Stochastic charging of dust grains in planetary rings: Diffusion rates and their effects on Lorentz resonances

Abstract

Dust grains in planetary rings acquire stochastically fluctuating electrical charges as they orbit through any corotating magnetospheric plasma. Here we investigate the nature of this stochastic charging and calculate its effect on the Lorentz resonance (LR). First we model grain charging as a Markov process, where the the transition probabilities are identified as the ensemble‐averaged charging fluxes due to plasma pickup and photoemission. We determine the distribution function P(t;N), giving the probability that a grain has N excess charges at time t. The autocorrelation function τq for the stochastic charge process can be approximated by a Fokker‐Planck treatment of the evolution equations for P(t;N). For a typical plasma, τq is approximately the charging time constant for the grain. Linear perturbation theory shows that the orbital variations of weakly charged dust grains satisfy forced harmonic oscillator equations. The forcing terms take the form Q(t)cos[ωt + ϕ], where Q(t) is linear in the charge q, the forcing frequency ω is related to the rate at which a grain samples spatial periodicities of the magnetic field, and ϕ is a phase shift. Large orbital evolution effects take place at LR's, radial locations where ω is close to a grain's orbital frequency. Since the charge q(t) is piecewise constant over the time interval between the arrivals of plasma particles or solar photon, we can iterate solutions to the perturbation equations over these intervals. For grains near a resonance the ensemble average of the oscillation amplitudes undergoes a long‐period sinusoidal cycle of growth and decay. Charge fluctuations cause slow transport in the phase space of the oscillator. This diffusion is driven by a random walk process; as plasma density n decreases, τq, and hence the size of the typical random step, increases. We calculate the mean square response 〈β²〉 to the stochastic fluctuations in the Lorentz force; for times longer than τq we find 〈β²〉 ∝ σ²τqt, where σ is the deviation of the charge from its mean value . Even when , so long as n > 0.1 cm−3, we find that transport in phase space is very small compared to the resonant increase in amplitudes due to the mean charge, over the timescale that the oscillator is resonantly pumped up. Therefore the stochastic charge variations cannot break the resonant interaction; locally, the Lorentz resonance is a robust mechanism for the shaping of ethereal dust ring systems. Slightly stronger bounds on plasma parameters are required when we consider the longer transit times between Lorentz resonances.