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 ωpo=4πe2no/me, where np=no is the constant proton density which is equal to the average electron density ne¯. This formula for ωpo is the basis for measurements of ne¯ in passive and active radio experiments located on spacecraft. We find that for cold plasma nonlinear cylindrical oscillations ne¯>np (i.e., a buildup of negative space charge near the axis of the cylinder). The resulting frequency of oscillations ωpe is greater than ωpo. The relation between ωpe and ne¯ is found to be logarithmic: ωpe=ωpo[1+ln(ne¯/no)/12] with 0.5% accuracy for the range 1⩽ne¯/np<6. For quasi-neutral plasma, when ne¯/np≈1, the logarithmic formula reduces to the linear one: ωpe=ωpo[11/12+(ne¯/np)/12]. For ne¯/np≫6, ωpe approaches an upper limit of 2ωpo. These results are expected to be helpful in diagnostics of ne¯ in the solar wind and in magnetospheric plasmas as well as in laboratory plasmas where cylindrical symmetry is present.