3 Kinematic and spatial analysis

Following the recommendations of Beers et al. (1990), we will characterize the velocity distribution of our cluster sample by means of the biweight estimators of central location (i.e., systemic velocity), $\overline{V}$, and scale (i.e., velocity dispersion), $\sigma$. We will assign errors to these estimates equal to the 68% bias-corrected bootstrap confidence intervals inferred from $10\,000$ resamplings. The program ROSTAT, kindly provided by T. Beers, will be used for all these calculations.

The ROSTAT program includes also a wide variety of statistical tests, which can be used to assess the consistency of the empirical line-of-sight velocity distribution of the A3733 members (see next subsection) with draws from a single Gaussian parent population. A fair representation of the overall results of the ROSTAT tests will be given by quoting the value of the statistic and associated probability for the canonical B₁ and B₂ tests, which measure, respectively, the skewness (asymmetry) and curtosis (elongation) of the velocity distribution, and for the Anderson-Darling A² omnibus test. Definitions of these tests can be found in Yahil & Vidal (1977) and D'Agostino (1986). The Gaussianity tests will be complemented by the Dip test of Hartigan & Hartigan (1985), which tests the hypothesis that a sample is drawn from a unimodal (though not necessarily Gaussian) parent distribution, and by the search of individual weighted gaps, $g_\ast$, in the velocity distribution of size 2.75 or larger (for a definition of weighted gap see, for instance, Beers et al. 1990). Individual weighted gaps this large are highly significant since they arise less than 1% of the time in random draws from a Gaussian distribution, independently of sample size. We refer the reader to the listed sources and references therein for a detailed explanation of these statistical techniques.

We will investigate also the presence of substructure in the spatial distribution of galaxies by means of two powerful tests. First, we will apply a 2D test developed by Salvador-Solé et al. (1993), hereafter referred to as the SSG test, which relies exclusively on the projected positions of galaxies on the sky (though velocity information is required to define strict cluster membership). This test produces two different estimates of the projected number density profile of the cluster, $N_{\rm dec}(r)$ and $N_{\rm d ir}(r)$, which are, respectively, sensitive and insensitive to the existence of correlation in the galaxy positions relative to the cluster background density. The subscript "dec'' identifies the density profile obtained via the deconvolution of the histogram of intergalaxy separations, while the subscript "dir'' applies to the density profile arising directly from the integral of the histogram of clustercentric distances of the cluster galaxies (Eqs. (4) and (6), respectively, in Salvador-Solé et al. 1993). The two profiles are convolved with a window of smoothing size $\lambda_{\rm min}$ corresponding to the minimum resolution-length imposed by the calculation of $N_{\rm dec}(r)$. The significance of substructure is estimated from the null hypothesis that $N_{\rm dec}(r)$arises from a Poissonian realization of some (unknown) theoretical density profile which has led to the observed radial distribution of galaxies. The probability of this being the case is calculated by means of the statistic:  

$$\chi^2 = {\left(N_{\rm dec}(0)-N_{\rm d ir}(0)\right)^2 \over{2S^2(0)}}\;,$$ (1)

for one degree of freedom. In Eq. (1), $N_{\rm dec}(0)$ and $N_{\rm d ir}(0)$ are the central values of the respective density profiles of the cluster, while S²(0) is the central value of the radial run of the variance of $N_{\rm d ir}(r)$ calculated from a set of simulated clusters convolved to the $\lambda_{\rm min}$ imposed by $N_{\rm dec}(r)$. The simulated clusters are generated by the azimuthal scrambling of the observed galaxy positions around the center of the cluster, i.e., by randomly shuffling between 0 and $2\pi$ the azimuthal angle of each galaxy, while maintaining its clustercentric distance unchanged. It must be stressed, however, that the sensitivity of the SSG test is not affected by deviations of the spatial distribution of the galaxy sample under scrutiny from circular symmetry (see Salvador-Solé et al. 1993). It is also worth noting that this test does not require a priori assumptions on the form of the true projected number density profile of the cluster, nor on the number and size of the subgroups that might be present in the data.

The second spatial substructure test that will be applied to our data is the 3D Dressler & Shectman (1988b; DS) test, which is sensitive to local kinematic deviations in the projected galaxy spatial distribution. The DS test assigns a local estimate of the velocity mean, $\overline{V}_{\rm loc}$, and dispersion, $\sigma_{\rm loc}$, to each galaxy with a measured radial velocity. These values are then compared with the values of the kinematical parameters for the entire sample. The statistic used to quantify the presence of substructure is the sum of the local kinematic deviations for each galaxy, $\delta_i$, over the N cluster members, which we will calculate through the expression:

$$\begin{eqnarray} \Delta & = & \sum_{i=1}^N\delta_i\nonumber \\ & = & \sum_{i=1}^... ...\rm loc},i}-\sigma\right)^2\right) \right]^{1/2}\;,\nonumber \\ &&\end{eqnarray}$$ (2)

where $N_{\rm kern}={\rm nint}(\sqrt N)$, ${\rm nint}(x)$stands for the integer nearest to x, maximizes the sensitivity of the DS test to significant substructure (Bird 1994). To avoid the formulation of any hypothesis on the form of the velocity distribution of the parent population, the DS test calibrates the $\Delta$ statistic by means of Monte-Carlo simulations that randomly shuffle the velocities of the galaxies while keeping their observed positions fixed. In this way any existing correlation between velocities and positions is destroyed. The probability of the null hypothesis that there are no local correlations between the position and velocity of the cluster members is given in terms of the fraction of simulated clusters for which their cumulative deviation, $\Delta_{ \rm sim }$, is smaller than the observed value, $\Delta_{\rm obs}$. Again, we refer the reader to the quoted references for further details on the two spatial substructure tests used in the present analysis.

3.1 The sample of cluster members

Before we can investigate the presence of substructure in A3733 we need to assign cluster membership to the galaxies in our sample. Examination of the radial velocities of the 112 galaxies listed in Table 1 allows the exclusion of 30 obvious interlopers (all background galaxies and groups), which are separated by more than 6500 $\mbox{km\,s}^{-1}$ from the main velocity group. Subsequent membership assignment for the remaining 82 galaxies is based on the their velocity distribution and projected positions, displayed in Figs. 1a and 1b, respectively. These figures show the existence of 8 objects with velocities smaller than $10\,500$ $\mbox{km\,s}^{-1}$ separated from the other galaxies by a gap in heliocentric velocity of $\sim 450$ $\mbox{km\,s}^{-1}$. Seven of these galaxies appear also to be concentrated on a small area of the sky. The cluster diagnostics described at the beginning of this section reveal that the above gap in velocity corresponds to an individually large normalized gap of size 3.39 in the 82 ordered velocities. The "per-gap'' probability for a weighted gap this size is only 0.001. This and the fact that the suspected foreground group of 7 galaxies has a velocity dispersion of only 73 $\mbox{km\,s}^{-1}$ suggest that it might constitute a separate dynamical entity. Accordingly, we chose to consider bona fide A3733 members the 74 galaxies in our sample with heliocentric velocities between $10\,500$ and $13\,000$ $\mbox{km\,s}^{-1}$. Note that we are excluding also from cluster membership the remaining foreground object with the lowest measured radial velocity. From the set of cluster members, we obtain $\overline{V}_{\rm hel}=11\,653^{+74}_{-76}$ $\mbox{km\,s}^{-1}$ and $\sigma =614^{+42}_{-30}$ $\mbox{km\,s}^{-1}$ after applying relativistic and measurement error corrections (Danese et al. 1980). These values are compatible, within the adopted uncertainties, with the values $\overline{V}_{\rm hel}=11\,716\pm 103$ $\mbox{km\,s}^{-1}$ and $\sigma =$ 522 $\pm$ 84 $\mbox{km\,s}^{-1}$ obtained in the previous analysis of this cluster by Stein (1997) from a sample containing 27 of the current cluster members. The mean heliocentric velocity calculated for A3733 results in a mean cosmological redshift of $\overline{z}_{ \rm C MB}=0.0380$after correction to the CMB rest frame (Kogut et al. 1993). At the cosmological distance of A3733, one Abell radius, $r_{\rm A}$ ($\equiv 1.5\; h^{-1}$ Mpc), is equal to 0.805 degrees. The subset of 82 galaxies with $V_{\rm hel}<13\,000$ $\mbox{km\,s}^{-1}$ has $\overline{V}_{\rm hel}=11\,532^{+94}_{-89}$ $\mbox{km\,s}^{-1}$, $\sigma =754^{+65}_{-48}$ $\mbox{km\,s}^{-1}$,$\overline{z}_{ \rm CMB}=0.0385$, and $r_{\rm A}=0.812$degrees.

  

Figure 1: a) Stripe density plot and velocity histogram of the galaxies with $V_{\rm hel}< 14\,000$ $\mbox{km\,s}^{-1}$in the A3733 sample. The arrow marks the location of a highly significant weighted gap (p=0.001) in the velocity distribution. b) Corresponding spatial distribution. The 7 members of a suspect foreground group are identified by open squares, while the asterisk marks the galaxy with the lowest $V_{\rm hel}$. Filled circles identify our choice of strict cluster members

The values of the kinematical parameters of the cluster have been calculated without taking into account its dynamical state. Indeed, the visual inspection of Fig. 1a yields suggestive indication of deviation of the velocity distribution from Gaussianity in the form of lighter tails and a hint of multimodality. The dictum of the B₂ statistic, which indicates the amount of elongation in a sample relative to the Gaussian, confirms the platycurtic behavior (i.e., B₂<3) of the velocity histogram giving only a 0.001 probability that it could have arisen by chance from a parent Gaussian population. Nevertheless, the results of the B₁ and A² tests do not indicate significant departures from normality. As for the possible multimodality, the Dip test cannot reject the unimodal hypothesis, nor we detect the presence of highly significant large gaps in the ordered velocities.

Comparable results are obtained if we remove from the sample of cluster members those galaxies with strong emission lines in their spectrum. Indeed, the spatial distribution and kinematic properties of these latter galaxies are similar to those of the galaxies for which only cross-correlation redshifts are available. Specifically, for the 12 cluster members with emission-line redshifts we find $\overline{V}_{\rm hel}=11\,416^{+256}_{-218}$ $\mbox{km\,s}^{-1}$ and $\sigma =694^{+164}_{-101}$ $\mbox{km\,s}^{-1}$, while the remaining 62 galaxies have $\overline{V}_{\rm hel}=$ $11\,694^{+80}_{-83}$ $\mbox{km\,s}^{-1}$ and $\sigma =594^{+44}_{-37}$ $\mbox{km\,s}^{-1}$.

3.2 The magnitude-limited sample

In order to mitigate the effects of incomplete sampling which may contaminate the results of the statistical tests, especially of those relying on local spatial information, we concentrate our subsequent analysis on the subset of 37 members of A3733 with $b_{\rm J}$ $\leq 18$, for which our original redshift sample contains 75% of the COSMOS galaxies. This magnitude limit is chosen as a compromise between defining a sample (nearly) free of sampling biases and simultaneously having a large enough number of objects for the detection of substructure not to be affected by Poissonian errors.

For this sample, the Gaussianity tests confirm essentially the results obtained for the whole set of cluster members: the B₂ test rejects the Gaussian hypothesis at the 6% significance level, while the B₁ and A² tests are consistent with a parent normal population. Remarkably, the results of the other two 1D tests are now substantially different: the Dip test rejects the hypothesis of unimodality at the 4% significance level, while a large gap of size roughly 230 $\mbox{km\,s}^{-1}$ ($g_\ast=3.14$, p=0.002) appears near the middle of the distribution ($V_{\rm hel}\sim 11\,500$ $\mbox{km\,s}^{-1}$) of velocities.

The kinematical complexity of the inner regions of A3733 suggested by these latter results is not reflected, however, on the spatial distribution of the galaxies. The SSG test gives, for 1000 realizations of the cluster generated by the azimuthal scrambling of the galaxy positions around the location of the cD (see Sect. 2), a 56% probability that there is no substructure, which is nonsignificant. The resulting $\lambda_{\rm min}$ of $16\hbox{$.\mkern-4mu^\prime$}7$ ($\equiv 0.52\,h^{-1}$ Mpc) puts an upper limit to the half-coherence length of any possible clump that may remain undetected in the central regions of A3733. This value is above the typical scale-length of $\sim 0.3\,h^{-1}$ Mpc of the clumps detected by Salvador-Solé et al. (1993) in the Dressler & Shectman's (1988a) clusters. This suggests that the presence of significant substructure in the magnitude-limited sample might be hidden by the large smoothing scale imposed by the calculation of $N_{\rm dec}(r)$. We have investigated this possibility by applying also the SSG test to the sample containing all the 74 cluster members, for which the minimum resolution-length reduces to only $3\hbox{$.\mkern-4mu^\prime$}69$ ($\equiv 0.12\,h^{-1}$ Mpc). In spite of the fact that this latter sample is biased towards the most populated regions of A3733, therefore emphasizing any possible clumpiness of the galaxy distribution on the plane of the sky, we still obtain a 14% probability for the null hypothesis.

  

Figure 2: a) Spatial distribution of the 37 galaxies belonging to the magnitude-limited sample ($b_{\rm J}$$\le 18$) of A3733 members. Galaxies with $V_{\rm hel}\leq 11\,500$ $\mbox{km\,s}^{-1}$ are identified by empty circles, while solid circles mark the location of the galaxies with $V_{\rm hel}\gt 11\,500$ $\mbox{km\,s}^{-1}$. Curves are equally spaced contours of the adaptive kernel density contour map for this sample. The contours range from $2.18\, 10^{-4}$ to $1.88\, 10^{-3} \mbox{\,galaxies\,arcmin}^{-2}$. The initial smoothing scale is set to $12\hbox{$.\mkern-4mu^\prime$}6$. b) Local deviations from the global kinematics as measured by the DS test. Open circles drawn at the position of the individual galaxies scale with the deviation of the local kinematics from the global kinematics, $\delta_i$, from which the test statistic $\Delta_{\rm obs}=\sum\delta_i$'s is calculated (see text). The adaptive kernel contour map is superposed (dashed lines). c) and d) Monte-Carlo models of the magnitude-limited sample obtained after 1000 random shufflings of the observed velocities: c model with the cumulative deviation $\Delta_{ \rm sim }$ closest to the median of the simulations; d model whose $\Delta_{ \rm sim }$ is closest to the value of the upper quartile. Spatial coordinates are relative to the cluster center (see text)

  

Table 2: Results of the statistical tests

The DS test also points to the lack of significant substructure in the magnitude-limited sample: more than 15% of the values of the statistic $\Delta_{ \rm sim }$ obtained in 1000 Monte-Carlo simulations of this sample are larger than $\Delta_{\rm obs}$. A visual judgment of the statistical significance of the local kinematical deviations can be done by comparing the plots in Figs. 2a-d. Figure 2a shows the spatial distribution of the galaxies superposed on their adaptive kernel density contour map (see Beers 1992 and references therein for a description of the adaptive kernel technique). The primary clump in this map is centered at the position of the cD galaxy and is elongated along the north-south axis; a mild density enhancement can be seen at the plot coordinates (17, -3). In this figure galaxies with $V_{\rm hel}\leq 11\,500$ $\mbox{km\,s}^{-1}$ are represented by empty circles, while solid circles mark the location of those with $V_{\rm hel}\gt 11\,500$ $\mbox{km\,s}^{-1}$. Although there is no strong spatial segregation among the galaxies belonging to each of these two velocity subgroups, the galaxies included in the second one dominate the central density enhancement. In Fig. 2b each galaxy is identified with a circle whose radius is proportional to $\exp(\delta_i)$, where $\delta_i$ is given by Eq. (2). Hence, the larger the circle, the larger the deviation from the global values (but beware of the insensitivity of the $\delta_i$'s to the sign of the deviations from the mean cluster velocity). The superposition of the projected density contours shows that most of the galaxies to the north of the density peak, and to a lesser extent those closest to the center of the eastern small density enhancement, have apparently large local deviations from the global kinematics. The remaining figures show two of the 1000 Monte-Carlo models performed: Fig. 2c corresponds to the model whose $\Delta_{ \rm sim }$ is closest to the median of the simulations, while Fig. 2d corresponds to the model with a $\Delta_{ \rm sim }$ closest to the value of the upper quartile. The comparison of Fig. 2b with these last two figures shows that the observed local kinematical deviations are indeed not significant.

As commented in the Introduction, Stein (1997) has not found either any evidence of significant clumpiness on his A3733 OPTOPUS data (see his Table 3). Nevertheless, we caution that this previous study is restricted to the innermost ($r\le 16\hbox{$^\prime$}$) regions of the cluster and that it uses, due to the small size of the sample, all the redshifts available without regard to their completeness.

The results of all the statistical tests applied to our magnitude-limited sample are summarized in Table 2, together with the results obtained from the whole sample of cluster members, for comparison. In Col. (1) we list the name of the sample and in Col. (2) the number of galaxies in it. Columns (3)-(14) give the values of the test statistic and associated significance levels for the B₁, B₂, A², Dip, SSG, and DS tests, respectively. The significance levels refer to the probability that the empirical value of a given statistic could have arisen by chance from the null hypothesis. Thus, the smaller the quoted probability the more significant is the departure from it.