Empirical Models for Dark Matter Halos. III. The Kormendy Relation and the log ρ_e-log R_e 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 γ (Paper I). By construction, the parameters of the Prugniel-Simien model equate to those of the Sérsic 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 (10¹² M_⊙) galaxy-sized dark matter halos in the ⟨μ⟩_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(ρ_e) = 0.5 - 2.5 log(R_e), in which ρ_e is the internal density at r = R_e. Simulated (10^14.5 M_⊙) cluster-sized dark matter halos and the gas component of real galaxy clusters follow the relation log(ρ_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.