A recommended procedure for estimating the cosmic-ray spectral parameter of a simple power law

A simple power law model with single spectral index α1 is believed to be an adequate description of the galactic cosmic-ray (GCR) proton flux at energies below 1013 eV. Two procedures for estimating α1—the method of moments and maximum likelihood (ML)—are developed and their statistical performance are compared. The ML procedure is shown to be the superior approach and is then generalized for application to real cosmic-ray data sets. Several other important results, such as the relationship between collecting power and detector energy resolution and inclusion of a non-Gaussian detector response function, are presented. These results have many practical benefits in the design phase of a cosmic-ray detector as they permit instrument developers to make important trade studies in design parameters as a function of one of the science objectives.