Clusters of stars

When we look at the spatial distribution of stars in our galaxy, we find that most of the light is concentrated in a thin disk. We are inside this disk, so we see it as a band of light on the sky, called the Milky Way. We will discuss this farther in Part III, but we will see in this chapter that location of stars in the galaxy can tell us something about those stars. In particular, some stars are confined to the thin disk of the Milky Way, while others form a more spherical distribution. In this chapter, we will discuss groupings of stars, called clusters, and see how they vary in size, content and galactic distribution.

13.1l Types of clusters

We distinguish between two types of star clusters -galactic clusters and globular clusters.

Galactic clusters are named for their confinement to the galactic disk. A selection of images of galactic clusters is in Fig. 13.1. A familiar galactic cluster, the Pleiades, is shown in Fig. 13.1(a). Note the open appearance in which individual stars can be seen. Because of this appearance, galactic clusters are also called open clusters. Galactic clusters typically contain <103 stars, and are less than ~10 pc across. Recent sensitive near IR surveys are showing more members than we had previously thought in many clusters. In the photograph, we see some starlight reflected from interstellar dust. Galactic clusters are sometimes associated with interstellar gas and dust.

Globular clusters are named for their compact spherical appearance (Fig. 13.2). They have 104 to

106 stars, and are 20 to 100 pc across. They seem to have no associated gas or dust. Some do have planetary nebulae, though. Globular clusters are not confined to the disk of the galaxy. Harlow Shapley used RR Lyrae stars and Cepheids to find the distances to globular clusters. This placed the globular clusters in three dimensions. It was found that the globular clusters form a spherical distribution with the Sun being about 10 kpc from the center. (This is still one of the best techniques for finding the distance to the galactic center.)

Before we look at the properties of the clusters themselves, we will look at an important technique for determining distances to relatively nearby galactic clusters.

13.2 I Distances to moving clusters

Let's assume that we have a star (or cluster) moving through space with a velocity v. The velocity makes an angle A with the line of sight. We can break the velocity into components parallel to the line of sight and perpendicular to the line of sight. The component parallel to the line of sight is the radial velocity vr and is responsible for the Doppler shift we observe. The component perpendicular to the line of sight is the transverse velocity vT. It is responsible for the motion of the star across the sky, called the proper motion.

From the right triangle in Fig. 13.3, we can see that these quantities are related by v2 = vr2 + vT2 (13.1)

Open clusters. (a) The Pleiades (M45) in Taurus. The nebulosity seen here is starlight reflected from interstellar dust. (b) M6, also known as the Butterfly Cluster. (c) M7 in Scorpius. Its distance is about 300 pc, and it is about 8 pc across. (d) M37 in Aurigi, at a distance of 1.5 kpc. [(a) Courtesy of 2MASS/UMASS/IPAC/NASA/JPL/Caltech; (b)-(d) NOAO/AURA/NSF]

Open clusters. (a) The Pleiades (M45) in Taurus. The nebulosity seen here is starlight reflected from interstellar dust. (b) M6, also known as the Butterfly Cluster. (c) M7 in Scorpius. Its distance is about 300 pc, and it is about 8 pc across. (d) M37 in Aurigi, at a distance of 1.5 kpc. [(a) Courtesy of 2MASS/UMASS/IPAC/NASA/JPL/Caltech; (b)-(d) NOAO/AURA/NSF]

ond, is just the transverse velocity divided by the distance to the star d:

The relationship between proper motion and tangential velocity is shown in Fig. 13.4. The proper motion p, expressed in radians per sec-

The greater the transverse velocity, the faster the star will appear to move across the sky. Also, the closer the star is, the greater the motion across the sky. In general, proper motions are very small, amounting to a few arc seconds per year, or less. (The largest proper motion is 10.3 arc sec/yr for Barnard's star.) For this reason, we would like to rewrite equation (13.5), expressing p in arc seconds per year, v in kilometers per second, and d in

Globular clusters. (a) M3, in Canes Venatici. (b) M5, at a distance of 0.8 kpc, is one of the most massive clusters in our galaxy. (c) MI5 in Pegasus, at a distance of 1 kpc. (d) M80. [(a), (b) NOAO/AURA/NSF; (c), (d) STScl/NASA]

parsecs. We can then rewrite equation (13.5) as vT ^ d

1 yr

In general, we can measure the radial velocity (from the Doppler shift) and we can also measure the proper motion. If we know the transverse velocity, we can find the distance from d = vT/4.74^ (13.7)

If, instead of vT, we know A, then we can use equation (13.4) in equation (13.7) to give d = vr tan A/4.74/U,



Space velocity.The velocity of the star is v, which makes an angle A with the line of sight.The radial and tangential components of the velocity are vr and vT, respectively.

The dependence of the proper motion on the distance d and tangential speed vT.We follow the motion of the star for a time At.

Vv Cluster a3 1

Convergent Point

With a cluster of stars, we can compare the proper motion with the rate at which the angular size of the cluster changes to find A. To see how this works, let us consider the case of a cluster moving away from us (positive radial velocity). As the cluster moves away (Fig. 13.5), A becomes smaller and approaches zero. As the cluster moves farther away, the proper motion approaches zero (by equation 13.5). Thus, the cluster appears to be heading toward a particular point in the sky. We


Convergent point.As the cluster moves farther away, the angle A between v and the line of sight approaches zero.That is A| > A2 > A3.As this happens, the cluster approaches one line of sight, the convergent point. Note that this figure is exaggerated to show the effect. Real clusters do not move that much over the times we could observe them.

call this point the convergent point of the cluster. We can see from the figure that the angle between the current line of sight and the line of sight to the convergent point is the current value of A. Similar reasoning applies to clusters that are moving toward us. They are moving away from their convergent point, so we find it by extrapolating their motion backwards in time.

To apply these ideas, we take a series of images a number of years apart. From the proper motion, and the change in angular size, we can find the convergent point, and therefore we know A. This is shown schematically in Fig. 13.5.

Convergent Point

Schematic representation of motions of stars within a cluster.Arrows represent total motions.

We measure the radial velocity vr and proper motion Since we know A, vr and ¡¡, we can find d from equation (13.8). The best determination of a distance to a moving cluster is the Hyades. This determination is an important cornerstone in our determination of distances to more distant objects in our galaxy and in other galaxies.

13.3 I Clusters as dynamical entities

In this section we look at the internal dynamics of star clusters. If the gravitational forces between the stars are sufficient to keep the cluster together, we say the cluster is gravitationally bound. (We have already discussed the idea of gravitational binding when we talked about binary stars, in Chapter 5 and Chapter 12.) However, gravity does more than assure the overall existence of the cluster. As stars move around within the cluster, pairs of stars will pass near each other. The gravitational attraction between the two stars in the pair will alter the motion of each star. The momentum and energy of each star will change in this gravitational encounter. Thus, these encounters alter the distribution of speeds, the number of stars traveling at a given speed. If there has been sufficient time for many encounters to occur, the distribution of speeds will reach some equilibrium. For every star that suffers a collision changing its speed from v1, there is another collision in which some other star has its speed changed to v1. (We refer to these gravitational encounters as collisions, even though the stars never actually get close enough for their surfaces to touch.) When a cluster has reached this stage, we say that it is dynamically relaxed.

13.3.1 The virial theorem

In a dynamically relaxed system, the kinetic and potential energies are related in a very specific way. This relationship is known as the virial theorem. In this section we derive it.

We begin with a collection of N particles. (We can think of each star in a cluster or each atom in a gas cloud being represented by a particle.) To simplify the calculation we assume that all particles have the same mass m. The final result would be the same if we allowed for different masses.

(See Problem 13.10.) We let the position of the ith particle, relative to some origin, be t». If we have two particles, i and j, the vector giving their separation is rj — r,, as shown in Fig. 13.7. We let F, be the net force on the ith particle. We can therefore write the equation of motion for this particle as

We are looking for a relationship between various types of energy. The vector dot product between force and distance will give us an energylike quantity. We therefore take the dot product of r with both sides of equation (13.9) and then sum the resulting quantities over all the particles to give

'dV dt2

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