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Astron. Astrophys. 331, 493-505 (1998)

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4. The fundamental plane

As discussed in Sect. 2, we want to investigate possible L -R - [FORMULA] correlations for the galaxy clusters in our sample. In the process, we also look at possible L -R, L - [FORMULA] and R - [FORMULA] relations. Note that all fits were performed on the sample of the 20 most contrasted clusters (see below). The cluster A0168 is the only one we have in common with S93.

4.1. Fitting techniques

We 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 [FORMULA]. 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 fitting

We performed the fitting of [FORMULA] vs. R only on the subsample of the 20 more contrasted clusters in order to avoid contaminating our sample with uncertain values of velocity dispersion. The contrast is defined as the percentage of all galaxies in the selected area which are cluster members. The number of background galaxies was deduced from the background density calculated by fitting the theoretical profiles (Paper VII). The nine less contrasted clusters excluded from the fitting analysis are A0087, A0168, A1069, A2362, A2480, A2800, A2911, A3128 and A3825.

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 [FORMULA] 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).

[FIGURE] Fig. 2. R - [FORMULA] relation for King profile fits. The dotted circles represent the more contrasted clusters while the dots denote the less contrasted ones.

On the other hand, R and L are correlated. The fitting of the L vs. R (and R vs. L) relations was done using two MIDAS regressions, viz. with (hereafter MIDWW) and without (hereafter MIDWOUTW) weights on the fitted quantities. The fits were also made using MINUIT (hereafter MWOUTW), without assigning any weight to the data. Results of the fits are given in Table 3, where the parameters [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are defined by: [FORMULA] = [FORMULA] and [FORMULA]. Note that only the King and Hubble radii were considered. The weight that we used here is the inverse of the square of the uncertainty in the R -determination. Fig. 3 shows the R vs. L scatter plots. In addition to the data, we also show the fitted L -R and R -L relations, as well as the bisector line which is the best estimator of a linear relation between two parameters, according to Isobe et al. (1990).


[TABLE]

Table 3. Fitted parameters for the two types of characteristic radius and the three fitting methods, for the L -R relation.


[FIGURE] Fig. 3. L -R relation for King (left) and Hubble (right) profiles (MIDWW calculations). The dotted circles represent the more contrasted clusters and the dots are the less contrasted clusters. The heavy line is the L -R relation, the intermediate line the R -L relation and the thin line is the bisector relation, which has a slope of 1.58 for the King profile and 1.55 for the Hubble profile.

A correlation is also found between [FORMULA] and L. We fitted the parameters [FORMULA], [FORMULA], [FORMULA] and [FORMULA] in the relations [FORMULA] = [FORMULA] and [FORMULA] (see Table 4 and Fig. 4), with [FORMULA] in km s-1. The weight that we used here is the inverse of the square of the uncertainty in the [FORMULA] -determination.


[TABLE]

Table 4. Fitted parameters for the two types of characteristic radius and the three fitting methods, for the L - [FORMULA] relation.


[FIGURE] Fig. 4. The L - [FORMULA] relation (MIDWW calculations). The dotted circles represent the more contrasted clusters and the dots the less contrasted ones. The heavy line is the L - [FORMULA] relation, the intermediate line the [FORMULA] -L relation and the thin line the bisector relation which has a slope of 1.56.

Finally we fitted the parameters [FORMULA] and [FORMULA] in the relation [FORMULA] = [FORMULA], with R in kpc and [FORMULA] in km s-1, for both the King and Hubble radii (see Fig. 5 and Table 5). We have made a fit which minimizes the r.m.s. deviation in the L direction. The weight used in the MIDWW regression is the inverse of the square of the error in R. From Fig. 5 one can see that the less contrasted clusters have a slightly larger dispersion around the fit than the more contrasted ones. We have verified that the parameters and the dispersion around the fit do not vary significantly if we slightly change the area in which we calculate the luminosity. In addition, the dispersion in the L -R - [FORMULA] relation for the King radius with a luminosity calculated within 4 King radii instead of 5, is smaller than when the luminosity is calculated within 4 Hubble radii.

[FIGURE] Fig. 5. L -R - [FORMULA] relation for King, Hubble and de Vaucouleurs profiles (MIDWW calculations). The dotted circles represent the more contrasted clusters and the dots are the less contrasted ones. The abscissa is equal to the logarithm of [FORMULA].

[TABLE]

Table 5. Fitted parameters for the two types of characteristic radius and the three fitting methods, for the L -R - [FORMULA] relation.


Following J [FORMULA] 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.


[TABLE]

Table 6. Fitted parameters for the cluster L -R - [FORMULA] relation, for the three minimization directions, using the King radius


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 [FORMULA] and the slope [FORMULA] in the relation between luminosity and velocity dispersion certainly do not obey [FORMULA]. However, it is well known that one has [FORMULA] = [FORMULA] only if the correlation coefficient between L and [FORMULA] is 1. In other words, one must have the same covariance between L and [FORMULA] as between [FORMULA] and L which is clearly not the case. On the other hand, the coefficients of the L -R - [FORMULA] relations do not depend significantly on the minimization direction. The dispersion in the 3-parameter relation is smaller than in the two 2-parameter relations, and the correlation coefficient is close to unity.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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