Download The effects of charged dust on Saturn`s rings

Charged rings
The effects of charged
T W Hartquist, O Havnes and G E Morfill report on how electric and magnetic forces on dust influence the
properties of and structures and dynamics in Saturn’s rings, to be observed with the Cassini spacecraft in 2004.
Abstract
1: Picture of Saturn’s rings taken
by Voyager 1, showing the large
differences in ring density.
Properties and features in Saturn’s rings
arising due to the charge on dust will be
observable in about one year with
instruments on the Cassini spacecraft.
Charge causes dust to respond to electric
and magnetic forces and plays a role in
electric field generation. Dust levitates
above some rings to form “spokes”, which
may have long-term effects on angular
momentum transport in those rings. A
tenuous ring can be supported by the
electric field caused by the charged dust.
Resonant interactions between the
oscillations of such a ring and the
magnetosphere rotating with the planet
may create structures in it. The electric
repulsion between a large solid object and
small grains may produce a bow wave with
characteristics that serve as diagnostics of
the ring plasma properties. The magnetic
force on charged dust is important for
supplying the material in one of the rings.
I
n July 2004 the Cassini spacecraft will
approach Saturn after journeying for seven
years (saturn.jpl.nasa.gov/cassini/). To reach
its destination, Cassini has flown by Venus
twice, then passed near the Earth, and swung by
Jupiter to be accelerated by the planetary gravitational fields. A primary aim of the Cassini
mission is a thorough investigation of Saturn’s
rings, some of which are shown in figure 1. The
Huygens probe will separate from the main
craft to fly to Titan, one of Saturn’s major
moons. Figure 2 shows the Cassini spacecraft.
Given that Cassini should provide information of unprecedented detail about Saturn’s
rings, they will be at the focus of a level of theoretical scrutiny exceeding even that given to
them by the Voyager 1 and 2 approaches in the
early 1980s. Some researchers have been speculating about the features that may be detected.
In anticipation of the Cassini data, we consider
some of the ways in which charged dust and
electric and magnetic fields may affect the structures and dynamics of Saturn’s rings.
The image of some of Saturn’s rings in figure
1 was produced with Voyager 1. The rings
showing clearly in the images are the most
opaque. More tenuous rings are not apparent
in the images. Figure 3 shows schematically the
5.26
locations of the various rings about Saturn. The
most tenuous rings are the E and G rings. The
properties of the solid particles in the rings are
mainly inferred from observations of how they
scatter sunlight. Voyager data are still the basis
for our rather incomplete knowledge of the ring
properties including their thicknesses, and the
densities and sizes of particles in them.
Observations of the rings by the Hubble Space
Telescope and ground-based optical telescopes,
especially when the rings are edge-on (parallel
to the line of sight from Earth) have provided
additional information about the rings and the
particles in them (Nicholson et al. 1995). Table
1 gives some properties of the rings. The F, G
and E rings are thought to possess the smallest
solid particles but the other rings may contain
small particles at least some of the time.
Most ring experts believe that many properties
of the rings with substantial populations of large
particles are governed primarily by the gravity
of the planet, gravitational perturbations by the
major moons and numerous minor moons, and
collisions between particles (e.g. Cuzzi et al.
1984). These properties include the velocity
dispersion of particles within a ring, its thickness, and the locations of the inner and outer
boundaries of a ring. Any plasma in one of the
optically thicker rings is usually assumed to be
adsorbed rapidly by the solid particles.
However, as described below, transient bursts of
plasma production triggered by impacts of
objects, such as meteors or possibly atmospheric
or ionospheric events, may give rise to electrostatic effects that have long-term consequences
for the properties of even the A and B rings.
Electrostatic and magnetic forces are likely to
have more immediately apparent effects on the
large-scale structures of the tenuous rings containing the smallest dust particles. The lower
opacities of these rings are due to the solid particles in them having less total surface area, on
to which any plasma in the rings might be
adsorbed. Thus the dust particles in the rings
are embedded in gases of atomic ions and electrons. The G and E rings are prime locations for
the electro-magnetic phenomena of plasmas to
be manifested. Though more opaque, the F ring
possesses very complicated and dynamic structures. Plasma processes contribute to the production and evolution of these structures.
Even in the less opaque rings some structures
may arise due to gravitational effects. For
instance, Kolvoord et al. (1990) have postulated
the existence of undetected moonlets to explain
some features of the F ring. Showalter (2000)
October 2003 Vol 44
Charged rings
dust on Saturn’s rings
3: The distances of
the various rings and
some of the major
moons shown in
relation to the
dimension of Saturn.
E-ring
Dione
Tethys
Enceladus
G
F
2: The Cassini spacecraft with its
instrumentation and Huygen’s Titan probe.
B
Self-consistent calculation of grain charge
Some aspects of the above discussion of the
process by which grains acquire charge were
based on the implicit assumption that the
October 2003 Vol 44
Enc
Cas k
s
ion
ivis ision
e d i div
in
C
has argued that many or all of the braids in the
F ring may arise due to excitation by the moon
Prometheus. Clumpy structures generated in the
moon–ring interaction possibly cause enough
back-reaction on the moon to generate a
pseudo-random component in the evolution of
its orbit (Showalter et al. 1999).
In the rings containing the smallest solid particles, those particles are affected by any electric force because they carry charge. Electrons
have thermal speeds that are of the order of one
hundred higher than those of the ions. Thus an
uncharged dust grain surrounded by a plasma
is more likely to collide with an electron than
with a positive ion. The grain will become progressively more negatively charged until the
effects of its electrostatic attraction of positive
ions and of the electrostatic repulsion of electrons are sufficiently large to counteract the
contribution of the electrons’ higher speeds to
the electron collision rate with the dust. In the
simplest cases, the number of electrons carried
by a spherical dust particle is proportional to
the plasma temperature and also to the radius
of the dust grain. Thus the charge-to-mass ratio
of a dust particle scales as the inverse square of
the radius. Consequently, the electric force on
a dust grain is stronger relative to the gravitational force the smaller that the grain is.
For electrostatic and magnetic effects to be
important for the dynamics of solid particles in
a ring, the ring must contain plasma and the
dust particles must be sufficiently small.
Mimas
A
D
Saturn
0
1
2
3
4
5
6
7
Table 1: Properties of Saturn’s rings
Ring
D
C
B
Cassini Division
A
F
G
E
Typical particle size
10–100 µm
1–500 cm
1–500 cm
1–750 cm
1–500 cm
0.1 µm–10 cm
< 0.03 µm
1 µm
magnitude of the total charge contained on
grains is very small compared to the magnitude
of the total charge carried by either the ions or
the electrons. When the total electron and ion
charges nearly balance one another, our previous statement concerning the dependence of the
average charge per grain on grain radius, rg , and
plasma temperature, T, is applicable. The average number of elementary charges on a grain,
Zg , is then given by
Zg = –2.51kB T(rg /e2 )
(1)
where cgs units are used, kB is the Boltzmann
constant, and e is the elementary charge equal
to that of a proton (Spitzer 1941). Equation 1
implies that the grains become charged to the
point that the electric potential energy per electron on the grain is comparable in magnitude
to the thermal energy per ion or electron in the
plasma. Equation 1 is based on the assumption
Optical depth
10–6–10–5
0.05–0.35
0.4–2.5
0.05–0.15
0.4–2
0.01–0.6
10–6
10–5
Mass (g)
unknown
1×1021
3×1022
5×1020
6×1021
0.1–10×1017
unknown
7×1011
that the plasma temperature is small enough
(T 10 eV) that secondary electron ejection
from the dust due to colliding thermal electrons
is unimportant. Photoelectric charging due to
solar radiation is also taken to be unimportant.
In many situations the total number of
charges on the grains is a substantial fraction of
the number of charges carried by ions. If the
dust is negatively charged, the number density
of electrons, ne , falls significantly below that of
ions as a consequence of an approximate
charge neutrality
Zi ni – ne + Zg ng ≈ 0
(2)
ni and ng are the number densities of ions and
grains and Zi is the number of charges carried
by an ion. If a significant number of dust particles are present, equation 2 is used in combination with calculations of the rates at which
ions and electrons strike the grain surface to
5.27
Charged rings
calculate Zg . The self-consistent calculation of
the magnitude of Zg gives a smaller value than
obtained with equation 1. A cloud of small dust
particles will also be charged up by the plasma,
resulting in an electric field throughout the dust
cloud which leads to a lifting force (or electrostatic pressure) on the charged dust. The magnitude of this electrostatic lifting force peaks
when the parameter
P = kB Tng rg / ne0 e2
(3)
(Havnes et al. 1990) is of the order of 1. Here
ne0 is the electron density above or below the
ring and ng is the dust density. For P 1
P ≈ ng Zg / ni Zi
(4)
In subsequent sections we describe the formation of spokes and self-consistent models of electrostatically supported rings where P is not
negligible. In contrast, when P is small, interesting “test particle” responses of grains to magnetospheric electric and magnetic effects occur.
Such responses are important in the E ring.
Spokes – evidence for the electrostatic
support of dust
Spokes were discovered in Saturn’s B ring with
the Voyager instruments (e.g. Eplee and Smith
1984, 1985). They are seen as dark shadows on
the opaque ring. A single spoke can form on the
illuminated side of the ring and then grow and
survive as the ring material orbits to the dark
side of the planet. Figure 4 contains images of the
rings when spokes were present and evolving.
The shadows are cast by submicron-sized dust
lifted above the B ring. The small dust grains are
most likely produced by the impact of objects,
including meteors, on the larger solid particles
in the rings. An impact also produces plasma,
and the grains become charged, as described
above. Electrostatic forces can lift the small
grains to form the spokes (Goertz and Morfill
1983, Morfill et al. 1983). The electric field
arises as a consequence of the ions, which are
light and hot, having a larger gravitational scale
height than the supermicron-sized solid particles. These larger solid particles, just as the submicron-sized particles do, carry negative charge.
Thus, there is some charge separation.
A prime advantage of a spokes model in
which electrostatic support plays a role is that
electrostatic support occurs for grains only up
to a certain size. Scattering properties of grains
in the spokes indicate that they are probably
submicron in size. A purely gravitational mechanism would not be size-selective and would lift
larger particles as well.
A dust particle elevated above a ring responds
to the Lorentz force due to its motion with
respect to the planetary magnetic field. Let rsyn
denote the radius of synchronous orbit, which
is the radius at which the Keplerian orbital
period around the planet is the same as the rotation period of the planet. At radii less than rsyn
the dust moves faster than the magnetic field
5.28
lines, which rotate around the planet with the
period at which the planet spins. At greater
radii the dust moves more slowly. The Lorentz
force drives the dust initially beyond rsyn to
greater radii and drives the dust initially nearer
to the planet than rsyn to smaller radii. Thus, the
spokes move radially as well as they orbit the
planet. They have been described as hydromagnetic waves (Tagger et al. 1991).
The radial motion causes angular momentum
transport, which may have substantial effects
on the structure of the B ring (Goertz et al.
1986, Shan and Goertz 1991). Angular momentum is transferred inwardly at radii greater
than rsyn and outwardly at radii smaller than
rsyn giving rise to a tendency of the ring to
become more extended. The outer boundary of
the B ring is probably limited by a gravitational
interaction with the moon Mimas which orbits
with twice the period at which the outer
boundary of the B ring orbits. Angular momentum is transferred to that moon in the resonate
interaction between it and the outer B ring
boundary, halting the outward expansion of
the ring material.
Various features in the B ring may originate
due to the transport of material and angular
momentum away from rsyn by the spokes. For
instance, a minimum in the opacity of the ring
exists just beyond rsyn . An instability in the
ring’s density distribution is associated with the
material and angular momentum transport in
the spokes, and the existence of observed
ringlets in the B ring may be due to it (Goertz
and Morfill 1988, Shan and Goertz 1991).
The A ring may well have spokes that were
undetectable with the Voyager instruments. If it
does or has had significant spoke activity in the
past, the spokes have almost certainly also had
effects on its structure and features.
Electrostatically supported rings
The typical solid particle size in some of
Saturn’s rings is about a micron or less. The
rings with small solid particles also contain
plasma as Voyager data demonstrated
(Richardson and Sittler 1990). The number density of electrons varies with distance from the
planet and height about the ring midplane but
is typically within an order of magnitude of
1–10 cm–3 away from the most opaque rings.
Because plasma is present and the solid particle
sizes are small, Havnes and Morfill (1984) suggested that electrostatic support of dust grains
is important in at least some tenuous planetary
rings and that it can be the main mechanism
determining the vertical thicknesses of some
rings. The G ring, in particular, is likely to have
a vertical structure (perpendicular to the ring
plane) governed by a balance of electrostatic
and gravitational forces.
Havnes and Morfill (1984) constructed a selfconsistent model of the vertical structure of a
4: The spoke phenomenon in the major rings
(A and B) of Saturn.
ring in which such a balance obtains. The input
parameters are the dust grain number densities
at midplane, the ion density above the ring, the
mass per ion, the radius and mass of a dust
grain, the plasma temperature, and the distance
from the planet. Obviously, the equations
include the one describing the balance between
electric force and gravitation force on the
grains. The charge on a dust grain appears
in that equation and is calculated on the
assumption that the number of ions striking a
grain per unit time is equal to the number of
electrons striking it per unit time. (As described
earlier, in equilibrium a grain will acquire sufficient negative charge for those collision rates
to be equal.) The number densities of ions
and electrons are assumed to obey Boltzmann
relations, i.e.
nα = nα0 exp(–Z αeV / kB T)
(5)
where the subscript α indicates which species is
being considered, the subscript 0 signifies that
the value of the quantity is that obtaining where
V = 0, and V is the electric potential (the gradient of which times –1 gives the electric field)
within the ring. The adoption of equation 5
implies that the gravitational force on the
plasma must be negligible. Though the electric
field balances the gravitational field, it is
assumed to be sufficiently modest that the magnitude of the net charge density at any point
(required to produce the electric field) is small
compared to the magnitude of the charge density associated with any of the ion, electron, or
dust components. For a mixture of dust particles with sizes between 1 µm and 1 cm the scale
height of ring will be several tens of kilometres
if the plasma number density and temperature
are about 10 cm–3 and 10 eV respectively.
The two most important predictions of a
model like that of Havnes and Morfill are that
a ring can never collapse to a monolayer as it
may do if only gravitational and collisional
effects are important and, in addition, if the
dust has a size distribution, the relative scale
heights of the dust will have a particular welldefined dependence on the dust size (Aslaksen
and Havnes 1992). The effects of dust grain
October 2003 Vol 44
S3
{ 1980
1980 S1
Charged rings
F
Saturn
D
C
B
G
A
n=3 4 5
E
54 3
2
1=n
co-rotation
distance
5: The distance from Saturn at which the major vertical magnetospheric–electrostatically resonances
will take place.
velocity dispersion have also been included in
the models of dust scale height – dust size relations in rings that are primarily or partially electrostatically supported (Aslaksen and Havnes
1992). The smallest dust particles may be confined to regions furthest above and below the
ring plane and may never cross it. Observations
with Cassini should allow the measurement of
scale heights of dust grains in tenuous rings as
functions of dust size.
Features arising from magnetic field –
vertical oscillation resonances
Within the context of the outer boundary of the
B ring we touched briefly on the fact that resonances between moons and the orbits of material in rings may be responsible for the
appearance of structures in the rings. Vertical
oscillations of dust that is supported electrostatically may be in resonance with the rotation
of Saturn’s magnetic field. Melandsø and
Havnes (1991) have predicted the locations at
which resonances between vertical oscillations
and the rotating planetary magnetosphere
should occur.
To calculate the frequencies associated with
vertical oscillations, Melandsø and Havnes
(1991) began with a model, like that due to
Havnes and Morfill (1984), for the vertical
structure of an electrostatically supported disk.
They then modified the force balance equation
to include the effect of a plasma environment
that changes with a frequency equal to that of
Saturn’s rotation, while the ring material moves
with the Kepler frequency. A change in the
plasma temperature or density will lead to a
change in the dust charges and the ring electric
potential. This will lead to a change in the vertical electric force, and the ring will oscillate. A
time-dependent mass-conservation equation
relating dust mass number density to the local
oscillation velocity was also included. The vertical motion away from the ring plane slows
down as the electric field decreases, due to the
decrease in dust density. When the ring thickness decreases the increased electric field halts
the contraction. In the analytical linear analysis
October 2003 Vol 44
of the vertical ring oscillations, the other equations remain the same as those used to construct
the static equilibrium model of the ring’s vertical structure.
Melandsø and Havnes (1991) then assumed
that the fluctuations of parameters, such as the
dust density, due to the oscillations have much
smaller magnitudes than the magnitudes of the
parameters in the equilibrium models. They
found that the fundamental vertical oscillation
is at an angular frequency of
ωf = √3ΩK
(6)
where ΩK is the local Keplerian orbital angular
frequency.
Figure 5 shows some of the estimated locations where resonances between the magnetospheric rotation and the vertical oscillations
occur. The locations inner to rsyn are determined
from the requirement that
n(ΩK – ωp) = √3ΩK
(7)
where n is a positive integer and ωp is the angular frequency of the planetary rotation. At a
location external to rsyn
–n(ΩK – ωp) = √3ΩK
(8)
Other locations where resonances may occur
may be found from the requirement that
±n(ΩK – ωp) = m√3ΩK
(9)
where m is a positive integer greater than unity.
Numerical tests including time-varying dust
charges showed that the major damping of the
oscillations was due to a phase difference
between the ring vertical oscillation and the
charge variation.
Melandsø and Havnes (1991) drew attention
to the fact that some of the resonance locations
given by equations 6 to 9 do show some evidence for features in the dusty component of the
rings or in the plasma properties. The Cassini
mission will be valuable for determining more
definitively whether the rings’ properties are
affected significantly by magnetosphericelectrostatically moderated vertical oscillations.
Mach cones as diagnostics of ring properties
When a supersonic rocket or plane passes
through air it leaves a wake. The outer boundary of a wake is called its Mach cone. The
opening angle of a Mach cone is the angle
between a line passing from the foremost tip of
the rocket to a distant downstream point in the
Mach cone and the relative velocity of the far
upstream air and the rocket. The sine of the
opening angle is given by the ratio of the speed
of sound in the air to the rocket’s speed.
Cassini would probably not be able to survive
a passage through one of the opaque rings and
its orbit will be above such rings. For many
regions the ring properties must therefore be
studied with remote observations of structures
in the rings and theoretical modelling of the
structures. Havnes et al. (1995) suggested that
the observation of wakes in the Saturn ring system will allow the inference of some properties
of the rings, the most directly determined of
which will be the speed of dust acoustic waves
(DAWs) in the region where the wake exists.
The sine of the opening angle of the Mach cone
will equal the ratio of the speed of the DAWs
to the speed of the object generating the wake.
Like a sound wave in a more familiar medium
like air, a DAW is a compressive wave. In a
DAW, dust grains take part in the wave motion.
In some places in a DAW, the density of grains
will be higher than in others, just as in an ordinary sound wave the density of material is
higher in some places than others.
In most of the rings, the solid particles have
a substantial size distribution. Even in the most
opaque rings, submicron-sized solid particles
exist, at least transiently in features like
spokes. At the other end of the size distribution, objects large enough to be considered
small moons exist. A wake may be driven by
an object as large as 50 m. The solid particles
in the wake will have sizes of about a micron
and below. Just as spokes show because of the
scattering of light by dust in them, wakes
should be detectable by the scattering of light
by dust in them.
The relative motion of the small dust particles
to that of a boulder is due to the competition
between gravitational forces and the force due
to the planetary magnetic field. A large solid
body or boulder will move in a Keplerian orbit.
A small charged dust grain will have an orbital
period close to the Keplerian period but it will
be modified by the magnetic field. Whether it
will be moving slower or faster than a Keplerian
body depends on the sign of the dust charge and
if the dust is outside or inside the synchronous
distance rsyn (Mendis et al. 1982). The velocity
difference between a boulder and finite sized
dust grain depends on both the radius of the
orbit and the size of the grain. The dependence
on the size of the grain is easily calculated, and
the size of the grains can be inferred from the
way that they scatter sunlight.
Mach cones created by boulders may be most
easily detected at positions just beyond and just
inside rsyn . At other radii, the relative speed of
5.29
Charged rings
the dust and boulder will be sufficiently large
for the kinetic energy of the relative motion to
overcome the electrostatic repulsion caused by
the negative charge on a boulder and the negative charge on the small dust particles and
absorption of dust by the boulders may be the
dominant effect. Mach cones may be observable
near rsyn only at times near those when spoke
activity is detectable because of a paucity of free
small particles at other times.
Other properties of the disturbance and its
Mach cone allow the inference of additional
information about the dust and plasma conditions (Havnes et al. 2001). For instance, the
disturbance will “stand-off” or be separated
from the leading point of a boulder by a measurable distance; that distance depends on the
dust size distribution. Also, the distance behind
a boulder to which a Mach cone is observable
will be determined by the dissipation rate of
DAWs, which in turn depends on the dusty
plasma parameters.
Unfortunately we are unable to be precise in
the prediction of the observational properties of
Mach cones in Saturn’s rings. One difficulty is
the absence of a fully developed theory of
DAWs or Mach cones in a dusty plasma with a
distribution of grain sizes. Our hope is that, for
the reasons outlined here, Mach cones in
Saturn’s ring system will prove detectable and
that their detection will stimulate the relevant
theoretical developments as well as lead to a
better understanding of the rings themselves.
As predicted by Havnes et al. (1996), Mach
cones have been observed in laboratory dusty
plasmas through which larger dust particles
pass (Samsonov et al. 1999). They have also
been simulated numerically for dust and plasma
parameters, typical of some regions in Saturn’s
rings (Brattli et al. 2002). Figure 6 shows the
results of one of the simulations and an image
of a Mach cone observed in an experiment.
The E ring
So far we have considered situations in which
P is non-negligible. In contrast, in the E ring,
motions of dust grains with sizes of about a
micron can be considered to be like that of a test
particle (e.g. Horányi 1996).
Several separate processes induce the orbit of
a test particle in the E ring to precess. The
oblateness of the planet is one cause of precession. The Lorentz force on a charged dust grain
due to its relative motion with respect to the
planetary magnetic field tends to drive the precession in the opposite direction to that caused
by the oblateness. For solid particles much
larger than one micron, oblateness drives the
precession, while for particles much smaller
than a micron the Lorentz force is the dominant cause of the precession. Micron-sized particles, for which the effects due to the
oblateness and the Lorentz force nearly cancel,
5.30
produced. The explanation of the abundances
of gaseous species in the E ring remains a challenge (e.g. Jurac et al. 2001). Further observations with Cassini of the plasma and neutral
species in the E ring will help in the refinement
of the dynamical model of its production.
Outlook
6: The upper figure shows a computed Mach
cone for conditions similar to those in
planetary rings (Brattli et al. 2002) while the
lower one is observed in dusty plasma
experiments (Samsonov et al. 1999).
precess because of the radiation pressure of
sunlight acting on them. The radiation pressure
also causes a periodic variation in the eccentricity of the orbit of a particle having a size in
the narrow range over which the effects of
oblation and the Lorentz force cancel. Solid
particles with sizes in this narrow range will fill
an extended torus. Other particles will not have
extended spatial distributions.
Hamilton and Burns (1994) suggested that the
high eccentricities that the micron-sized particles in the E ring periodically obtain in response
to the radiation pressure allow the E ring to sustain itself. Any moon in a circular orbit is bombarded by that fraction of the dust that is in
highly eccentric orbits. The bombardment produces more dust, which escapes from the moon
and enters the ring. Those escaped dust particles with sizes around one micron refill the
torus, and within about 20 years they also collide with a moon.
As escaped dust particles collide with moons
to produce more dust particles, one might think
that the amount of dust in the E ring should continuously increase. This is not the case as too
high a level of dust in the rings would result in
grain–grain collisions preventing the particles
from reaching highly eccentric orbits. Thus there
is an upper limit to the amount of dust that the
E ring gains from the sputtering of moons.
Though the E ring contains a number of
moons, the theoretical models point towards
Enceladus as the primary source of the E ring
particles. It is a major moon and lies in the most
prominent part of the E ring.
As mentioned earlier, collisions between solid
particles produce plasma if the impact speed is
high enough. Neutral gaseous material is also
A period in excess of two decades has passed
since the Voyager vehicles flew past Saturn.
During it some observations with Earth-based
and near-Earth facilities, such as the Hubble
Space Telescope, have added to our knowledge
of Saturn’s rings. In particular, data obtained
when the rings were edge-on in 1995 have
proved valuable. However, research on Saturn’s
rings has for a long time now been hampered
by the absence of enough new observations.
The Cassini mission will change that. The
craft carries a greater variety of instrumentation
than the Voyagers did. Cassini has the additional advantage that it will make many orbits
around Saturn. Thus we expect discoveries of
phenomena not previously seen, including some
related to the ideas and predictions described
here, which may lead to a better understanding
of how the subtle effects of dust charging can
affect the ring dynamics and evolution. ●
T W Hartquist, Dept of Physics and Astronomy,
University of Leeds, Leeds LS2 9JT, United
Kingdom; O Havnes, Dept of Physics, University of
Tromsø, N–9037 Tromsø, Norway; G E Morfill,
Max-Planck-Institut für extraterrestrische Physik,
D–85740 Garching, Germany. The authors thank
the referee, Richard Durisen, for constructive comnents which improved the paper.
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