# A fractal origin for the mass spectrum of interstellar clouds. II. Cloud models and power-law slopes

## Abstract

Three-dimensional fractal models on grids of ∼2003 pixels are generated from the inverse Fourier transform of noise with a power-law cutoff and exponentiated to give a lognormal distribution of density. The fractals are clipped at various intensity levels, and the mass and size distribution functions of the clipped peaks and their subpeaks are determined. These distribution functions are analogous to the cloud mass functions determined from maps of the fractal interstellar medium using various thresholds for the definition of a cloud. The model mass functions are found to be power laws with powers ranging from -1.6 to -2.4 in linear mass intervals as the clipping level increases from ∼0.03 to ∼0.3 of the peak intensity. The low clipping value gives a cloud-filling factor of ∼10% and should be a good model for molecular cloud surveys. The agreement between the mass spectrum of this model and the observed cloud and clump mass spectra suggests that a pervasively fractal interstellar medium can be interpreted as a cloud/ intercloud medium if the peaks of the fractal intensity distribution are taken to be clouds. Their mass function is a power law even though the density distribution function in the gas is a lognormal. This is because the size distribution function of the clipped clouds is a power law, and with clipping, each cloud has about the same average density. A similar result would apply to projected clouds that are clipped fractals, giving nearly constant column densities for power-law mass functions. The steepening of the mass function for higher clip values suggests a partial explanation for the steeper slope of the mass functions for star clusters and OB associations, which sample denser regions of interstellar gas. The mass function of the highest peaks is similar to the Salpeter initial mass function, suggesting again that stellar masses may be determined in part by the geometry of turbulent gas.