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Astron. Astrophys. 358, 347-352 (2000)

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Appendix A: the spatial diffusion coefficients

The coefficient of parallel diffusion [FORMULA] is for a mixture of Alfvén and fast magnetosonic waves given by:

[EQUATION]

Here the Fokker-Planck coefficient

[EQUATION]

describes the resonant cyclotron interactions of particles with slab Alfvén waves (see Schlickeiser (1989), Eq. (59a)), where [FORMULA] denotes the relativistic gyrofrequency and [FORMULA], [FORMULA] as well as [FORMULA] are the Alfvénic spectral index, minimum wavenumber and the plasma wave magnetic field component, respectively.

Schlickeiser & Miller (1998) calculated transport and acceleration parameters for cosmic ray particles interacting resonantly with undamped fast magnetosonic waves. Since fast mode waves are compressional the corresponding Fokker-Planck coefficient (compare Schlickeiser & Miller (1998), Eq. (27))

[EQUATION]

consist of contributions caused by gyroresonant wave-particle interactions as well as transit-time damping. Here [FORMULA], [FORMULA] and [FORMULA] are the fast mode spectral index, the corresponding minimum wavenumber and the fast mode magnetic field component. The cosine of the angle between the propagation direction of the fast mode waves and the background magnetic field [FORMULA] is denoted by [FORMULA]. Since [FORMULA] is the condition for particles in order to gain energy by the process of transit-time damping, the integral in Eq. A.1 can be split into two seperate integrals in order to obtain

[EQUATION]

where we have used symmetry conditions of the Fokker-Planck coefficients with respect to the pitch-angle. [FORMULA] and [FORMULA] denote the contributions resulting from transit-time damping ([FORMULA]) and gyroresonant interactions ([FORMULA]).

It is obvious that, considering the limit [FORMULA], Eq. (A.4) yields the Alfvénic coefficient of spatial diffusion (see Schlickeiser (1989), Eq. (74)). Because [FORMULA], we obtain the following expression:

[EQUATION]

Using [FORMULA] and considering low-energetic particles, we get, disregarding constants, the dependence on radius r and momentum p

[EQUATION]

On the other hand, the case of a vanishing intensity of Alfvén waves leads to the spatial diffusion coefficient for particles propagating in a fast magnetosonic turbulence (see also Schlickeiser & Miller (1998), Eq. (100))

[EQUATION]

where [FORMULA] is Riemann's zeta function. With the radial dependence of the magnetic field ratio mentioned above, one obtains from Eq. (A.7)

[EQUATION]

From a mathematical point of view, the treatment of the mixed turbulence, i.e. [FORMULA] as well as [FORMULA], is more difficult than the pure Alfvénic and magnetosonic case. Following the calculations by Schlickeiser & Miller (1998) one finds, apart from a factor 2, after straightforward algebra the formula

[EQUATION]

where we have introduced the abbreviation

[EQUATION]

Considering, for simplicity, the case that Alfvén and fast magnetosonic waves have equal intensities, i.e. [FORMULA], identical spectral indices and scales, that means [FORMULA] and [FORMULA], one obtains the expression

[EQUATION]

with the correspondingly simplified abbreviation [FORMULA]. Since [FORMULA], one derives again dependences in r and p of the following form:

[EQUATION]

Eqs. (A6), (A8) and (A12) give the values [FORMULA], [FORMULA] and [FORMULA] used in Sect. 2 above.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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