The Effects of Charged Dust on Saturn's Rings - University of Leeds

<strong>Charged</strong> ringscalculate Z g . <strong>The</strong> self-consistent calculation <strong>of</strong>the magnitude <strong>of</strong> Z g gives a smaller value thanobtained with equation 1. A cloud <strong>of</strong> small dustparticles will also be charged up by the plasma,resulting in an electric field throughout the dustcloud which leads to a lifting force (or electrostaticpressure) on the charged dust. <strong>The</strong> magnitude<strong>of</strong> this electrostatic lifting force peakswhen the parameterP = k B Tn g r g / n e0 e 2 (3)(Havnes et al. 1990) is <strong>of</strong> the order <strong>of</strong> 1. Heren e0 is the electron density above or below thering and n g is the dust density. For P 1P ≈ n g Z g / n i Z i (4)In subsequent sections we describe the formation<strong>of</strong> spokes and self-consistent models <strong>of</strong> electrostaticallysupported rings where P is notnegligible. In contrast, when P is small, interesting“test particle” responses <strong>of</strong> grains to magnetosphericelectric and magnetic effects occur.Such responses are important in the E ring.Spokes – evidence for the electrostaticsupport <strong>of</strong> dustSpokes were discovered in Saturn’s B ring withthe Voyager instruments (e.g. Eplee and Smith1984, 1985). <strong>The</strong>y are seen as dark shadows onthe opaque ring. A single spoke can form on theilluminated side <strong>of</strong> the ring and then grow andsurvive as the ring material orbits to the darkside <strong>of</strong> the planet. Figure 4 contains images <strong>of</strong> therings when spokes were present and evolving.<strong>The</strong> shadows are cast by submicron-sized dustlifted above the B ring. <strong>The</strong> small dust grains aremost likely produced by the impact <strong>of</strong> objects,including meteors, on the larger solid particlesin the rings. An impact also produces plasma,and the grains become charged, as describedabove. Electrostatic forces can lift the smallgrains to form the spokes (Goertz and Morfill1983, Morfill et al. 1983). <strong>The</strong> electric fieldarises as a consequence <strong>of</strong> the ions, which arelight and hot, having a larger gravitational scaleheight than the supermicron-sized solid particles.<strong>The</strong>se larger solid particles, just as the submicron-sizedparticles do, carry negative charge.Thus, there is some charge separation.A prime advantage <strong>of</strong> a spokes model inwhich electrostatic support plays a role is thatelectrostatic support occurs for grains only upto a certain size. Scattering properties <strong>of</strong> grainsin the spokes indicate that they are probablysubmicron in size. A purely gravitational mechanismwould not be size-selective and would liftlarger particles as well.A dust particle elevated above a ring respondsto the Lorentz force due to its motion withrespect to the planetary magnetic field. Let r syndenote the radius <strong>of</strong> synchronous orbit, whichis the radius at which the Keplerian orbitalperiod around the planet is the same as the rotationperiod <strong>of</strong> the planet. At radii less than r synthe dust moves faster than the magnetic fieldlines, which rotate around the planet with theperiod at which the planet spins. At greaterradii the dust moves more slowly. <strong>The</strong> Lorentzforce drives the dust initially beyond r syn togreater radii and drives the dust initially nearerto the planet than r syn to smaller radii. Thus, thespokes move radially as well as they orbit theplanet. <strong>The</strong>y have been described as hydromagneticwaves (Tagger et al. 1991).<strong>The</strong> radial motion causes angular momentumtransport, which may have substantial effectson the structure <strong>of</strong> the B ring (Goertz et al.1986, Shan and Goertz 1991). Angular momentumis transferred inwardly at radii greaterthan r syn and outwardly at radii smaller thanr syn giving rise to a tendency <strong>of</strong> the ring tobecome more extended. <strong>The</strong> outer boundary <strong>of</strong>the B ring is probably limited by a gravitationalinteraction with the moon Mimas which orbitswith twice the period at which the outerboundary <strong>of</strong> the B ring orbits. Angular momentumis transferred to that moon in the resonateinteraction between it and the outer B ringboundary, halting the outward expansion <strong>of</strong>the ring material.Various features in the B ring may originatedue to the transport <strong>of</strong> material and angularmomentum away from r syn by the spokes. Forinstance, a minimum in the opacity <strong>of</strong> the ringexists just beyond r syn . An instability in thering’s density distribution is associated with thematerial and angular momentum transport inthe spokes, and the existence <strong>of</strong> observedringlets in the B ring may be due to it (Goertzand Morfill 1988, Shan and Goertz 1991).<strong>The</strong> A ring may well have spokes that wereundetectable with the Voyager instruments. If itdoes or has had significant spoke activity in thepast, the spokes have almost certainly also hadeffects on its structure and features.Electrostatically supported rings<strong>The</strong> typical solid particle size in some <strong>of</strong>Saturn’s rings is about a micron or less. <strong>The</strong>rings with small solid particles also containplasma as Voyager data demonstrated(Richardson and Sittler 1990). <strong>The</strong> number density<strong>of</strong> electrons varies with distance from theplanet and height about the ring midplane butis typically within an order <strong>of</strong> magnitude <strong>of</strong>1–10 cm –3 away from the most opaque rings.Because plasma is present and the solid particlesizes are small, Havnes and Morfill (1984) suggestedthat electrostatic support <strong>of</strong> dust grainsis important in at least some tenuous planetaryrings and that it can be the main mechanismdetermining the vertical thicknesses <strong>of</strong> somerings. <strong>The</strong> G ring, in particular, is likely to havea vertical structure (perpendicular to the ringplane) governed by a balance <strong>of</strong> electrostaticand gravitational forces.Havnes and Morfill (1984) constructed a selfconsistentmodel <strong>of</strong> the vertical structure <strong>of</strong> a4: <strong>The</strong> spoke phenomenon in the major rings(A and B) <strong>of</strong> Saturn.ring in which such a balance obtains. <strong>The</strong> inputparameters are the dust grain number densitiesat midplane, the ion density above the ring, themass per ion, the radius and mass <strong>of</strong> a dustgrain, the plasma temperature, and the distancefrom the planet. Obviously, the equationsinclude the one describing the balance betweenelectric force and gravitation force on thegrains. <strong>The</strong> charge on a dust grain appearsin that equation and is calculated on theassumption that the number <strong>of</strong> ions striking agrain per unit time is equal to the number <strong>of</strong>electrons striking it per unit time. (As describedearlier, in equilibrium a grain will acquire sufficientnegative charge for those collision ratesto be equal.) <strong>The</strong> number densities <strong>of</strong> ionsand electrons are assumed to obey Boltzmannrelations, i.e.n α = n α0 exp(–Z α eV / k B T) (5)where the subscript α indicates which species isbeing considered, the subscript 0 signifies thatthe value <strong>of</strong> the quantity is that obtaining whereV = 0, and V is the electric potential (the gradient<strong>of</strong> which times –1 gives the electric field)within the ring. <strong>The</strong> adoption <strong>of</strong> equation 5implies that the gravitational force on theplasma must be negligible. Though the electricfield balances the gravitational field, it isassumed to be sufficiently modest that the magnitude<strong>of</strong> the net charge density at any point(required to produce the electric field) is smallcompared to the magnitude <strong>of</strong> the charge densityassociated with any <strong>of</strong> the ion, electron, ordust components. For a mixture <strong>of</strong> dust particleswith sizes between 1 µm and 1 cm the scaleheight <strong>of</strong> ring will be several tens <strong>of</strong> kilometresif the plasma number density and temperatureare about 10 cm –3 and 10 eV respectively.<strong>The</strong> two most important predictions <strong>of</strong> amodel like that <strong>of</strong> Havnes and Morfill are thata ring can never collapse to a monolayer as itmay do if only gravitational and collisionaleffects are important and, in addition, if thedust has a size distribution, the relative scaleheights <strong>of</strong> the dust will have a particular welldefineddependence on the dust size (Aslaksenand Havnes 1992). <strong>The</strong> effects <strong>of</strong> dust grain5.28 October 2003 Vol 44

<strong>Charged</strong> ringsSaturnvelocity dispersion have also been included inthe models <strong>of</strong> dust scale height – dust size relationsin rings that are primarily or partially electrostaticallysupported (Aslaksen and Havnes1992). <strong>The</strong> smallest dust particles may be confinedto regions furthest above and below thering plane and may never cross it. Observationswith Cassini should allow the measurement <strong>of</strong>scale heights <strong>of</strong> dust grains in tenuous rings asfunctions <strong>of</strong> dust size.Features arising from magnetic field –vertical oscillation resonancesWithin the context <strong>of</strong> the outer boundary <strong>of</strong> theB ring we touched briefly on the fact that resonancesbetween moons and the orbits <strong>of</strong> materialin rings may be responsible for theappearance <strong>of</strong> structures in the rings. Verticaloscillations <strong>of</strong> dust that is supported electrostaticallymay be in resonance with the rotation<strong>of</strong> Saturn’s magnetic field. Melandsø andHavnes (1991) have predicted the locations atwhich resonances between vertical oscillationsand the rotating planetary magnetosphereshould occur.To calculate the frequencies associated withvertical oscillations, Melandsø and Havnes(1991) began with a model, like that due toHavnes and Morfill (1984), for the verticalstructure <strong>of</strong> an electrostatically supported disk.<strong>The</strong>y then modified the force balance equationto include the effect <strong>of</strong> a plasma environmentthat changes with a frequency equal to that <strong>of</strong>Saturn’s rotation, while the ring material moveswith the Kepler frequency. A change in theplasma temperature or density will lead to achange in the dust charges and the ring electricpotential. This will lead to a change in the verticalelectric force, and the ring will oscillate. Atime-dependent mass-conservation equationrelating dust mass number density to the localoscillation velocity was also included. <strong>The</strong> verticalmotion away from the ring plane slowsdown as the electric field decreases, due to thedecrease in dust density. When the ring thicknessdecreases the increased electric field haltsthe contraction. In the analytical linear analysisOctober 2003 Vol 44DCBAFn = 3 4 5 5 4 3 2 1 = nco-rotationdistance1980 S3{ 1980 S15: <strong>The</strong> distance from Saturn at which the major vertical magnetospheric–electrostatically resonanceswill take place.GE<strong>of</strong> the vertical ring oscillations, the other equationsremain the same as those used to constructthe static equilibrium model <strong>of</strong> the ring’s verticalstructure.Melandsø and Havnes (1991) then assumedthat the fluctuations <strong>of</strong> parameters, such as thedust density, due to the oscillations have muchsmaller magnitudes than the magnitudes <strong>of</strong> theparameters in the equilibrium models. <strong>The</strong>yfound that the fundamental vertical oscillationis at an angular frequency <strong>of</strong>ω f = √3Ω K (6)where Ω K is the local Keplerian orbital angularfrequency.Figure 5 shows some <strong>of</strong> the estimated locationswhere resonances between the magnetosphericrotation and the vertical oscillationsoccur. <strong>The</strong> locations inner to r syn are determinedfrom the requirement thatn(Ω K – ω p )=√3Ω K (7)where n is a positive integer and ω p is the angularfrequency <strong>of</strong> the planetary rotation. At alocation external to r syn–n(Ω K – ω p )=√3Ω K (8)Other locations where resonances may occurmay be found from the requirement that±n(Ω K – ω p )=m√3Ω K (9)where m is a positive integer greater than unity.Numerical tests including time-varying dustcharges showed that the major damping <strong>of</strong> theoscillations was due to a phase differencebetween the ring vertical oscillation and thecharge variation.Melandsø and Havnes (1991) drew attentionto the fact that some <strong>of</strong> the resonance locationsgiven by equations 6 to 9 do show some evidencefor features in the dusty component <strong>of</strong> therings or in the plasma properties. <strong>The</strong> Cassinimission will be valuable for determining moredefinitively whether the rings’ properties areaffected significantly by magnetosphericelectrostaticallymoderated vertical oscillations.Mach cones as diagnostics <strong>of</strong> ring propertiesWhen a supersonic rocket or plane passesthrough air it leaves a wake. <strong>The</strong> outer boundary<strong>of</strong> a wake is called its Mach cone. <strong>The</strong>opening angle <strong>of</strong> a Mach cone is the anglebetween a line passing from the foremost tip <strong>of</strong>the rocket to a distant downstream point in theMach cone and the relative velocity <strong>of</strong> the farupstream air and the rocket. <strong>The</strong> sine <strong>of</strong> theopening angle is given by the ratio <strong>of</strong> the speed<strong>of</strong> sound in the air to the rocket’s speed.Cassini would probably not be able to survivea passage through one <strong>of</strong> the opaque rings andits orbit will be above such rings. For manyregions the ring properties must therefore bestudied with remote observations <strong>of</strong> structuresin the rings and theoretical modelling <strong>of</strong> thestructures. Havnes et al. (1995) suggested thatthe observation <strong>of</strong> wakes in the Saturn ring systemwill allow the inference <strong>of</strong> some properties<strong>of</strong> the rings, the most directly determined <strong>of</strong>which will be the speed <strong>of</strong> dust acoustic waves(DAWs) in the region where the wake exists.<strong>The</strong> sine <strong>of</strong> the opening angle <strong>of</strong> the Mach conewill equal the ratio <strong>of</strong> the speed <strong>of</strong> the DAWsto the speed <strong>of</strong> the object generating the wake.Like a sound wave in a more familiar mediumlike air, a DAW is a compressive wave. In aDAW, dust grains take part in the wave motion.In some places in a DAW, the density <strong>of</strong> grainswill be higher than in others, just as in an ordinarysound wave the density <strong>of</strong> material ishigher in some places than others.In most <strong>of</strong> the rings, the solid particles havea substantial size distribution. Even in the mostopaque rings, submicron-sized solid particlesexist, at least transiently in features likespokes. At the other end <strong>of</strong> the size distribution,objects large enough to be consideredsmall moons exist. A wake may be driven byan object as large as 50 m. <strong>The</strong> solid particlesin the wake will have sizes <strong>of</strong> about a micronand below. Just as spokes show because <strong>of</strong> thescattering <strong>of</strong> light by dust in them, wakesshould be detectable by the scattering <strong>of</strong> lightby dust in them.<strong>The</strong> relative motion <strong>of</strong> the small dust particlesto that <strong>of</strong> a boulder is due to the competitionbetween gravitational forces and the force dueto the planetary magnetic field. A large solidbody or boulder will move in a Keplerian orbit.A small charged dust grain will have an orbitalperiod close to the Keplerian period but it willbe modified by the magnetic field. Whether itwill be moving slower or faster than a Keplerianbody depends on the sign <strong>of</strong> the dust charge andif the dust is outside or inside the synchronousdistance r syn (Mendis et al. 1982). <strong>The</strong> velocitydifference between a boulder and finite sizeddust grain depends on both the radius <strong>of</strong> theorbit and the size <strong>of</strong> the grain. <strong>The</strong> dependenceon the size <strong>of</strong> the grain is easily calculated, andthe size <strong>of</strong> the grains can be inferred from theway that they scatter sunlight.Mach cones created by boulders may be mosteasily detected at positions just beyond and justinside r syn . At other radii, the relative speed <strong>of</strong>5.29

<strong>Charged</strong> ringsthe dust and boulder will be sufficiently largefor the kinetic energy <strong>of</strong> the relative motion toovercome the electrostatic repulsion caused bythe negative charge on a boulder and the negativecharge on the small dust particles andabsorption <strong>of</strong> dust by the boulders may be thedominant effect. Mach cones may be observablenear r syn only at times near those when spokeactivity is detectable because <strong>of</strong> a paucity <strong>of</strong> freesmall particles at other times.Other properties <strong>of</strong> the disturbance and itsMach cone allow the inference <strong>of</strong> additionalinformation about the dust and plasma conditions(Havnes et al. 2001). For instance, thedisturbance will “stand-<strong>of</strong>f” or be separatedfrom the leading point <strong>of</strong> a boulder by a measurabledistance; that distance depends on thedust size distribution. Also, the distance behinda boulder to which a Mach cone is observablewill be determined by the dissipation rate <strong>of</strong>DAWs, which in turn depends on the dustyplasma parameters.Unfortunately we are unable to be precise inthe prediction <strong>of</strong> the observational properties <strong>of</strong>Mach cones in Saturn’s rings. One difficulty isthe absence <strong>of</strong> a fully developed theory <strong>of</strong>DAWs or Mach cones in a dusty plasma with adistribution <strong>of</strong> grain sizes. Our hope is that, forthe reasons outlined here, Mach cones inSaturn’s ring system will prove detectable andthat their detection will stimulate the relevanttheoretical developments as well as lead to abetter understanding <strong>of</strong> the rings themselves.As predicted by Havnes et al. (1996), Machcones have been observed in laboratory dustyplasmas through which larger dust particlespass (Samsonov et al. 1999). <strong>The</strong>y have alsobeen simulated numerically for dust and plasmaparameters, typical <strong>of</strong> some regions in Saturn’srings (Brattli et al. 2002). Figure 6 shows theresults <strong>of</strong> one <strong>of</strong> the simulations and an image<strong>of</strong> a Mach cone observed in an experiment.<strong>The</strong> E ringSo far we have considered situations in whichP is non-negligible. In contrast, in the E ring,motions <strong>of</strong> dust grains with sizes <strong>of</strong> about amicron can be considered to be like that <strong>of</strong> a testparticle (e.g. Horányi 1996).Several separate processes induce the orbit <strong>of</strong>a test particle in the E ring to precess. <strong>The</strong>oblateness <strong>of</strong> the planet is one cause <strong>of</strong> precession.<strong>The</strong> Lorentz force on a charged dust graindue to its relative motion with respect to theplanetary magnetic field tends to drive the precessionin the opposite direction to that causedby the oblateness. For solid particles muchlarger than one micron, oblateness drives theprecession, while for particles much smallerthan a micron the Lorentz force is the dominantcause <strong>of</strong> the precession. Micron-sized particles,for which the effects due to theoblateness and the Lorentz force nearly cancel,6: <strong>The</strong> upper figure shows a computed Machcone for conditions similar to those inplanetary rings (Brattli et al. 2002) while thelower one is observed in dusty plasmaexperiments (Samsonov et al. 1999).precess because <strong>of</strong> the radiation pressure <strong>of</strong>sunlight acting on them. <strong>The</strong> radiation pressurealso causes a periodic variation in the eccentricity<strong>of</strong> the orbit <strong>of</strong> a particle having a size inthe narrow range over which the effects <strong>of</strong>oblation and the Lorentz force cancel. Solidparticles with sizes in this narrow range will fillan extended torus. Other particles will not haveextended spatial distributions.Hamilton and Burns (1994) suggested that thehigh eccentricities that the micron-sized particlesin the E ring periodically obtain in responseto the radiation pressure allow the E ring to sustainitself. Any moon in a circular orbit is bombardedby that fraction <strong>of</strong> the dust that is inhighly eccentric orbits. <strong>The</strong> bombardment producesmore dust, which escapes from the moonand enters the ring. Those escaped dust particleswith sizes around one micron refill thetorus, and within about 20 years they also collidewith a moon.As escaped dust particles collide with moonsto produce more dust particles, one might thinkthat the amount <strong>of</strong> dust in the E ring should continuouslyincrease. This is not the case as toohigh a level <strong>of</strong> dust in the rings would result ingrain–grain collisions preventing the particlesfrom reaching highly eccentric orbits. Thus thereis an upper limit to the amount <strong>of</strong> dust that theE ring gains from the sputtering <strong>of</strong> moons.Though the E ring contains a number <strong>of</strong>moons, the theoretical models point towardsEnceladus as the primary source <strong>of</strong> the E ringparticles. It is a major moon and lies in the mostprominent part <strong>of</strong> the E ring.As mentioned earlier, collisions between solidparticles produce plasma if the impact speed ishigh enough. Neutral gaseous material is alsoproduced. <strong>The</strong> explanation <strong>of</strong> the abundances<strong>of</strong> gaseous species in the E ring remains a challenge(e.g. Jurac et al. 2001). Further observationswith Cassini <strong>of</strong> the plasma and neutralspecies in the E ring will help in the refinement<strong>of</strong> the dynamical model <strong>of</strong> its production.OutlookA period in excess <strong>of</strong> two decades has passedsince the Voyager vehicles flew past Saturn.During it some observations with Earth-basedand near-Earth facilities, such as the HubbleSpace Telescope, have added to our knowledge<strong>of</strong> Saturn’s rings. In particular, data obtainedwhen the rings were edge-on in 1995 haveproved valuable. However, research on Saturn’srings has for a long time now been hamperedby the absence <strong>of</strong> enough new observations.<strong>The</strong> Cassini mission will change that. <strong>The</strong>craft carries a greater variety <strong>of</strong> instrumentationthan the Voyagers did. Cassini has the additionaladvantage that it will make many orbitsaround Saturn. Thus we expect discoveries <strong>of</strong>phenomena not previously seen, including somerelated to the ideas and predictions describedhere, which may lead to a better understanding<strong>of</strong> how the subtle effects <strong>of</strong> dust charging canaffect the ring dynamics and evolution. ●T W Hartquist, Dept <strong>of</strong> Physics and Astronomy,University <strong>of</strong> Leeds, Leeds LS2 9JT, UnitedKingdom; O Havnes, Dept <strong>of</strong> Physics, University <strong>of</strong>Tromsø, N–9037 Tromsø, Norway; G E Morfill,Max-Planck-Institut für extraterrestrische Physik,D–85740 Garching, Germany. <strong>The</strong> authors thankthe referee, Richard Durisen, for constructive comnentswhich improved the paper.ReferencesAslaksen T K and Havnes O 1991 JGR 97 19 175–85.Brattli A et al. 2002 Phys. <strong>of</strong> Plasmas 9 958–63.Cuzzi J N et al. 1984 in Planetary Rings eds R Greenberg and ABrahic, Univ. <strong>of</strong> Arizona Press 73–199.Eplee R E and Smith B A 1985 Icarus 59 188–98.Eplee R E and Smith B A 1985 Icarus 63 304–11.Goertz C K et al. 1986 Nature 320 141–43.Goertz C K and Morfill G 1983 Icarus 53 219–29.Goertz C K and Morfill G 1988 Icarus 74 325–30.Hamilton D P and Burns J A 1994 Science 264 550–53.Havnes O and Morfill G E 1984 Adv. 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