  This paper is one in a series that explores the importance of radius as a
second parameter in galaxy evolution. The topic investigated here is the
relationship between star formation rate (SFR) and galaxy radius ($R_{\rm e}$)
for main-sequence star-forming galaxies. The key observational result is that,
over a wide range of stellar mass and redshift in both CANDELS and SDSS, there
is little trend between SFR and $R_{\rm e}$ at fixed stellar mass. The
Kennicutt-Schmidt law, or any similar density-related star formation law, then
implies that smaller galaxies must have lower gas fractions than larger
galaxies (at fixed $M_{\ast}$), and this is supported by observations of local
star-forming galaxies. We investigate the implication by adopting the
equilibrium "bathtub" model: the ISM gas mass is assumed to be constant over
time and the net star formation rate is the difference between the accretion
rate of gas onto the galaxy from the halo and the outflow rate due to winds. To
match the observed null correlation between SFR and radius, the bathtub model
requires that smaller galaxies at fixed mass have weaker galactic winds. Our
hypothesis is that galaxies are a 2-dimensional family whose properties are set
mainly by halo mass and concentration. Galaxy radius and accretion rate
plausibly both depend on halo concentration, which predicts how wind strength
should vary with $R_{\rm e}$ and SFR.
