PROTON WHISTLER INTERACTIONS NEAR THE EQUATOR IN THE RADIATION BELTS

The interactions of energetic protons with whistlers propagating near the quasi-electrostatic limit, are investigated using a test particle, Hamiltonian formalism. We assume that wave packets exist with finite bandwidths of frequencies, which are close to the equatorial electron gyrofrequency and propagate obliquely with respect to the geomagnetic field. Near the equator the protons interact with the waves which appe ar Doppler shifted to some harmonic of their cyclotron frequency. In a n inhomogeneous geomagnetic field the spacing between cyclotron harmon ic resonances is very small. The Hamiltonian equations of motion are s olved including multiple, independent harmonics for each resonance. Th e wave frequency varies as a function of the distance along the field line, with only one frequency being resonant at a given point. Thus th e inhomogeneity of the magnetic field is compensated by the frequency variation. The proton whistler interactions satisfy the conditions for second-order resonances for all the harmonics. The resonances may als o overlap in phase space, leading to significant changes in the proton s energies and pitch angles. The combined contributions of positive an d negative harmonics allow protons to diffuse toward smaller pitch ang les. Numerical calculations applying this formalism to parameters rele vant to the plasmasphere and controlled VLF transmission experiments a re presented.