An Elliptical Galaxy Luminosity Function and Velocity Dispersion Sample of Relevance for Gravitational Lensing Statistics

Yu-Chung N. Cheng 1 and Lawrence M. Krauss 2
Department of Physics, Case Western Reserve University, 10900 Euclid Ave.,
Cleveland, OH 44106-7079, USA

Abstract
We have selected 42 elliptical galaxies from the literature and estimated their ve-
locity dispersions at the effective radius (Re) and at 0.54 effective radii (0.54Re).
We find by a dynamical analysis that the normalized velocity dispersion of the dark
halo of an elliptical galaxy DM is roughly Re multiplied by a constant, which is
almost independent of the core radius or the anisotropy parameter of each galaxy.
Our sample analysis suggests that ∗DM lies in the range 178-198 km s−1. The power
law relation we find between the luminosity and the dark matter velocity dispersion
measured in this way is (L/L∗) = (DM/∗DM), where      is between 2-3. These
results are of interest for strong gravitational lensing statistics studies.
In order to determine the value of ∗DM, we calculate M∗BT(0) in the same BT (0)
band in which ∗DM has been estimated. We select 131 elliptical galaxies as a com-
plete sample set with apparent magnitudes BT (0) between 9.26 and 12.19. We find
that the luminosity function is well fitted to the Schechter form, with parameters
M∗BT(0) = -19.66 + 5· log10 h±0.30, = 0.15±0.55, and the normalization constant
∗ = (1.34 ± 0.30) × 10−3h3 Mpc−3, with the Hubble constant Ho = 100 h km
s−1 Mpc−1. This normalization implies that morphology type E galaxies make up
(10.8 ± 1.2) per cent of all galaxies.