We derive a physical model for the observed relations between star formation rate (SFR) and molecular line (CO and HCN) emission in galaxies and show how these observed relations are reflective of the underlying star formation law. We do this by combining 3D non-LTE radiative transfer calculations with hydrodynamic simulations of isolated disk galaxies and galaxy mergers. We demonstrate that the observed SFR-molecular line relations are driven by the relationship between molecular line emission and gas density and anchored by the index of the underlying Schmidt law controlling the SFR in the galaxy. Lines with low critical densities (e.g., CO J = 1–0) are typically thermalized and trace the gas density faithfully. In these cases, the SFR will be related to line luminosity with an index similar to the Schmidt law index. Lines with high critical densities greater than the mean density of most of the emitting clouds in a galaxy (e.g., CO J = 3–2, HCN J = 1–0) will have only a small amount of thermalized gas and consequently a superlinear relationship between molecular line luminosity (Lmol) and mean gas density (). This results in an SFR-line luminosity index less than the Schmidt index for high critical density tracers. One observational consequence of this is a significant redistribution of light from the small pockets of dense, thermalized gas to diffuse gas along the line of sight, and prodigious emission from subthermally excited gas. At the highest star formation rates, the SFR-Lmol slope tends to the Schmidt index, regardless of the molecular transition. The fundamental relation is the Kennicutt-Schmidt law, rather than the relation between SFR and molecular line luminosity. Our model for SFR-molecular line relations quantitatively reproduces the slopes of the observed SFR-CO (J = 1–0), CO (J = 3–2), and HCN (J = 1–0) relations when a Schmidt law with index of ~1.5 describes the SFR. We use these results to make imminently testable predictions for the SFR-molecular line relations of unobserved transitions.
“Molecular Star Formation Rate Indicators in Galaxies” Narayanan, D., Cox, T.J., Shirley, Y., Davé, R., Hernquist, L., Walker, C.K., 2008, ApJ, 684, 996