Power-Law Spectra from Fermi Acceleration at Relativistic Shocks

Abstract

The theory of diffusive acceleration at shock fronts, which applies only if the fluid speed is nonrelativistic, yields a simple formula for the power-law index s of accelerated particles: s = 3r/(r – 1), where r is the compression ratio of the shock front. Although the acceleration process depends on there being effective pitch-angle scattering of the particles in both the upstream and downstream regions, no property associated with this process appears in the formula. Unfortunately, if the velocity of the fluid through the shock front is relativistic, as seems to be the case in the central engines of AGN's, and also in some hot-spots in the outerparts of their jets, this attractive property ceases to hold. To find the index s, it becomes necessary to develop specific models describing the scattering process. The physical reason for this is that the particle distribution close to a relativistic shock is anisotropic. The exact type of anisotropy depends on the properties of the pitch angle scattering and determines both the average energy gain per shock crossing, as well as the probability of escape downstream. In this paper, results are presented for the pitch angle diffusion resulting from scattering in a weakly turbulent plasma with a Kolmogorov spectrum of Alfvén waves moving parallel and antiparallel to the magnetic field. This kind of spectrum has been employed in nonrelativistic models of hot spots [1]. However, the results obtained tend not to vary too dramatically as a function of the turbulence spectrum, being less than about 0.2 in the resulting s (0.1 in the predicted synchroton spectral index) for turbulence spectra between k−1 and k−2 [3].