Cluster velocity dispersion

We have estimated the line–of–sight velocity dispersions using the bi–weight estima- tor (Beers et al. 1990). This algorithm is usually employed in astronomy (see also Sec. 6.4) and represents a robust method for the determination of the location (the

mean) and the scale (dispersion). It gives higher weight to points that are closer to the centre of the distribution and is therefore insensitive to the presence of outliers. For a Gaussian distribution, the bi–weight location estimator reduces to the conven- tional mean and the bi–weight scale estimator reduces to the conventional standard deviation.

For each cluster, we simply take all the galaxies within the redshift interval_±0.015 from the BCG and compute the line–of–sight velocity with respect to the cluster centre of mass as:

v_|| = V||− ¯

V_||

1 + ¯V_||/c

whereV_|| is the recessional velocity of the galaxy (redshift times the speed of light), ¯

V_|| is the median recessional velocity of the cluster members and c is the speed of light.

These values are then used to determine a robust estimate of the line–of–sight velocity dispersion. Errors on the measured values are determined using the bootstrap method (Press et al. 1992). 1000 Monte Carlo realizations are carried–out for each cluster.

Figs. 5.8 and 5.9 show the velocity histograms for all five clusters used in this anal- ysis, in bins of 500 km s−1. In each panel, the red arrow marks the median recessional velocity of the cluster members, while the blue arrow marks the recessional velocity of the BCG. The red Gaussians correspond to the measured velocity dispersions. In each panel we also give the number of spectroscopically confirmed members and the estimated velocity dispersions, together with the 68 per cent confidence levels, estimated using the bootstrap method.

Note that this subset of clusters covers a wide range in σ, from systems with rela- tively low velocity dispersion (like cl1040–1155) to systems with a very high velocity dispersion (like cl1216–1201 and cl1232–1250). Note, however, that all the veloc- ity histograms show clear signs of deviation from a pure Gaussian. This is a clear evidence of a complex dynamical structure. We will investigate in more detail the presence of dynamical substructures in the following section.

5.6.2 3D–substructures

In order to check for the presence of substructure in the three-dimensional space, we combine velocity and positional information by computing the statistics devised by Dressler & Shectman (1988). The test works in the following way: for each galaxy with a measured radial velocity, the ten nearest neighbours are found, and the local velocity mean and velocity dispersion are computed from this sample of 11 galaxies. These quantities are compared to the global dynamical parameters computed for the

5.6 Dynamical analysis

Figure 5.8: Recessional velocity histograms for the four clusters in the high redshift

bin. The red arrows mark the median recessional velocity of all the galaxies in the spectroscopic sample whose redshifts differ from the redshift of the BCG by±0.15, while the blue arrows mark the recessional velocity of the BCGs. The red Gaussians correspond to the measured velocity dispersion.

clusters (see previous section) by defining the deviationδ as:

δ2= (11/σ2)h(¯vlocal−v¯)2+ (σlocal−σ)2 i

whereσ and ¯vare the global dynamical parameters and σlocal and ¯vlocal are the local mean recessional velocity and velocity dispersion, determined using the 10 closest galaxies with measured radial velocities.

Dressler & Shectman also define the cumulative deviation ∆ as the sum of theδfor all the cluster membersNg. Note that the ∆ statistic is similar to aχ2: if the cluster

Figure 5.9: As in Fig. 5.8 but for the cluster cl1232–1250.

velocity distribution is close to Gaussian and the local variations are only random fluctuations, then ∆ will be of the order ofNg.

The technique devised by Dressler & Shectman does not allow a direct identification of galaxies belonging to a detected substructure. However, it can roughly identify the positions of substructures. This can be achieved by plotting the spatial distribution of the galaxies using symbols whose size is proportional to the parameter δ, that quantifies the local deviation from the global dynamical properties of the cluster. This is done in Fig. 5.10 for the four clusters in the high redshift bin and in Fig. 5.11 for the cluster in the low redshift bin. In each panel, the size of the symbols is proportional toeδ and the red cross marks the position of the BCG. A visual inspection of these figures suggests that a complex structure is present in all the clusters studied in the present analysis.

The ∆ statistic can be used to give a quantitative estimate of the significance of substructure. Note that the actual value of this number does not have great statistical significance, since individual points are not statistically independent. In order to calibrate the ∆ statistic, we perform 1000 Monte Carlo models for each cluster by randomly shuffling the velocities of the galaxies used for the analysis. This is done in order to remove any significant substructure in the 3–D space. The significance

5.6 Dynamical analysis

Figure 5.10: Spatial distribution of all the galaxies in the spectroscopic sample

whose redshifts differ from the spectroscopic redshift of the BCG by ±0.015, for the four clusters in the high redshift bin. The size of the symbols is proportional toeδ_{, whereδ is the parameter that quantifies the local deviation from the cluster}

dynamical properties (see text for details). The red cross in each panel marks the position of the BCG.

of the occurrence of dynamical substructure can be then quantified using the ratio

P between the number of simulations in which a value of ∆ that is larger than the observed value occurs, and the total number of simulations.

In Table 5.2 we list, for each of the clusters used in this analysis, the number of spectroscopic members, the measured ∆ statistic, and the significanceP.

For three out of the five clusters, a ∆ statistic higher than that measured occurs in fewer than 10 per cent of the Monte Carlo realizations carried–out. In one case (cl1054–1245), it occurs in fewer than 20 per cent of the realizations and for the last

Figure 5.11: As in Fig. 5.10 but for the cluster cl1232–1250. Cluster name Ng ∆ P cl1232–1250 54 96.57 0.04 cl1040–1155 30 55.11 0.08 cl1054–1146 48 71.77 0.60 cl1054–1245 37 108.87 0.15 cl1216–1201 67 135.58 0.03

Table 5.2: Number of spectroscopic members (Ng), value of the ∆ statistic, and its

significance for each of the cluster used in the present analysis (see text for details).

case (cl1054–1146), a value of ∆ higher than that measured occurs in 60 per cent of the realizations. On the basis of spectroscopic and positional data then, all but one of the clusters used in this analysis exhibit a significant amount of substructure.