$$\require{cancel}$$

# 12.22 Stars of Different Sizes

As astronomers studied the properties of large samples of stars, they found stars with many different combinations of luminosity and temperature. Most of these stars had properties that placed them at different positions on the main sequence of the H-R diagram, slanting from high luminosity and high temperature to low luminosity and low temperature. However, some stars had properties dissimilar from any main sequence star.

Hertzsprung-Russell diagram showing color and size of stars. Click here for original source URL

Some red stars with spectral types K and M have much higher luminosity than the faint, main-sequence K and M stars. They radiate more light than the fainter stars of the same temperature. According to the Stefan-Boltzmann law, they must have larger areas. Hertzsprung named them giant stars. Most are called red giants, since their outer regions are cool and thus look reddish.

Other stars are brighter than the giants or the main-sequence O stars. Application of the Stefan-Boltzmann law showed that they were larger than giant; they came to be called super giant stars. For example, the enormous red giant Antares in the constellation Scorpius, would dwarf the Sun. Antares is only ten times the mass of the Sun but it is so bloated it is almost 900 times the size; the Sun would fit into Antares 700 million times over!

Still other stars have combinations of high temperature and low luminosity that place them below the main sequence of the H-R diagram. By using the same radiation law to derive their sizes, astronomers recognized that these dim stars were unusually small. They came to be called white dwarf stars. A white dwarf the mass of the Sun would be a hundred times smaller than the Sun, so a million white dwarfs would fit inside the Sun. Combining the last two comparisons a thousand trillion white dwarfs would fit inside a red super giant! The size range of stars is enormous.

Hertzsprung, Russell, and their followers showed the systematic properties of different types of stars: one type was the main sequence, and the giants, super giants, and white dwarfs were distinctly different types of stars. These types refer to the luminosity class of a star. The main sequence — seen as stars whose properties place them on a diagonal of the H-R diagram — is the set of stars converting hydrogen into helium, a set that includes the Sun. Half of the most prominent stars in the sky are giants or super giants. Although the whales among the fishes are rare, they account for many of the familiar stars. Theire combinations of size and luminosity means they must have a different mechanism for energy production than the Sun and other mani sequence stars. Further work showed that these groups occur all over the sky and at many distances from the Sun. This meant that universal physical processes could be sought to explain the different non-main-sequence stars.

Occasionally astronomers find groups of stars at about the same distance from us and formed at about the same time. These star clusters offer, so to speak, living H-R diagrams in the sky. Just by studying the colors and brightness of stars in a cluster, astronomers can see the pattern of the H-R diagram. Most of the stars are on the main sequence; the brightest ones are blue-white, and the faintest ones are red.

The H-R diagram only shows temperature and luminosity. However, we can use the Stefan-Boltzmann law to calculate stellar radius and show it on the H-R diagram as well. The important concept is this: any point on the H-R diagram corresponds to a star of a certain radius. The reasoning should be familiar. Any point on the diagram corresponds to a certain temperature T. According to the Stefan-Boltzmann law, any surface at temperature T must radiate a certain amount of energy per square meter each second. But any point on the diagram also corresponds to a particular luminosity L, which is the total amount of energy radiated each second. This luminosity fixes the number of square meters involved; giving us the total area and thus the radius.

The size of Betelgeuse compared to our solar system. Click here for original source URL.

There is an enormous range in the physical properties of stars in the H-R diagram. Astronomers have the practical problem that stars with the same color might be quite different in their other properties. To see this, consider stars of different luminosity that have the same surface temperature (stars aligned vertically in the H-R diagram). Arcturus is over ten times larger than the Sun but has the same color. Betelgeuse is more than 10,000 times larger than Proxima Centauri but they both have a reddish color. Without knowing distances, is there any way to distinguish main sequence stars, giants, and super giants of the same color? Yes. The more diffuse atmosphere in the larger stars gives less atomic collisions and a narrower absorption line. Spectroscopy can be sufficient to establish the luminosity class (main sequence, giant, supergiant) of a star. This allows a distance to be estimated, even if no parallax  measurement is possible. However, spectral diagnostics apply to the outer layers of a star so they are indirect; the main differences between these star types stems from their masses and their energy production mechanisms.