^ x5/2 P(fo, rs/ l fJ ' WoP S V3 (r') l * (r')J ' '
Here fo is an arbitrary normalization frequency, ne is the electron density, and £p - foP(fo, rs /ßS << 1 is a dimensionless constant. All quantities indexed by s refer to the values at the solar surface. When deriving the spectrum, conservation of mass and quasi-neutrality in an electron-proton plasma are used, i.e., Aneu = const.
In the special case, considered by Vainio et al. (2003a), when Pf, rs) = £pßS If, the initial dimensionless spectrum becomes I(x,0) = x . In this case the power spectrum of the AlfVen waves can approximate by
where the spectral break point frequency fc (r) = fo /r(r) decreases with heliocentric distance (the spectrum is a broken power law with a spectral index of -1 and -5/3 below and above fc (r); such a form of the power spectrum is supported by observations in the solar wind according to Horbury, 1999).
2.19.3. Determining of the energetic particle mean free path
Vainio et al. (2003a) show that in the case of a wave-heated solar wind the power spectrum of the Alfven waves determines the mean free path A(v, r) of energetic particles with velocity v (excluding electrons). Taking the Alfven waves to be linearly polarized quasi-parallel propagating waves, the mean free path will be:
where Dm is the pitch-angle diffusion coefficient over pitch-angle cosine p:
Here Q and fr are the (angular) particle cyclotron and resonant wave frequency, respectively. Substituting the form of the power spectrum described by Eq. 2.19.5 gives
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