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.

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.

The different types of pulsating stars

The pulsating stars (not to be confused with pulsars, which are rotating not pulsating), are recognized by their periodic changes in brightness accompanied by periodic variations of their radial velocities. We distinguish several classes of pulsating stars. The most frequent type are the δ Cephei stars, which have periods of light variation of a few days to a few weeks. Another very common type is the RR Lyrae star, named after the first-discovered variable of this kind. The RR Lyrae stars vary with periods of the order of half a day. Other types of short period variables are the δ Scuti stars, which are population I variables like the longer period δ Cephei stars. The δ Cephei stars are generally of spectral types F and G, and are supergiants, while the δ Scuti stars are A stars of luminosity class IV to V. The amplitudes of the δ Cephei variables are of the order of one magnitude, i.e., the light output changes by a factor of 2 or 3, while the light variations of the δ Scuti stars are so small that they are hard to detect.

The RR Lyrae stars are frequently found in globular clusters; therefore, they belong to population II. The amplitudes of their light variations are also of the order of one magnitude, and can therefore be recognized fairly easily.

There are a few stars in our neighborhood whose spectra show a different chemical composition for their photospheres. These stars were previously known as subdwarfs. The reason for this name was their position in the color magnitude diagram: they appeared below the main sequence, which means they either are too faint for their color or they are too blue for their brightness. A spectrum analysis showed that the latter is the case. It turned out that, for these stars, the relative abundances of the heavy elements with respect to one another are quite similar to the ones observed for the sun, but the overall abundances of the heavy elements with respect to hydrogen and helium are considerably reduced by up to a factor of 500, though most of them have much smaller abundance reductions. In these metal-poor stars, the metallic lines are much weaker than for normal stars of the same temperature. Since spectral lines are generally stronger in the blue, and especially in the ultraviolet, than in the red, the lines take more energy out of the ultraviolet and blue spectral region than out of the red. If the lines are weakened in the metal-poor stars, more energy is restored to the ultraviolet and blue spectral region than in the red and the stars therefore look bluer, especially in the ultraviolet. They show an ultraviolet excess which can, in fact, be used to determine their metal deficiencies.

The topic of this volume is stellar astronomy or more accurately stellar astrophysics. We call it astrophysics because all our knowledge about stars is based on the application of the laws of physics to the stars. We want to find out how big the stars are, how much mass they have, what material they are made of, how hot they are, how they evolve in time, and how they are distributed in space. The last question does not strictly belong to the field of stellar astrophysics but knowledge of stellar structure and evolution will provide a means by which to determine their distances. There are also important correlations, for instance, between the location and motion of the stars in our Galaxy and their physical properties.

In Volume 1, we shall be concerned mainly with finding out about the global properties of stars, such as brightnesses, colors, masses and radii. Brightnesses and colors can be measured directly for all stars, for masses and radii we have to study binaries. Parallax measurements can give us distances to nearby stars. We shall first discuss the majority of stars which we call normal stars. In the latter parts of this volume we shall also look at stars which seem to be different, the so-called ‘peculiar’ stars.

How can we get information, for instance, about the physical properties of the stars such as their temperatures, pressures, and chemical compositions?