Astron. Astrophys. 333, 452-458 (1998)

Appendix

A.1. Energy dependent acceleration and escape

In 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 s, defined by is related to the timescales by , whereas in the case of energy independent acceleration and escape .

© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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