STOCHASTIC CHARGING OF DUST GRAINS IN PLANETARY RINGS - DIFFUSION RATES AND THEIR EFFECTS ON LORENTZ RESONANCES

Dust grains in planetary rings acquire stochastically fluctuating elec trical charges as they orbit through any corotating magnetospheric pla sma. Here we investigate the nature of this stochastic charging and ca lculate 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 tau(q) for the stochastic charge proces s can be approximated by a Fokker-Planck treatment of the evolution eq uations for P(t; N). For a typical plasma, tau(q) is approximately the charging time constant for the grain. Linear perturbation theory show s that the orbital variations of weakly charged dust grains satisfy fo rced harmonic oscillator equations. The forcing terms take the form Q( t) cos[omega t + phi], where Q(t) is linear in the charge q, the forci ng frequency omega is related to the rate at which a grain samples spa tial periodicities of the magnetic field, and phi is a phase shift. La rge orbital evolution effects take place at LR's, radial locations whe re omega 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 per turbation equations over these intervals. For grains near a resonance the ensemble average of the oscillation amplitudes undergoes a long-pe riod sinusoidal cycle of growth and decay. Charge fluctuations cause s low transport in the phase space of the oscillator. This diffusion is driven by a random walk process; as plasma density n decreases, tau(q) , and hence the size of the typical random step, increases. We calcula te the mean square response (beta(2)) to the stochastic fluctuations i n the Lorentz force; for times longer than tau(q) we find (beta(2)) pr oportional to sigma(2) tau(q)t, where sigma is the deviation of the ch arge from its mean value ($) over bar N. Even when ($) over bar N simi lar to 1, so long as n > 0.1 cm(-3), we find that transport in phase s pace is very small compared to the resonant increase in amplitudes due to the mean charge, over the timescale that the oscillator is resonan tly 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 stro nger bounds on plasma parameters are required when we consider the lon ger transit times between Lorentz resonances.