The scatter in the galaxy size versus stellar mass ( M _ { \star } ) relation gets largely reduced when , rather than the half-mass radius R _ { e } , the size at a fixed surface density is used .
Here we address why this happens .
We show how a reduction is to be expected because any two galaxies with the same M _ { \star } have at least one radius with identical surface density , where the galaxies have identical size .
However , the reason why the scatter is reduced to the observed level is not trivial , and we pin it down to the galaxy surface density profiles approximately following Sersic profiles with their R _ { e } and Sersic index ( n ) anti-correlated ( i.e. , given M _ { \star } , n increases when R _ { e } decreases ) .
Our analytical results describe very well the behavior of the observed galaxies as portrayed in the NASA Sloan Atlas ( NSA ) , which contains more than half a million local objects with 7 < \log ( M _ { \star } / M _ { \odot } ) < 11.5 .
The comparison with NSA galaxies also allows us to find the optimal values for the mass surface density ( 2.4 _ { -0.9 } ^ { +1.3 } M _ { \odot } { pc } ^ { -2 } ) and surface brightness ( r -band 24.7 \pm 0.5 { mag arcsec ^ { -2 } } ) that minimize the scatter , although the actual values depend somehow on the subset of NSA galaxies used for optimization .
The physical reason for the existence of optimal values is unknown but , as \citet 2020arXiv200102689T point out , they are close to the gas surface density threshold to form stars and thus may trace the physical end of a galaxy .
Our NSA-based size–mass relation agrees with theirs on the slope as well as on the magnitude of the scatter .
As a by-product of the narrowness of the size–mass relation ( only 0.06 dex ) , we propose to use the size of a galaxy to measure its stellar mass .
In terms of observing time , it is not more demanding than the usual photometric techniques and may present practical advantages in particular cases .