Introduction

In Volume 1 we saw that the surface of the sun is not smooth, but that we see bright granules separated by darker intergranular lanes. The structures appear to have dimensions of the order of 500 km diameter and therefore can only be recognized under conditions of very good seeing (1 arcsec corresponds to 700 km on the sun). There may be even smaller structures which we cannot resolve because of the atmospheric seeing and because the solar observation satellites so far do not have mirrors large enough to resolve such small-scale structures. When we discussed these structures on the solar surface, we pointed out that in the bright regions the motions, measured by the Doppler shift, are mainly directed outwards, while in the dark intergranulum the motions are mainly downwards. These motions and temperature inhomogeneities seen in the granulation pattern are due to the hydrogen convection zone just below the solar photosphere. These motions in the hydrogen convection zone are believed to be the source of the mechanical energy flux which heats the solar chromosphere and corona. Similar convection zones in other stars are believed to be responsible for the heating of the stellar chromospheres and coronae whose spectra are observed in the ultraviolet and in the X-ray region by means of satellites. Before we can discuss these outer layers of the stars, we have to discuss briefly the reason for these convection zones and the velocities expected to be generated by this convection, and how these and the mechanical energy flux generated by these motions are expected to vary for different stars with different Teff, gravity, and with different chemical abundances.

The spectral sequence

If we look at the spectra of the stars we find that most spectra show a series of very strong lines which can now be identified as being due to absorption by hydrogen atoms. They are called the Balmer lines. Since these lines appeared to be strong in almost all stellar spectra, astronomers started to classify spectra according to the strengths of these lines, even though they did not know what caused them. The spectra with the strongest Balmer lines were called A stars, those with somewhat fainter lines were called B stars, and so on down the alphabet. It later turned out that within one such class there were spectra which otherwise looked very different; there were also stars within one spectral class which had very different colors. Taking this into account, the spectra were then rearranged mainly according to the B – V colors of the stars which did, however, turn out to give a much better sequence of the spectra. Spectra in one class now really looked quite similar. Now the spectra with the strongest Balmer lines are in the middle of the new sequence. The bluest stars have spectral type O, the next class now has spectral type B, and then follow the A stars, the F stars, and then the G, K, and M stars. In Fig. 2.1 we again show the spectral sequence. The bluest stars – the O stars – are at the top. Their Balmer lines are rather weak, as the letter O indicates.

The apparent magnitudes

The easiest quantity to measure for a star is its brightness. It can be measured either by intercomparison of the brightness of different stars or by a quantitative measurement of the energy received on Earth. For a bright enough star – the sun, for instance – the latter could be done in a fundamental but not very accurate way by having the sun shine on a well-insulated bowl of water for a certain length of time and then measuring the increase in temperature. Knowing how much water is in the bowl and knowing the surface area of the water, we can calculate the amount of heat energy received from the sun per cm2 s. The amount received will, of course, depend on the direction of the light beam from the sun with respect to the plane of the water surface (see Fig. 1.1). If the angle between the normal to the water surface and the direction of the beam is δ (see Fig. 1.1a), then the effective cross-section of the beam t is smaller than the actual surface area of the water by a factor of cos δ. In Fig. 1.1a, the amount of heat energy received per cm2 s will therefore be smaller by this factor than in Fig. 1.1b, where the sunlight falls perpendicularly onto the bowl of water.

For stars this method in general will not work; we would have to wait too long before we could measure an increase in temperature, and it is very difficult to prevent the water surface from cooling again in the meantime. We need much more sensitive instruments to measure radiation.

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.