Electrostatic instability excited by an electron beam trapped in the magnetic mirror of the magnetosphere

A dispersion relation which takes into account the nonuniformity of the magnetic field as well as the plasma density along the field lines is obtained for an electrostatic wave propagating parallel to the magnetic field. This dispersion relation is solved for a particular case in which a group of electrons with a monochromatic distribution in magnetic moment is mixed with a low energy plasma. Such electrons are shown to excite ion acoustic waves carried by the low-energy plasma component near multiples of the bounce frequency of these electrons. The theoretical results are applied to explain electrostatic oscillations with a period of approximately fifty seconds observed in the high energy electron fluxes at synchronous altitude.