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astronomers use the virial theorem, knowing that it is a technique that may be off by a factor of two or three. But this can be very useful for measuring the masses of a variety of astronomical systems. We will look more at virial masses when we talk about the masses of interstellar clouds (Chapter 14) and the masses of clusters of galaxies (Chapter 18).

13.4 I HR diagrams for clusters

By studying the HR diagram for a cluster, we are studying a group of stars with a common distance. We can study their relative properties without knowing what their actual distance is. If we do know the distance to the cluster, we plot directly the absolute magnitudes on the HR diagram. If we don't know the distance, we plot the apparent magnitudes. We then see how many magnitudes we would have to shift the diagram up or down to calculate the right absolute magnitudes for each spectral type. The amount of shift gives the distance modulus for the cluster, and therefore the distance. This procedure is known as main sequence fitting. It is like doing a spectroscopic parallax measurement, but it uses the information from all of the main sequence stars in the cluster. This is more accurate than studying a single star.

The HR diagram for a group of galactic clusters is shown schematically in Fig. 13.9. Note that the lower (cooler or later) part of the main sequence is the same for all the clusters shown. For each cluster, there is some point at which the main sequence stops. Beyond that point, no hotter stars are seen on the main sequence. The hotter stars all appear to be above the main sequence. The point at which this happens for a given cluster is called the turn-off point. Stars of earlier spectral type (hotter than) the turn-off point appear above the main sequence, meaning that they are more luminous, and therefore larger than main sequence stars of the same spectral type. Each cluster has its turn-off point at a different spectral type. Data for one cluster are shown in Fig. 13.10, to see the scatter in the points.

We interpret this behavior as representing stellar aging, in which stars use up their basic fuel supply, as described in Chapter 10. Hotter, more massive, stars evolve faster than the cooler, low mass stars, and leave the main sequence sooner. We assume that the stars in a cluster were formed

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