The star formation law in a multifractal ISM

The surface density of the star formation rate in different galaxies, as well as in different parts of a single galaxy, scales non-linearly with the surface density of the total gas. This observationally established relation is known as the Kennicutt-Schmidt star formation law. The slope of the star formation law has been shown to change with the density of the gas against which the star formation rate is plotted. This dependence implies a non-linear scaling between the dense gas and the total gas surface densities within galaxies. Here, we explore a possible interpretation of this scaling as a property of the geometry of the interstellar medium (ISM), and we find that it arises naturally if the topology of the ISM is multifractal. Under the additional assumption that, at very high densities, the star formation time-scale is roughly constant, the star formation law itself can also be recovered as a consequence of the multifractal geometry of the ISM. The slope of the scaling depends on the width of the global probability density function (PDF), and is between 1.5 and 1.6 for wide PDFs relevant to high-mass systems, while it is higher for narrower PDFs appropriate for lower mass dwarf galaxies, in agreement with observations.