JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A09218, doi:10.1029/2004JA010577, 2005
Tenuous ring formation by the capture of
interplanetary dust at Saturn
C. J. Mitchell,1 J. E. Colwell, and M. Horányi1
Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, Colorado, USA
Received 6 May 2004; revised 18 January 2005; accepted 11 April 2005; published 15 September 2005.
[1] We examine the capture of interplanetary dust particles by Saturn using both a simple
two-dimensional code as well as a three-dimensional code with more detailed charging
currents and forces. The incoming grain’s energy may be lost to the magnetosphere
due to the grain’s finite charging time as it passes through the magnetospheric plasma. The
two-dimensional code uses an equilibrium grain potential as a function of radial position
and a charging time as a function of particle size. The ratio of prograde to retrograde
captures depends on the shape of the equilibrium potential as a function of distance from
the planet, with more prograde captures observed for more negative potentials. In the
three-dimensional simulations, we observe a thick ring of dust particles captured into
retrograde orbits. This dust, which has a typical size of 0.1 mm, forms a tenuous ring
around the planet extending from the edge of the main rings to 9 planetary radii
with a thickness of 3 planetary radii. We predict that the peak density for retrograde
particles is 3 1015 cm3, which should be observed by the Cassini spacecraft. We
expect that with the appropriate viewing geometry, about 20 grains will be detected by the
Cosmic Dust Analyzer during the first 4 years of the mission.
Citation: Mitchell, C. J., J. E. Colwell, and M. Horányi (2005), Tenuous ring formation by the capture of interplanetary dust at
Saturn, J. Geophys. Res., 110, A09218, doi:10.1029/2004JA010577.
1. Introduction
[2] Capture of dust grains by planets was first studied by
Artem’ev [1968], where the incoming dust grains were
assumed to undergo an instantaneous charging event upon
entering the magnetosphere. Subsequent studies by Hill and
Mendis [1979, 1980, 1981] invoked electrostatic disruption
of large grains to calculate the spatial distribution of dust
entering the Jovian magnetosphere for various initial conditions and at various times after the captures took place.
This was then used to examine the ratio of leading to
trailing collisions with the Galilean satellites.
[3] In addition to electrostatic disruption, charging by the
plasma and interactions with the corotational electric field
can cause a grain’s energy to decrease [Horányi and
Cravens, 1996]. As a dust grain enters a planetary magnetosphere, it is subjected to changing plasma densities and
temperatures which cause the charge on the grain to
fluctuate. If there are regions in the magnetosphere where
the equilibrium potential on a grain increases with radial
distance, then energy can be lost by the particle as it moves
about the planet. As the grain falls inward, the equilibrium
potential decreases, though the actual charge on the grain is
greater than this value due to the finite charging time.
Conversely, as the grain moves away from the planet, its
charge is less than its equilibrium value. Since Saturn’s
corotational electric field points outward, the work done by
the field as the grain moves outward is less than the work
done on the grain as the particle moves inward, causing the
grain to lose energy. It should be noted that magnetospheres
typically have both positive and negative gradients for the
equilibrium potential of embedded dust grains. Regions
where the potential is decreasing radially will add energy
to a particle and so whether or not a particle is captured is
sensitive to the exact shape of the potential function.
[4] This effect has been evoked to predict the capture of
dust from comet Shoemaker-Levy 9 into orbit about Jupiter
in the vicinity of Io [Horányi, 1994]. More recently, a
population of dust grains moving in a retrograde sense
about Jupiter has been observed by the Galileo spacecraft.
This dust is captured from the interplanetary medium
through interactions with the planetary magnetosphere
[Colwell et al., 1998]. Numerical simulations by Colwell
and Horányi [1996] showed that particles are more likely to
be captured in retrograde orbits than in prograde ones about
Jupiter. Here we expand this work to include dust capture at
Saturn and examine the asymmetry as a function of the
charge on the dust grains.
2. Equations of Motion
1
Also at Department of Physics, University of Colorado, Boulder,
Colorado, USA.
Copyright 2005 by the American Geophysical Union.
0148-0227/05/2004JA010577$09.00
[5] The equations of motion for a dust grain moving in
planetary magnetospheres have been thoroughly studied by
a variety of authors. Here we use the same model as Juhász
and Horányi [2002] for their E ring calculations, except that
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ignore the magnetotail, hence magnetosphere is azimuthally
symmetric. An equilibrium potential (Feq) for the dust
grains is prescribed as a piecewise smooth function of the
grain’s radial position inside the magnetosphere. Beyond
12 planetary radii (RS), there is little plasma and charging
is dominated by the photoelectric effect. We therefore
choose the equilibrium potential to be 3 volts outside this
radius. We fix the potential at the origin to 2 volts to
represent the lack of plasma near the planet. Finally, we take
the potential at 7 RS to be a variable, Fdip. We then connect
these three points (the origin, 7 RS, and 12 RS) with linear
potentials. Fdip can now be used as a parameter describing
the plasma properties in the magnetosphere (see Figure 1).
[8] Charging of the grain is described by the equation
Figure 1. The equilibrium potential for Fdip = 11 V.
1
F_ ¼ Feq F ;
t
we ignore all forms of drag (e.g., ion, neutral, and PoyntingRobertson) due to the short lifetimes of the particles
(<10 years). A dust grain of mass m moving in a planetary
magnetosphere in inertial coordinates with the origin at the
planet’s center has the equation of motion
where t is the characteristic charging time, which we take to
be inversely proportional to the grain radius and F is the
particle’s potential in volts given by
F¼
2
m
d r
¼ FL þ FG þ FRP þ FD ;
dt2
ð1Þ
where FL is the Lorentz force including both the magnetic
force and the force caused by the corotational electric field.
FG, FRP, and FD are the forces due to gravity, radiation
pressure, and drag, respectively. The force due to radiation
pressure (neglecting terms of order v/c) is given by
FRP ¼
SA
QPR^rS ;
c
ð2Þ
where S is the energy flux from the Sun, A is the crosssectional area of the particle, ^rS is the unit vector pointing
along the flux, and QPR is an efficiency factor averaged over
the solar spectrum which depends on the size of the grain as
well as its material properties [Burns et al., 1979]. We ignore
this effect for our simple model and assume the particles are
olivine for the three-dimensional (3-D) simulations.
[6] The charging of the dust grains is described by
dq
¼
dt
X
Ik ;
ð3Þ
k
where the charging currents Ik are functions of the plasma
parameters at the grain’s position as well as velocity, size,
and material properties. We use the plasma parameters
provided by Richardson [1995] to calculate collection
currents from hot and cold electrons, as well as hydrogen
and oxygen ions. We also include currents from secondary
electrons and photoelectrons as per Juhász and Horányi
[2002].
q
;
700eam
ð5Þ
where e is the elementary charge and am is the particle radius
in microns. For a 1 mm grain, a charging time of 1000 s was
selected.
[9] We restrict ourselves to equatorial orbits by assuming
the magnetic field is a dipole aligned with the planetary
rotation axis and located at the center of the planet, a good
approximation for Saturn [Connerney, 1993]. We neglect
planetary oblateness for simplicity, as well as any forces due
to drag or radiation pressure. For our magnetosphere, the
magnetic field is perpendicular to the equatorial plane;
therefore the corotational electric field has no out of plane
component. Particles started in the equatorial plane which
have no velocity component along the normal will therefore
remain in the plane.
[10] The equation of motion given by equation (1) now
reduces to
r ¼ rf_ 2 m
W _
fW 2
2
r
r
ð6Þ
and
¼ r_ W 2f_ ;
f
r r3
ð7Þ
where we use polar coordinates. W = 1.64 104s1 is the
planet’s rotation frequency, m = 3.8 1022 cm3/s2 is the
planet’s mass times the gravitational constant, and W is
given by
W¼
3. 2-D Simulations
[7] To gain insight into the capture mechanism and the
history of a dust grain’s charge, we start with a highly
simplified model of Saturn’s magnetosphere. We assume the
plasma corotates everywhere inside the magnetosphere and
ð4Þ
qB0 3
R ;
mc S
ð8Þ
where q is the charge on the grain, B0 = 0.215 Gs is the
magnetic field on the equator (using southward as
positive), m is the grain’s mass, c is the speed of light,
and RS = 6.03 109 cm is the planetary radius.
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Figure 2. A typical captured orbit for a 97 nm dust grain in the two-dimensional code with Fdip = 11 V.
(a) Radial distance versus time, (b) trajectory, (c) charge versus distance, and (d) force as a function of
the distance from the planet after the grain has settled into an eccentric orbit with constant semimajor
axis (arbitrary units). The grain moves counterclockwise around the upper loop and clockwise around
the lower loop. The energy gained in the lower loop is equal to the energy lost in upper loop.
[11] The specific energy and canonical angular momentum of the grain are now given by
E¼
m WW
1 2
r_ þ r2 f_ 2 2
r
ð9Þ
and
J ¼ r2 f_ þ
W
:
r
ð10Þ
If the charge on the grain remains constant, then these are
constants of the motion and it is impossible for the grain to
lose energy and be captured into orbit about the planet.
However, when q is allowed to vary, neither the energy nor
the angular momentum are conserved. If the equilibrium
potential of a grain increases with the distance from the
planet, then dust grains will lose energy unless they are on
circular orbits [Horányi and Cravens, 1996]. Owing to the
grain’s finite charging time, its actual charge is greater than
the Feq while moving radially inward. The opposite is true
when it moves outward, so the work done on the grain due to
the electric field over one orbital period is negative. If Feq
decreases with distance, the particle will gain energy. Since
magnetospheres are expected to have regions of both
increasing and decreasing Feq, the fate of a dust grain
depends on which regions it visits. Additionally, while
particles may settle into circular orbits in regions where Feq is
increasing, circular orbits are unstable where Feq is
decreasing. Stable eccentric orbits are possible if the particle
spends time in both regions, assuming the grain gains as
much energy where Feq is decreasing as it loses where Feq is
increasing.
[12] Although neither the energy nor angular momentum
are conserved when the grain charge varies, if the magnetosphere is corotating, then it can be readily shown that the
Jacobi constant, given by
H ¼ E WJ ;
ð11Þ
is conserved [Northrop and Hill, 1983; Schaffer and Burns,
1994]. This quantity is the Hamiltonian in the frame of
reference rotating with the planet, where the electric field
vanishes and no net work can be done on the particle due to
charge fluctuations.
[13] To explore the effect of the magnetosphere on the
orbital evolution of the dust grains, we chose 18 values of
Fdip for our simulations, from 3 V to 14 V (for Saturn,
Fdip 11 V). For each simulation, 4000 particles in the
size range from 10 < rg < 200 nm were started on a line
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Figure 3. Grain radius vs impact parameter for particles followed in the two-dimensional simulations
for four values of Fdip (upper left is 2 V, upper right is 3 V, lower left is 9 V, and lower right is 14 V).
Particles represented by red dots collide with the planet, dark blue escape by going beyond 200 RS, light
blue are captured into prograde orbits, and yellow are captured into retrograde orbits. See color version of
this figure at back of this issue.
located parallel to the x-axis and 100 RS away along the
y-axis from planet’s center, which was located at the origin.
Initial x-coordinates ranged from 35 to 35 RS, and the
grains’ velocities were chosen to be the local escape
velocity at the particles’ positions pointing in the negative
y direction. Dust grains were followed until they collided
with the planet, got further than 200 RS away from the
planet, or settled into an orbit with a constant eccentricity.
The Jacobi constant was calculated during each integration
as a check on the code’s accuracy.
[14] Dust grains normally make tens of revolutions about
the planet before their semimajor axis would stop decreasing. A typical trajectory of a captured grain is shown in
Figure 2, which also shows the history of radial position for
the particle, as well as the charge on the particle and force
due to the corotational electric field as functions of the
radial position. Once the particle settles into an orbit with
constant semimajor axis and eccentricity, the energy gained
by the particle in the region of negative sloping equilibrium
potential is equal to that lost in the region with positive
slope, as can be seen in Figure 2d.
[15] Figure 3 shows the results of our simulations, where
we have plotted particle size versus impact parameter for
the grains used in the simulations for four different values
of Fdip, the color of each data point indicating the fate of
the particle. For positive values Fdip, more particles are
captured into retrograde orbits than prograde, while the
opposite is true for negative Fdip. This is partially due to
the particles which collide with the planet. For retrograde
orbits, negatively charged particles are pulled into the
planet while positively charged ones are repelled by the
magnetic field. The opposite is true for prograde orbits.
This effect can be seen in Figure 3, as the red particles
which hit the planet initially start off in the prograde
region but move into the retrograde region as Fdip
decreases. The ratio of retrograde to prograde captures
varies from 5 at Fdip = 3 V to 0.4 at Fdip = 14 V.
Figure 4 shows the number of captures for retrograde and
prograde orbits as a function of Fdip.
4. 3-D Simulations
[16] For our three-dimensional code, the charge is followed using equation (3), instead of the simplified version,
equation (4). It uses the Z3 model [Connerney, 1993] for
Figure 4. The number of captures in the two-dimensional
simulations for retrograde (stars) and prograde (diamonds)
orbits as a function of Fdip.
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Figure 5. Equilibrium potentials for particles orbiting in
the equatorial plane assuming Keplerian velocities for
prograde (solid) and retrograde (dashed) orbits. The
difference is mainly due to the modulation of the ion
current as a function of relative velocity.
Saturn’s magnetic field and includes solar radiation pressure,
as well as the J2 term in the gravitational potential. The
plasma parameters were derived from those provided by
Richardson [1995]. There is no magnetotail in this model,
although we do include the interplanetary magnetic field,
with the field orientation flipping every 14 days. Effects
from Saturn’s shadow were included with radiation pressure
and the photoelectric effect turning off when the grains are in
the shadow.
[17] We approximate the effects of sputtering by assuming a linear decrease in grain radius [Juhász and Horányi,
2002], with sputtering only occurring inside the magnetosphere (r 20 RS). We take the lifetime of a 1 mm grain to
be 50 years [Jurac et al., 2001]. As the size of a grain
decreases, the Lorentz force becomes increasingly more
important as the particle’s charge to mass ratio increases.
[18] Figure 5 shows the equilibrium particle potential,
calculated by assuming a circular Kepler orbit in the
equatorial plane, either prograde or retrograde, and then
integrating the charging currents until equilibrium was
reached. As in the two-dimensional simulations, energy loss
occurs in regions with a positive slope while energy is
gained in regions with a negative slope.
[19] The initial conditions for the dust grains were based
on the interplanetary dust model by Divine [1993]. Particles
were chosen from the asteroid population, since no captures
from the interstellar or Oort cloud populations were observed in our initial runs. Eccentricities for the grains’
orbits about the Sun were randomly chosen, uniformly
distributed in the range from 0 to 0.3. Inclinations were
then chosen between 0 and 10 degrees, with the distribution
uniform in cos(i). Finally, semimajor axes were chosen
from the range of values that crossed Saturn’s orbit for an
eccentricity of e = 0.3. Dust particles which intersected
Saturn’s orbit were then given a small, randomly generated
impact parameter (0 < b < 0.15AU, 1AU 2450RS), and
the grain’s position and velocity was calculated on the
surface of a sphere with a radius of 50 planetary radii.
[20] Individual particles were followed until they moved
further than 500 RS from Saturn, crashed into the planet,
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collided with the ring, or sputtered down to less than 10 nm
in radius. Figure 6 shows the results of a simulation where
104 particles from 10 to 500 nm in radius were started
simultaneously and followed for 2 years. From the steep
initial drop in the number of particles, the amount of time
for uncaptured particles to leave the system can be seen to
be about 170 days. Particles which have survived in the
simulation longer than 170 days may therefore be considered temporarily captured by Saturn.
[21] Figure 7 shows a trajectory for a typical captured
dust particle in this code, while Figure 8 shows the grain’s
radial position and its radius as functions of time. The grain
gets captured and begins to lose energy. The particle’s orbit
can not circularize due to a combination of radiation
pressure and sputtering. This has a tendency to pull the
grain into the planet when in regions that have an increasing
Feq, since the energy lost causes a decrease in the semimajor axis. Once the particle enters a region where Feq
decreases with distance, the eccentricity begins to grow due
to the energy gain until the particle starts losing enough
energy in a region of increasing Feq to offset the energy
gain. This occurs at about t = 8 years in Figure 7. The
grain’s orbit then changes little until about t = 14 years, at
which point the grain is about 0.1 mm in radius, and the
grain’s vertical and radial motions begin to grow. Its radial
motion eventually takes the grain out of the magnetosphere,
where it is then removed by the interplanetary electric field.
This particle, and others like it, leave Saturn travelling at of
few tens of kilometers per second and are a minor contribution to the stream particles being ejected from the Saturn
system [Horányi, 2000].
[22] To calculate the density of captured dust, particles
with a uniform size distribution (10 nm < rg < 500 nm) were
injected into the system at a steady rate of 1 per 40 min
of simulation time for 12 years (the amount of time
necessary to reach equilibrium), a total number of 1.6 105 particles, corresponding to a simulated flux of 9.2 1021 m2s1. Thirteen runs were completed, differing only
in the seeds for the random number generators, with about
Figure 6. The number of particles surviving in a
simulation where 105 particles are simultaneously injected
into the system. The first decrease is caused by particles
crashing into either the rings or planet on their first pass.
The second, less abrupt decrease ending at about 170 days
is caused by particles leaving the system at r = 500 RS.
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Figure 7. A typical trajectory for a captured particle projected into (a) the (x, y) plane and (b) the (r, z)
plane. (c) The energy of the particle first decreases as it is captured into orbit but eventually begins to
increase as the particle visits regions where Feq decreases with distance. (d) The potential on the grain
also shows large variations as the particle begins to undergo large radial motions.
2700 particles remaining at the end of each run (see
Figure 9). After this initial period of 12 years, we continued
to follow the trajectories of these particles with a sampling
time (t) of 1 week with the particles’ positions being
recorded at a sampling rate (n) of 10 min; hence a single
grain contributed nt = 1008 data points. For every time
step, the particles’ positions were binned into tori with axes
of symmetry coinciding with the planet’s rotation axis and a
square cross section 0.2 RS on a side. During each time step,
every dust grain made a contribution to the number of
particles in the bin it was located. A grain’s real contribution
(NR) to the number of particles in a particular torus was
calculated by multiplying the number of times the grain was
in that torus (NC) by the ratio of the Divine model’s flux
(FD) to the simulated flux for the corresponding grain size
(FC) and then dividing by the number of data points a grain
could contribute. The contribution for each particle to the
number of particles in the ith torus was therefore
Figure 8. (a) Distance from Saturn and (b) grain size as a
function of time for the particle in Figure 6.
Figure 9. The number of particles averaged over all runs
with ages greater than 170 days.
NRi ¼ NCi
FD 1
:
FC nt
ð12Þ
Summing the contributions of all the particles in each
torus and then dividing the by its volume gave the density
of dust grains in that torus. By counting only dust particles
with retrograde motion, we were able to produce the
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forces become important and the particles are ejected
from the system. We found that this ring has a density
of 3 1015 cm3, one order of magnitude less dense
than Jupiter’s (4 1014 cm3). The dust grains are
considerably smaller due to the inability of Saturn’s
magnetosphere to capture large particles.
[25] Using the average density, the Cassini trajectory, and
the detector area for the Cosmic Dust Analyzer, we predict
that Cassini will observe 20 of these particles during the
spacecraft’s 4 year mission, using the spacecraft’s trajectory
through the ring and assuming the best possible viewing
geometry for the CDA. Though these particles are in the
same region of space as the E ring, they will be easily
distinguishable due to their retrograde motion. Additionally,
there may be compositional differences between the captured dust and E ring particles, which can be measured by
the chemical analyzer on the CDA.
Figure 10. Density of retrograde captured dust particles
about Saturn. The two maxima are at the radial turning
points for the average dust grain. The densities are in units
of 1015 cm3. See color version of this figure at back of
this issue.
density profile for captured retrograde dust ring as shown
in Figure 10.
[23] The ring extends from the outer edge of the main
rings to about 9 RS, with a thickness of 3 RS. It has a peak
density of 3 1015 cm3 with a maxima at 3 RS. It is
mainly composed of particles from 50 to 150 nm in
radius. The average lifetime for a retrograde particle is
about 2 years, which is set by sputtering and shorter than
that resulting from collisions with the E ring particles,
which is approximately 10 years. Though it is impossible
to see in Figure 10 due to the azimuthal averaging, the
ring does exhibit an asymmetry similar to that of the E
ring [Juhász and Horányi, 2004] due to radiation pressure, where dust grains are pushed out of the ring plane.
The density of prograde captured dust is similar to that of
the retrograde dust, which is to say far lower than the E
ring’s density of 74 m3 in the same region [Juhász and
Horányi, 2002]. The background density of uncaptured
dust is approximately one order of magnitude lower than
the peak density of the captured dust.
5. Conclusions
[24] We have shown that as at Jupiter, interplanetary
dust grains are captured at Saturn by interactions with the
magnetosphere. Unlike at Jupiter, the small size of the
dust grains captured (0.8 mm at Jupiter compared with
0.1 mm at Saturn) causes the effects of sputtering to be
important to the evolution of a grain’s orbit. The sputtering does not actually evaporate the particles completely
but rather reduces them to a size where electromagnetic
[26] Acknowledgments. We are grateful for the support from the
Cassini project. We would also like to thank the reviewers, A. Juhasz and
A. Krivov, for their useful comments.
[27] Arthur Richmond thanks Antal Juhasz and Alexander V. Krivov
for their assistance in evaluating this paper.
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J. E. Colwell, M. Horányi, and C. J. Mitchell, Laboratory for
Atmospheric and Space Physics, University of Colorado, Campus Box
0392, Boulder, CO 80309, USA. ( josh.colwell@lasp.colorado.edu;
horanyi@colorado.edu; colin.mitchell@lasp.colorado.edu)
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Figure 3. Grain radius vs impact parameter for particles followed in the two-dimensional simulations
for four values of Fdip (upper left is 2 V, upper right is 3 V, lower left is 9 V, and lower right is
14 V). Particles represented by red dots collide with the planet, dark blue escape by going beyond
200 RS, light blue are captured into prograde orbits, and yellow are captured into retrograde orbits.
Figure 10. Density of retrograde captured dust particles about Saturn. The two maxima are at the radial
turning points for the average dust grain. The densities are in units of 1015 cm3.
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