The Volumetric Extended-Schmidt Law: A Unity Slope

We investigate the extended-Schmidt (ES) law in volume densities ($\rho_{\rm SFR}$ $\propto$ $(\rho_{\rm gas}\rho_{\rm star}^{0.5})^{\alpha^{\rm VES}}$) for spatially-resolved regions in spiral, dwarf, and ultra-diffuse galaxies (UDGs), and compare to the volumetric Kennicutt-Schmidt (KS) law ($\rho_{\rm SFR}$ $\propto$ $\rho_{\rm gas}^{\alpha^{\rm VKS}}$). We first characterize these star formation laws in individual galaxies using a sample of 11 spirals, finding median slopes $\alpha^{\rm VES}$=0.98 and $\alpha^{\rm VKS}$=1.42, with a galaxy-to-galaxy rms fluctuation that is substantially smaller for the volumetric ES law (0.18 vs 0.41). By combining all regions in spirals with those in additional 13 dwarfs and one UDG into one single dataset, it is found that the rms scatter of the volumetric ES law at given x-axis is 0.25 dex, also smaller than that of the volumetric KS law (0.34 dex). At the extremely low gas density regime as offered by the UDG, the volumetric KS law breaks down but the volumetric ES law still holds. On the other hand, as compared to the surface density ES law, the volumetric ES law instead has a slightly larger rms scatter, consistent with the scenario that the ES law has an intrinsic slope of $\alpha^{\rm VES} \equiv$1 but the additional observational error of the scale height increases the uncertainty of the volume density. The unity slope of the ES law implies that the star formation efficiency (=$\rho_{\rm SFR}$/$\rho_{\rm gas}$) is regulated by the quantity that is related to the $\rho_{\rm star}^{0.5}$.