A new tool to derive simultaneously exponent and extremes of power-law distributions

Abstract

ABSTRACT Many experimental quantities show a power-law distribution p(x) ∝ x−α. In astrophysics, examples are: size distribution of dust grains or luminosity function of galaxies. Such distributions are characterized by the exponent α and by the extremes xminxmax where the distribution extends. There are no mathematical tools that derive the three unknowns at the same time. In general, one estimates a set of α corresponding to different guesses of xminxmax. Then, the best set of values describing the observed data is selected a posteriori. In this paper, we present a tool that finds contextually the three parameters based on simple assumptions on how the observed values xi populate the unknown range between xmin and xmax for a given α. Our tool, freely downloadable, finds the best values through a non-linear least-squares fit. We compare our technique with the maximum likelihood estimators for power-law distributions, both truncated and not. Through simulated data, we show for each method the reliability of the computed parameters as a function of the number N of data in the sample. We then apply our method to observed data to derive: (i) the slope of the core mass function in the Perseus star-forming region, finding two power-law distributions: α = 2.576 between $1.06\, \mathrm{M}_{\odot }$ and $3.35\, \mathrm{M}_{\odot }$, α = 3.39 between $3.48\, \mathrm{M}_{\odot }$ and $33.4\, \mathrm{M}_{\odot }$; (ii) the slope of the γ-ray spectrum of the blazar J0011.4+0057, extracted from the Fermi-LAT archive. For the latter case, we derive α = 2.89 between 1484 MeV and 28.7 GeV; then we derive the time-resolved slopes using subsets of 200 photons each.