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

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5. Discussion

We have confirmed the result of Schaeffer et al. (1993) that clusters of galaxies populate a Fundamental Plane (FP). The orientation of the FP of clusters can be determined reasonably well from our ENACS data, but the details depend somewhat on the 'direction' in which we minimize the r.m.s. deviations from the FP. In particular, we find a slightly steeper dependence of L on [FORMULA] if we minimize in the [FORMULA] -direction, although the difference is not significant. In Fig. 5 we show what is essentially a sideways view of the FP by projecting the observations in a direction parallel to the FP, onto a plane that is orthogonal to the FP. The reality of the inclination of the FP in the (log L, log R, log [FORMULA])-space is evident already in Figs. 2,  3 and  4, which represent projections along the coordinate axes. In Fig. 6 we show an attempt at a 3-dimensional visualisation of the FP in the (log L, log R, log [FORMULA])-space. This figure is for King profile fits.

[FIGURE] Fig. 6. Two projections of the Fundamental Plane of clusters, in the (log L, log [FORMULA], log R)-space, which show the distribution of clusters in the plane (left) and the dispersion around the plane (right).

The comparison with the FP parameters obtained by S93, for a different sample of clusters, requires some care because S93 used a de Vaucouleurs profile fit. Therefore, we have also repeated our analysis for de Vaucouleurs-profile fits for our clusters. However, we stress once again that this profile provides a fit to the data that is inferior to that of a King or Hubble profile (see also Paper VII).

In Table 7 we give the FP parameters obtained from the ENACS clusters for the 3 types of profile just mentioned. Although the effect is not highly significant in view of the errors, there is a tendency for the L-dependence on R to decrease from King to Hubble to de Vaucouleurs profile. However, there is no apparent change of the dependence of L on [FORMULA] along the same sequence. We also show in the same table the FP parameters derived by S93. There is a hint that the relation between L and [FORMULA] (for the de Vaucouleurs profile) may be steeper in S93 than it is in our data, but the difference of about 30% in [FORMULA] is within one sigma. It is therefore evident that the present results qualitatively agree well with those of S93, although there are some quantitative differences. This provides additional evidence of the reality of the galaxy cluster FP, because both the data-samples and the methods used are different.


Table 7. Fundamental Plane parameters for galaxy clusters and for elliptical galaxies.

We note in passing that West et al. (1989) derived an L -R relation, using the same data as S93, that also agrees quite well with our result.

For clusters in virial equilibrium, which have a fixed value of M /L, one expects [FORMULA], i.e. [FORMULA] and [FORMULA]. Clearly, our [FORMULA] -value is different from the virial prediction. In this respect, there is a similarity between galaxy clusters and elliptical galaxies. The deviation from the pure virial relation, as apparent from the cluster FP, may not seem very large but it is quite significant. This is apparent from the orthogonal dispersion of the 20 clusters around the virial relation of 0.12, which is more than two times larger than the dispersion around our best-fitting FP of 0.05 (see also Table 8). For the sake of the present discussion, we will therefore assume that the FP-fit to our data provides a considerably better description of the [FORMULA] -relation of galaxy clusters than does the virial relation.


Table 8. Orthogonal dispersions around the three fitted relations for the MIDWW coefficients. and

The virial theorem can be expressed as follows (Kormendy & Djorgovski, 1989):


In this equation, S is a parameter related to the internal structure of the system, and [FORMULA] is the ratio of the potential to the kinetic energy of the system, and it measures the degree of relaxation of the system. The deviations of the ([FORMULA], [FORMULA]) coefficients from the (1,2) virial prediction could be the result of a non-constant value for the product of S, [FORMULA] and M /L for the different clusters.

As discussed in S93, the case of non-constant M /L has a special solution if [FORMULA] ; then one has [FORMULA]. For the FP fit in S93, with [FORMULA] and [FORMULA], this special case could indeed apply. If we take our result for de Vaucouleurs fits to the density profiles ([FORMULA] and [FORMULA]) we would again conclude that the data are consistent with this special case. However, we must stress again that the de Vaucouleurs profile fits are notably worse than the King profile fits. Comparing therefore the values [FORMULA] and [FORMULA] that we found for the King profile fits, we conclude that M /L is likely not to have a simple power-law dependence on L.

This would then imply that probably also different S and [FORMULA] are needed for different clusters to explain the deviation of the cluster FP from the virial relation. Another way to summarize the situation could be to say that the structure of clusters is such that the simple equilibrium density laws either do not fit the data very well (de Vaucouleurs), while M /L can be assumed proportional to L or M, or they fit quite nicely (King) but then it is unlikely that M /L has a simple power-law dependence
on M.

We can set an upper limit to the variation of M /L among clusters, if we assume the virialization state and internal structure of all clusters to be identical. In that case M /L [FORMULA], or approximately M /L [FORMULA] because [FORMULA] 1. I.e., systems with larger [FORMULA] have larger M /L. The M /L ratio of rich clusters would then be expected to vary at most a factor of 2 to 3, given the observed [FORMULA] -range of rich galaxy clusters (see Paper II).

If the M /L ratio of rich clusters indeed is more or less proportional to [FORMULA], this could fundamentally affect the determination of the density parameter of the Universe from clusters which are acting as gravitational lenses. These clusters are generally among the most massive ones so they have a large [FORMULA]. Therefore they could have atypically high M /L ratios which would lead to an overestimation of the density parameter (e.g. Bonnet et al. 1994).

It is of interest to compare the cluster FP with that of elliptical galaxies. The results of Djorgovski & Davis (1987), Bender et al. (1992), Guzman et al. (1993), Pahre et al. (1995) and J [FORMULA] rgensen et al. (1996), are listed in Table 7. As was done by J [FORMULA] rgensen et al. we translated, if necessary, the published FP solution to the form we used ([FORMULA]) by using the relation: [FORMULA]. Of course, this is not exactly identical as we determined L, [FORMULA] and R independently, whereas [FORMULA] and [FORMULA] were derived from the same luminosity profile, and are therefore somewhat correlated. We have also used the global, overall velocity dispersion, while for ellipticals it is the central velocity dispersion that is used. However, the velocity dispersion profiles of galaxy clusters are in general quite flat (see e.g. den Hartog and Katgert 1996). Note that the result of Pahre et al. refers to the near-infrared, while the other results refer to the optical.

The [FORMULA] -values found for ellipticals by the various authors, in the optical and near-infrared, appear quite consistent. The [FORMULA] -values found by the different groups are also quite similar, although the value found in the near-infrared (Pahre et al., 1995) is slightly high compared to the optical values. This is even more evident in the recent result by Pahre et al. (1996) who find, again in the near-infrared, [FORMULA] = 0.67 and [FORMULA] = 2.21.

Comparing the various determinations of [FORMULA] and [FORMULA] in Table 7, we cannot be certain that the values of [FORMULA] are significantly different for ellipticals and for galaxy clusters. This does not mean that they are identical. As we said before, the agreement in [FORMULA] is good for de Vaucouleurs fits to the cluster profiles, but much less (about 2 [FORMULA]) for the King profile that describes our cluster observations much better. The situation appears much clearer for [FORMULA]: independently of the type of profile used, the [FORMULA] -value for the clusters appears significantly lower ([FORMULA] 2 [FORMULA]) than that of the ellipticals and much more when compared to the result described by Pahre (1996).

With caveats of the previous paragraphs in mind, it seems safe to conclude from Table 7 that the FP's of clusters of galaxies and of elliptical galaxies have different orientations. It is therefore quite unlikely that the two types of system share a common, universal FP, and we do not think that our data support the conclusion to this effect in S93. If our result is confirmed, this will have very important implications for the formation of gravitationally bound systems on different scales. A more accurate determination of the cluster FP is needed in order to quantify the differences between the FP's of galaxy clusters and of elliptical galaxies more accurately before one can start to investigate the implications in more detail.

For the moment, we can only speculate about the possible implications. Taking the values in Table 7 at face value, we are struck by the fact that the deviations from the virial prediction seem to be different for galaxy clusters and ellipticals. For the ellipticals it seems that the value of [FORMULA] may be quite close to the virial value (especially in the near-infrared). Or, if one gives more weight to the optical data, the ellipticals at least seem to obey the relation [FORMULA] quite well. None of that is true for the clusters, for which it seems quite likely that [FORMULA] agrees with the virial prediction (which probably is not the case for ellipticals !), while the value of [FORMULA] seems definitely at odds with the virial expectation.

Therefore, it is possible (if not likely) that the deviations from the virial prediction have quite different origins for ellipticals and for clusters. For ellipticals the deviations may well be due to non-homology and non-constant M/L ratio's. For clusters the deviations may be due primarily to their relative youth or, in other words: to an absence of real equilibrium. Even though we selected the most regular and apparently relaxed clusters for the present analysis, it is unlikely that they have really attained equilibrium except in their very centres (see e.g. den Hartog and Katgert 1996, and Paper III). If this is true, it is not immediately clear why we should still find a well-defined FP for these clusters which probably are not fully relaxed.

The orthogonal dispersion of the 20 well-contrasted clusters around the FP that they define is 0.05. This is supported by visual inspection of the upper left-hand panel of Fig. 5. However, inspection of that same figure also reminds one of the fact that the other 10 clusters are distributed less narrowly around the FP. For the moment we will assume that these less contrasted and less regular clusters are probably not as relaxed as the other 20 clusters, so that one cannot expect them to define an FP as narrowly as the more contrasted and more regular clusters. It is interesting to observe in Fig. 5 that the dispersion around the FP is indeed smallest for the King profile, somewhat larger for the Hubble profile and largest for the de Vaucouleurs profile. This supports in a rather independent way the conclusion in Paper VII that de Vaucouleurs profiles do not provide good fits to the galaxy distributions in clusters.

We have tried to estimate how much of the scatter in the FP is due to measurement errors and how much to the intrinsic width of the FP. To that end we have done the following experiment. We have assumed the solution for the FP, with [FORMULA] and [FORMULA] = 0.91, to be correct. Therefore, we have taken the orthogonal projections of the observed points onto that FP as starting points for the following simulations. For the set of 20 points in the FP obtained thus, we simulated 500 artificial sets in which uncorrelated errors are added in L, R and [FORMULA] to each of the 20 points, according to Gaussian distributions with dispersions as given in Table 1 and Sect. 3.3. We assumed an error of 10% for L.

For each of the 500 sets of 20 artificial points, the FP was solved in exactly the same way as it was done for the observations with the MIDWW method. As a result we obtain 500 pairs ([FORMULA], [FORMULA]). The distributions of the 500 values of [FORMULA] and [FORMULA] yield average values and dispersions of [FORMULA] and [FORMULA] for [FORMULA] and [FORMULA] respectively. These values must be compared to the values and their estimated errors from the FP fit to the observations of [FORMULA] and [FORMULA]. The estimated errors in the latter values are calculated from the assumed errors in L, R and [FORMULA].

The average values of [FORMULA] and [FORMULA] in the 500 artificial sets are not identical to the input values (the most probable values found in the FP fit to the data) but they are quite close. On the other hand, the dispersions in the [FORMULA] and [FORMULA] values of the 500 artificial sets are significantly smaller than the estimated uncertainties found in the FP fit to the data. We interpret this as evidence that the apparent width of the FP of 0.05 (for the King profile fits) has a important contribution from the intrinsic width of the FP, i.e. that it is not exclusively due to measurement errors.

It is not trivial to estimate how much of the apparent width of the observed FP is intrinsic. One can get some idea from the average dispersion around the 500 artificial datasets of 20 'observed' points, generated as described above. The average dispersion around these 500 individual FP's is 0.06, i.e. larger than the value 0.05 for the 'optimum' FP, as it should be. This provides a crude estimate of the noise contribution to the width of the FP of about 0.03-0.04, which then implies a similar range of values for the intrinsic width of the FP.

In Table 8 we show the dispersions around the other fits for all relations that we studied in this paper, and with the present dataset. Clearly, the dispersions around the FP are smaller than those around the L -R and L - [FORMULA] -relations, for all types of profile fitted. The differences in the dispersions around the L -R - [FORMULA] -relations are striking. They show that the King profile not only provides the best description of the central, regular parts of galaxy clusters, but that it also provides a significantly narrower FP than any other of the currently popular profiles.

One might think that the large dispersion in the L - [FORMULA] relation could be partly due to systematic errors in [FORMULA] that are due to contaminating groups of galaxies projected onto the cluster core. Such galaxies could be falling into the cluster and would not be virialized in the cluster potential. However, that is not likely to be an important effect, because the velocity dispersions vary by only a few percent if we exclude the emission-line galaxies, which are the ones suspected to be on fairly radial, infalling orbits (see, e.g., Paper III). Apparently our selection of the most regular (i.e. probably virialized) cluster cores, has already minimized this effect. The fact that the scatter around the FP appears to increase when lower-contrast and less regular clusters are included (see Fig. 5) could partly be due to such contaminating galaxies.

On the other hand, the increase of the scatter when lower-contrast and less regular clusters are included could equally well be the result of the fact that these are apparently less well relaxed. The larger dispersion around the FP for the latter clusters is not very likely to be due to relatively larger formal uncertainties in L, R or [FORMULA], as inspection of Table 1 will show.

We conclude that a significant part of the dispersion of clusters around the FP is intrinsic. As we discussed earlier, there are many physical effects that may be responsible for the intrinsic width of the FP: structural differences among clusters, differences in virialization state, and peculiar velocities with respect to a uniform Hubble flow.

Deviations from a pure Hubble flow result in errors in the distance of clusters, [FORMULA], which translate into an error [FORMULA] /L = 2 [FORMULA] and an error [FORMULA] /R = [FORMULA]. Therefore the orthogonal intrinsic dispersion around the FP of 0.03-0.04 translates into a dispersion in the L -direction of about 0.05. As our clusters are at an average cz of 20000 km s-1 , it is thus unlikely that they have peculiar velocities that greatly exceed 1000 km s-1 , as those would induce a larger dispersion around the FP than we observe. This is in agreement with S93 and with the recent estimates based on Tully-Fisher galaxy distances by Bahcall & Oh (1996).

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998