On the Functional Form of the Universal Star Formation Law

We study the functional form of the star formation law, using the Vaschy-Buckingham Pi theorem. We find that that it should have a form $\rm \dot{\Sigma}_{\star} \propto \sqrt{\frac{G}{L}}\Sigma_{gas}^{3/2}$, where L is a characteristic length that is related with an integration scale. With a reasonable estimation for L, we find that galaxies from different types and redshifts, including Low Surface Brightness galaxies, and individual star-forming regions in our galaxy, obey this single star formation law. We also find that depending on the assumption for L, this star formation law adopt different formulations of $\rm \dot{\Sigma}_{\star}$ scaling, that are widely studied in the literature: $\rm \Sigma_{gas}^{3/2}, \Sigma_{gas}/t_{orb}, \Sigma_{gas}/t_{ff} \, and \, \Sigma_{gas}^{2}/v_{turb}$. We also study secondary controlling parameters of the star formation law, based on the current evidence from numerical simulations and find that for galaxies, the star formation efficiency should be controlled, at least, by the turbulent Toomre parameter, the sonic and Alfvenic Mach numbers.