The black body

If we want to analyse the radiation of stars we must have laboratory light sources with which we can compare the radiation. For instance, if we want to determine the temperature of the stellar gas, we have to know how the radiation of a gas changes with temperature. We also need a light source whose radiation properties do not depend on the kind of material of which it is made, since a priori we do not know what the stars are made of. Such an ideal light source is the so-called black body.

What is a black body? We call something black if it does not reflect any light falling on it. In the absence of any radiation coming from the black body itself, it then looks black because no light falling on it is redirected or scattered into our eyes. If we want to determine temperatures from a comparison with an ideal light source, then this light source must have the same temperature everywhere. This means it must be in thermodynamic equilibrium, which means that it has reached a final state of equilibrium such that nothing will change in time. Such an ideal light source is best realized by a volume of gas inside a well-insulated box with a tiny hole in it. This hole is nearly a perfect black body because any light beam falling into this tiny hole will be reflected back and forth on the walls of this box (see Fig. 3.1) until it is finally absorbed either by the wall or by the gas in the box. The chances of the light getting out of this tiny hole again are extremely small.

The radiative transfer equation

In the outer layers of the stars, heat transport must be by radiation, since there is no other means of transporting heat into the vacuum surrounding the star. (The extremely low density interstellar material cannot provide any other method of heat transport comparable to the radiative energy loss of a star. It therefore can be considered to be a vacuum in this context.) These outer layers, where heat transport is by radiation only, may be shallow in some stars and may extend almost to the center of the star in others, as we shall see later. In the following discussion, we shall consider the case in which these outer layers have a very large optical depth τλ (see Chapter 1) but a geometrical height which is small compared to the radius of the star. In the case of the sun, for instance, a layer with τλ = 10 at visual wavelength has a geometrical height of about 500 km, which is certainly small in comparison with the solar radius of 700 000 km. In this case the radius of curvature is much larger than the height of the layer and we can consider it as plane parallel, as we did with the Earth's atmosphere in Chapter 1.

As in Chapter 1, we consider the flow of radiative energy through this outer layer, which we call the atmosphere of the star. In contrast to the Earth's atmosphere, there is no solid star underneath, since the stellar temperatures are much too high to permit solidification – except perhaps in such exotic objects as the white dwarfs.

In Volume 2 of Introduction to Stellar Astrophysics we will deal mainly with stellar atmospheres. What are stellar atmospheres? We have seen in Volume 1 that stars have temperatures starting at about 3000 K for the coolest stars up to somewhere around 40 000 K for the hottest stars. With such high temperatures stars certainly cannot be solid; they must all be in a gaseous phase. Therefore, the atmosphere cannot be defined as a gaseous layer on top of a solid core as on the Earth; there are no solid cores in the stars. Instead, astronomers define the atmosphere as those layers of the star from which we get the radiation. This means, of course, that this is the layer of the star about which we can obtain direct information. We see no photons from beneath the layer we call the atmosphere. All the radiation which originally came from deeper layers has been absorbed once or many times by atoms in the overlying layers and is finally emitted by an atom in the stellar atmosphere. The photons we receive tell us directly only about the condition of the atoms from which they were last emitted and those are the atoms in the stellar atmosphere. This is why we devote all of Volume 2 to stellar atmospheres.

How thick is this stellar atmosphere? When we discussed absorption in the Earth's atmosphere in Volume 1, we saw that the intensity of a light beam passing through a gas is diminished by a factor e-τ, where τλ is the so-called optical depth of the layer of gas along the beam of light.

The Balmer jump and the hydrogen lines

Hot stars

In Section 8.2 we saw that the Balmer discontinuity is determined by the ratio of the continuous absorption coefficients on both sides of this discontinuity. For hot stars we derived that the discontinuity is determined by the ratio of the number of hydrogen atoms in the second quantum level (absorbing shortward of 3647 Å) to the number of hydrogen atoms in the third quantum level (absorbing on the long wavelength side of 3647 Å). This ratio is completely determined by the temperature. For the B stars the temperature can in principle be determined from the Balmer discontinuity alone.

The pressure can be determined from the electron density ne. For late B stars, for instance, the hydrogen is completely ionized, while the helium is not. Therefore, we know that ne = H+ and N = ne + H+ + He = ne + H+ + 01H+ if the abundance of helium is 10% by number of atoms. With this we find

and, of course,

As we saw in the previous chapter, the number of free electrons can be determined from the hydrogen lines in two ways: either from the number of visible Balmer lines by means of the Inglis–Teller formula (equation 11.1), or by means of the hydrogen line wings. In Section 10.1 we saw that the line depth for optically thin lines is given by equation (10.8):

For the hot stars the continuous absorption coefficient Kc in the visual is due to the Paschen continuum, which means that it is due to absorption from the third level of the hydrogen atom.

Powers of ten notation

If you are familiar with powers of ten or exponential notation for very large or very small numbers you can skip this section.

In everyday life we have hardly any occasion to use really big numbers. For quite a lot of transactions the ten fingers are enough. We speak of having one house, two cars, three children, of buying ten apples, or of earning some thousands of pounds or dollars a year. All of these numbers can be written with a few digits. On the other hand, financial newspapers or a finance Minister deal in millions and billions of pounds or dollars.

Science extends far beyond the everyday domain of our senses, and doesn't shrink from large numbers. There is no practical word, at once precise and in general use, to describe the number of synapses in the brain, the number of stars in our galaxy or the number of molecules in a litre of water. To express such quantities we use powers of ten. They have their origin in mathematics, but you don't need to know too much in order to understand their principle.

Quite simply: ten to the power of one is ten or 10; the second power is ten multiplied by itself twice, that is a hundred or 100.

The universe is a huge place. Three objects in the night sky, visible to the naked eye, can give us some impression of the dizzying depths of space. These three objects, the Moon, the Pole Star and the Andromeda galaxy belong respectively to the planetary, stellar and extragalactic domains. Light, travelling at 300,000 kilometres (186,000 miles) per second, takes one and a quarter seconds to reach us from the Moon, six hundred years from the Pole Star, and two million years to journey from the Andromeda Galaxy.

The universe is also ancient. Its past history is a series of overlapping epochs. Just a few thousand years encompass the historic past, and a few million take us back to the dawn of prehistory. Geologic history extends a few billion years into the past, whereas the cosmological history of the universe, takes us back fifteen billion years to the Big Bang itself.

The universe is full of delights. There are spiral galaxies and gaseous nebulas, faintly glowing mists set against the backdrop of deep space, multiple stars spewing out fantastic arcs of matter, and the fabulous landscapes of planets and their satellites. Humans have walked on the nearest object, the Moon. It always fills me with amazement when I see it in the early evening, between the first quarter and Full Moon, dominating the clear sky, as dusk begins.

Quantum uncertainty

Spin

I want to start this section with a description of particle spin. In addition to the usual properties of mass, charge, position and velocity, particles have a further parameter which describes their state of rotation. Physicists call this ‘spin’. We have already seen that in quantum physics it is impossible to measure simultaneously, in a precise way, both speed and position (recall the Heisenberg uncertainty relations). The same considerations apply to spin, which cannot be absolutely determined with arbitrary precision. The things that we can in principle find out are the rate of rotation (that is, the number of revolutions per second) and the component of this rotation in a given direction in space (for example, the angle of the axis of rotation relative to the direction of motion).

It is important to understand that these two knowable quantities can take only certain quantised values. This is an aspect of quantum physics that we have already encountered when describing wave motion. We can use as an analogy the stable vibrations that can be sustained on a piano string: the basic note is generated when the centre of the string vibrates to its maximum extent while the two ends are (necessarily) fixed.

The trail to life

In the last chapter we said that the Big Bang and the appearance of the universe from the vacuum were two very striking facts. Now we want to turn to a third, very striking, impression, which emerges from this long history of the universe: the fate of the universe seems quite fantastic. The destiny of the universe seems mischievously entwined with a lot of disconnected events. Some of these took place unimaginably quickly, such as the burst of activity during the first 10-32 second of inflation, or the sudden intervention of a cosmic domain creating havoc in its path. Others seem rather protracted affairs, such as the lethargic progress that followed the first quarter of an hour, and the launching of the universe on a never-ending expansion.

Key events like these arise from microphysical properties, such as the Heisenberg uncertainty relations, and from processes operating on the largest scale, such as the expansion of space as described by Einstein's equations.

Among the decisive events, some had direct action on the course of the universe and, therefore, on our place within it. The eventual disappearance of matter, through proton decay, would have major consequences for the universe and in particular for us: no matter, no humans.

Relativistic space

Four-dimensional spacetime

The twentieth century has given rise to two great theories in physics: relativity and the quantum theory. They gave mankind a radically different view of the nature of the universe. We have to use these ideas in order to understand more clearly the meaning of the quick look at the universe that we ran through in Chapter 3. Relativity particularly has provided a complete and coherent history of the universe from 0.01 seconds after the Big Bang right through to the present age of 15 billion years. When you see the majestic unfolding of this immensely rich tapestry for the first time it takes your breath away.

In seeking perfection we find that Einstein's general theory of relativity replaces Newton's law. In doing so, it replaces the Newtonian gravitational force with a completely different concept: gravitation results from the curvature of space created by masses located in space. This curvature guides the motion of particles, and makes them follow trajectories that correspond to the orbits of Newton's theory. This establishes the general framework for our investigation.

General relativity is a theory of gravitation which followed the results of the special theory of relativity. Some years earlier Einstein had completed the special theory, which is essentially a questioning and redefinition of the nature of space and time.

Chapters 4 and 5 tackled the relativistic and quantum aspects of the universe, in accordance with the two major theories of the age, general relativity and quantum theory. Even though these have enjoyed much success in explaining numerous experiments, as well as observational data, they both have shortcomings and need a more complete synthesis. Supersymmetry, supergravity and even Grand Unification are promising ways to approach this goal but they lack experimental support, being only at the stage of preliminary outlines. However, the cosmologist Dennis Sciama has made this encouraging remark: ‘It is hard to imagine that everything is wrong or illusory. We are witnessing the beginning of a new and imaginative scenario for understanding the universe’.

In Chapter 4 we played the game of trying to understand the universe, and took some risks. It is fun to launch out on a promising track, avoiding the pitfalls for the unwary and sidestepping the dead ends, in order to see if the chosen route will open up new horizons or lead to an impasse. In any case, to accompany a scientific mind voyaging a slightly dangerous but rational course is an interesting pursuit.

Weighing up the Big Bang

The Big Bang scenario, which has already been discussed in this book, has plenty of positive features.

Some Roman cosmology…

I'm going to use a wonderful text, written in Rome at the end of the reign of Emperor Augustus by the poet Marcus Manilius, to summarise the thousand-year enigmas of the cosmos. The extracts are taken from his poem on astronomy. This is more than two thousand years old and lay forgotten until the tenth century. The first French translation by Pingré was published in Paris in 1786. I know all this not because I am a historian, but because I stumbled across the book by sheer chance on the stand of a bookseller by the Seine. With my love of astronomy, the Seine, and old books I just could not resist such a find. It was my first introduction to the works of Manilius.

Manilius himself was probably not an astronomer. Instead he drew his knowledge from a variety of Greek and Roman authors. The special fascination of his work is that he gives an extensive review of astronomy, and this summary was made at a pivotal moment during the evolution of western philosophy. Astronomy flourished in the millennium before Manilius: the diameter of the Earth had been determined, and the idea that the Earth was isolated in space was taken seriously.

Night, our window on the universe…

The night is our window on the universe. In the daytime the blueness of the sky prevents us from seeing into space. The blue light is caused when the intense sunlight is scattered by oxygen molecules in the upper atmosphere. Beneath this atmospheric layer, which is a few tens of miles deep, the situation is somewhat like looking through net curtains: if a searchlight were illuminating the fabric, it would be impossible to see anything apart from the searchlight itself. By day we can see only the Sun and the Moon (if it has risen), which is so much fainter by comparison that many people think incorrectly that it is invisible during the day. But when we switch off the searchlight, the surrounding landscape can be discerned: the stars, the planets…

If the Earth did not have an atmosphere, the stars would be visible in broad daylight. The dozen Apollo astronauts who landed on the Moon experienced just such a spectacle, despite the dazzle of sunlight. However, things could be worse. The planet Venus is shrouded in a dense atmosphere and the surface pressure is one hundred times that of the Earth's atmosphere. This blanket is so thick that not a single star is visible at night; even in daytime the Sun itself is invisible and only a feeble glimmer of light reaches the surface.

Where has our exploration of the enigma of the universe left us, after all these pages? Clearly, we have encountered facts that are profoundly significant. In terms of spatial dimensions, we have probed the cosmos from the Planck length to quasars and the cosmological horizon. In time we have gone from the Planck time to the Dyson age. We have looked at structures ranging from the trio of quarks lost inside a proton, like three viruses in a volume the size of the Sun, through to the filamentary structure of the universe at large. Our story has embraced particles as subtle as the neutrino and as hypothetical as massive and destructive magnetic monopoles. The recession of the galaxies and the cosmic microwave background are relics of its explosive beginning. Using relativity, with its fusion of space and time, we can show how this Big Bang leads to what I term the grandiose fresco, or the golden moment that started the universe as we know it. As it aged from one second to fifteen billion years, the universe first experienced fifteen minutes of frenetic nuclear activity, followed by a lengthy period of lethargy, lasting a hundred million years when relatively little happened.