Comparing stars

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# 1 The Hertzsprung-Russell diagram

## 1.1 Constructing the H-R diagram

Three properties which are suitable for comparing stars are temperature, luminosity and radius. However, we don't need all three.

### Question 1

Why not?

Since stars emit like black bodies, temperature, luminosity and radius are related, via Equation A:

where L, R and T are respectively the luminosity, radius and temperature of the star, and σ is a constant. Thus, if we know any two, we can obtain the third.

Temperature and luminosity are more directly measurable for a far greater number of stars than radius, and so it is these two properties that are used, as shown in Figure 1. Each point displays the temperature and luminosity of a particular star: you should check that the values given for the Sun are in accord with the values given earlier. Note the logarithmic scales on both axes, and that temperature increases to the left.

Such a diagram is called a Hertzsprung-Russell diagram, or H-R diagram, after the Danish astronomer Ejnar Hertzsprung and the US astronomer Henry Norris Russell.

Figure 1 The Hertzsprung-Russell diagram for the Sun and a few nearby stars.

### Ejnar Hertzsprung (1873-1967) and Henry Norris Russell(1877-1957)

Ejnar Hertzsprung (Figure 2a) born in Denmark, initially chose chemical engineering as a career because of the poor financial prospects in astronomy. However, after developing his astronomical skills as a private astronomer, he became Assistant Professor of Astronomy at Göttingen Observatory in Germany and then Professor and Director of Leiden Observatory in the Netherlands. He proposed the concept of the absolute magnitude of a star as its magnitude at a distance of 10 parsecs. In 1906 he plotted a graph of the relationship between the absolute magnitudes and colour of stars in the Pleiades and coined the terms red giant and red dwarf. He published his work in a photographic journal without the diagrams and they were unknown to other astronomers.

In 1913 Henry Norris Russell (Figure 2b), then Director of the University Observatory at Princeton, plotted Annie Cannon's spectral classification against absolute magnitude and found that most stars lay in certain regions of the diagram. The diagram, which became a fundamental tool of modern stellar astronomy, was eventually called the Hertzsprung-Russell diagram in recognition of their independent work. One of the first applications of the H-R diagram was in the development of spectroscopic parallax by Hertzsprung, using observations of Cepheid variable stars made by Henrietta Leavitt.

Figure 2 (a) Ejnar Hertzsprung and (b) Henry Norris Russell. ((a) Royal Astronomical Society; (b) Science Photo Library)

### Question 1

Where, in the H-R diagram, do the following types of star appear: hot, high luminosity stars; hot, low luminosity stars; cool, low luminosity stars; cool, high luminosity stars?

The positions in the H-R diagram of these types of star are as follows:

 hot, high luminosity stars top left hot, low luminosity stars bottom left cool, low luminosity stars bottom right cool, high luminosity stars top right

The H-R diagram in Figure 1 contains too few stars to give us an overall picture. Before we examine a diagram containing many more stars we can speculate on what we might find. Will we find that the stars are fairly uniformly peppered over the diagram, with, for example, as many hot, high luminosity stars as any other kind? Or will we find that certain combinations of luminosity and temperature are more common than others? In any general population there are usually more small things than big things, more faint things than bright, and more cool things than hot. Therefore we might expect there to be more stars towards the bottom of the H-R diagram and more towards the right. To some extent these explanations are borne out but with some surprises. When more data are plotted, more stars are found towards the bottom right of the H-R diagram (Figure 3) but there are also noticeable empty zones, and a striking locus from hot bright to cool faint stars. The shaded regions show where stars tend to concentrate: the darker the shading, the greater the concentration. Each concentration defines a particular class of stars, and we shall shortly examine each main class in more detail, but first let's add stellar radius to Figure 3.

Figure 3 An H-R diagram, showing where stars tend to concentrate.
Figure 4 An H-R diagram, showing where stars of solar radius lie.

From the relationship between radius, temperature and luminosity in Equation A, we see that at each point in the H-R diagram there is a unique stellar radius, given by R = [L/(4σT4)]1/2. Let's now add to the diagram lines of constant radius. For example, consider stars with a radius equal to that of the Sun, R. From Equation A we see that any other star with the same radius will have its luminosity and temperature related by L ≈ (4R2σ)T4. Thus, as T increases, L also increases since for a given radius, the hotter the star the more power it radiates. With T increasing to the left in the H-R diagram, this gives a line sloping upwards from lower right to upper left, as in Figure 4. The line is straight because we are using logarithmic scales.

Figure 5 The H-R diagram in Figure 3, with the addition of stellar radii, and other information. (Adapted from Seeds, 1984)

Figure 5 is the H-R diagram in Figure 3 with several lines of constant radius added, and you can see that there are some classes of stars that are considerably smaller and some that are considerably larger than the Sun. These relative sizes are reflected in the names given to many of the classes, as shown in Figure 5; white dwarfs, red giants, supergiants. As you might expect, white dwarfs are small, red giants are large, and supergiants even larger.

### Question 2

In terms of the Earth's radius, and the Earth's distance from the Sun, how large are white dwarfs and red giants?

From Figure 5, we see that white dwarfs have radii of order 0.01R, which is about the radius of the Earth. Likewise, we see that red giants have radii of order 30R, which is about 3000 times the Earth's radius, or about a tenth of the distance of the Earth from the Sun. Note that the ranges of radii for white dwarfs, and particularly for red giants, are large.

The class names are descriptive in ways other than size.

### Question 2

Why white dwarfs, red giants?

White dwarfs have temperatures that result in yellowish-white to bluish-white colours. Red giants have tints towards the red end of the visible spectrum, embracing orange-white and yellowish-white.

We have added to Figure 5 an indication of the colour associated with each temperature. However, to the unaided eye, star colours are in many cases not very striking. This is partly because too little light is being received for our colour vision to be strongly excited, and partly because the colours are, in any case, rather weak. However, stellar colours can be emphasized photographically (as you will see in Figure 14).

The H-R diagram can be represented in a number of different ways; the colour index of a star is a measure of its temperature and the spectral classification scheme is also a temperature sequence. Another way of representing stellar luminosity is through the absolute visual magnitude. It should be clear therefore that the luminosity axis of the H-R diagram could equally well be plotted as absolute visual magnitude and the temperature axis with spectral type or colour index. These kinds of diagrams are often used by astronomers as these are quantities that are obtained more directly from observation. A particular type of H-R diagram, called a colour-magnitude diagram, shows Mv against B - V. Figure 5 illustrates the approximate values of absolute visual magnitude and colour index as well as spectral type. The exact appearance of the H-R diagram will be slightly different when these alternative axes are used (e.g. the absolute visual magnitude Mv is directly related to the luminosity in the V band, Lv, and not the total luminosity L). Also, spectral type depends weakly on luminosity.

Let's now look at the main classes of stars in more detail.

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