is the non-relativistic Alfven speed, and ne is the electron density of the plasma. Negative (positive) frequencies denote right (left) handed polarization and the sign of m/k fixes the propagation direction of the wave relative to the background magnetic field direction. Assuming that (va/c)2 << 4Qp/|Qe| = 0.00218, Vainio and
Schlickeiser (1999a) write the dispersion relation described by Eq. 2.20.1 in the dimensionless form
K = kva/pe\, f = m/lQe|; Op =Qp/pe\ = 1/1836. (2.20.5)
The wave frequency f takes values between -1 < f < O p ; and the sign fixes the wave propagation direction relative to the background magnetic field. When \f| << Op the dispersion relation Eq. 2.20.4 describes Alfven waves. At positive frequencies the Alfven waves are converted to proton-cyclotron waves as f ^ Op . At negative frequencies they are first converted to whistlers at f ~ -O p and finally to electron-cyclotron waves as f ^ -1.
Finally, the gyro-resonance condition between the CR and the parallel/antiparallel waves is f v//!va =O , (2.20.6)
where K and f' are the dimensionless resonant wave number and wave frequency, v is the CR particle speed and v// is the particle velocity parallel to the background magnetic field, O = qB/(fc|Qe|) is the (signed) dimensionless gyro-frequency, q is the charge, y is the Lorenz factor, and m is the mass of the CR particle.
Vainio and Schlickeiser (1999) treat Oo =Oy = meq/(m|qe|) as constant. The combining Eq. 2.20.4 and Eq. 2.20.6 allows the writing down of an equation for the phase speed w = vaf'/K of the waves resonant with CR particles of fixed v as a function of v// in a parametric form w(f ')=±vc
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