Energetic particle distributions in space physics represented by the Pearson differential equation for the kappa distribution

The kappa distribution has been used for decades as a numerical fitting function to represent the energetic particle distributions measured with satellite instruments. The distributions are generally either a three dimensional distribution in the particle velocity or the one dimensional distribution in a velocity component in a particular direction. The theoretical basis for the origin of the kappa functional form is incomplete. The current paper considers published data for one dimensional distributions and provides a unique method of analysis to determine whether the measured distributions are well represented by the kappa distribution. The analysis is based on the Pearson differential equation which arises from the steady distribution of a Fokker-Planck equation.