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 Newton's
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
$\langle k_e\rangle =5.15\pm0.09$ and $\langle k_e\rangle =4.01\pm0.18$,
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 $M\propto(\sigma_*^2R_e)^{0.92}$. 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 $R_e = a\sqrt{1-\epsilon}$. 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.
