Revisiting the Size-Luminosity Relation in the Era of Ultra Diffuse Galaxies

Galaxies are generally found to follow a relation between their size and luminosity, such that luminous galaxies typically have large sizes. The recent identification of a significant population of galaxies with large sizes but low luminosities ("ultra diffuse galaxies", or UDGs) raises the question whether the inverse is also true, that is, whether large galaxies typically have high luminosities. Here we address this question by studying a size-limited sample of galaxies in the Coma cluster. We select red cluster galaxies with sizes $r_{\mathrm{eff}} > 2 \ \mathrm{kpc}$ down to $M_{g} \sim -13 \ \mathrm{mag}$ in an area of $9 \ \mathrm{deg}^2$, using carefully-filtered CFHT images. The sample is complete to a central surface brightness of $\mu_{g,0}\approx 25.0 \ \mathrm{mag\,arcsec}^{-2}$ and includes 90% of Dragonfly-discovered UDGs brighter than this limit. Unexpectedly, we find that red, large galaxies have a fairly uniform distribution in the size-luminosity plane: there is no peak at the absolute magnitude implied by the canonical size-luminosity relation. The number of galaxies within $\pm 0.5$ magnitudes of the canonical peak ($M_g = -19.69$ for $2<r_{\mathrm{eff}}<3$ kpc) is a factor of $\sim 9$ smaller than the number of fainter galaxies with $-19<M_g<-13$. Large, faint galaxies such as UDGs are far more common than large galaxies that are on the size-luminosity relation. An implication is that, for large galaxies, size is not an indicator of halo mass. Finally, we show that the structure of faint large galaxies is different from that of bright large galaxies: at fixed large size, the S\'ersic index decreases with magnitude following the relation $\log_{10} n \approx -0.067M_g-0.989$.