Dependence of frequency of nonlinear cold plasma cylindrical oscillations on electron density

Abstract

For cold plasma, the frequency of small amplitude Langmuir oscillations along one Cartesian coordinate is ω_po=√4πe²n_o/m_e , where n_p=n_o is the constant proton density which is equal to the average electron density n_e¯. This formula for ω_po is the basis for measurements of n_e¯ in passive and active radio experiments located on spacecraft. We find that for cold plasma nonlinear cylindrical oscillations n_e¯>n_p (i.e., a buildup of negative space charge near the axis of the cylinder). The resulting frequency of oscillations ω_pe is greater than ω_po. The relation between ω_pe and n_e¯ is found to be logarithmic: ω_pe=ω_po[1+ln(n_e¯/n_o)/12] with 0.5% accuracy for the range 1⩽n_e¯/n_p<6. For quasi-neutral plasma, when n_e¯/n_p≈1, the logarithmic formula reduces to the linear one: ω_pe=ω_po[11/12+(n_e¯/n_p)/12]. For n_e¯/n_p≫6, ω_pe approaches an upper limit of √2 ω_po. These results are expected to be helpful in diagnostics of n_e¯ in the solar wind and in magnetospheric plasmas as well as in laboratory plasmas where cylindrical symmetry is present.