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

Many experimental quantities show a power-law distribution p(x) proportional to x(-alpha). In astrophysics, examples are: size distribution of dust grains or luminosity function of galaxies. Such distributions are characterized by the exponent alpha and by the extremes x(min)x(max) 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 alpha corresponding to different guesses of x(min)x(max). 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 x(i) populate the unknown range between x(min) and x(max) for a given alpha. 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: alpha = 2.576 between and , alpha = 3.39 between and ; (ii) the slope of the gamma-ray spectrum of the blazar J0011.4+0057, extracted from the Fermi-LAT archive. For the latter case, we derive alpha = 2.89 between 1484 MeV and 28.7 GeV; then we derive the time-resolved slopes using subsets of 200 photons each.