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), , and scale (i.e., velocity dispersion), . We will assign errors to these estimates equal to the 68% bias-corrected bootstrap confidence intervals inferred from 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 *B _{1}* and

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, and
, 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
corresponding to the minimum resolution-length
imposed by the calculation of . The significance of
substructure is estimated from the null hypothesis that 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:

(1) |

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, , and dispersion, , 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, , over the *N*
cluster members, which we will calculate through the expression:

(2) |

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 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 separated from the other galaxies by a gap in heliocentric
velocity of . 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 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 and . 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
and after applying relativistic and measurement error corrections
(Danese et al. 1980). These values are
compatible, within the adopted uncertainties, with the values
and 522 84 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 after correction to the CMB rest frame
(Kogut et al. 1993). At the cosmological distance of A3733, one Abell
radius, ( Mpc), is equal to
0.805 degrees. The subset of 82 galaxies with has
, ,, and degrees.

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 and , while the remaining 62 galaxies have and .

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 , 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 *B _{2}* test rejects
the Gaussian hypothesis at the 6% significance level, while the

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 of ( 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 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 . 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 ( 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.

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 obtained in 1000 Monte-Carlo simulations of this sample are larger than . 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 are represented by empty circles, while solid circles mark the location of those with . 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 , where 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 '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 is closest to the median of the simulations, while Fig. 2d corresponds to the model with a 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 () 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 *B _{1}*,

Copyright The European Southern Observatory (ESO)