A Millimeter Continuum Size-Luminosity Relationship for Protoplanetary Disks

We present a sub-arcsecond resolution survey of the 340 GHz dust continuum emission from 50 nearby protoplanetary disks, based on new and archival observations with the Submillimeter Array. The observed visibility data were modeled with a simple prescription for the radial surface brightness profile. The results were used to extract intuitive, empirical estimates of the emission "size" for each disk, $R_{\rm eff}$, defined as the radius that encircles a fixed fraction of the total continuum luminosity, $L_{\rm mm}$. We find a significant correlation between the sizes and luminosities, such that $R_{\rm eff} \propto L_{\rm mm}^{0.5}$, providing a confirmation and quantitative characterization of a putative trend that was noted previously. This correlation suggests that these disks have roughly the same average surface brightness interior to their given effective radius, ~0.2 Jy arcsec$^{-2}$ (or 8 K in brightness temperature). The same trend remains, but the 0.2dex of dispersion perpendicular to this relation essentially disappears, when we account for the irradiation environment of each disk with a crude approximation of the dust temperatures based on the stellar host luminosities. We consider two (not mutually exclusive) explanations for the origin of this size-luminosity relationship. Simple models of the growth and migration of disk solids can account for the observed trend for a reasonable range of initial conditions, but only on timescales that are much shorter than the nominal ages present in the sample. An alternative scenario invokes optically thick emission concentrated on unresolved scales, with filling factors of a few tens of percent, that are perhaps manifestations of localized particle traps.