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Period (days)

Distribution of periods for Cepheids. Note that there are two distinct groupings.

Leavitt, who was studying Cepheids in the Large and Small Magellanic Clouds, two small galaxies near the Milky Way. The advantage of studying Cepheids in either of these galaxies is that the Cepheids are all at the same distance. For the Small Magellanic Cloud it was found that there is a relationship between the period of the Cepheid and its mean apparent magnitude. Since all of the stars are at essentially the same distance, this means that there is a relationship between the period and the mean absolute magnitude.

If we know the exact relationship between period and absolute magnitude, then, when we observe a Cepheid, we can measure its period and convert that into an absolute magnitude. We can always measure the apparent magnitude. The difference m - M is the distance modulus, and gives us the distance. This technique is important because Cepheids are bright enough to be seen in other galaxies, providing us with distances to those galaxies.

However, before we can use the period-luminosity relationship, it must be calibrated. We need independent methods of measuring distances to some Cepheids. This is difficult, since there are none nearby. Statistical studies have been used to achieve this calibration. More recently, the Hiparcos satellite, which was designed to provide more accurate trigonometric parallaxes than had previously been available, made great strides on this problem.

If we plot a histogram indicating how many Cepheids have various periods (Fig. 10.7), we find an interesting result. The distribution has two peaks in it. This suggests that there are actually two different types of Cepheids. The group with the shorter periods are typical of those studied in the Magellanic Clouds, and are called classical Cepheids. The group with longer periods are called type II Cepheids or W Virginis stars (named after their prototype). Type II Cepheids are found in globular clusters in our galaxy (see Chapter 13 for a discussion of clusters). In general, a type II Cepheid is 1.5 mag fainter than a classical Cepheid of the same period. Also, the period-luminosity relation is slightly different for the two types.

The original calibration of the Cepheid distance scale was carried out for type II Cepheids, since we can study them in our galaxy. However, when we look at a distant galaxy, we can more easily study the brighter classical Cepheids. Therefore, the Cepheids studied in other galaxies were 1.5 mag brighter than assumed. This means that the galaxies are farther away than originally assumed.

Example 10.1 Cepheid distance scale By how much does the calculated distance to a galaxy change when we realize that we are looking at classical, rather than type II, Cepheids?

solution

We have already seen that the Cepheids originally studied are 1.5 mag brighter than assumed. This increases the distance modulus, m — M, by 1.5 mag. By equation (2.18), this increases the distance by a factor of io(15/5) = 2. Thus, these galaxies are twice being driven away. The actual mechanism for driv-

as far away as originally thought. The difference between the two types of Cepheids was realized in the 1950s, and people talked about the size of the universe doubling.

Another type of variable star that is useful in distance determinations is the RR Lyrae variable. These are found in globular clusters and are sometimes called cluster variables. They have short periods, generally less than one day. The absolute magnitudes of all RR Lyrae stars are very close to zero. Actually, they fall between zero and unity, and obey a weak period-luminosity relation of their own. The absolute magnitudes were established by using clusters whose distances were known from other techniques. Once the absolute magnitudes are calibrated, we can use RR Lyrae stars as distance indicators.

It should not be surprising that stars with pulsations have period-luminosity relations. For radial oscillations, we expect the period to be roughly equal to (Gp)~1/2, where p is the average density of the star. We can understand this qualitatively by noting that a star pulsating under its own gravity is like a large pendulum. The period of a pendulum is 2^ (L/g)1/2. For a star, L = R and g = GM/R2, so the period is approximately (GM/R3)~1/2, and M/R3 is approximately the density. Therefore, since the period is related to p (which is approximately M/R3), and the luminosity is related to the radius, the period should be related to the luminosity (see Problem 10.4). (Gp)~1/2 is also approximately the period of a satellite orbiting near the surface of a mass M, or the period of a small mass dropped through a hole in a larger mass. In short, if gravity dominates, (Gp)~1/2 is the time scale.

ing material away is still not fully understood. It may involve pressure waves moving radially outward. It may also involve radiation pressure. Photons carry

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