A two-phase model of galaxy formation: II. The size-mass relation of dynamically hot galaxies
In Paper-I we developed a two-phase model to connect dynamically hot galaxies (such as ellipticals and bulges) with the formation of self-gravitating, turbulent gas clouds (SGC) associated with the fast assembly of dark matter halos. Here we explore the implications of the model for the size-stellar mass relation of dynamically hot galaxies. Star-forming sub-clouds produced by the fragmentation of the SGC inherit its spatial structure and dynamical hotness, which produces a tight and 'homologous' relation, $r_{\rm f}\approx\, 100 r_{\rm bulge}$, between the size of a dynamically hot galaxy ($r_{\rm bulge}$) and that of its host halo assembled in the fast assembly regime ($r_{\rm f}$), independent of redshift and halo mass. This relation is preserved by the 'dry' expansion driven by dynamical heating when a galaxy becomes gas-poor due to inefficient cooling, and is frozen during the slow assembly regime when the bulge stops growing in mass and dynamical heating is no longer effective. The size-stellar mass relation is thus a simple combination of the galaxy-halo homology and the non-linear relation between stellar mass and halo mass. Using a set of halo assembly histories we demonstrate that this model can reproduce all properties in the observed size-mass relation of dynamically hot galaxies, including the flattening of the relation in the low-mass end and the upturn in the very massive end. The predicted evolution of this relation matches observational data currently available to redshift $z \approx 4$, and can be tested in the future at higher $z$. Our results indicate that the sizes of dynamically hot galaxies are produced by the dissipation and collapse of gas in dark matter halos to establish self-gravitating systems of sub-clouds in which stars form.