Particle Acceleration by Cosmic-Ray Viscosity in Relativistic Jet Shear Flows

Abstract:

We study energetic particle acceleration in relativistic radio-jet shear flows, taking into account particle scattering both into and out of the shear flow region. This involves solving a mixed Dirichlet-Von Neumann boundary value problem at radius r=r₂ at the edge of the jet. The flow velocity profile of the jet u=u(r) e_z is directed along the z-axis of the jet, and depends only on cylindrical radius r about the jet axis, and u(r) is a monotonic decreasing function of r. The mean scattering time τ is a power law in momentum p in the scattering frame or fluid frame, inside the jet at 0<r<r₂, i.e. τ = τ₀ (p/p₀)^α.. At r>r ₂ τ = τ₀ (p/p₀)^α (r/r₂)^s where s is a positive constant and α is a constant. The Green’s function solution is obtained as an eigen-function expansion in terms of J₀ Bessel functions (which depend on r) and power law functions of p. The solutions describe particle acceleration by shear in jets. A time dependent model is used to assess the effects of cosmic ray inertia in limiting the upper particle momentum at time t. The competition between particle energy gains due to momentum space diffusion and synchrotron losses is studied. Application of the estimates of the maximum momentum change formula is used to assess the acceleration of cosmic rays to TeV energies in active galactic nucleii radio- jets. A study of the relationship between momentum p₀, of particles at r→∞, and the momentum p of particles at position r, as described by the particle propagator or Green’s function is investigated.