Spectral analysis for wide energy channels
For energetic particle measurements whose spectra follow a power law, it is often challenging to define a characteristic ("effective") energy of an energy channel. In order to avoid time-consuming calculations, the geometric mean is often used as an approximation for the effective energy. This approximation is considered to be pretty good. It is, however, potentially inadequate in cases with wide energy channels and soft spectral slopes. In order to determine the limits of the goodness of the approximation, we derive formulas to calculate the deviation of the effective energy, phase space density and energy density based on the geometric-mean approximation from those based on the power law. The results show that the geometric-mean approximation is usually adequate and that corrections are needed only in extraordinary cases.