Acceleration and Spectral Redistribution of Cosmic Rays in Radio-jet Shear Flows

Abstract

A steady-state, semi-analytical model of energetic particle acceleration in radio-jet shear flows due to cosmic-ray viscosity obtained by Webb et al. is generalized to take into account more general cosmic-ray boundary spectra. This involves solving a mixed Dirichlet–Von Neumann boundary value problem at the edge of the jet. The energetic particle distribution function f _0 ( r , p ) at cylindrical radius r from the jet axis (assumed to lie along the z -axis) is given by convolving the particle momentum spectrum ${f}_{0}(\infty ,p^{\prime} )$ with the Green’s function $G(r,p;p^{\prime} )$ , which describes the monoenergetic spectrum solution in which ${f}_{0}\to \delta (p-p^{\prime} )$ as r → ∞ . Previous work by Webb et al. studied only the Green’s function solution for $G(r,p;p^{\prime} )$ . In this paper, we explore for the first time, solutions for more general and realistic forms for ${f}_{0}(\infty ,p^{\prime} )$ . The flow velocity u = u ( r ) e _z is along the axis of the jet (the z -axis). u is independent of z , and u ( r ) is a monotonic decreasing function of r . The scattering time $\tau {(r,p)={\tau }_{0}(p/{p}_{0})}^{\alpha }$ in the shear flow region 0 0 in the region r > r _2 is outside the jet. Other original aspects of the analysis are (i) the use of cosmic ray flow lines in ( r , p ) space to clarify the particle spatial transport and momentum changes and (ii) the determination of the probability distribution ${\psi }_{p}(r,p;p^{\prime} )$ that particles observed at ( r , p ) originated from r → ∞ with momentum $p^{\prime} $ . The acceleration of ultrahigh-energy cosmic rays in active galactic nuclei jet sources is discussed. Leaky box models for electron acceleration are described.

Keywords

• Radio jets
• High-energy cosmic radiation
• Ultra-high-energy cosmic radiation
• Relativistic jets