Magnetic Field Effects on the Motion of Circumplanetary Dust
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Abstract
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.