## Open Star Clusters

Open clusters usually contain from a few tens to a few hundreds of stars. The kinetic energy of the cluster members, the differential rotation of the Milky Way (Sect. 17.3) and external gravitational disturbances tend to gradually disperse the open clusters. Still, many of them are fairly permanent; for example, the Pleiades is many hundreds of millions of years old, but nevertheless, quite a dense cluster.

The distances of star clusters - and also of associations - can be obtained from the photometric or spectroscopic distances of their brightest members. For the nearest clusters, in particular for the Hyades, one can use the method of kinematic parallaxes, which is based on the fact that the stars in a cluster all have the same average space velocity with respect to the Sun. The proper motions in the Hyades are shown in Fig. 16.4a. They all appear to be directed to the same point. Figure 16.4b explains how this convergence can be understood as an

Hannu Karttunen et al. (Eds.), Star Clusters and Associations.

In: Hannu Karttunen et al. (Eds.), Fundamental Astronomy, 5th Edition. pp. 339-345 (2007) DOI: 11685739_16 © Springer-Verlag Berlin Heidelberg 2007

Fig. 16.1a-c. Open clusters. (a) The Hyades slightly to the lower left in the photograph. Above them to the right the Pleiades (Photograph M. Korpi). (b) The Pleiades photographed with the Metsahovi Schmidt camera. The diameter of the cluster is about 1 °. Reflection nebulae are visible around some of the stars (Photograph M. Poutanen and H. Virtanen, Helsinki University). (c) Misan or h and x Persei, the double cluster in Perseus. The separation between the clusters is about 25'. Picture taken with the Metsahovi 60-cm Ritchey Chretien telescope (Photograph T. Markkanen, Helsinki University)

effect of perspective, if all cluster members have the same velocity vector with respect to the observer. Let 6 be the angular distance of a given star from the convergence point. The angle between the velocity of the star and the line of sight will then also be 6. The velocity components along the line of sight and at right angles to it, vr and vt, are therefore given by vr = v cos 0 , vt = v sin 0 .

Thus the distance can be calculated:

The radial velocity vr can be measured from the Doppler shift of the stellar spectrum. The tangential velocity vt is related to the proper motion x and the distance r:

By means of this method, the distances of the individual stars can be determined from the motion of the cluster as a whole. Since the method of (ground-based) trigonometric parallaxes is reliable only out to a distance of 30 pc, the moving cluster method is an indispensable way of determining stellar distances. The distance of the Hyades obtained in this way is about 46 pc. The Hyades is the nearest open cluster.

The observed HR diagram or the corresponding colour-magnitude diagram of the Hyades and other nearby star clusters show a very well-defined and narrow main sequence (Fig. 16.5). Most of the cluster mem-

mmmm

Fig. 16.2. The globular cluster w Centauri. The picture was taken with the Danish 1.5-m telescope at La Silla, Chile. Thanks to the excellent seeing, one can see through the entire cluster in some places. (Photograph T. Korhonen, Turku University)

Fig. 16.4. (a) Proper motions of the Hyades. The vectors stars move in the same direction, their tangential velocity show the movement of the stars in about 10,000 years. (van components appear to be directed towards the convergence Bueren, H. G. (1952): Bull. Astr. Inst. Neth. 11). (b) If all point K

Fig. 16.4. (a) Proper motions of the Hyades. The vectors stars move in the same direction, their tangential velocity show the movement of the stars in about 10,000 years. (van components appear to be directed towards the convergence Bueren, H. G. (1952): Bull. Astr. Inst. Neth. 11). (b) If all point K

bers are main sequence stars; there are only a few giants. There are quite a few stars slightly less than one magnitude above the main sequence. These are apparently binary stars whose components have not been resolved. To see this, let us consider a binary, where both components have the same magnitude m and the same colour index. If this system is unresolved, the colour index will still be the same, but the observed magnitude will be m — 0.75, i. e. slightly less than one magnitude brighter.

The main sequences of open clusters are generally located in the same section of the HR or colour-magnitude diagram (Fig. 16.6). This is because the material from which the clusters formed has not varied much, i. e. their initial chemical composition has been fairly constant. In younger clusters the main sequence extends to brighter and hotter stars and earlier spectral types. Usually one can clearly see the point in the diagram where the main sequence ends and bends over towards the giant branch. This point will depend very strongly on the age of the cluster. It can therefore be used in determining the ages of open clusters. Star clusters are of central importance in the study of stellar evolution.

The colour-magnitude diagrams of star clusters can also be used to determine their distances. The method is called main sequence fitting. By means of multicolour photometry the reddening due to interstellar dust can be removed from the observed colours B — V of the stars, yielding the intrinsic colours (B — V)0. Most star clusters are so far away from us that all cluster members can be taken to be at the same distance. The distance modulus mv0 - My = 5lg

10 pc

will then be the same for all members. In (16.4), mVo is the apparent, Mv the absolute visual magnitude of a star, r

Fig. 16.6. Schematic colour-magnitude diagrams of star clusters. M3 is a globular cluster; the others are open clusters. Cluster ages are shown along the main sequence. The age of a cluster can be told from the point where its stars begin to turn off the main sequence. (Sandage, A. (1956): Publ. Astron. Soc. Pac. 68, 498)

Fig. 16.6. Schematic colour-magnitude diagrams of star clusters. M3 is a globular cluster; the others are open clusters. Cluster ages are shown along the main sequence. The age of a cluster can be told from the point where its stars begin to turn off the main sequence. (Sandage, A. (1956): Publ. Astron. Soc. Pac. 68, 498)

and r the distance. It has been assumed that the extinction due to interstellar dust AV has been determined from multicolour photometry and its effect removed from the observed visual magnitude m V:

When the observed colour-magnitude diagram of the cluster is plotted using the apparent magnitude mV0 rather than the absolute magnitude MV on the vertical axis, the only change will be that the position of the main sequence is shifted vertically by an amount corresponding to the distance modulus. The observed (mV0 , (B — V)0) diagram may now be compared with the Hyades (MV, (B — V)0) diagram used as a standard. By demanding that the main sequences of the two diagrams agree, the distance modulus and hence the distance can be determined. The method is very accurate and efficient. It can be used to determine cluster distances out to many kiloparsecs.

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