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

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3. Parameters of the fundamental plane

3.1. Characteristic radii

Using Cosmos galaxy positions, we have fitted four kinds of density profiles to the galaxy distribution of each cluster. The background density is determined for each cluster and for each density profile. We used the profiles given by King (1962), by Hubble (e.g. Bruzual & Spinrad 1978), by de Vaucouleurs (1948) and by Navarro et al. (1996, which we will refer to as the NFW profile). These profiles were generalized by allowing the exponents to vary. Each profile has its characteristic scale, which we determined by Maximum Likelihood fitting (see Paper VII for details). Although we used all four types of profile, it was found that only the King and Hubble profiles provide good fits to the observed clusters density profiles. The de Vaucouleurs and NFW profiles have cusps and do not provide a good fit to the projected galaxy density in the central regions of clusters, which do not show a significant central cusp. Therefore, we mostly use the characteristic radii of the King and Hubble profiles in the present discussion; these are listed (with their errors) in Table 1.


Table 1. Data for the sample of 29 clusters. Col.(1) lists the cluster name, col.(2) the velocity dispersion, cols.(3) and (4) the King (K) and Hubble (H) radius, cols (5) and (6) [FORMULA] and [FORMULA], and cols (7) and (8) the fitted center of the clusters. In clusters marked by a [FORMULA] interlopers have been removed with the method of den Hartog & Katgert (1996).

As we need to adopt an area within which to define the cluster properties, we chose to consider a square of 10 core radii size centered on the cluster center assuming a King profile. The cluster center we adopted was determined iteratively by fitting a King profile. The centers of the clusters used in the present analysis are given in Table 1. The chosen area is expected to include a large fraction of the true "physical cluster", as the galaxy volume density at 5 King core radii is about 1 % of the central density. For the other profiles the contrast between the central density and that at 5 King core radii is at least as large as for the King profile.

3.2. Velocity dispersions

The velocity dispersions were obtained from the ENACS data-base. All 29 clusters considered here have more than 10 galaxies in the selected area and in the main group identified in radial velocity space (see Paper I). While in Paper I the groups were defined using a fixed gap of 1000 km s-1 in radial velocity space, here we slightly modify this criterion to account for the fact that a fixed gap can overestimate the number of groups when the number density of galaxies is too low (simply because it is more likely to find larger gaps in sparse data-sets). The new "variable" gap, which we will call density gap, is based on simulations of the occurrence of gaps of a given size in distributions of varying number of objects, drawn from the same Gaussian distribution. The density gap follows from the expression: 500 (1  +  exp(-(n-6)/33)) km s [FORMULA] where n is the number of galaxies in the redshift survey of a given cluster.

We stress that the criterion used in Paper I works very well for the ENACS datasets, since the number of galaxies does not vary too much. However, when we consider cluster datasets drawn from the literature, large differences in the number densities may occur. This is the case when one includes, e.g., the Virgo or the Coma clusters, for which redshifts are known for more than 500 galaxies. In these cases a fixed gap-size fails to identify the main cluster structure, and merges systems which are likely to be separate entities. For the present paper, we could have maintained the fixed gap definition, but since we will in the future also include large datasets such as that of Coma, we prefer to use the density gap already in this discussion. We stress that using this gap definition, the membership of the ENACS main systems hardly changes, compared to Paper I.

Similarly to what we did in Paper II, we identified interlopers in the systems by using both the spatial and the velocity information. More specifically, we applied the technique developed by den Hartog and Katgert (1996) to clusters with at least 50 galaxies left after the gapping in velocity space. These clusters are marked in Table 1. The effect of the interloper removal was discussed at length in Paper II. For systems with less than 50 galaxies the method becomes unreliable so we have not applied interloper rejection to these systems.

For the clusters thus defined, we calculate the velocity dispersion by a biweight technique using the ROSTAT package (Beers et al. 1990), which is ideally suited for poorly sampled and/or non-Gaussian distributions. As it is difficult to say whether our velocity distributions are truly Gaussian (in some cases we only have 10 galaxies), a classical velocity dispersion estimator might not give a reliable value. Lax (1985) has shown, from simulations, that the biweight estimator gives better results in that case. Errors were estimated from 1000 bootstrap resamplings for each cluster (see Stein 1996).

The reliability of our velocity dispersion values can be checked by a comparison with previous investigations. In Paper II we obtained velocity dispersions by using a fixed gap criterion, and using all galaxies in the ENACS regions. The present velocity dispersion estimates were correlated with those in Paper II, and the best-fit straight line has a slope of 0.98 [FORMULA] and an offset of 113 [FORMULA], clearly consistent with a slope of 1 and an offset of 0. Comparing the present velocity dispersions with those of Fadda et al. (1996), who used a combination of ENACS and literature data, we find a best-fit line with a slope of 0.90 [FORMULA] and an offset of 235 [FORMULA], which again is consistent with the hypothesis that the two estimates are equivalent.

3.3. Luminosity

In order to determine the cluster luminosities, we used the Cosmos data, and followed the procedure described below.

  1. We summed up the individual apparent luminosities of all galaxies in the selected area, after K-correcting the Cosmos [FORMULA] magnitudes following Frei & Gunn (1994) and assuming that all galaxies lie at the average redshift of the main system (as determined from the ENACS data). We then converted from apparent to absolute magnitudes (using the standard cosmological formulae, see, e.g. Lang 1980), assuming an [FORMULA] solar magnitude of 5.53 (Lang, 1980, Gullixson et al. 1995). The result is [FORMULA]
  2. We applied a correction to the integrated luminosities for the contamination by fore- and background galaxies, by making the assumption that the fraction [FORMULA] = [FORMULA] of the luminosity of cluster galaxies in the selected field, is the same in the Cosmos sample as it is in the ENACS sample, i.e assuming that [FORMULA] = [FORMULA] = [FORMULA]. This is only approximately true, because this fraction changes with limiting magnitude and the Cosmos limit is 0.5 to 1.0 magnitude fainter than the ENACS limit. However, for some clusters that were very well sampled in the ENACS data we have verified that [FORMULA] is not significantly biased.

    This allows us to take advantage of the fact that all ENACS galaxies have a measured redshift, so that membership assignment is relatively straightforward (while of course many Cosmos galaxies do not have such information available). We can then separate the luminosity of the cluster members from that of fore- and background galaxies. We calculate the value of [FORMULA] for each cluster, which we assume to be equal to [FORMULA]. We find the mean value of [FORMULA] = 0.83 if we consider only clusters with more than 15 galaxies. For the clusters with relatively poor statistics, the individual corrections are not considered reliable. Therefore, we apply the mean correction to all clusters by taking the mean value [FORMULA] [FORMULA] [FORMULA] = 0.83; this introduces an uncertainty in the luminosities of the order of the dispersion in the [FORMULA] values, which is 0.10. We thus have:


  3. We correct for the incompleteness effect due to the Cosmos magnitude limit as follows. We adopt a completeness magnitude for the Cosmos catalogue of [FORMULA]. Following Ellis et al. (1996), we take into account all fainter cluster members by adopting the following Schechter (1976) luminosity function:


    with [FORMULA], [FORMULA] and L the absolute luminosity.

    The fraction of the total luminosity that we have observed is then equal to:

    F = [FORMULA]

    where the value of [FORMULA] follows for each cluster from the median redshift and [FORMULA] = 20. Finally we have:


    We have checked the dependence of [FORMULA] on [FORMULA]. For the same limiting magnitude and a redshift equal to 0.07 (the typical distance of our clusters), we have a variation of less than 10% of [FORMULA] when [FORMULA] is changed from 0.5 to 1.3. There is thus only a weak dependence of the total luminosity on [FORMULA] when this parameter is in the commonly employed range of values (e.g. Carlberg et al. 1996, Lumsden et al. 1997).

  4. There is an additional source of incompleteness at bright luminosities. In fact, we have found that in some cases, bright ENACS galaxies have no Cosmos counterpart (see Paper V). In those cases, we simply add [FORMULA], i.e. the sum of the ENACS luminosity (scaled to the [FORMULA] system) of the galaxies which inadvertently were not included in [FORMULA]:


  5. Finally, we correct for galactic extinction [FORMULA] by using the Burstein & Heiles map (1982). We use the formula [FORMULA]. The correction to the luminosity is [FORMULA] = [FORMULA]. We have:


All individual corrections are given in Table 2; note however that the values of [FORMULA] are given for information only, as we applied [FORMULA] [FORMULA] [FORMULA] = 0.83 for all clusters.


Table 2. Luminosities (in units of [FORMULA]) and corrections for the 29 clusters.

Estimating the errors on the luminosities [FORMULA] is not easy. From the uncertainty in the slope of the luminosity function, we estimate that [FORMULA] is uncertain by 10 %. To this we should probably add other error sources but it is very difficult to make quantitative estimates of those. Therefore we take the optimistic view of an error of 10 % in the luminosities.

To check the general consistency of our luminosities, we compare them with those obtained by Carlberg et al. (1996). They found, in Gunn r, a median K-corrected luminosity for 16 distant ([FORMULA]) clusters of 1.95 [FORMULA] h-2 [FORMULA], corresponding, to a value of 9.36 1011 h-2 [FORMULA], where we assumed [FORMULA] and [FORMULA]. Our mean cluster luminosity is 7.65 [FORMULA] h-2 [FORMULA]. Given that Carlberg et al. (1996) considered larger areas than we do (larger than 1.0 h-1 Mpc vs. 0.9 h-1 Mpc ), the agreement can be considered quite satisfactory.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998