Subspace approximations to the cosmic ray Fokker-Planck equation

In the theory of cosmic rays we use diffusive transport equations to model the motion of energetic particles through space. A more fundamental description can be achieved if the cosmic ray Fokker-Planck equation is used instead of the usual diffusive transport equation. The Fokker-Planck equation allows for a pitch-angle dependent description of the transport. An analytical solution of this equation is desired in several applications ranging from solar modulation studies to investigations of particle acceleration at astrophysical shocks such as supernova remnants. Furthermore, analytical solutions of the Fokker-Planck equation are required in the development of theories for particle transport across a mean magnetic field. Previous solutions were either pure numerical solutions or methods which are half numerical and half analytical. For some applications, however, one needs simple and pure analytical solutions. In this paper we employ a subspace approximation method to the Fokker-Planck equation. In particular the second-order solution derived here, will be very useful for applications and agrees well with pure numerical solutions for later times.