Disentangling the physical parameters of gaseous nebulae and galaxies

We present an analysis to disentangle the connection between physical quantities that characterize the conditions of ionized HII regions -- metallicity ($Z$), ionization parameter ($U$), and electron density ($n_\mathrm{e}$) -- and the global stellar mass ($M_\ast$) and specific star formation rate ($\mathrm{sSFR}=\mathrm{SFR}/M_\ast$) of the host galaxies. We construct composite spectra of galaxies at $0.027 \le z \le 0.25$ from Sloan Digital Sky Survey, separating the sample into bins of $M_\ast$ and sSFR, and estimate the nebular conditions from the emission line flux ratios. Specially, metallicity is estimated from the direct method based on the faint auroral lines [OIII]$\lambda$4363 and [OII]$\lambda\lambda$7320,7330. The metallicity estimates cover a wide range from $12+\log\mathrm{O/H}\sim7.6\textrm{--}8.9$. It is found that these three nebular parameters all are tightly correlated with the location in the $M_\ast$--sSFR plane. With simple physically-motivated ans\"atze, we derive scaling relations between these physical quantities by performing multi regression analysis. In particular, we find that $U$ is primarily controlled by sSFR, as $U \propto \mathrm{sSFR}^{0.43}$, but also depends significantly on both $Z$ and $n_\mathrm{e}$. The derived partial dependence of $U \propto Z^{-0.36}$ is weaker than the apparent correlation ($U\propto Z^{-1.52}$). The remaining negative dependence of $U$ on $n_\mathrm{e}$ is found to be $U \propto n_\mathrm{e}^{-0.29}$. The scaling relations we derived are in agreement with predictions from theoretical models and observations of each aspect of the link between these quantities. Our results provide a useful set of equations to predict the nebular conditions and emission-line fluxes of galaxies in semi-analytic models.