Propagation of Low-Energy Cosmic Rays in Molecular Clouds: Calculations   in Two Dimensions
  We calculate cosmic ray transport with a collisional Boltzmann Transport Equation, including E-M forces. We apply the Crank-Nicholson Method to solve this equation. At each time step, the spatial distribution of cosmic rays is applied to the ZEUS 2D magnetohydrodynamics model, which is then utilized to calculate the resulting E-M field. Finally, the field is applied to the initial equation. This sequence is repeated over many time steps until a steady state is reached. We consider results from t = 0 until steady state for an isotropic low energy cosmic ray flux, and also for an enhanced cosmic ray flux impinging on one side of a molecular cloud. This cosmic ray flux is used to determine an ionization rate of interstellar hydrogen by cosmic rays, zeta. Astrochemical implications are briefly mentioned. 
Paul Rimmer and Eric Herbst
astro-ph.HE