ar X iv : a st ro - p h / 97 07 19 8 v 1 1 7 Ju l 1 99 7 On the 3 D Velocity Reconstruction of Clusters of Galaxies

1. The reconstruction of the 3D-velocities of the clusters of galaxies and their large-scale motions remains one of the central problems of observational cosmology (see [2,7] and references therein). The reason is clear: these properties should contain crucial information on the mechanisms of formation of the filaments and even on more early phases of evolution of the Universe, as well as on the present values of the cosmological parameters. Below we are formulating the problem of the 3D-velocity reconstruction of the cluster of galaxies based essentially on the measured redshift i.e. 1D-velocity distribution of the galaxies within the cluster. The procedure we propose includes the following steps: a) the determination of the physical cluster; b) the obtaining of the galaxy redshift distribution of that cluster; c) the reconstruction of the mean 3D velocity distribution of the cluster from the redshift distribution of galaxies. The first step can be performed by S-Tree [4,5] or similar technique enabling one the separation of the physically interacting galaxies from the 2D-image of the cluster area. We illustrate the proposed reconstruction procedure by numerical simulations, thus revealing the possibilities and the limitations posed by the observational parameters of the clusters of galaxies. 2. In 1935 Ambartsumian [1] has solved the stellar dynamical problem of reconstruction of 2D and 3D velocity distributions based on the observed line-of-sight velocity distributions of stars. The main assumption made was the independence of the distribution functions on the spatial regions (directions). Ambartsumian's formula relating the 3D velocity distribution function φ(v x , v y , v z) with the observed line-of-sight velocity distribution f (v r , l, b) has the form: 1 8π 2 dW 1 W d dW cos b dl db η(l,b) f (v x cos l cos b + v y sin l cos b + v z sin b + W, l, b) (1) where W = v r − x cos α − y sin α in some frame. Computer experiments show that the direct application of the this formula is hardly possible for 1