Hypervelocity impacts on satellites or ring particles replenish

circumplanetary dusty rings with grains of all sizes. Due to

interactions with the plasma environment and sunlight, these grains

become electrically charged. We study the motion of charged dust

grains launched at the Kepler orbital speed, under the combined

effects of gravity and the electromagnetic force.

We conduct numerical simulations of dust grain trajectories, covering

a broad range of launch distances from the planetary surface to beyond

synchronous orbit, and the full range of charge-to-mass ratios from

ions to rocks, with both positive and negative electric

potentials. Initially, we assume that dust grains have a constant

electric potential, and, treating the spinning planetary magnetic

field as an aligned and centered dipole, we map regions of radial

instability (positive grains only), where dust grains are driven to

escape or collide with the planet at high speed, and vertical

instability (both positive and negative charges) whereby grains

launched near the equatorial plane and are forced up magnetic field

lines to high latitudes, where they may collide with the planet.

We derive analytical criteria for local stability in the equatorial

plane, and solve for the boundaries between all unstable and stable

outcomes. Comparing our analytical solutions to our numerical

simulations, we develop an extensive model for the radial, vertical

and azimuthal motions of dust grains of arbitrary size and launch

location. We test these solutions at Jupiter and Saturn, both of whose

magnetic fields are reasonably well represented by aligned dipoles, as

well as at the Earth, whose magnetic field is close to an anti-aligned

dipole.

We then evaluate the robustness of our stability boundaries to more

general conditions. Firstly, we examine the effects of non-zero launch

speeds, of up to 0.5 km s$^{-1}$, in the frame of the parent

body. Although these only weakly affect stability boundaries, we find

that the influence of a launch impulse on stability boundaries

strongly depends on its direction.

Secondly, we focus on the effects of higher-order magnetic field

components on orbital stability. We find that vertical stability

boundaries are particularly sensitive to a moderate vertical offset in

an aligned dipolar magnetic field. This configuration suffices as a

model for Saturn's full magnetic field. The vertical instability also

expands to cover a wider range of launch distances in slightly tilted

magnetic dipoles, like the magnetic field configurations for Earth and

Jupiter. By contrast, our radial stability criteria remain largely

unaffected by both dipolar tilts and vertical offsets.

Nevertheless, a tilted dipole magnetic field model introduces

non-axisymmetric forces on orbiting dust grains, which are exacerbated

by the inclusion of other higher-order magnetic field components,

including the quadrupolar and octupolar terms. Dust grains whose

orbital periods are commensurate with the spatial periodicities of a

rotating non-axisymmetric magnetic field experience destabilizing

Lorentz resonances. These have been studied by other authors for the

largest dust grains moving on perturbed Keplerian ellipses. With

Jupiter's full magnetic field as our model, we extend the concept of

Lorentz resonances to smaller dust grains and find that these can

destabilize trajectories on surprisingly short timescales, and even

cause negatively-charged dust grains to escape within weeks. We

provide detailed numerically-derived stability maps highlighting the

destabilizing effects of specific higher-order terms in Jupiter's

magnetic field, and we develop analytical solutions for the radial

locations of these resonances for all charge-to-mass ratios.

We include stability maps for the full magnetic field configurations

of Jupiter, Saturn, and Earth, to compare with our analytics. We

further provide numerically-derived stability maps for the tortured

magnetic fields of Uranus and Neptune.

Relaxing the assumption of constant electric charges on dust, we test

the effects of time-variable grain charging on dust grain motion in

two distinct environments. Firstly, we examine orbital stability in

the tenuous plasma of Jupiter's main ring and gossamer ring where

sunlight, the dominant source of grain charging, is periodically

interrupted by transit through the planetary shadow. This dramatically

expands dynamical instabilities to cover a large range of grain

sizes. Secondly, we study the motion of dust grain orbits in the dense

plasma environment of the Io torus. Here dust grain charges deviate

little from equilibrium, and our stability map conforms closely to

that of constant, negatively-charged dust grains.

Finally, we focus on the poorly understood spokes in Saturn's B ring,

highlighting the observational constraints on spokes, and present our

hypothesis for spoke formation.