If we substitute this into the virial theorem (equation 13.47), the mass is given by

It is important to understand which motions we are talking about. The cluster has some overall motion of its center of mass, shared by all the stars in the cluster. The stars have individual motions within the cluster (with respect to the center of mass of the cluster). The net motion of each star is the vector sum of these two motions, and that is what we observe. In equation (13.47) the quantity {v2} is the average of the square of the star velocities with respect to the center of mass of the cluster.

The best way for us to measure the velocities of individual stars is through their Doppler shifts. However, this only gives us the component of the velocity along the line of sight. This means that we are measuring {vj?} rather than {v2}. However, if the internal motions of the stars are random, we can relate these two quantities.

Suppose we resolve the motion of any star into its components in an (x, y, z) coordinate system. The velocity can then be written in terms of its components as v = vxX + vyy + vzz (13.48)

where X, y and z are the unit vectors in the three directions, respectively. To be definite, we can let the x-direction correspond to the line of sight. The square of v, which is v • v, is simply the sum of the squares of the components, v2 = vj + v? + vj (13.49)

If we then take the average of both sides of the equation, we have

However, if the motions are random, the averages of the squares of the components should be the same for all directions. This means that v2} = v2} = v2} (13.50)

Using this, equation (13.49) becomes

and since the x-direction is the one corresponding to the line of sight, vr = vx, so

Example 13.4 Virial mass of cluster

Find the virial mass of a cluster with {vr} = 10 km/s and R = 5 pc.


From equation (13.52) we have

When we talked about binary stars (Chapter 5), we noted that the best way to measure the mass of an object, or a group of objects, is to measure their gravitational effects on other objects. The gravitational effects are independent of how bright the objects are; they depend only on how massive they are. Using virial masses is an extension of these ideas. The more massive the cluster, the greater the internal motions that we will observe. To determine {vr}, it is not even necessary to measure radial velocities for all the stars in the cluster. We just need a representative sample.

What are the limitations of this method? An important one is that we don't know if any particular cluster is dynamically relaxed, or even gravitationally bound. We may measure large internal motions in an unbound system and mistake them for bound motions in a more massive system. This can introduce errors that are off by as much as a few orders of magnitude. Our calculation of the relaxation time suggests that all but the youngest clusters should be relaxed. We will talk about indicators of age of a cluster in the next section. Another limitation can be from geometric effects. We may observe clusters that are elliptical, rather than spherical. Or we may observe clusters that don't have a uniform density. The most likely variation is having a higher density in the center. The effect of these geometric effects can be to produce errors of order unity (see Problem 13.16). For most applications,

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