The Kennicutt-Schmidt (KS) relationship between the surface density of the star formation rate (SFR) and the gas surface density has three distinct power laws that may result from one model in which gas collapses at a fixed fraction of the dynamical rate. The power law slope is 1 when the observed gas has a characteristic density for detection, 1.5 for total gas when the thickness is about constant as in the main disks of galaxies, and 2 for total gas when the thickness is regulated by self-gravity and the velocity dispersion is about constant, as in the outer parts of spirals, dwarf irregulars, and giant molecular clouds. The observed scaling of the star formation efficiency (SFR per unit CO) with the dense gas fraction (HCN/CO) is derived from the KS relationship when one tracer (HCN) is on the linear part and the other (CO) is on the 1.5 part. Observations of a threshold density or column density with a constant SFR per unit gas mass above the threshold are proposed to be selection effects, as are observations of star formation in only the dense parts of clouds. The model allows a derivation of all three KS relations using the probability distribution function of density with no thresholds for star formation. Failed galaxies and systems with sub-KS SFRs are predicted to have gas that is dominated by an equilibrium warm phase where the thermal Jeans length exceeds the Toomre length. A squared relation is predicted for molecular gas-dominated young galaxies.
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