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

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Appendix

A.1. Energy dependent acceleration and escape

In general, one might expect the acceleration time [FORMULA] and escape time [FORMULA] introduced in Eq. (1) to be functions of particle energy. In this case the solution Eq. (3) is modified. Defining

[EQUATION]

which is positive in the range of interest, one can write

[EQUATION]

The solution of this equation subject to the boundary condition [FORMULA] is easily found using Laplace transforms:

[EQUATION]

where

[EQUATION]

In addition to the straightforward case [FORMULA] constant, [FORMULA] constant dealt with above, it is also interesting to consider the case in which both of these quantities are linearly proportional to [FORMULA]. This would arise in modelling diffusive acceleration with a `gyro-Bohm' spatial diffusion coefficient (e.g., Kirk et al.  1994). The solution is

[EQUATION]

with

[EQUATION]

and where we have written

[EQUATION]

Note that in this case the power-law index s, defined by [FORMULA] is related to the timescales by [FORMULA], whereas in the case of energy independent acceleration and escape [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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