Particle Acceleration in Relativistic Shearing Flows: Energy Spectrum

TL;DR

This paper investigates how charged particles gain energy in relativistic shearing flows, revealing a universal power-law spectrum at high speeds and showing how different flow profiles influence the spectral slope, with implications for astrophysical jet sources.

Contribution

It provides numerical and analytical analysis of particle acceleration in relativistic shear flows, establishing a universal spectral slope and exploring the effects of flow profiles on particle energy spectra.

Findings

At high Lorentz factors, the energy spectrum approaches a universal power law with slope 1+α.

Mildly relativistic flows produce softer spectra sensitive to flow profiles.

Power-law velocity profiles yield harder spectra, with slope around 2 for Γ₀≈3.

Abstract

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.