The relative distances to the Virgo, Fornax and Coma clusters of galaxies through the Dn-σ and the Fundamental Plane relations

We derive the relative distances to the Virgo, Fornax, and Coma clusters ofgalaxies by applying the Dn-sigma and the Fundamental Plane (FP) relations tothe data of the homogeneous samples of early--type galaxies studied by Caonetal, Lucey etal, and Jorgensen etal. The two distance indicators giveconsistent results, the relative distance moduli to Fornax and Coma withrespect to Virgo being Dmu(FV)=0.45+-0.15 and Dmu(CV)=3.55+-0.15) respectively.The formal error on D(m-M) may be as small as 0.07 mag (~3% in distance),provided that all the sources of bias are taken into account and a correctstatistical approach is used. Much of the actual uncertainty in the relativedistance of the clusters (12-15%), is due to the existence of systematicdepartures in the measurements of the velocity dispersions among the variousdatasets, and to the corrections for aperture effects. The above result for theFornax cluster is supported by the L-sigma-mu relation and, with lesseraccuracy, by the log(m)-log(re) relations. Our value of Dmu(FV) is in fairagreement with the one derived using planetary nebulae and SNe-Ia, while is inopen contrast with that coming from surface brightness fluctuations, globularclusters luminosity function, and infrared Tully-Fisher relation. In our dataComa appears slightly nearer than indicated by the other distance indicators,but now a better agreement with the Tully-Fisher relation seems to exist.Finally we found that there aren't correlations of the residuals of theDn-sigma with the effective surface brightness, the total luminosity of thegalaxies, with the ellipticity and the shape parameter a4. Instead acorrelation seems to exists with maximum Vrot, with (V/sigma) and with theexponent m of the r^1/m fit to major axis light of galaxies.