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Astron. Astrophys. 323, 271-285 (1997)

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Appendix A: anisotropic growth rate for Alfvén waves in the non relativistic regime

The combination of a mono-energetic pair distribution function (Eq. (22)) the resonant momentum (Eq. (40)) leads to a growth rate (Melrose (1968))

[EQUATION]

where we recall the resonant momentum [FORMULA]. The next step is done by expressing the function [FORMULA] as an explicit function of µ and by transforming the [FORMULA] function of p to [FORMULA], with [FORMULA]. Therefore we can integrate successively over µ and over p. Using the Eq. (1) transforming the derivatives over µ and p to derivatives over [FORMULA], the integral reads as

[EQUATION]

After the integration over µ, [FORMULA] and p are linked by the relation [FORMULA]. This leads to complete expression

[EQUATION]

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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