The use of Minimal Spanning Tree to characterize the 2D cluster galaxy distribution

We use the Minimal Spanning Tree to characterize the aggregation level of given sets of points. We test 3 distances based on the histogram of the MST edges to discriminate between the distributions. We calibrate the method by using artificial sets following Poisson, King or NFW distributions. The distance using the mean, the dispersion and the skewness of the histogram of MST edges provides the more efficient results. We apply this distance to a subsample of the ENACS clusters and we show that the bright galaxies are significantly more aggregated than the faint ones. The contamination provided by uniformly distributed field galaxies is neglectible. On the other hand, we show that the presence of clustered groups on the same cluster line of sight masked the variation of the distance with the considered magnitude.Comment: 9 pages, 7 postscript figures, LateX A\{&}A, accepted in A\{&}

Group analysis in the SSRS2 catalog

We present an automated method to detect populations of groups in galaxy redshift catalogs. This method uses both analysis of the redshift distribution along lines of sight in fixed cells to detect elementary structures and a friend-of-friend algorithm to merge these elementary structures into physical structures. We apply this method to the SSRS2 galaxy redshift catalog. The groups detected with our method are similar to group catalogs detected with pure friend-of-friend algorithms. They have similar mass distribution, similar abundance versus redshift, similar 2-point correlation function and the same redshift completeness limit, close to 5000 km/s. If instead of SSRS2, we use catalogs of new generation, it would lead to a completeness limit of z$\sim$0.7. We model the luminosity function for nearby galaxy groups by a Schechter function with parameters M*=(-19.99+/-0.36)+5logh and alpha=-1.46 +/- 0.17 to compute the mass to light ratio. The median value of the mass to light ratio is 360 h M/L and we deduce a relation between mass to light ratio and velocity dispersion sigma (M/L=3.79 +/- 0.64)sigma -(294 +/- 570)). The more massive the group, the higher the mass to light ratio, and therefore, the larger the amount of dark matter inside the group. Another explanation is a significant stripping of the gas of the galaxies in massive groups as opposed to low mass groups. This extends to groups of galaxies the mild tendency already detected for rich clusters of galaxies. Finally, we detect a barely significant fundamental plane for these groups but much less narrow than for clusters of galaxies.Comment: 8 pages, 5 figures, accepted in A&A, shortened abstrac

Cluster luminosity function and n^th ranked magnitude as a distance indicator

We define here a standard candle to determine the distance of clusters of galaxies and to investigate their peculiar velocities by using the n^{th} rank galaxy (magnitude m$_n$). We address the question of the universality of the luminosity function for a sample of 28 rich clusters of galaxies ($cz \simeq 20000 km/s$) in order to model the influence on $m_n$ of cluster richness. This luminosity function is found to be universal and the fit of a Schechter profile gives $\alpha = -1.50 \pm 0.11$ and $M_{bj}* = -19.91 \pm 0.21$ in the range [-21,-17]. The uncorrected distance indicator $m_n$ is more efficient for the first ranks n. With n=5, we have a dispersion of 0.61 magnitude for the (m$_n$,5log(cz)) relation. When we correct for the richness effect and subtract the background galaxies we reduce the uncertainty to 0.21 magnitude with n=15. Simulations show that a large part of this dispersion originates from the intrinsic scatter of the standard candle itself. These provide upper bounds on the amplitude $\sigma_v$ of cluster radial peculiar motions. At a confidence level of 90%, the dispersion is 0.13 magnitude and $\sigma_v$ is limited to 1200 km/s for our sample of clusters.Comment: 9 pages, 7 postscript figures, LateX A&A, accepted in A&

Piecewise Extended Chebyshev Spaces: a numerical test for design

Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all of the same dimension, we join them via convenient connection matrices without increasing the dimension. The global space is called a Piecewise Extended Chebyshev (PEC) Space. In such a space one can count the total number of zeroes of any non-zero element, exactly as in each EC-section-space. When this number is bounded above in the global space the same way as in its section-spaces, we say that it is an Extended Chebyshev Piecewise (ECP) space. A thorough study of ECP-spaces has been developed in the last two decades in relation to blossoms, with a view to design. In particular, extending a classical procedure for EC-spaces, ECP-spaces were recently proved to all be obtained by means of piecewise generalised derivatives. This yields an interesting constructive characterisation of ECP-spaces. Unfortunately, except for low dimensions and for very few adjacent intervals, this characterisation proved to be rather difficult to handle in practice. To try to overcome this difficulty, in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given PEC-space is an ECP-space. This procedure is then translated into a numerical test, whose usefulness is illustrated by relevant examples