A physically motivated definition for the size of galaxies in an era of ultra-deep imaging

Abstract: Present-day multi-wavelength deep imaging surveys allow to characterise the outskirts of galaxies with unprecedented precision. Taking advantage of this situation, we define a new physically motivated measurement of size for galaxies based on the expected location of the gas density threshold for star formation. Employing both theoretical and observational arguments, we use the stellar mass density contour at 1 $M_{\rm \odot}$ pc$^{-2}$ as a proxy for this density threshold for star formation. This choice makes our size definition operative. With this new size measure, the intrinsic scatter of the global stellar mass ($M_{\rm \star}$) - size relation (explored over five orders of magnitude in stellar mass) decreases to $\sim$0.06 dex. This value is 2.5 times smaller than the scatter measured using the effective radius ($\sim$0.15 dex) and between 1.5 and 1.8 times smaller than those using other traditional size indicators such as $R_{\rm 23.5,i}$ ($\sim$0.09 dex), the Holmberg radius $R_{\rm H}$ ($\sim$0.09 dex) and the half-mass radius $R_{\rm e,M_{\star}}$ ($\sim$0.11 dex). Moreover, galaxies with 10$^7$ $M_{\rm \odot} <$ $M_{\star} < 10^{11}$ $M_{\rm \odot}$ increase monotonically in size following a power-law with a slope very close to 1/3, equivalent to an average stellar mass 3D density of $\sim$4.5$\times$10$^{-3}$ $M_{\rm \odot}$ pc$^{-3}$ for galaxies within this mass range. Galaxies with $M_{\rm \star}$$>$10$^{11}$ $M_{\rm \odot}$ show a different slope with stellar mass, which is suggestive of a larger gas density threshold for star formation at the epoch when their star formation peaks.