In Figs. 2 and 3 the results have been presented so as to make evident the dependence on different parameters. It is obvious that the star density distribution in real galaxies is not a universal quantity. It will depend on the nature of the galaxy itself and on its age, as theoretical evolutive models also show (see e.g. Murphy et al. 1991). Therefore, to compare our results to those derived by Paltani & Courvoisier 1997 we have reported in Fig. 4 the luminosity and variability values corresponding to a variety of different pairs of k and , namely all those present in Figs. 2 and 3 plus a series of pairs (k,) corresponding to density configurations having a collision rate . A straight line with the same slope as that Paltani & Courvoisier 1997 derived as approximative trend of as a function of has been also reported in Fig. 4. This straight line has been arbitrarily shifted since, as described in the introduction, it is the trend which is important in this comparison. On the other hand a detailed comparison between our results and those by Paltani & Courvoisier 1997 would imply the non-evident task of relating the luminosity and the light curve observed at a specific wave length with the spectral integrated luminosity and light curve derived here.
The straight line with slope -0.08 is compatible with the points drawn in Fig. 4. It is however important to understand how much this result depends on the parameter choice. In this figure different pairs () have been reported for a restricted choice of the collision rate value () and we know from Sect. 5 that the results are not influenced by other free parameters of the model. Hence the only possibility is that the allowed range in , or equivalently that in , causes the compatibility between the points of Fig. 4 and the straight line with slope -0.08 derived by Paltani & Courvoisier 1997.
To test if the compatibility is real, we have added to the pairs previously computed the results obtained for the case of sixteen different configurations having the same collision rate . For these sixteen additional cases, marked in Fig. 4 by asterisks, the Spearman's correlation coefficient is equal to -0.63. The probability to get such a high correlation coefficient for 16 pairs of quantities belonging to two uncorrelated groups is less than 0.011. We conclude that the correlation is most probably real. This correlation proves that the compatibility of all points drawn in Fig. 4 with the line of slope -0.08 does not depend on the choice of values, since such a compatibility still exists for a single value.
© European Southern Observatory (ESO) 2000
Online publication: June 26, 2000