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?

Introduction

Even though this volume is dedicated only to stellar observations and the theory of stellar structure, we have to talk a little about the interstellar material, which means the material which is between the stars, because the light of the stars, especially of those which are very far away, has to pass through this interstellar material before it reaches us. Just as in the Earth's atmosphere, the star light is influenced by interstellar absorption, also called interstellar extinction. We cannot observe all the light which is emitted from the star because the interstellar gas and the interstellar grains, called dust, have absorbed part of the light. Just as for the Earth's atmosphere, we have to correct for interstellar absorption. For the Earth's atmosphere we can determine the extinction if we observe the star at different zenith distances, which means for different path lengths, through the atmosphere. For interstellar extinction we cannot do this since the path length through the interstellar medium is always the same. We have to find other ways to determine the influence of the interstellar extinction.

Basically there are two components of interstellar extinction: the absorption by the grains and the absorption by the gas. These components are usually, but not necessarily, associated. In the following paragraphs, we will discuss both components and see how we can find out about interstellar absorption.

The spectral sequence

So far we have talked only about global properties of the stars and about their brightnesses in broad wavelength bands. We get, of course, more information when we reduce the widths of the wavelength bands in which we study the radiative energy emitted by the stars. If we reduce the bandwidth to the order of a few Å or even a fraction of an Å, and cover all wavelengths, we talk of stellar spectra. If we compare spectra of different stars, we see that there are many different kinds of spectra. Most of them can be ordered in a continuous sequence of spectra, the so-called spectral sequence. In Fig. 10.1 we show the sequence as it is used now, and as it was established finally by Morgan, Keenan and Kellman (1943).

If we plot the energy distribution in a spectrum as a function of wavelength, we get plots as shown in Figs 10.2. Basically, we see a continuous energy distribution (Fig. 10.2(a)), but there are many wavelengths for which the energy is reduced by varying amounts, the so-called spectral lines, see Fig. 10.2(b). If we look at these wavelengths in a spectrum, they look dark because there is little energy at these wavelengths. Such dark lines are called absorption lines. There are also some spectra which have bright lines, i.e., for which there is more energy at these wavelengths.

Introduction

The sun is, of course, the star nearest to us and is, therefore, the best studied star. We have mentioned the sun several times as an example when we talked about distances of stars, effective temperatures and masses of stars, as well as angular radii. For all these studies we considered the sun to be just one of the normal stars, which it most probably is. It is the most thoroughly studied normal star. The sun is also the only star for which we can get high spatial resolution, which enables us to observe fine details on the surface which we will not be able to observe on other stars, at least not in the foreseeable future. These high spatial resolution studies of the sun reveal many features and processes which may well be also going on in other stars, but which we are not able to study in any other celestial object. Some of these features are the solar chromosphere and corona, though ultraviolet observations by means of satellites now permit us also to study global properties of these outer layers of stars other than the sun.

Another such phenomenon, which we can study in detail only in the sun, is the solar activity, which means flares, sunspots, and the whole solar cycle of activities. Again, observations with very high resolution and very sensitive receivers now permit us to study global effects of activity in other stars which seem to show activities similar to the sun.

General discussion of binaries

For the stars for which angular diameters have been measured we can, of course, determine the radii by multiplying the angular radius by the distance to the star, provided we know the distance. For stars further away than about 20 pc we cannot measure trigonometric parallaxes accurately, we can therefore determine distances only in indirect ways. Star stream parallaxes (see for instance Becker, 1950) have been used to determine the distance to the Hyades cluster, but it turned out that the photometric parallaxes were more accurate. Fortunately, we can also determine radii for stars in special binary systems, namely in eclipsing binaries. For these binaries we can also determine the masses of the stars. We shall therefore devote this section to the discussion of binaries in general, and the next sections to the discussion of the special types of binaries for which stellar radii can be determined, namely eclipsing binaries, and for those binaries for which stellar masses can also be determined, again the eclipsing binaries and also the visual binaries.

We can determine stellar masses for binaries using Kepler's third law. Let us briefly discuss the mechanics of a binary system.

From the solar system we are accustomed to the case that one body, the sun, has a much larger mass than the other bodies, the planets. For binaries with two stars we have to take into account that both bodies have elliptical (or perhaps circular) orbits around their center of gravity S (see Fig. 9.1).

General discussion

We are all accustomed to the fact that our Earth has a magnetic field whose shape comes rather close to that of a dipole field, with the magnetic axis not being very different from the rotational axis of the Earth. This raises the suspicion that the magnetism is due or at least related to the rotation of the Earth. We may therefore wonder whether stars which have much higher rotational velocities than the Earth might have much stronger magnetic fields than the Earth. It seems very interesting to check. The question arises: how can we measure magnetic fields on stars? We clearly cannot take a magnetometer to the star's surface. All we can get from the star is its light. Fortunately, nature has provided an effect of a magnetic field on the light which can be used to measure magnetic fields of stars. This is the Zeeman effect, named after its discoverer.

The Zeeman effect

Zeeman discovered that for a laboratory light source which emits an emission line spectrum in a laboratory magnetic field, the spectral lines generally split into several components. If you observe the light in an arbitrary direction with respect to the direction of the magnetic field, you will, in the simplest case, see a line split into three components, the socalled Lorentz triplet. The central component remains at the original wavelength λ0, which the line had without a magnetic field.

The coordinate system

If we want to study stars, the first thing to look at might be their positions in the sky. This in itself does not tell us much about the nature of the stars, but it is very helpful when we want to find a particular star or a group of stars in the sky. We have to have a reference point with respect to which we can describe the position of the star in which we are interested. We all know or have at least heard about the constellations of stars which in earlier times were extremely helpful in describing the positions of stars with respect to a given star in a particular constellation. We still name the brightest stars according to the constellations in which they are found, but we like to have a more general description of the positions. When looking at the sky we can measure the positions only as projected against the sphere of the sky, i.e., against a two-dimensional surface. We can therefore describe the positions of the stars by two quantities. Since the surface against which we measure the positions is a sphere, we use spherical polar coordinates. Since our telescopes are fixed on the Earth, we use a coordinate system which is fixed with respect to the Earth. The Earth is rotating, but we do not want to have a rotating coordinate system, which would cause many problems.

Color magnitude diagrams of nearby stars

For nearby stars, say within 20 pc, we can determine the distances from trigonometric parallaxes. From the apparent magnitudes and the distances we can calculate the absolute magnitudes, i.e., the magnitudes which the stars would have if they were at a distance of 10 pc. This means that for absolute magnitudes we compare the brightness the star would have if it were at a distance of 10 pc with the actual brightness of Vega at its actual distance, i.e. with its apparent brightness. It turns out to be quite instructive to plot the absolute magnitudes of the stars as a function of their B – V colors. In Fig. 5.1 we do this for the nearby stars. While we might have expected that stars with a given color could have quite different absolute magnitudes, it turns out that this is generally not the case. Most of the stars with a given B – V color have the same absolute magnitude. Most of the stars fall along one line in the color magnitude diagram. This line is called the main sequence. The intrinsic brightnesses and the colors of these stars are obviously determined by just one parameter, since they follow a one-dimensional sequence. It turns out, as we shall see in Volume 3, that this one parameter is the mass of the star.