in terms of the 'constant' speed v and Lorentz factor y (9 denotes the gyro-phase of the particle, and p and Oo, the norm of its momentum and non-relativistic gyro-frequency, respectively). Integrating the time-averaged Eq. 2.21.8 on a time-scale short enough to keep the particles rigidity constant, and in the range of pitch-angles /< va/v where the scattering by resonant processes is known to be the most deficient, it can be shown that the particles are in fact linearly pushed out of the small p range by the low-frequency waves. Indeed, i io ((- to )EJ dk ^V]
with = ak + (tji/(ya/v)- j)vato constant on the time interval [to, t], if we only keep in the spectrum the wave numbers smaller than KM = min((, AkF), K being the largest wave-number satisfying the condition: t-to <<(n/2)ic^K))/Opo), and AkF the actual width of the FMW spectrum. The contribution to the variation of p from the wave-numbers larger than K is negligibly small, because it is given by the integral of an oscillating sine function of time and k. It was checked that there exists a time interval t - to such that K is larger than the lower boundary of the spectrum, and /-/o ~va/v, i.e., that the Eq. 2.21.9 holds until the particles leave the small-p range. Once they have reached the boundary va/v, they are efficiently scattered away by the transit-time damping interaction with the fast magneto-sound waves (in accordance with Schlickeiser and Miller, 1998; Ragot, 1999b).
Averaging on many successive passages through this small-p region, i.e., over the phases a'k, one can estimate the average exit time t, and an equivalent
'diffusion' coefficient Duunr = (> )V(t). Assuming that the spectrum is a simple power law, i.e., « k~q above km , one can write:
t œ r(va/v) ( (va/v)mpkc Vq-l)/2 8bQ0^q -1 ^ Ymkm ,
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