Kelvin-Helmholtz hypothesis of energy generation by contraction (see§3.2.2) could not possibly be true and stellar energy must be produced by sub atomic processes. Nuclear physics, however, was still in its infancy and details of how the stellar energy is produced could not be worked out at that time. With the rapid advances in nuclear physics within the next few years, it became possible to work out the details of energy-producing nuclear reactions inside stars. To build sufficiently detailed and realistic models of stars and stellar evolution, a good understanding of energy generation mechanisms is essential. We turn to this subject now.

Let us consider a nucleus of atomic mass A and atomic number Z. It is made of Z protons and A Z neutrons. The mass mnuc of the nucleus is always found to be less than the combined mass of these protons and neutrons.

Introduction

We have pointed out in §6.1 that astronomers in the early twentieth century thought that our Milky Way Galaxy is the entire Universe! Even a small telescope shows many nebulous objects in the sky. The great German philosopher Kant already conjectured in the eighteenth century that some of these nebulae could be island universes outside our Galaxy (Kant, 1755). However, astronomers at that time knew no way of either establishing or refuting this conjecture. In 1920 the National Academy of Sciences of USA arranged a debate on this subject – Shapley arguing that these nebulae are within our Galaxy and Curtis arguing that they are extragalactic objects (Shapley, 1921; Curtis, 1921). We discussed in §6.1.2 how the distances of Cepheid variable stars can be determined. Using the newly commissioned Mount Wilson telescope, which was much more powerful than any previous telescope, Hubble (1922) resolved some Cepheid variables in the Andromeda Galaxy M31 and estimated its distance, clearly showing that it must be lying far outside our Milky Way Galaxy. Our current best estimate of the distance of M31 is about 740 kpc. It soon became clear that many of the spiral nebulae are galaxies outside our Galaxy, heralding the subject of extragalactic astronomy and establishing that galaxies are the building blocks of the Universe.

Normal galaxies

Light coming from a typical simple galaxy seems like a composite of light emitted by a large number of stars.

Introduction

We have seen in the previous two chapters that the gravitational attraction inside a normal star is balanced by the thermal pressure caused by the thermonuclear reactions taking place in the stellar interior. Eventually, however, the nuclear fuel of the star is exhausted and there is no further source of thermal pressure to balance gravity. We have pointed out in §4.5 that such a star keeps on contracting – unless some kind of pressure other than thermal pressure is eventually able to balance gravity again. The aim of this chapter is to discuss the possible end configurations of stars which have nonuclear fuel left in them.

We have to make use of one very important property of Fermi particles. In a unit cell of volume h3 in the six-dimensional position-momentum phase space, there cannot be more than two Fermi particles (one with spin up and the other with spin down). The electrons inside the stellar matter make up a Fermi gas, and when the density inside the contracting star becomes sufficiently high, this electron gas becomes ‘degenerate’. This means that the theoretical limit of twoparticles per unit cell of phase space is almost reached. We shall show in §5.2 that such a degenerate Fermi gas exerts what is known as the degeneracy pressure. White dwarf stars discussed in §3.6 are believed to represent stellar configurations in which the inward pull of gravity is balanced by the degeneracy pressure of the electron gas.

Introduction

Since gravity is a long-range attractive force, any star in a galaxy attracts all the other stars in the galaxy all the time. For simplicity, we can regard the stars as point particles. Then a galaxy or a star cluster can be regarded as a collection of particles in which all the particles are attracting each other through an inverse square law of force. The aim of stellar dynamics is to study the dynamics of such a system of self-gravitating particles. We, of course, know that there is also gas between the stars in a galaxy, which can add extra complications. However, it is generally believed that stellar dynamics holds the key to understanding the structure of galaxies or star clusters.

We have discussed our Galaxy in Chapter 6 and shall discuss external galaxies in Chapter 9. Although some galaxies are irregular in appearance, we shall see in §9.2 that most galaxies have very regular shapes. The fundamental question of stellar dynamics is: why do collections of self-gravitating mass particles tend to take certain particular configurations in preference to many other possible configurations? A fully satisfactory answer to this question is still not known. Hence the subject of galactic structure is on a much less firm footing compared to the subject of stellar structure. We know that the gravitational attraction of the stars has to be balanced by their motions, to ensure that the stars do not all fall towards the centre of the stellar system together due to their mutual gravitational attraction.

The shape and size of our Galaxy

When we look around at the night sky, we find that the stars are not distributed very uniformly. There is a faint band of light – the Milky Way – going around the celestial sphere in a great circle. Even a moderate telescope reveals that the Milky Way is a collection of innumerable faint stars. Herschel (1785) offered an explanation of the Milky Way by suggesting that we are near the centre of a flat disk-like stellar system. When we look in the plane of the disk, we see many more stars than what we see in the other directions, thus producing the band of the Milky Way. After the development of photography, it became much easier to record distributions of stars in different directions. In the beginning of the twentieth century, Kapteyn attempted to put Herschel's view on a firm footing, by undertaking a huge programme of counting stars in different directions and measuring their proper motions with a view of estimating distances. From a painstaking statistical analysis of these data, it was inferred that we are at the centre of an oblate stellar disk with a thickness of a few hundreds of pc and a disk radius of about a few kpc (Kapteyn and van Rhijn, 1920; Kapteyn, 1922). This model is usually referred to as the Kapteyn Universe, since it was believed at that time that this was the whole Universe!

Introduction

At the beginning of §2.4, we pointed out the scope of the subject stellar interior. It appears from observational data (to be discussed in detail later) that various quantities pertaining to stars have some relations amongst each other.For example, a more massive star usually has a higher luminosity and also a higher surface temperature. To explain such observed relations theoretically, we have to figure out the equations which should hold inside a star and then solve them to construct models of stellar structure.

The years ≈1920–1940 constituted the golden period of research in this field, when theoretical developments led to elegant explanations of a vast mass of observational data pertaining to stars. Ever since that time, the subject of stellarinterior or stellar structure has remained a cornerstone of modern astrophysics and improved computational powers have ledto more detailed models. This is a subject in which theory and observations are intimately combined together to build up an imposing edifice. While presenting a subject like this, the first question that a teacher or a writer has to face is this: from a purely pedagogical point of view, is it better to start with a discussion of observational data or with a discussion of basic theoretical ideas?

It follows from simple theoretical considerations that there must be objects like stars, provided energy can be generated by some mechanism in the central regions.