Surfing and generation of cosmic rays in relativistic shock waves

We consider the problem of cosmic-ray generation through the surfing acceleration of charged particles in relativistic magnetosonic shock waves (the branch of fast magnetic sound) propagating in magnetized space plasmas. The dependence of the particle surfing acceleration efficiency on the angle θ _Bn between the normal to the shock front plane and the magnetic field vector in the plasma upstream of the shock is analyzed in detail. We show that for angles satisfying the condition χ = βΓ tan θ _Bn ⩾ 1, where β = U/c, Γ = (1 − β)^{2 −1/2}, U is the shock velocity, and c is the speed of light, the particles can theoretically be accelerated through surfing for an unlimited time and can gain an unlimited energy. For angles satisfying the condition χ < 1, the kinetic energy ℰ of the particles is limited by ℰ = 2mc ²χ²/(1 − χ²) (m is the particle rest mass). Our main conclusion is that the generation of cosmic rays through the surfing acceleration of particles in the front of a relativistic shock wave for Γ ≫ 1 is also efficient when the angle θ _Bn is very small, i.e., it differs significantly from a right angle. Estimates for the energies of particles accelerated through surfing in relativistic jets are provided.