4. Structural parameters of the clusters
To demonstrate whether there is a variation in the distribution of the various mass stars we performed the following analysis:
Fig. 8 shows the radial density profiles, (vs) , of SL 666 derived from star counts on our CCD frames, in different magnitude bins (see e.g. Kontizas et al., 1987a). We defined as f the number of stars per unit area at a radius r, corrected (i) for incompleteness (Sect. 2) and (ii) for the `background' contribution. The background values were found from a series of plots showing the number of stars per unit area (vs) distance (the counts were performed in concentric annuli) at each magnitude range. After a certain radius the density becomes constant and this is the adopted `background'. This was repeated for all magnitude ranges.
It should be noted that in order to have a representative sample of cluster stars in our star counts we only used main sequence stars, since they constitute the overwhelming majority of the stellar content of these clusters (Sect. 3). Error propagation in all values of star count, densities and logarithmic values was always taken into consideration. Therefore the error bars represent the maximum uncertainties.
The isotropic single-mass tidally truncated models of King (1966) have been traditionally used for the description of the structure of star clusters. Here, we shall follow this approach (although these models may well be inadequate for the description of the particular clusters), as a way of parametrising the main structural characteristics of our clusters.
The best fitting single mass King models overlayed on the density profiles give estimates of the core radius, and concentration parameter for SL 666 and NGC 2098 respectively (Table 2). It is important to clarify that the best fit to King profiles is found from photographic plates, where star counts can be performed over a large area around each cluster. However the photographic plates have not the resolution and photometric accuracy of the present CCD data. On the other hand the CCD data at the outer cluster areas represent only a fraction of the cluster region (a semi-periphery, or even one quarter only). The effect is responsible for the noise observed in the profiles of Fig.8. We do not therefore think that King models can be reliably fitted to these diagrams, theoretically not expected to be the case anyway from the dynamical point of view. However a cumulative profile from the photographic plates reaching , where all rings around the cluster are measured gives a smoother profile, which fits reasonably well the model of as a first approximation (Fig. 8m & 8n). The parametric values corresponding to this profile provide the structural dynamical quantities given in Table 2.
Table 2. Derived dynamical parameters of the Clusters
Based on these determinations of and C, we have also derived (Table 2) a number of other dynamical parameters, i.e. the half mass radius, , the dynamical mass, M, and the relaxation and crossing times. The Spitzer radius (1958) of the clusters was also calculated at two different R magnitudes, 21.00 and 18.00 mag. This characteristic radius is defined as the median of the distances of the cluster stars from its centre. It also shows that the brightest stars are more centrally located.
One way of investigating the profiles for the various magnitude intervals and parametrising their potential differences is to approximate them with the relation (e.g. Subramaniam et al., 1993) and compare the resulting values of the slopes , shown in Table 3, along with the corresponding errors. This linear approximation is valid in principle only for the outer regions of the clusters. However, the flattening of the profiles usually occurs closer to the centre than reached by our observations. We repeated this analysis for the B magnitudes, with similar results. The values are plotted against magnitude (B and R) in figure 9. We applied a statistical test to verify the significance of the differences in the slope . We performed a chi-square linear fitting to the data of figure 9 and we found a slope for the R and for the B magnitude. The goodness of fitting for the former is 0.68, whereas for the last is 0.81. We also computed the Spearman correlation coefficient (s) between the slopes and the corresponding magnitude. For the R magnitude we found with a probability for the data of not being correlated and for the B magnitude with a probability of not correlated data . So we believe that the relation (vs) magnitude does follow a correlated trend. This trend is indicative of mass segregation in the case of SL 666. In the case of NGC 2098 the data are not as good,but there is a systematic trend.
Table 3. The values of the slopes of the linear relation (vs) for the density profiles, with the corresponding errors.
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998