Maximum duration of discrete very low frequency emissions

A number of mechanisms have been considered for the generation of VLF emissions. These are Cerenkov radiation [Ellis, 1957], the traveling wave tube (TWT) analogy of Gallet and Helliwell [1959], Doppler-shifted cyclotron radiation from protons [MacArthur, 1959], amplification by a proton beam [Kimura, 1961], Doppler-shifted cyclotron radiation from electrons [Dowden, 1962], and the transverse resonance plasma instability for electrons [Brice, 1963; Bell and Buneman, 1964]. For nonrelativistic protons, the frequency of the radiation emitted in the direction of the earth's magnetic field is found from [Kimura, 1961] ((1))where υ‖ is the velocity of the protons parallel to the earth's magnetic field, υp is the phase velocity of the emitted wave, ƒ is the wave frequency, and ƒI is the ion (proton) gyrofrequency. For frequencies of interest ((2))so that we can write ((3))