## Appendix## A.1. Energy dependent acceleration and escapeIn general, one might expect the acceleration time and escape time introduced in Eq. (1) to be functions of particle energy. In this case the solution Eq. (3) is modified. Defining which is positive in the range of interest, one can write The solution of this equation subject to the boundary condition is easily found using Laplace transforms: where In addition to the straightforward case constant, constant dealt with above, it is also interesting to consider the case in which both of these quantities are linearly proportional to . This would arise in modelling diffusive acceleration with a `gyro-Bohm' spatial diffusion coefficient (e.g., Kirk et al. 1994). The solution is with and where we have written Note that in this case the power-law index © European Southern Observatory (ESO) 1998 Online publication: April 20, 1998 |