Hemispherical transport equation: modeling of quasiparallel collisionless shocks

Using a so-called hemispherical model we derive a general transport equation for cosmic ray and thermal particles scattered in pitch angle by magnetic inhomogeneities in a moving collisionless plasma. The weak scattering through 90 degrees results in isotropic particle distributions in each hemisphere. The consideration is not limited by small anisotropies and by the condition that particle velocities are higher than characteristic flow velocity differences. For high velocities and small anisotropies the standard cosmic ray transport equation is recovered. We apply the equations derived for investigation of injection and acceleration of particles by collisionless shocks.