Evidence for a non-universal Kennicutt-Schmidt relationship using hierarchical Bayesian linear regression

For investigating the relationship between the star formation rate and gas surface density, we develop a Bayesian linear regression method that rigorously treats measurement uncertainties and accounts for hierarchical data structure. The hierarchical Bayesian method simultaneously estimates the intercept, slope and scatter about the regression line of each individual subject (e. g. a galaxy) and the population (e. g. an ensemble of galaxies). Using synthetic data sets, we demonstrate that the method accurately recovers the underlying parameters of both the individuals and the population, especially when compared to commonly employed ordinary least squares techniques, such as the bisector fit. We apply the hierarchical Bayesian method to estimate the Kennicutt-Schmidt (KS) parameters of a sample of spiral galaxies compiled by Bigiel et al. We find significant variation in the KS parameters, indicating that no single KS relationship holds for all galaxies. This suggests that the relationship between molecular gas and star formation differs from galaxy to galaxy, possibly due to the influence of other physical properties within a given galaxy, such as metallicity, molecular gas fraction, stellar mass and/or magnetic fields. In four of the seven galaxies the slope estimates are sublinear, especially for M51, where unity is excluded at the 2 sigma level. We estimate the mean index of the KS relationship for the population to be 0.84, with 2 sigma range [0.63, 1.0]. For the galaxies with sublinear KS relationships, a possible interpretation is that CO emission is tracing some molecular gas that is not directly associated with star formation. Equivalently, a sublinear KS relationship may be indicative of an increasing gas depletion time at higher surface densities, as traced by CO emission. The hierarchical Bayesian method can account for all sources of uncertainties, including variations in the conversion of observed intensities to star formation rates and gas surface densities (e. g. the X-CO factor), and is therefore well suited for a thorough statistical analysis of the KS relationship.