The power-law spectrum of shock-accelerated relativistic particles

An alternative formalism is developed for calculating the power-law index of relativistic particles (for example, cosmic rays) accelerated by shock waves. It is noted that this formalism, based on classical macroscopic thermodynamics, reproduces the results found by other investigators. In addition, higher order effects are easily calculated, and examples of such generalizations are provided. It is found, for example, that the higher velocity shock waves tend to produce flatter (higher mean energy) spectra. The fundamental expectation of a power-law spectrum per se is reconfirmed.