The discrepancy between dynamical and stellar masses in massive compact galaxies traces non-homology

For many massive compact galaxies, their dynamical masses (M_dyn ∝ σ²r_e) are lower than their stellar masses (M_⋆). We analyse the unphysical mass discrepancy M_⋆/M_dyn > 1 on a stellar-mass-selected sample of early-type galaxies (M_⋆ ≳ 10¹¹ M_⊙) at redshifts z ∼ 0.2 to z ∼ 1.1. We build stacked spectra for bins of redshift, size and stellar mass, obtain velocity dispersions, and infer dynamical masses using the virial relation M_dyn ≡ K σ_e² r_e/ G with K = 5.0; this assumes homology between our galaxies and nearby massive ellipticals. Our sample is completed using literature data, including individual objects up to z ∼ 2.5 and a large local reference sample from the Sloan Digital Sky Survey (SDSS). We find that, at all redshifts, the discrepancy between M_⋆ and M_dyn grows as galaxies depart from the present-day relation between stellar mass and size: the more compact a galaxy, the larger its M_⋆/M_dyn. Current uncertainties in stellar masses cannot account for values of M_⋆/M_dyn above 1. Our results suggest that the homology hypothesis contained in the M_dyn formula above breaks down for compact galaxies. We provide an approximation to the virial coefficient K ∼ 6.0[r_e/(3.185 kpc)]^-0.81[M_⋆/(10¹¹ M_⊙)]^0.45, which solves the mass discrepancy problem. A rough approximation to the dynamical mass is given by M_dyn ∼ [σ_e/(200 km s^-1)]^3.6[r_e/(3 kpc)]^0.352.1 × 10¹¹ M_⊙.