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