The Loss Cone Driven Instability for Langmuir Waves in an Unmagnetized Plasma

Analytic and numerical results are presented for the growth rate of Langmuir waves due to a loss-cone distribution of energetic electrons. The effect of the magnetic field on the wave-particle interaction is ignored, and the resonance condition is described in terms of a resonance hyperboloid in momentum space. The collisional evolution of a distribution of magnetically trapped electrons is followed numerically to show how a `gap' distribution develops. The growth is most favorable for an intermediate sized loss cone (α ∼ 45 °) and a `gap' distribution in which the mean energy of the suprathermal electrons is much larger than the thermal energy of the background electrons. It is plausible that loss-cone `gap' distributions do develop in the solar corona, and that they should lead to second harmonic plasma emission weakly polarized in the x-mode.