On the Limited Sizes of Dusty Starbursting Regions at High Redshifts
  Using the far-infrared data obtained by the Herschel Space Observatory, we
study the relation between the infrared luminosity (L_IR) and the dust
temperature (T) of dusty starbursting galaxies at high redshifts (high-z). We
focus on the total infrared luminosity from the cold-dust component
(L_IR^(cd)), whose emission can be described by a modified black body (MBB) of
a single temperature (T_mbb). An object on the (L_IR^(cd), T_mbb) plane can be
explained by the equivalent of the Stefan-Boltzmann law for a MBB with an
effective radius of R_eff. We show that R_eff is a good measure of the combined
size of the dusty starbursting regions (DSBRs) of the host galaxy. In at least
one case where the individual DSBRs are well resolved through strong
gravitational lensing, R_eff is consistent with the direct size measurement. We
show that the observed L_IR-T relation is simply due to the limited R_eff (<~ 2
kpc). The small R_eff values also agree with the compact sizes of the DSBRs
seen in the local universe. However, previous interferometric observations to
resolve high-z dusty starbursting galaxies often quote much larger sizes. This
inconsistency can be reconciled by the blending effect when considering that
the current interferometry might still not be of sufficient resolution. From
R_eff we infer the lower limits to the volume densities of the star formation
rate ("minSFR3D") in the DSBRs, and find that the $L_{IR}$-$T$ relation
outlines a boundary on the (L_IR^(cd), T) plane, below which is the "zone of
avoidance" in terms of minSFR3D.
