Power-law spectrum of shock-accelerated relativistic particles

We develop an alternative formalism for calculating the power-law index of relativistic particles (e.g., cosmic rays) accelerated by shock waves. This formalism, based on classical macroscopic thermodynamics, reproduces the results found by Bell and by Blandford and Ostriker. Moreover, higher order effects are easily calculated (e.g., effects of order v/c), and examples of such generalizations are provided. We find, for example, that the higher velocity shock waves tend to produce flatter (higher mean energy) spectra. We also reconfirm the fundamental expectation of a power-law spectrum per se