## 6. Correlations between velocity and positionIn 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 that is considerably smaller than the overall value and with significant velocity offsets (with regard to the non-ELG), and another with 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 distributionsIn 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 . Note that a velocity limit of has been applied, as before. A 2 - D KS-test (Fasano & Franceschini 1987) gives a probability 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 that is 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.
For a more quantitative discussion of this point we consider the
distributions of and
for In Fig. 12 we show the normalized distributions of and (i.e. ) 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 for ELG-ELG pairs; similarly, the larger global of the ELG causes a wider 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 and , and thus show what fraction of the structure in Fig. 11 is real. The shaded histograms in Fig. 12 represent the and distributions for scrambled versions of the ELG and non-ELG datasets.
In principle, the scrambling of the (r,v)-datasets can be done in
three ways. First, one may leave the values of
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 -profile intact. However, in
that case the galaxies near the centre of a system (with small values
of , and consequently also small values of
) globally retain their
relative velocities, and the scrambling will be far from perfect.
Secondly, one may apply velocity scrambling. In that case, the
-profile is not conserved; however, the average
decrease of over 1 h 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
0.2 h We are thus led to a picture in which a fairly small fraction of the galaxies are in 'real' pairs with small and , with the fraction of ELG in such pairs probably slightly larger () than that of non-ELG (). 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 's of these pairs are not very small, whereas in Sect. 4.3 we found that 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 v and , valid for a gaussian, to hold for those putative groups, one derives typical masses of several times 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. SubstructureIt 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, , 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 for a given galaxy implies a high probability for it to be located in a spatially compact subsystem, which has either a v that differs from the overall cluster mean, or a different , or both. We have applied this test to the 25 systems with . 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 that a value of as large as the one observed is obtained by chance. When this probability is low, one thus has strong evidence for subclustering. The probability 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.
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 less than 0.01 have an average number of galaxies of , whereas for the other 17 systems this number is . Perhaps more significantly, the 8 systems with signs of
substructure have ELG, and the other 17
systems only ELG on average. This might lead
one to suspect that the ELG are a very important, if not 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 , we have compared the distributions of the individual values of 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 . Note that this conclusion does not depend critically on the 8 clusters with clear evidence of substructure. When we exclude these clusters, the 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 , are more likely to reside in substructure, than among the galaxies that are not likely to belong to substructure; for galaxies with , and , for galaxies with . We conclude therefore that in substructures the ELG occur
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