A two-phase model of galaxy formation – II. The size–mass relation of dynamically hot galaxies

Abstract

ABSTRACT 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 gas clouds (SGCs) associated with the fast assembly of dark matter haloes. Here, we explore the implications of the model for the size–stellar mass relation of dynamically hot galaxies. Star-forming sub-clouds resulting from the fragmentation of the turbulent SGC inherit its spatial structure and dynamical hotness, producing a ‘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 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 due to the stop of bulge growth during the slow assembly regime of the halo. The size–stellar mass relation is thus a simple combination of the galaxy–halo homology and the non-linear stellar mass–halo mass relation. Using a set of halo assembly histories, we reproduce all properties in the observed size–mass relation of dynamically hot galaxies, including the flattening in the low-mass end and the upturn in the massive end. The prediction matches observational data currently available to $z \approx 4$, 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 haloes to establish SGCs in which stars form.