## 4. The fundamental planeAs discussed in Sect. 2, we want to investigate possible
## 4.1. Fitting techniquesWe employ three fitting methods. First, we used the ESO MIDAS-package which has an integrated fitting procedure that we used with and without weights. The method consists of a classical least squares fit, either unweighted, or weighted by the inverse of the square of the error in velocity dispersion or characteristic radius. We also used the MINUIT package to do a least squares fitting by minimizing . This method has been developed to fit particle trajectories and it is recognized to provide very good results. The two MINUIT minimization methods are Simplex (Nelder et al. 1965) and Migrad (Fletcher,1970). These two methods do not use derivatives. Our strategy was to use Simplex to approach the final parameter values and Migrad to solve for the parameters and estimate their errors. As we show in Paper VII, Simplex systematically underestimates the errors. If Migrad does not converge, we take only the Simplex values without errors. The strategy of using Migrad after Simplex in practice gives good results and is commonly used, for example in the minimization routines of the Greg numerical package. ## 4.2. Results of the fittingWe performed the fitting of vs. The clusters are indicated in Figs. 3, 4 and 5 by dots and dotted circles, to distinguish the 9 less contrasted from the 20 more contrasted clusters. We checked for the existence of a relation between and R, where we considered only King and Hubble radii, as discussed in Sec 3. None of the various fitting routines finds a relation between radius and velocity dispersion (see e.g. Fig. 2, for King profile fits), in agreement with the conclusion of Girardi et al. (1996).
On the other hand,
A correlation is also found between and
Finally we fitted the parameters and
in the relation =
, with
Following J rgensen et al. (1996), we have also minimized the r.m.s. deviations in the two other directions (Table 6). We ran only a MIDWW fit for the King profile. We find no significant variations with minimization direction. This supports the reliability of our fitting results.
The differences between the values of the parameters obtained with
different methods are within the fitting errors. There is a slight
apparent inconsistency in the results of the 2-parameter fits when
minimized in the two parameter directions. For example, the slope
and the slope in the
relation between luminosity and velocity dispersion certainly do not
obey . However, it is well known that one has
= only if the
correlation coefficient between © European Southern Observatory (ESO) 1998 Online publication: February 16, 1998 |