Structure, mass and distance of the Virgo cluster from a Tolman-Bondi model

12 pages, 7 figures. Astronomy and Astrophysics, in press (accepted May 31, 2001)We have applied a relativistic Tolman-Bondi model of the Virgo cluster to a sample of 183 galaxies with measured distances within a radius of 8 degrees from M87. We find that the sample is significantly contaminated by background galaxies which lead to too large a cluster mean distance if not excluded. The Tolman-Bondi model predictions, together with the HI deficiency of spiral galaxies, allows one to identify these background galaxies. One such galaxy is clearly identified among the 6 calibrating galaxies with Cepheid distances. As the Tolman-Bondi model predicts the expected distance ratio to the Virgo distance, this galaxy can still be used to estimate the Virgo distance, and the average value over the 6 galaxies is 15.4 +- 0.5 Mpc. Well-known background groups of galaxies are clearly recovered, together with filaments of galaxies which link these groups to the main cluster, and are falling into it. No foreground galaxy is clearly detected in our sample. Applying the B-band Tully-Fisher method to a sample of 51 true members of the Virgo cluster according to our classification gives a cluster distance of 18.0 +- 1.2 Mpc, larger than the mean Cepheid distance. Finally, the same model is used to estimate the Virgo cluster mass, which is M = 1.2 10^{15} Msun within 8 degrees from the cluster center (2.2 Mpc radius), and amounts to 1.7 virial mass