6. Relation between velocity dispersion, luminosity, and a physical scale
Another way to investigate similarities among different globular cluster populations is to look at the relation between velocity dispersion, luminosity, and a physical size scale. In such a parameter space, systematic differences between two systems of globular clusters, in terms of structure or ratio, would result in either different mean relations or in different degrees of scatter about a given relation. We do not attempt here to derive mean relations through bivariate fits (e.g. Djorgovski 1995) as they turn out to be rather unstable because of both the small number of data points as well as the weak correlation between the physical size parameter and the luminosity and velocity dispersion. Instead, we fit to the data the relations expected from the Virial theorem, or from the King models. Assuming a constant ratio, the Virial Theorem predicts the relation
between the global velocity dispersion , the half-light radius , and the absolute visual magnitude . Similarly, King models predict
where is the central velocity dispersion, µ a dimensionless parameter which varies with cluster concentration c, the core radius, and the absolute visual magnitude. Fig. 5 shows the relations between vs. (uppermost row of panels), vs. (second row), and vs. (third row), for different data sets. For the Galactic clusters, we use velocity dispersion measurements based on radial velocities of individual stars, taken from the compilation of Pryor and Meylan (1993). For the other clusters, we use velocity dispersions derived from integrated-light observations, taken from Dubath et al. (1996a) for 8 old Magellanic clusters, from Dubath et al. (1992) for 3 clusters belonging to the Fornax dwarf spheroidal galaxy, from the present study for 9 M 31 clusters, and from Dubath (1994) for 10 Centaurus A clusters. The continuous lines represent the relation (uppermost row), and the relations (5) and (6), in the second and third rows respectively, with constants derived by fitting the Galactic cluster data. The dashed lines show the relations obtained when central velocity dispersions (extrapolated for these clusters by Pryor and Meylan  using King models) are considered instead of the global velocity dispersion.
It is worth mentioning that correlations resulting from bivariate fits do not differ much from the relations expected from the Virial theorem (Djorgovski 1995), and that Fig. 5 would look very similar if slightly different projections were used. The scatter of the data points probably results to a large extent from measurement errors, and it is quite remarkable that this scatter is of the same order in all panels of Fig. 5. The similarity of the different panels of Fig. 5 indicates both that there is no large systematic differences in globular cluster ratios between one galaxy to another, and that measurement errors are comparable, and even smaller, for extragalactic clusters than for Galactic clusters.
In the first row of Fig. 5, we expect a higher degree of scatter since the physical scale of each cluster (which acts as a second parameter) is not taken into account. Notably, a few large-size/low-concentration Galactic clusters lie well below the other clusters in the upper left panel. The fact that similar large clusters are not present in our M 31 and Cen A samples is due to a selection bias which favors brighter and generally more compact clusters. In any case, the relatively small spread of the data points around the straight lines in the uppermost panels points to an additional similarity in terms of physical size range between the clusters of these different galaxies, with the possible exception of the old Magellanic clusters. This is particularly interesting in the case of the Centaurus A clusters since these clusters are brighter than the Galactic ones, but their physical scales have not yet been measured.
For the old Magellanic clusters, the products µ are systematically smaller than the Galactic average. This can perfectly explain why the Magellanic clusters appear above the Galactic relation in the upper panel in Fig. 5, while the Magellanic and Galactic middle panels are similar. The lower panel is, however, rather puzzling. Unexpectedly, the data points all lie above the Galactic relation, as if the half-light radii used here and taken from van den Bergh (1994) were about 40% too large. This point is further discussed in another paper (Dubath et al. 1996b).
The two M 31 clusters (Bo158 & 225) with large ratio estimates from last section stand out above the Galactic relation in Fig. 5. As already pointed out, these clusters only have ground-based measurements of which may be overestimates. The scatter of the other clusters is remarkably small compared to the scatter of Galactic clusters.
6.1. Individual globular clusters as extragalactic distance indicators?
As illustrated in Fig. 5, the absolute magnitude of an individual globular cluster can be derived from its velocity dispersion and physical scale with an accuracy of 0.5 magnitude. For a given parent galaxy, providing there is no systematic differences in , an accurate distance modulus can in principle be computed using the mean distance modulus of a moderate number (10 - 20) of its associated globular clusters. In other words, plotting apparent instead of absolute magnitude in Fig. 5, the difference between the distance modulii of two parent galaxies is given by the horizontal shift required to bring the two (dereddened) data sets in agreement. Combining the capabilities of the HST and of 10-m class, ground-based telescopes, this method could be applied for galaxies out as far as the Virgo cluster.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998