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Astron. Astrophys. 321, 84-104 (1997)

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6. Correlations between velocity and position

In Sect.  4 and Sect.  5 we discussed separately the kinematics and spatial distribution of ELG and non-ELG and the differences between them. From the discussion in Sect.  4.3 we concluded that there is evidence for two ELG populations, one with a [FORMULA] that is considerably smaller than the overall value and with significant velocity offsets (with regard to the non-ELG), and another with [FORMULA] larger than the overall value and without significant velocity offsets. This result immediately raises the question of possible correlations between velocity and position or, in other words: of structure in phase-space. Do the characters of the phase-space distributions of ELG and non-ELG differ and if so, in what way. What evidence do we have on substructure, i.e. on the existence of spatially and/or kinematically compact groups, and are there differences between ELG and non-ELG in that respect.

6.1. The phase-space distributions

In Fig. 11 we show adaptive kernel maps (see e.g. Merritt & Gebhardt 1995) of the distributions of both ELG and non-ELG with regard to normalized-velocity (see Sect.  4.3) and clustercentric distance, for the synthetic cluster constructed from the 75 systems with [FORMULA]. Note that a velocity limit of [FORMULA] [FORMULA] has been applied, as before. A 2 - D KS-test (Fasano & Franceschini 1987) gives a probability [FORMULA] that the two distributions are drawn from the same parent distribution. This is hardly surprising in view of the fact that we found a less centrally concentrated spatial distribution for the ELG than for the non-ELG, as well as a [FORMULA] that is [FORMULA] 20% larger for the (majority of the) ELG than it is for the non-ELG. Both effects are clearly visible in Fig. 11. However, it is very difficult to tell which features in the distributions in Fig. 11 represent real substructure, if only because the distributions represent sums over all 75 clusters. It is equally difficult to estimate from Fig. 11 what fraction of the galaxies is in real substructure that is compact both in position and velocity.

[FIGURE] Fig. 11. Adaptive-kernel maps of the 2-dimensional distribution w.r.t normalized velocity and clustercentric distance for the non-ELG (left panel) and ELG (right panel) in the synthetic cluster constructed from the 75 systems with [FORMULA].

For a more quantitative discussion of this point we consider the distributions of [FORMULA] [FORMULA] and [FORMULA] for pairs of galaxies (rather than individual galaxies) and, in particular, pairs of nearest neighbours from the same class. For the non-ELG we use all 75 systems in sample 3 (with [FORMULA]) which contain 3150 galaxies in total. The number of non-ELG nearest-neighbour pairs is 2219. This is less than the number of galaxies because when B is the nearest neighbour of A and, at the same time, A happens to be the nearest neighbour of B, the pair A-B is used only once. For the ELG we have considered only the 18 systems with [FORMULA] 10 (for reasons that will become apparent); these 18 systems contain 306 ELG (3 ELG were removed in the [FORMULA] clipping) with which we have formed 207 nearest-neighbour pairs.

In Fig. 12 we show the normalized distributions of [FORMULA] [FORMULA] and [FORMULA] (i.e. [FORMULA]) for nearest neighbours, for non-ELG (upper two panels) and ELG (lower two panels). The global differences between the two sets of distributions are not unexpected: the lower surface density of ELG gives rise to larger [FORMULA] [FORMULA] for ELG-ELG pairs; similarly, the larger global [FORMULA] of the ELG causes a wider [FORMULA] distribution for the ELG-ELG pairs. In order to get a more quantitative estimate of the amount of real, compact substructure in Fig. 11, we have compared these distributions with scrambled versions of the same. The scrambled data should give the number of accidental pairs with given values of [FORMULA] [FORMULA] and [FORMULA] , and thus show what fraction of the structure in Fig. 11 is real. The shaded histograms in Fig. 12 represent the [FORMULA] [FORMULA] and [FORMULA] distributions for scrambled versions of the ELG and non-ELG datasets.

[FIGURE] Fig. 12. The observed distributions of [FORMULA] and [FORMULA] for all nearest-neighbor pairs of non-ELG (top) and ELG (bottom) are shown as full-drawn line histograms. The shaded histograms represent the same distributions obtained from the observations after scrambling with regard to radial velocity and azimuthal angle (see text for details).

In principle, the scrambling of the (r,v)-datasets can be done in three ways. First, one may leave the values of [FORMULA] and v intact, and only reassign the value of the azimuthal angle of each galaxy randomly. This will keep both the radial density profile as well as the [FORMULA] -profile intact. However, in that case the galaxies near the centre of a system (with small values of [FORMULA], and consequently also small values of [FORMULA] [FORMULA]) globally retain their relative velocities, and the scrambling will be far from perfect. Secondly, one may apply velocity scrambling. In that case, the [FORMULA] -profile is not conserved; however, the average decrease of [FORMULA] over 1 h-1 Mpc is modest (see, e.g. den Hartog and Katgert 1996), and we do not consider the non-conservation of the [FORMULA] -profile a serious problem.

However, if one does not scramble the azimuthal angle at the same time, velocity scrambling only makes sense if the number of galaxies in a system is quite large. If that is not the case, there will be an important amount of 'memory' between the pairs in the original and in the scrambled data. Therefore, we applied both velocity- and azimuth scrambling. Even then, the scrambled ELG distribution may have significant memory of the observed distribution in view of the small average number of ELG (and therefore ELG-ELG nearest-neighbour pairs) in a system. To minimize this effect (which will lead to an underestimation of the amount of real small-scale structure) we have used for the ELG only the 20 systems with at least 10 ELG (remember that for the non-ELG we used the 75 systems with at least 20 members).

From Fig. 12 we conclude that both for the non-ELG and the ELG there is an excess of nearest-neighbour pairs with [FORMULA] [FORMULA] 0.2 h-1 Mpc, viz. of about 7% for the non-ELG and about 15% for the ELG. Moreover, for the non-ELG there appears to be a small excess (of about 4%) of nearest-neighbour pairs with [FORMULA] [FORMULA] 0.6. For the ELG the excess is about 7 % , but the values of [FORMULA] are between [FORMULA] 0.5 and 1.2. The number of excess pairs in the [FORMULA] distribution is about half that in the [FORMULA] [FORMULA] distribution, for ELG as well as non-ELG. This must mean that there is more 'memory' about velocity than about position in the scrambled datasets. Nevertheless, it seems safe to conclude from Fig. 12 that the ELG show more small-scale structure than the non-ELG. However, whereas the non-ELG excess pairs have small [FORMULA] [FORMULA] as well as small [FORMULA] , the ELG excess pairs have small [FORMULA] [FORMULA] but fairly large [FORMULA] 's.

We are thus led to a picture in which a fairly small fraction of the galaxies are in 'real' pairs with small [FORMULA] [FORMULA] and [FORMULA] , with the fraction of ELG in such pairs probably slightly larger ([FORMULA]) than that of non-ELG ([FORMULA]). Interestingly, the estimated fraction of ELG in pairs is quite consistent with the value derived in Sect.  4.3. It is a bit puzzling that we now find that the [FORMULA] 's of these pairs are not very small, whereas in Sect.  4.3 we found that [FORMULA] for these ELG must be quite small. If one assumes these ELG pairs to be in groups, and if one assumes the relation between the average [FORMULA] v and [FORMULA] , valid for a gaussian, to hold for those putative groups, one derives typical masses of several times [FORMULA] solar masses (using the projected virial mass estimator for isotropic orbits, see Heisler, Tremaine & Bahcall 1985). This implies that the real ELG pairs could be in small groups of a few to several ELG, depending on the average mass of the ELG in question.

6.2. Substructure

It is interesting to find out whether the groups of ELG (and, to a lesser extent, non-ELG) that we 'detected' in the analysis in Sect.  6.1, are detectable as substructure in the velocity-position databases of individual clusters as well. In order to investigate this we have applied the test (due to Dressler & Shectman 1988, but with the modifications proposed by Bird 1994) for the presence of substructure. This test compares the value of a substructure parameter, [FORMULA], for a cluster, with the distribution of values of the same parameter that one obtains in 1000 Monte Carlo randomizations of the cluster data-set. A large value of [FORMULA] for a given galaxy implies a high probability for it to be located in a spatially compact subsystem, which has either a [FORMULA] v [FORMULA] that differs from the overall cluster mean, or a different [FORMULA] , or both.

We have applied this test to the 25 systems with [FORMULA]. These contain a sufficiently large number of galaxies (on average 86 of which 14 are ELG) that for these systems the test may be expected to produce significant results. An additional advantage of this selection is that from all these systems interlopers were removed. In Table 6 we list the probability [FORMULA] that a value of [FORMULA] as large as the one observed is obtained by chance. When this probability is low, one thus has strong evidence for subclustering. The probability [FORMULA] was calculated separately for all galaxies (ELG+non-ELG) (col.3), and for the non-ELG only (col.4), i.e. with the ELG removed.


Table 6. The Dressler & Shectman test for substructure

In 8 systems we find evidence for substructure at the 0.99 conf.level, using all galaxies (i.e. for A548W, A548E, A3094, A3122, A3128, A3354, A3562 and A3695). In addition, there are 2 systems with substructure at the 0.98 conf.level, viz. A514 and A3651. One might suspect that the systems with a substructure signal are preferentially found among the systems with the largest number of galaxies, as for those it will be relatively easier to detect deviations from uniformity. From Table 6 it indeed appears that there is small effect of this kind: the 8 systems with [FORMULA] less than 0.01 have an average number of galaxies of [FORMULA], whereas for the other 17 systems this number is [FORMULA].

Perhaps more significantly, the 8 systems with signs of substructure have [FORMULA] ELG, and the other 17 systems only [FORMULA] ELG on average. This might lead one to suspect that the ELG are a very important, if not the, cause of substructure. However, there is no evidence that that is so; among the 8 systems with [FORMULA] for all galaxies, there are 6 for which [FORMULA] is still less than 0.01 if the ELG are excluded. Therefore, it is very unlikely that the presence of ELG is a requirement for the occurrence of substructure.

However, there is an indication that the ELG preferentially occur in substructure, if the system to which the ELG belong indeed does have substructure. This presumes that the substructure is probably delineated primarily by the non-ELG (and/or the dark matter), and that the ELG so to speak 'follow' the substructure that is present. This conclusion is based on the following evidence.

Combining all systems with [FORMULA], we have compared the distributions of the individual values of [FORMULA] of the 1808 non-ELG and 340 ELG in these 25 systems. According to a KS-test, the probability that the distributions are drawn from the same population is [FORMULA]. Note that this conclusion does not depend critically on the 8 clusters with clear evidence of substructure. When we exclude these clusters, the [FORMULA] distributions of ELG and non-ELG are still different at the 0.994 conf.level. Using the total sample of galaxies in the 25 clusters, we find that the fraction of ELG is almost twice as large among the galaxies that, according to their value of [FORMULA], are more likely to reside in substructure, than among the galaxies that are not likely to belong to substructure; [FORMULA] for galaxies with [FORMULA], and [FORMULA], for galaxies with [FORMULA].

We conclude therefore that in substructures the ELG occur relatively more frequently than the non-ELG. In a fairly small fraction of the systems they may even account for most of the substructure; however, in general the ELG seem to follow the substructure rather than that they define it. As we saw in Sect.  3.2 there is a clear tendency for the fraction of ELG to be larger in smaller systems. It is thus not totally unexpected to find that the ELG are relatively more associated with substructure since, to some extent, the ELG can be regarded as low-richness groups within richer systems. While ELG are more frequently found in subclusters than non-ELG, their average velocities are seldom different from the cluster ones; therefore, groups containing ELG cannot be rapidly infalling into the cluster, unless the infall of these groups is more or less isotropic.

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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998