Tenuous ring formation by the capture of interplanetary dust at Saturn

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, 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. Horányi 1 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 10 15 cm 3, 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,, 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 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 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 [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 1of7

Figure 1. The equilibrium potential for F dip = 11 V. we ignore all forms of drag (e.g., ion, neutral, and Poynting- Robertson) 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 m d2 r dt 2 ¼ F L þ F G þ F RP þ F D ; where F L is the Lorentz force including both the magnetic force and the force caused by the corotational electric field. F G, F RP, and F D 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 F RP ¼ SA c Q PR^r S ; where S is the energy flux from the Sun, A is the crosssectional area of the particle, ^r S is the unit vector pointing along the flux, and Q PR 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 dt ¼ X k where the charging currents I k 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]. 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 I k ; ð1þ ð2þ ð3þ ignore the magnetotail, hence magnetosphere is azimuthally symmetric. An equilibrium potential (F eq ) for the dust grains is prescribed as a piecewise smooth function of the grain s radial position inside the magnetosphere. Beyond 12 planetary radii (R S ), 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 R S to be a variable, F dip. We then connect these three points (the origin, 7 R S, and 12 R S ) with linear potentials. F dip 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 _F ¼ 1 t F eq F ; ð4þ 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 F ¼ q ; 700ea m where e is the elementary charge and a m 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 and r ¼ r _ f 2 W r 2 f _ m W r 2 ð5þ ð6þ f ¼ _r W r r 3 2 f _ ; ð7þ where we use polar coordinates. W = 1.64 10 4 s 1 is the planet s rotation frequency, m =3.8 10 22 cm 3 /s 2 is the planet s mass times the gravitational constant, and W is given by W ¼ qb 0 mc R3 S ; where q is the charge on the grain, B 0 = 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 R S = 6.03 10 9 cm is the planetary radius. ð8þ 2of7

Figure 2. A typical captured orbit for a 97 nm dust grain in the two-dimensional code with F dip = 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 E ¼ 1 2 _r2 þ r 2 f _ 2 m WW ð9þ r and J ¼ r 2 _ 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 F eq 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 F eq decreases with distance, the particle will gain energy. Since magnetospheres are expected to have regions of both increasing and decreasing F eq, the fate of a dust grain depends on which regions it visits. Additionally, while particles may settle into circular orbits in regions where F eq is increasing, circular orbits are unstable where F eq is decreasing. Stable eccentric orbits are possible if the particle spends time in both regions, assuming the grain gains as much energy where F eq is decreasing as it loses where F eq 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 F dip for our simulations, from 3 V to 14 V (for Saturn, F dip 11 V). For each simulation, 4000 particles in the size range from 10 < r g < 200 nm were started on a line 3of7

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. 4 of 7

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 R S ). 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 2450R S ), 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 R S from Saturn, crashed into the planet, collided with the ring, or sputtered down to less than 10 nm in radius. Figure 6 shows the results of a simulation where 10 4 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 F eq, since the energy lost causes a decrease in the semimajor axis. Once the particle enters a region where F eq 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 F eq 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 < r g < 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 10 5 particles, corresponding to a simulated flux of 9.2 10 21 m 2 s 1. 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 10 5 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 R S. 5of7

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 6 of 7

forces become important and the particles are ejected from the system. We found that this ring has a density of 3 10 15 cm 3, one order of magnitude less dense than Jupiter s (4 10 14 cm 3 ). 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 10 15 cm 3. 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 R S, with a thickness of 3 R S. It has a peak density of 3 10 15 cm 3 with a maxima at 3 R S.Itis 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 m 3 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. 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D. (1995), An extended plasma model for Saturn, Geophys. Res. Lett., 95, 12,019 12,031. Schaffer, L., and J. A. Burns (1994), Charged dust in planetary magnetospheres: Hamiltonian dynamics and numerical simulations for highly charged grains, J. Geophys. Res., 99, 17,211 17,223. 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) 7of7