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.
