A Recommended Procedure for Estimating the Cosmic-Ray Spectral Parameter of a Simple Power Law With Applications to Detector Design

A simple power law model consisting of a single spectral index alpha-1 is believed to be an adequate description of the galactic cosmic-ray (GCR) proton flux at energies below 10(exp 13) eV. Two procedures for estimating alpha-1 the method of moments and maximum likelihood (ML), are developed and their statistical performance compared. It is concluded that the ML procedure attains the most desirable statistical properties and is hence the recommended statistical estimation procedure for estimating alpha-1. The ML procedure is then generalized for application to a set of real cosmic-ray data and thereby makes this approach applicable to existing cosmic-ray data sets. Several other important results, such as the relationship between collecting power and detector energy resolution, as well as 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. This is particularly important for space-based detectors where physical parameters, such as dimension and weight, impose rigorous practical limits to the design envelope.