Wave packet propagation in an amplifying medium and its application to the dispersion characteristics and to the generation mechanisms of Pc 1 events

The formulae which give the propagation characteristics of a wave packet in a dispersive and amplifying medium, are established. Application is made to the propagation of Pc 1 elements through a magnetosphere constituted of a cold plasma and a high energy proton population. It is shown that the spectral shape, in a frequency-time coordinate system, of the Pc 1 elements is related to two terms : v = d²ω/dk², which represents the variation of the group velocity with frequency and which depends only on the cold plasma characteristics, and μ = -d²γ/dk², in which γ is the amplification coefficient depending on the frequency and which is related to the high energy particle distribution function. When v ? μ, only the usual dispersion effects occur, but a new method is found for determining the line of force on which the micropulsations are generated, without making any assumption about the cold plasma density distribution inside the magnetosphere. It is also possible to deduce some characteristics about the high energy proton distribution. Theoretical computations are presented, which give the frequency variation of the amplification coefficient as a function of the e-folding energy and the anisotropy factor of these high energy protons. Applications are made to ~30 pearl events which are analysed in detail according to this theory. When μ ? v, other effects do appear. After a preliminary phase, the pearl elements can become parallel for a while, or even re-erect before lying again; the duration of each element gives an indication about the number of interacting particles. The conditions for the validity of the quasi-linear theory, and some other non-linear effects related with the interpretation of Pc 1 micropulsations are also discussed.