5 Velocity analysis

Our sample contains 69 velocities in the direction of Abell 970. Figure 5 shows a wedge velocity diagram in the direction of the cluster in right ascension (up) and declination (down), and indicates that most of the velocities are between 15000 and 20000 kms^-1. A histogram of the velocity distribution is displayed in Fig. 6. In this section we will discuss the velocity distribution, looking for non-equilibrium effects.

Figure 5: Wedge velocity diagram in right ascension (up), and declination (down) for the measured galaxies in Abell 970 with radial velocities smaller than 25000  $~{\rm km\,s^{-1}}$.

5.1 The velocity distribution

Since the usual recursive $3 \sigma$-clipping (Yahil & Vidal 1977) failed to simultaneously remove the low ( $v \sim 12\,000~{\rm km\,s^{-1}}$) and high velocity ( $v \sim 22\,000~{\rm km\,s^{-1}}$) tails of the distribution, we decided to analyse the radial velocity distribution of the cluster by constructing four different data samples to which we applied several statistical tests in order to assess the normality of their parent distributions. These samples are: A, comprising all galaxies within $11\,500~{\rm km\,s^{-1}}< v < 22\,000~{\rm km\,s^{-1}}$; B, which is identical to sample A except that the high- and low-velocity tails have been removed, thus covering the range $16\,200~{\rm km\,s^{-1}}< v < 19\,600~{\rm km\,s^{-1}}$; C, which is similar to sample B added with the high-velocity tail, that is $16\,200~{\rm km\,s^{-1}}< v < 22\,000~{\rm km\,s^{-1}}$; D, comprising only galaxies in the range $16\,200~{\rm km\,s^{-1}}< v < 18\,500~{\rm km\,s^{-1}}$: this sample has been considered in view of a significant gap in the data occurring at $v \sim 18\,500~{\rm km\,s^{-1}}$ (see below).

The analysis was made using the ROSTAT statistical package (Beers et al. 1990; Bird & Beers 1993), which proposes various statistical tests based on the empirical distributions of samples. We roughly distinguish 3 categories of normality tests among those included in the ROSTAT package. The first one contains the so-called omnibus tests, which try to quantify the overall deviation of the velocity distribution from a Gaussian, such as the Cramer von-Mises W² test, the Watson U² test and the Anderson-Darling A² test (see Beers et al. 1991, for references). The Kolmogorof-Smirnov (KS) test, which directly calculates the consistency of the observed distribution with a Gaussian, may also be included in this class of tests. The second group of tests are devised to measure the shape of the outskirts of the distribution, such as the kurtosis test (the B2 test) and its robust counterpart, the Tail Index (TI) test (see Bird & Beers 1993, for a discussion), or to test its tail population, such as the a and the W tests, which are most sensitive to the tail of the underlying populations and the u test, which is sensitive to contamination by extreme values (see Yahil & Vidal 1977, for a discussion on these tests). Finally, there are tests which measure the asymmetry of the distribution: the skewness test (B1 test) and its robust version, the Asymmetry Index (AI) test (Bird & Beers 1993). For each of these tests, ROSTAT computes its statistics as well as their associated probabilities p.

The ROSTAT package also provides two statistical tests helping to identify kinematical substructures in the velocity distributions. These are the gap analysis (Wainer & Shacht 1978) and the Dip test unimodality (Hartigan & Hartigan 1985). The Dip test compares the observed distribution against an uniform one and as so is a conservative test for the unimodality of sample. The gap analysis estimates the probability that a gap of a given size and location, between the ordered velocities, may be produced by random sampling from a Gaussian population. A gap is considered significant if this probability is less than 0.03. As mentioned above, our radial velocity sample shows a significant gap at $v \sim 18\,500~{\rm km\,s^{-1}}$. This could be an indication that the distribution is bimodal. However, the Dip test failed to reject the unimodality at significance levels better than 10% for any of the samples.

In Table 3 we list, for each of the samples described above, the values of the statistics and the associated probabilities, p, for the tests discussed above. The B1 and B2 tests were discarded in favour of their robust versions, the AI and the TI tests. Moreover, because the results of the omnibus tests systematically agreed with the KS test, only the results for this last test have been quoted.

   

Table 3: Normality tests for the velocity distribution.

Sample	N	a	p(a)	u	p(u)	W	p(W)	$p({\rm KS})$	AI	$p({\rm AI})$	TI	$p({\rm TI})$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
A	65	0.639	0.01	5.005		0.838	0.01	0.01	0.069		1.292	0.05
B	56	0.780		3.777	0.02	0.954	0.06		0.081		1.080
C	59	0.711	0.01	4.730		0.881	0.01	0.01	1.057	0.04	1.273	0.05
D	48	0.829		3.403	0.01	0.940	0.02		-0.514		1.015

We quote only those p values indicating a rejection of the null
hypothesis at significance levels better than 10% (that is $p \leq 0.1$).

As can be seen from this Table, the normality hypothesis is rejected for samples A and C at significance levels better than 3% for all the statistical tests, except the u test. The values of the statistics of the a and W tests, as well as the TI values, indicate long-tailed underlying distributions, with sample C being significantly skewed towards high velocities, a consequence of removing the low-velocity tail from sample A. Removing both tails of sample A produces sample B, which seems nearly consistent with normality, as indicated by most of the tests. Although both the u and the W tests reject normality at high significance levels for this sample, their results seem contradictory, for the u statistics suggests a cutoff of the underlying distribution whereas the W statistics indicates it is long tailed. Notice that both the AI and TI tests are consistent with a normal underlying distribution for sample B. Very similar results were also obtained for sample D, but now the distribution seems slightly, but not significantly, skewed towards low velocities ( $p({\rm AI}) \sim 0.16$). This is not unexpected for, even if the marginal indication of bimodality of the distribution given by the gap analysis were confirmed, there would be no way, at this level of analysis, to disentangle galaxies belonging to one or to the other underlying distributions, that is, to B or D. Since the Dip statistics failed to reject unimodality for any of the samples, we will not consider this possibility for now, adopting sample B as representative of the radial velocity distribution of the cluster. We will return to this point at the end of this section.

We will thus assume that the cluster galaxies have radial velocities in the range between $16\,200$ and $19\,600~{\rm km\,s^{-1}}$. It is interesting to note that a low-velocity tail at $v \sim 12\,000~{\rm km\,s^{-1}}$, similar to the one found here, also affected the velocity distribution of the cluster Abell 979 (Proust et al. 1995), which is the nearest neighbour cluster of Abell 970 (at about $3^{\circ}$ NE from its centre), both belonging to the same supercluster. This suggests the existence of a large foreground structure projected in this region of the sky.

Figure 6: The radial velocity distribution for the Abell 970 sample of galaxies. The continuous curve shows the Gaussian distribution corresponding to the mean velocity and velocity dispersion quoted in the text (normalized to the sample size and range). The inset displays the velocity distribution between 10000 and 25000 $~{\rm km\,s^{-1}}$.

Considering only the 56 galaxies within this velocity range, the cluster mean velocity is $\overline{V}=17\,600\,\pm\,118 ~{\rm km\,s^{-1}}$ (corresponding to z = 0.0587). For comparison, the radial velocity of the E/D galaxy located at the centre of main cluster is $17\,525 \pm 52$, near that of the whole cluster, as should be expected if this is the dominant cluster galaxy. The cluster velocity dispersion, corrected following Danese et al. (1980) is $\sigma_{\rm corr} = 845_{-69}^{+92} ~{\rm km\,s^{-1}}$ (at a confidence level of 68%). Figure 6 presents the radial velocity distribution of the cluster galaxies, as well as a Gaussian curve with the same mean velocity and dispersion observed for these galaxies. Note that this value of $\sigma_{\rm corr}$ is well above the value favoured by the $\sigma{-}T_{\rm X}$relation, $\sim$ $700~~{\rm km\,s^{-1}}$ (cf. Sect. 4).

If we consider the morphological types, the mean velocities and corrected velocity dispersions are: $\overline{V}= 17\,631~{\rm km\,s^{-1}}$ and $\sigma = 846_{-87}^{+125}~{\rm km\,s^{-1}}$ for E + S0 galaxies (35 objects), and $\overline{V}= 17\,655~{\rm km\,s^{-1}}$ and $\sigma = 841_{-112}^{+185}~{\rm km\,s^{-1}}$ for S + I galaxies (19 objects). Hence, contrary to what is observed in most clusters, where the velocity dispersion of the late type population tends to be larger than that of the early type population (Sodré et al. 1989; Stein 1997; Carlberg et al. 1997; Adami et al. 1998), in Abell 970 we do not see any significant difference between the velocity dispersion of these two populations. This might be another indication - besides the presence of a substructure - that Abell 970 is not in overall dynamical equilibrium.

5.2 Substructures in the galaxy distribution

Let us now consider again the substructure, as well as the peak of the galaxy distribution (cf. Fig. 2), taking into account the galaxy velocities. This analysis will be done with galaxies brighter than $b_{\rm J}^{\rm cosmos}=19.0$, the magnitude where the completeness of our velocity catalogue in the central regions of the cluster is 75%.

The substructure NW of the main cluster has, within a 3 arcmin ( 274 h₅₀ kpc) circular region centered on the brightest S0/S galaxy, 7 cluster galaxies brighter than $b_{\rm J}^{\rm cosmos}=19.0$. Together, these galaxies have a low velocity dispersion, $\sigma_{\rm NW} = 378_{-77}^{+120} ~{\rm km\,s^{-1}}$, more typical of that of loose groups. The mean velocity is $\overline{V}_{\rm NW} = 17\,834 \pm 135~{\rm km\,s^{-1}}$, significantly higher than the overall mean velocity of the cluster. Our velocity catalogue contains 2 galaxies fainter that $b_{\rm J}^{\rm cosmos}=19.0$ inside this region. Their inclusion does not significantly change the value of the mean velocity, although it increases the velocity dispersion to $\sigma_{\rm NW} = 525_{-87}^{+160} ~{\rm km\,s^{-1}}$, a value significantly lower than the cluster overall velocity dispersion. These results are consistent with the suggestion that this clump of galaxies forms a loose group infalling towards the cluster main central condensation. Arguing against the reality of such a group, we notice that its dominant S0/S galaxy is also the lowest velocity member, with $v = 17\,184~{\rm km\,s^{-1}}$, but this is not statistically significant.

The central cluster condensation has a N-S elongation (see Fig. 2). A closer examination of the galaxy distribution indicates that this region is dominated by two small clumps of galaxies, which we will denote by A and B (see also Fig. 7 below). Considering circular regions of 1 arcmin ($\sim$ $91 \ {\rm kpc\,h_{50}^{-1}}$), the central clump, A, is tightly concentrated around the E/D galaxy, having 6 galaxies brighter than $b_{\rm J}^{\rm cosmos}=19.0$ with $\overline{V_{\rm A}} = 17\,624~{\rm km\,s^{-1}}$, and a dispersion $\sigma_{\rm A} = 816 ~{\rm km\,s^{-1}}$. The other clump, B, is about 1.5 arcmin NW of clump A and is more sparse, with only 4 galaxies, of which 3 are tightly packed in velocity space with velocity dispersion $\sigma_{\rm B} =711~{\rm km\,s^{-1}}$ and mean velocity $\overline{V}_{\rm B} = 19\,227~{\rm km\,s^{-1}}$. The fourth galaxy that is, apparently, a member of this clump has, however, a very discrepant radial velocity, $16\,847~{\rm km\,s^{-1}}$. Since it is not apparent in Fig. 2, it is not clear if B is a real substructure or a fortuitous projected group of cluster galaxies.

5.3 Velocity gradients

Figures 7 and 8 display, respectively, the adaptive kernel maps for the mean velocity and the mean velocity dispersion of the sample of cluster galaxies with measured radial velocities brighter than $b_{\rm J}^{\rm cosmos} =18.9$. These maps were calculated from the local kernel weighted averages, with initial kernel size usually larger - in our case by a factor of 3, as a compromise between signal-to-noise and spatial resolution - than the optimal size prescribed by Silverman (1986), as suggested by Biviano et al. (1996). Significance regions for each map were obtained by a bootstrap, in a fashion similar to that applied to the projected density maps.

The mean velocity map of Fig. 7 clearly indicates the existence of a velocity gradient across the field, roughly in the E-W direction. This occurs because galaxies with $v > 18\,500~{\rm km\,s^{-1}}$ populate predominantly the East-side of the field. The mean velocity of the NW group discussed above is consistent with this gradient. Interestingly, this gradient is also consistent with the general gradient one would obtain by considering all galaxies of our catalogue with velocities between $12\,000$and $22\,000~{\rm km\,s^{-1}}$, suggesting that the cluster may be part of a larger structure running more or less in the E-W direction within, at least, this velocity range. In fact a (smaller) velocity gradient, in this same general direction, is also depicted by the mean redshifts of the supercluster members (cf. Einasto et al. 1997).

Figure 7: Mean velocity map of the galaxies kinematically linked to the cluster and brighter than $b_{\rm J}^{\rm cosmos} =18.9$. The positions of COSMOS galaxies brighter than this limit are also plotted, with symbols following Fig. 2. All regions of this map are significant. A and B correspond to the galaxy clumps discussed in Sect. 5.2.

The mean velocity dispersion map displayed in Fig. 8 indicates that there is a radial gradient of the velocity dispersion. This was confirmed by direct calculations of the velocity dispersions within concentric regions centered in the E/D galaxy. Such a gradient is expected if the cluster grows through the capture of low velocity dispersion groups by the central, main galaxy concentration.

The above discussion points to the complexity of the velocity field of Abell 970. It is possible that the high velocity tail of the cluster velocity distribution displayed in Fig. 6 may be contaminated by another component with mean velocity , which reveals itself trough the peculiarities of its spatial distribution. The fact that the velocity distribution shows some signs of bimodality, as pointed out at the beginning of this section, reinforces this suggestion. If real, this component could be interpreted as a diffuse halo located at the East-side of the cluster, probably infalling into its dark matter potential well. If correct, such a scenario may be revealed by some X-ray emission features typical of gas shocks produced during this infall.

Figure 8: Mean velocity dispersion map of the galaxies kinematically linked to the cluster and brighter than $b_{\rm J}^{\rm cosmos} =18.9$. The positions of COSMOS galaxies brighter than this limit are also plotted, with symbols following Fig. 2. The dashed contours delineate the boundaries of the 99% significance levels regions of the map.