Four‐parameter fits to the initial mass function using stable distributions

We propose the use of the statistically well‐founded Pareto–Levy family of stable distributions for the characterization of the mass distributions of stellar and pre‐stellar populations. This family of distributions, which includes the Gaussian, Levy and Cauchy distributions, can be shown to obey the central limit theorem. This makes them statistically justifiable models for populations resulting from the combination of many independent variables. Applying publicly available software to raw data sets we find probability distributions which fit the data to 95 per cent confidence levels. This automatic process is much simpler, faster and more objective than the normal procedure of deciding upon bin sizes and thresholds, and only four parameters are required to define the best‐fitting stable distribution. As there is only one function to describe the whole mass range, there is no need to speculate upon the physical significance of ‘thresholds’ between different subranges. Using stable distribution fits we demonstrate that the modal peaks and widths of the peaks for core mass functions from Ophiuchus, Aquila and Orion are measurably different, width increasing with peak mass. The slopes of the high‐mass power laws are indistinguishable.