The Faber-Jackson relation for early-type galaxies: Dependence on the
  magnitude range
  We take a sample of early-type galaxies from the Sloan Digital Sky Survey
(SDSS-DR7, $\sim$ 90 000 galaxies) spanning a range of approximately 7 $mag$ in
both $g$ and $r$ filters and analyse the behaviour of the Faber-Jackson
relation parameters as functions of the magnitude range. We calculate the
parameters in two ways: i) We consider the faintest (brightest) galaxies in
each sample and we progressively increase the width of the magnitude interval
by inclusion of the brighter (fainter) galaxies
(increasing-magnitude-intervals), and ii) we consider narrow-magnitude
intervals of the same width ($\Delta M = 1.0$ $mag$) over the whole magnitude
range available (narrow-magnitude-intervals). Our main results are that: i) in
both increasing and narrow-magnitude-intervals the Faber-Jackson relation
parameters change systematically, ii) non-parametric tests show that the
fluctuations in the values of the slope of the Faber-Jackson relation are not
products of chance variations. We conclude that the values of the Faber-Jackson
relation parameters depend on the width of the magnitude range and the
luminosity of galaxies within the magnitude range. This dependence is caused,
to a great extent by the selection effects and because the geometrical shape of
the distribution of galaxies on the $M - \log (\sigma_{0})$ plane depends on
luminosity. We therefore emphasize that if the luminosity of galaxies or the
width of the magnitude range or both are not taken into consideration when
comparing the structural relations of galaxy samples for different wavelengths,
environments, redshifts and luminosities, any differences found may be
misinterpreted.
