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Present-day multi-wavelength deep imaging surveys allow to characterise theoutskirts of galaxies with unprecedented precision. Taking advantage of thissituation, we define a new physically motivated measurement of size forgalaxies based on the expected location of the gas density threshold for starformation. Employing both theoretical and observational arguments, we use thestellar mass density contour at 1 $M_{\rm \odot}$ pc$^{-2}$ as a proxy for thisdensity threshold for star formation. This choice makes our size definitionoperative. With this new size measure, the intrinsic scatter of the globalstellar mass ($M_{\rm \star}$) - size relation (explored over five orders ofmagnitude in stellar mass) decreases to $\sim$0.06 dex. This value is 2.5 timessmaller than the scatter measured using the effective radius ($\sim$0.15 dex)and between 1.5 and 1.8 times smaller than those using other traditional sizeindicators 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-lawwith a slope very close to 1/3, equivalent to an average stellar mass 3Ddensity of $\sim$4.5$\times$10$^{-3}$ $M_{\rm \odot}$ pc$^{-3}$ for galaxieswithin this mass range. Galaxies with $M_{\rm \star}$$>$10$^{11}$ $M_{\rm\odot}$ show a different slope with stellar mass, which is suggestive of alarger gas density threshold for star formation at the epoch when their starformation peaks.

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