A stochastic simulation of the propagation of Galactic cosmic rays reflecting the discreteness of cosmic ray sources. Age and path length distribution

The path length distribution of Galactic cosmic rays (GCRs) is the fundamental ingredient for modeling the propagation process of GCRs based on the so-called weighted slab method. We try to derive this distribution numerically by taking into account the discreteness in both space and time of occurrences of supernova explosions where GCRs are suspected to be born. We solve numerically the stochastic differential equations equivalent to the Parker diffusion-convection equation which describes the propagation process of GCR in the Galaxy. We assume the three-dimensional diffusion is an isotropic one without any free escape boundaries. We ignore any energy change of GCRs and the existence of the Galactic wind for simplicity. We also assume axisymmetric configurations for the density distributions of the interstellar matter and for the surface density of supernovae. We have calculated age and path length of GCR protons arriving at the solar system with this stochastic method. The obtained age is not the escape time of GCRs from the Galaxy as usually assumed, but the time spent by GCRs during their journey to the solar system from the supernova remnants where they were born. The derived age and path length show a distribution spread in a wide range even for GCR protons arriving at the solar system with the same energy. The distributions show a cut-off at a lower range in age or path length depending on the energy of GCRs. These cut-offs clearly come from the discreteness of occurrence of supernovae. The mean age of GeV particles obtained from the distributions is consistent with the age obtained by direct observation of radioactive secondary nuclei. The energy dependence of the B/C ratio estimated with the path length distribution reproduces reliably the energy dependence of B/C obtained by recent observations in space.