Electrostatic Landau pole for κ-velocity distributions

In this paper, the analytical solution of the linear electrostatic Vlasov dispersion relation is obtained for non-Maxwellian equilibrium distributions of particle velocities (κ distributions). The unphysical singularities for certain values of the parameter κ, recovered by several authors in solving the Landau integral, are discussed in detail, and a way to cancel these singularities and get the correct solution for Langmuir waves is proposed. The solution of the electrostatic dispersion relation presented in this paper provides a theoretical prediction for the oscillation frequency and the damping rate of Langmuir waves, for real values of κ>1∕2 and in particular in the range 1∕2<κ⩽3∕2, where previous analytical solutions fail. Velocity distributions with small values of κ have been frequently observed in solar wind plasmas; therefore, the results presented in this paper are relevant in the interpretation of the solar wind experimental data. Eulerian Vlasov numerical simulations have been performed to substantiate the analytical results; the numerical results are in very good agreement with the theoretical predictions.