The Virial Relation and Intrinsic Shape of Early-Type Galaxies

Early-type galaxies (ETGs) are supposed to follow the virial relation $M = k_e sigma_*^2 R_e / G$, with $M$ being the mass, $sigma_*$ being the stellar velocity dispersion, $R_e$ being the effective radius, $G$ being Newtons constant, and $k_e$ being the virial factor, a geometry factor of order unity. Applying this relation to (a) the ATLAS3D sample of Cappellari et al. (2013) and (b) the sample of Saglia et al. (2016) gives ensemble-averaged factors $langle k_erangle =5.15pm0.09$ and $langle k_erangle =4.01pm0.18$, respectively, with the difference arising from different definitions of effective velocity dispersions. The two datasets reveal a statistically significant tilt of the empirical relation relative to the theoretical virial relation such that $Mpropto(sigma_*^2R_e)^{0.92}$. This tilt disappears when replacing $R_e$ with the semi-major axis of the projected half-light ellipse, $a$. All best-fit scaling relations show zero intrinsic scatter, implying that the mass plane of ETGs is fully determined by the virial relation. Whenever a comparison is possible, my results are consistent with, and confirm, the results by Cappellari et al. (2013). The difference between the relations using either $a$ or $R_e$ arises from a known lack of highly elliptical high-mass galaxies; this leads to a scaling $(1-epsilon) propto M^{0.12}$, with $epsilon$ being the ellipticity and $R_e = asqrt{1-epsilon}$. Accordingly, $a$, not $R_e$, is the correct proxy for the scale radius of ETGs. By geometry, this implies that early-type galaxies are axisymmetric and oblate in general, in agreement with published results from modeling based on kinematics and light distributions.