from which an implicit form, i.e., v// = v// (w), may be derived straightforwardly. In

Fig. 2.20.1 and Fig. 2.20.2 are plotted the solutions of Eq. 2.20.7 for two values of O corresponding to non-relativistic protons and mildly relativistic electrons.

Fig. 2.20.1. Phase speed w, as a function of parallel particle velocity, of parallel-propagating transverse waves resonant with CR particles having constant dimensionless gyro-frequency of o = op (left) and O = -1/3 (right). According to Vainio and Schlickeiser (1999).

In Fig. 2.20.1 and Fig. 2.20.2 there are indicated what values the wave frequency f' takes in each branch of the curves. The curves are plotted for parallel-propagating waves; for anti-parallel waves, both v// and w change signs for constant f', and a complete figures would include the negative w axes with curves obtained by rotating the plots in Fig. 2.20.1 and Fig. 2.20.2 about their origins by 180°.

Fig. 2.20.2. The same as in Fig. 2.20.1 but for ® = -1/3. According to Vainio and Schlickeiser (1999).

2.20.4. Bulk motion of the CR in space plasma

The scattering by waves which all move with the same phase speed, e.g., parallel Alfven waves, tends to make the CR particle distribution isotropic in the wave frame, i.e., the coordinate system moving with the phase speed of the waves relative to the plasma. This results in a plasma-frame bulk motion of the CR with the phase speed of the waves. If waves with several speeds are present the situation is a bit more involved, but the use of quasi-linear theory with the diffusion approximation for CR propagation gives the plasma-frame bulk speed of the particles in form (Schlickeiser, 1989)

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