Astron. Astrophys. 324, 505-522 (1997)

9. Relation between velocity dispersion, luminosity, and a physical scale

Galactic globular clusters show tight correlations between velocity dispersion, luminosity and a physical scale. These correlations, which are analogous to the fundamental plane correlations for elliptical galaxies, have already been discussed by several authors (e.g., Meylan & Mayor 1986, Paturel & Garnier 1992, Djorgovski & Meylan 1994). Djorgovski (1995) shows that the scaling law, corresponding to the best fit to the currently available data for Galactic globular clusters, is consistent with the scaling law expected from the Virial Theorem. This suggests that globular clusters are virialized systems, with a universal and constant M/L ratio to within the measurement errors.

With a constant M/L ratio, the Virial Theorem predicts the following relation:

between the global velocity dispersion , the half-light radius , and the absolute V magnitude , whereas the King (1966) models lead to:

where is the central velocity dispersion, µ the dimensionless mass, the core radius, and the absolute V magnitude. Figure 16 shows the relations between vs. (first row), and vs. (second row), for different data sets. In panels a and b, we plot clusters with a global velocity dispersion measurement, based on radial velocities of individual stars, taken from the compilation of Pryor and Meylan (1993). In panels c, d, e, we plot clusters with a core velocity dispersion derived from integrated-light measurements in the present study. Half-light radii are taken from Trager et al. (1993), and cluster concentrations and core radii are from Pryor and Meylan (1993). In panels a and b, the continuous lines represent the relations (6) and (7), respectively, with best fitted constants. The dashed lines represent the relations obtained when central velocity dispersions (extrapolated for these clusters from King models by Pryor and Meylan 1993) are considered instead of the global velocity dispersion. For the purpose of comparison, the dotted line of panel a is drawn in panel c, and the dotted lide of panel b is reproduced in panels d and e.

Fig. 16. Fundamental plane correlations for Galactic and Magellanic globular clusters: this figure shows a combination of the velocity dispersion with a physical scale (half-light radius on the first row and the product of the dimensionless mass µ and the core radius on the second) as a function of the absolute visual magnitude . In panels a and b, we use the global velocity dispersions, based on radial velocities of individual stars. In panels c, d, e, we use the core velocity dispersions derived in the present study. The open square in panel e represents NGC 121, the only SMC cluster. The straight lines represent the relations derived from the Virial Theorem in panel a, and from the King models in panel b, using global (continuous lines) or central (dashed lines) velocity dispersions. The dashed lines in panels a and b are reproduced in panels c, d, and e, for the purpose of comparison.

In Fig. 16, when using global velocity dispersions based on radial velocities of individual stars, the standard deviations around the relation expected from the Virial Theorem is (panel a), while the standard deviations around the relation expected from the King models is (panel b). These standard deviations are comparable to the observational uncertainties (Pryor & Meylan 1993): the data do not show significant deviations from the Virial Theorem or the King models. Note that in panel b, it would be more appropriate to use central velocity dispersions instead of global ones.

When using core velocity dispersions from integrated-light measurements, the dispersions are somewhat larger in panels c and d (Galactic clusters). The standard deviation in panel c is 0.84, while the corresponding value is 0.49 in panel a. This increase is probably due, in part, to the larger uncertainties of the velocity dispersions derived from integrated light, and also to a possibly larger intrinsic scatter since this panel displays core velocity dispersions of high-concentration (collapsed?) globular clusters. A similar degradation is observed from panel b to panel d. Part of the scatter in panel d is probably due to the observational difficulties of measuring the very small core radius of the very high-concentration clusters.

The scatter observed in panel e for Magellanic clusters is similar to the scatter observed for Galactic clusters with velocity dispersions based on radial velocities of individual stars. This shows that our measurements for Magellanic clusters are reliable, and that their M/L ratios may be similar to the M/L ratios of Galactic clusters.

It is worth noticing that our simulations (see Sect. 6) show that statistical errors due to small samples lead, on average, to underestimates of the velocity dispersion. Therefore, if these errors were completely dominant, the Galactic clusters should lie below the expected relations in panels c and d. This is not observed, which, again, suggests that our statistical error estimates from Sect. 6 are somewhat pessimistic.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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