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

We derive the relative distances to the Virgo, Fornax and Coma clusters of galaxies by applying the D_n-sigma and the Fundamental Plane (FP) relations to the data of the homogeneous samples of early-type galaxies studied by Caon et al., Lucey et al. and Jorgensen et al. The two distance indicators give consistent results, the relative distance moduli to Fornax and Coma with respect to Virgo being Deltamu_FV=(0.45+/-0.15) mag and Deltamu_CV=(3.55+/-0.15) mag respectively. The formal error on Deltamu may be as small as 0.07 mag (~3 per cent in distance), provided that all the sources of bias are taken into account and a correct statistical approach is used. Unfortunately, much of the actual uncertainty in the relative distance of the clusters (~12-15 per cent) is due to the existence of systematic departures in the measurements of the velocity dispersions among the various data sets, and to the corrections for aperture effects. The above result for the Fornax cluster is supported by the L-sigma-mu relation and, with lesser accuracy, by the log(m)-log(r_e) relation. Our value of Deltamu_FV is in fair agreement with the one derived using planetary nebulae and SNe Ia, while it is in open contrast to that coming from surface brightness fluctuations, the luminosity function of globular clusters, and the infrared Tully-Fisher relation. In our data Coma appears slightly nearer than indicated by the other distance indicators, but now a better agreement with the Tully-Fisher relation seems to exist. We show that for the galaxies of the Virgo and Fornax clusters the residuals of the D_n-sigma relation do not correlate with the effective surface brightness <mu>_e. There is also no correlation of the residuals of the D_n-sigma and FP relations with the total luminosity of the galaxies, or with the ellipticity, or with the isophotal shape parameter a_4. Instead, a correlation seems to exist with the maximum rotation velocity of the galaxies, V_m, with the (V/sigma) ratio, and with the exponent m of the r^1/m fit to the major axis light profiles of the galaxies. If confirmed, these effects introduce a systematic bias in both relations when used as distance indicators.