1 Introduction

The distribution of galaxies in the universe is seen as a foam with bubbles and voids. This picture was predicted by Zel'dovich (1970) and seen by Joeveer et al. (1978). In the near universe these structures appear as large 2D filaments or large 3D walls (de Lapparent et al. 1986; Haynes & Giovanelli 1986). Indeed, the 3D distribution of galaxies built from their position and their radial velocity clearly shows these kinds of structures. In Fig. 1 we plotted galaxies with known radial velocities in a slice of $\pm 10$ Mpc around the plane defined by the closest superclusters (Paturel et al. 1988). The polar direction, perpendicular to this plane is about $l=52^{\circ}$ and $b=16^{\circ}$ in galactic coordinates according to Di Nella & Paturel (1994).

Figure 1: Distribution of galaxies seen from their positions and radial velocities within a slice of $\pm 10$ Mpc around the plane defined by near superclusters. The distances are in Mpc assuming a Hubble constant of 70 km s-1 Mpc-1.

Pratton et al. (1997) showed that the velocity field around clusters could generate an apparent distortion which appears as tangential structures or radial filaments ("Finger of God''), similar to observed ones. A remarkable result shown by Rauzy et al. (1992) is that infall velocity does not affect the observed cosmological radial velocity for galaxies located on a sphere (hereafter the Rauzy sphere) having a diameter with ends at the position of the observer and at the center of the attractive galaxy cluster (on the Rauzy sphere, the infall direction is perpendicular to the line of sight). When plotting the distribution of galaxies with distances calculated from their observed radial velocities and a given Hubble constant (d=v/H), an artificial density enhancement on the Rauzy sphere is produced. This is illustrated in Fig. 2.

Figure 2: Illustration of the apparent density enhancement around a cluster. A galaxy in position G1will be placed at position G'1 (because its radial velocity is augmented by the projection $\Delta V$ of its infall velocity. So the displacement $G_1G'_1=\Delta V / H$), while a galaxy in position G2 will be placed at position G'2. The galaxies are placed closer to the Rauzy sphere. On the contrary, in the direction of the cluster center the galaxies are placed far from the Rauzy's sphere because their infall velocities are very high.

This description, applied to near clusters, could lead to the scheme given in Fig. 3. This resembles the observed distribution of galaxies (Fig. 1).

Figure 3: Illustration of the density enhancement around near clusters. The apparent density enhancement is shown for each cluster placed as in Fig. 1. Beyond the doted circle, the selection function on apparent magnitudes contributes to give the circular aspect (doted circle). The approximate mean radial velocity of each cluster is given in parenthesis under the name.

A kinematical model is needed to give a more quantitative description. The linearized model made by Peebles (1976) is a simple way to get the infall velocities from a limited set of parameters. It leads to the following equation of the infall velocity:

$$v_{\rm infall} = \frac{C}{r^{\gamma - 1}}\cdot$$ (1)

In the present paper we want to get realistic values for parameters C and $\gamma$. The determination of accurate extragalactic distances is the fundamental step for achieving this goal. Let us explain the process to go from extragalactic distances to the parameters of the Peebles' model. From an extragalactic distance and the Hubble constant one can predict the cosmological radial velocity at this distance ( $V_{\rm predicted}=H.d$). The comparison with the observed radial velocity directly gives the line of sight component of the infall velocity for this galaxy. On the other hand, if the distances of the cluster and of the considered galaxy are known, all the geometrical elements are also known to calculate both $v_{\rm infall}$ and r entering Eq. (1).

The spiral galaxies are good tracers of the velocity field because they are located, on average, in the outskirts of clusters. On the other hand, the relation between the absolute magnitude of a spiral galaxy and the rotation velocity of its disk (Tully & Fisher 1977) is the best known distance indicator. The method of sosie galaxies (look-alike galaxies) is a particular application of the Tully-Fisher method which bypasses some practical problems (Paturel 1984). However, the method does not correct for the Malmquist bias (Malmquist 1922; Sandage & Tammann 1975; Teerikorpi 1975) which has to be taken into account. The Spaenhauer diagram (Spaenhauer 1978; Sandage 1994) can be used to find galaxies not affected by this bias. In a recent paper (Paturel et al. 1998), the method of sosie galaxies and Spaenhauer diagram were presented for two calibrators (M 31 and M 81). The limited number of sosie galaxies didn't allow the present study. Here, we extend the method to 21 calibrators for which the distance has been recently calculated from the Cepheid Period-Luminosity relation using two independent zero-point calibrations. We use both B- and I-band magnitudes. The sample is deep enough to check directly Peebles' model.

In Sect. 2, we will select the calibrating galaxies and in Sect. 3 we will search for sosie galaxies of these selected calibrators. Then, in Sect. 4 we apply the Spaenhauer diagram method in order to select unbiased galaxies from which we analyze (Sect. 5) the Hubble constant in different regions around Virgo. In the direction of Virgo the comparison is made with predictions by Peebles' model.