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 $Phisim M/R_e$, and surface mass density $Sigmasim M/R_e^2$. 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.
The well-established correlations between the mass of a galaxy and the properties of its stars are considered evidence for mass driving the evolution of the stellar population. However, for early-type galaxies (ETGs), we find that $g-i$ color and ste
llar metallicity [Z/H] correlate more strongly with gravitational potential $Phi$ than with mass $M$, whereas stellar population age correlates best with surface density $Sigma$. Specifically, for our sample of 625 ETGs with integral-field spectroscopy from the SAMI Galaxy Survey, compared to correlations with mass, the color--$Phi$, [Z/H]--$Phi$, and age--$Sigma$ relations show both smaller scatter and less residual trend with galaxy size. For the star formation duration proxy [$alpha$/Fe], we find comparable results for trends with $Phi$ and $Sigma$, with both being significantly stronger than the [$alpha$/Fe]-$M$ relation. In determining the strength of a trend, we analyze both the overall scatter, and the observational uncertainty on the parameters, in order to compare the intrinsic scatter in each correlation. These results lead us to the following inferences and interpretations: (1) the color--$Phi$ diagram is a more precise tool for determining the developmental stage of the stellar population than the conventional color--mass diagram; and (2) gravitational potential is the primary regulator of global stellar metallicity, via its relation to the gas escape velocity. Furthermore, we propose the following two mechanisms for the age and [$alpha$/Fe] relations with $Sigma$: (a) the age--$Sigma$ and [$alpha$/Fe]--$Sigma$ correlations arise as results of compactness driven quenching mechanisms; and/or (b) as fossil records of the $Sigma_{SFR}proptoSigma_{gas}$ relation in their disk-dominated progenitors.
