  Stellar population parameters correlate with a range of galaxy properties,
but it is unclear which relations are causal and which are the result of
another underlying trend. In this series, we quantitatively compare trends
between stellar population properties and galaxy structural parameters in order
to determine which relations are intrinsically tighter, and are therefore more
likely to reflect a causal relation. Specifically, we focus on the galaxy
structural parameters of mass $M$, gravitational potential $\Phi\sim M/R_e$,
and surface mass density $\Sigma\sim M/R_e^2$. In Barone et al. (2018) we found
that for early-type galaxies the age-$\Sigma$ and [Z/H]-$\Phi$ relations show
the least intrinsic scatter as well as the least residual trend with galaxy
size. In this work we study the ages and metallicities measured from full
spectral fitting of 2085 star-forming galaxies from the SDSS Legacy Survey,
selected so all galaxies in the sample are probed to one effective radius. As
with the trends found in early-type galaxies, we find that in star-forming
galaxies age correlates best with stellar surface mass density, and [Z/H]
correlates best with gravitational potential. We discuss multiple mechanisms
that could lead to these scaling relations. For the [Z/H]--$\Phi$ relation we
conclude that gravitational potential is the primary regulator of metallicity,
via its relation to the gas escape velocity. The age--$\Sigma$ relation is
consistent with compact galaxies forming earlier, as higher gas fractions in
the early universe cause old galaxies to form more compactly during their
in-situ formation phase, and may be reinforced by compactness-related quenching
mechanisms.
