On the Functional Form of the Universal Star Formation Law

TL;DR

This paper derives a universal star formation law using dimensional analysis, showing it applies across galaxy types and redshifts, and explores how different assumptions about characteristic scales influence its form.

Contribution

It introduces a new functional form for the star formation law based on the Vaschy-Buckingham Pi theorem, linking it to a characteristic length scale and unifying various formulations.

Findings

Galaxies and star-forming regions follow the derived star formation law.

Different assumptions about the length scale L lead to known star formation scaling relations.

Star formation efficiency is influenced by turbulence parameters like the Toomre and Mach numbers.

Abstract

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.