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

We have selected 42 elliptical galaxies from the literature and estimated their velocity dispersions at the effective radius (${\sigma }_{\text{re}}$) and at 0.54 effective radii ($\text{vff}$). We find by a dynamical analysis that the normalized velocity dispersion of the dark halo of an elliptical galaxy $\text{vdm}$ is roughly ${\sigma }_{\text{re}}$ multiplied by a constant, which is almost independent of the core radius or the anisotropy parameter of each galaxy. Our sample analysis suggests that ${\text{vdm}}^{\ast }$ 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 $(L\mathrm{/}{L}^{\ast })=(\text{vdm}\mathrm{/}{\text{vdm}}^{\ast }{)}^{\gamma }$, where $\gamma$ is between 2-3. These results are of interest for strong gravitational lensing statistics studies. In order to determine the value of ${\text{vdm}}^{\ast }$, we calculate $\text{mstar}$ in the same $\text{bt}$ band in which ${\text{vdm}}^{\ast }$ has been estimated. We select 131 elliptical galaxies as a complete sample set with apparent magnitudes $\text{bt}$ between 9.26 and 12.19. We find that the luminosity function is well fitted to the Schechter form, with parameters $\text{mstar}$ = -19.66 + 5$\cdot\log_{10}h \pm 0.30$,$\alpha = 0.15 \pm 0.55$,andthenormalizationconstant$\phi^{*} = (1.34 \pm 0.30) \times 10^{-3} h^{3}$Mpc$^{-3}$,withtheHubbleconstant$\hnot$=100$h$kms$^{-1}$Mpc$^{-1}$. This normalization implies that morphology type E galaxies make up (10.8 $\pm$ 1.2) per cent of all galaxies.