2023ApJ...958..169W

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 f₀(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 < r < r₂, and $\tau {(r,p)={\tau }_{0}(p/{p}_{0})}^{\alpha }{(r/{r}_{2})}^{s}$, where s > 0 in the region r > r₂ 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.