Empirical models for dark matter halos.: III.: The Kormendy relation and the log ρe-log Re relation

We have recently shown that the three-parameter density profile model from Prugniel & Simien provides a better fit to simulated galaxy- and cluster-sized dark matter halos than an Navarro-Frenk-White-like model with arbitrary inner profile slope gamma (Paper I). By construction, the parameters of the Prugniel-Simien model equate to those of the Sersic R-1/n function fitted to the projected distribution. Using the Prugniel-Simien model we are therefore able to show that the location of simulated (10(12) M-circle dot) galaxy-sized dark matter halos in the (e) -log R-e diagram coincides with that of the brightest cluster galaxies, i.e.; the dark matter halos appear consistent with the Kormendy relation defined by luminous elliptical galaxies. These objects are also seen to define the new, and equally important, relation log (rho(e)) 0.5 - 2.5 log (R-e), in which rho(e) is the internal density at r = R-e. Simulated (10(14.5) M-circle dot) cluster-sized dark matter halos and the gas component of real galaxy clusters follow the relation log (rho(e)) = 2.5[1 - log (R-e)] . Given the shapes of the various density profiles, we are able to conclude that while dwarf elliptical galaxies and galaxy clusters can have dark matter halos with effective radii of comparable size to the effective radii of their baryonic component, luminous elliptical galaxies cannot. For increasingly large elliptical galaxies, with increasingly large profile shapes n, to be dark-matter-dominated at large radii requires dark matter halos with increasingly large effective radii compared to the effective radii of their stellar components.