Particle Acceleration in Relativistic Shearing Flows: Energy Spectrum

We consider the acceleration of charged particles in relativistic shearing flows, with Lorentz factor up to $\Gamma_0 \sim 20$. We present numerical solutions to the particle transport equation and compare these with results from analytical calculations. We show that in the highly relativistic limit the particle energy spectrum that results from acceleration approaches a power law, $N(E)\propto E^{-\tilde{q}}$, with a universal value $\tilde{q}=(1+\alpha)$ for the slope of this power law, where $\alpha$ parameterizes the power-law momentum dependence of the particle mean free path. At mildly relativistic flow speeds, the energy spectrum becomes softer and sensitive to the underlying flow profile. We explore different flow examples, including Gaussian and power-law-type velocity profiles, showing that the latter yield comparatively harder spectra, producing $\tilde{q}\simeq 2$ for $\Gamma_0 \simeq 3$ and Kolmogorov turbulence. We provide a comparison with a simplified leaky-box approach and derive an approximate relation for estimating the spectral index as a function of the maximum shear flow speed. These results are of relevance for jetted, high-energy astrophysical sources such as active galactic nuclei, since shear acceleration is a promising mechanism for the acceleration of charged particles to relativistic energies and is likely to contribute to the high-energy radiation observed.