The Dust ballerina skirt is a set of well defined streams composed of nanometric sized dust particles that escape from the Jovian system and may be accelerated up to ≥200 km/s. The source of this dust is Jupiter's moon Io, the most volcanically active body in the Solar system. The escape of dust grains from Jupiter requires first the escape of these grains from Io. This work is basically devoted to explain this escape given that the driving of dust particles to great heights and later injection into the ionosphere of Io may give the particles an equilibrium potential that allow the magnetic field to accelerate them away from Io. The grain sizes obtained through this study match very well to the values required for the particles to escape from the Jovian system.
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 The early detections of dust by the Pioneer 10 and 11 missions in 1973 and 1974 as they passed near Jupiter motivated the search for dust sources in the Jovian system. This, next to the discovery of Io's amazing volcanism in 1979 by the Voyager missions posed the possibility of Io as dust source [Johnson et al., 1980]. Io is now known to be an important source of dust grains for the Jovian System and for the interplanetary space as well [Zook et al., 1996; Horanyi et al., 1997; Graps et al., 2000].
 Most –if not all- of the material deposited onto Io's surface or injected into its surroundings is done through the volcanoes, be it lava flows or plumes. The mean height of the more than a dozen plumes detected on Io is above 100 km [McEwen et al., 1998a], but some are above 300 km and one of them -Pele- registered a record height of 460 km [Spencer et al., 1997].
 Models have already been developed to explain successfully and with detail the dynamics and charging of dust grains in the plasma torus and their later ejection outwards the Jovian system [Horanyi et al., 1997; Krüger et al., 2003b], Nevertheless the actual escaping of dust grains from Io still remains unexplained. The discovery of the ionosphere of Io poses new possibilities for the explanation of this fact which we explore in this work.
 The Ionian Ionosphere was detected by the Galileo spacecraft on 7 December 1995. Galileo passed near Io (0.5 RIo ≈ 900 km) and measured a 10 eV plasma at rest with the satellite with average ion density of about 18000 cm−3. According to Frank et al. , S+, S2+, O+, O2+ and SO2+ ions are moving at 10 km/s, which is a speed faster than the escape speed from Io's surface (2.56 km/s). This plasma layer reaches 400 km above the surface of the satellite matching the height of some plumes. This means that all dust grains driven to such heights will be injected into this plasma where they will be exposed to different charging mechanisms and may obtain certain equilibrium potential. Depending on their size and composition grains will follow ballistic trajectories and fall back to Io's surface or will be dragged away by the electromagnetic forces and escape from Io. Our interest along this work will be to define the dominant charging mechanism, to calculate the equilibrium potential of the grains and the interval in which this charging takes place, as well as the grain sizes and surface potentials involved in the escape.
2. Charging of the Grains
 When grains are injected into the ionosphere they are at or near the top of their plumes, therefore it is reasonable to suppose that they are at rest with the plasma. Two types of particles are considered, sulphur/sulphur dioxide particles and silicate particles based on, at least, three lines of evidence that indicate the composition of the gas and dust ejected from the volcanoes and through the plumes: First, in the plasma torus it is possible to find sulfur, oxygen and sodium ions [Nash et al., 1986] and near Io, sulfur oxide and dioxide atoms and ions [Frank et al., 1996; Kivelson et al., 1996]. Second, Spectroscopy says that the thin atmosphere of Io is dominated by SO2 and its subproducts SO, O and S, that plumes like Pele contain S2 [Spencer et al., 2000] and also that Io is covered by a thick layer of sulfur and sulfur dioxide; Third, Io's crust and upper mantle should be composed of silicates, possibly, enriched with magnesium (olivine) to explain such a high temperature volcanism and superheated lava flows [McEwen et al., 1998b] and part of these silicates must be swept by the sulphur/sulphur dioxide ejections in its way out as fine clast or dust particles.
 The ejected gases through Io's volcanoes expand very fast and the temperature and pressure inside the plume drop also fast. According to Kieffer , sulphur may crystallize at ∼380 K at a pressure of ∼10−5 bar which may be the conditions of some sections of the new plumes right after the gases have decompressed. On the other hand Belton  states that when the temperature of the plume lowers to ∼110 K sulphur dioxide gases crystallize. Zhang et al.  state that, for the near 400 km plume of Pele, temperatues are below 400 K above a ∼50 km radius from the vent, and below ∼155 K above a ∼50 km radius from the vent. This means –for Pele- that dust condenses less than a couple of minutes after ejection and decouples from the gas at low heights near the vent at near ejection speed. For any of these volcanoes, the gases are decompressed violently and thus, for less energetic ejections the condensation temperatures are reached at lower heights. Silicates on the other hand may be ejected already in the solid phase since they may crystallize above 1000 K.
 Also according to observation most of the grains that condense in Io's plumes are micrometric or sub micrometric, but those of sub micrometric size dominate [Collins, 1981]. On the whole, the size of these particles produced must be small (micrometric or submicrometric) due to the low density of the plumes (1012 to 1017 molecules/m3 [Zhang et al., 2003]).
 Basically four mechanisms would be present in the ionosphere: Electron capture, Ion capture, Secondary electron emission and Photoelectron emission. But for Io's ionosphere only one dominates due to its own properties, the physical properties of the dust grains and the ambient conditions in which these grains are injected. Since most ions are singly ionized, we can take the density of both species as equal. But even if the electron and ion densities are similar, the large speed of the electrons, ve = = 2 × 106 m/s, enables them to interact more frequently with the grains than ions. If grains are at rest with the plasma, the encounters with ions are reduced, making the electron capture a far more efficient charging mechanism than the ion capture.
 It is relatively direct to see that electron secondary emission and photoionization are not significant effects as well. In the case of the secondary electron emission, we use the Sternglass  approximation of the ratio of emitted to incident electrons –the yield-: δ(E) = 7.4δM(E/EM) exp [−2(E/EM)1/2]. Experimentally it is known that the yield δ(E) has a maximum value δM (from 1 to 10) for an optimum incident energy EM (∼100 to ∼1000 eV). Draine and Salpenter  obtained δM = 2.9 and EM = 420 eV for SOx (silicate) particles and Horanyi et al.  obtained δM = 3 and EM = 300 eV for SOx (sulphide) particles. We see that, for SOx, the yield has its maximum around 3 for incident energies E between 270 and 350 eV for SiOx particles, the maximum is around 2 for incident energies between 263 and 338 eV. In our case, the yield for the 10 eV electrons, δ ≈ 0.5, 0.16 sulphur and silicate particles respectively. We say then that this plasma is too cold to produce an important amount of secondary electrons.
 As for the photoionization, from Horanyi's  expression, Iν = 2.5 × 1010πr2e(χ/R2) if ϕ < 0 surface potential, says that the photo-effect becomes less efficient as particles become smaller. For example, for a r = 0.1 μm (particle a typical large grain from Io's plumes) at Io's heliocentric distance R = 5.2 AU, the current of photoelectrons would be Iν ≈ 10−5 and 10−6 e/s for silicate and sulphur/sulphur dioxide particles respectively (χ, the efficiency factor, is close to 1 for conductors just like silicates and close to 0.1 for dielectrics just like sulphides). At this low rate, for example, a SiOx particle would reach a ϕ = 1 Volt potential in: τ = (4πɛ0rgϕe)/Iν ≈ 81 days. For a S/SOx particle the charging time would be around 2 years. For both cases the time is quite long. We conclude that the photoionization and the secondary electron emission are non-efficient mechanism for sub-micrometric particles in the ionosphere of Io. Therefore it is possible to discard them as important mechanisms.
 For the electron capture the situation is quite different: Let us consider, for S+ and O+ plasmas, which are the main components of the ionospheric plasma of Io, β = 3.6 and 3.9 respectively [Horanyi, 1996]. If other mechanisms are neglected, only thermal currents are considered and then the equilibrium potential may be obtained through the Spitzer  equation with Te = Ti = T. Then for a sub micrometric grain injected into 10 eV ionospheric plasma, the equilibrium potential will be around:
 If the dust particle has r = 0.1 μm, it would need to capture ne ≈ 700rμϕ = 2380 electrons to reach the ϕ–34 V equilibrium surface potential. For this effect, the time in which the particle reaches its equilibrium potential τequil or captures those 2380 electrons may be inferred from τ = ∣Q/e · I∣, that is:
 Which is indeed a very short time that still confirms that electron capture dominates among all charging mechanisms. Let us remark that this −34 V potential corresponds to a surface electric field −34 V/0.1 × 10−6 m = −3.4 × 109 V/m which is of the order of the negative threshold given for the electron field emission by Graps and Grün , Ef = −109 V/m, that limits the charging of dust particles.
 Let us also keep in mind that if particles have a radius r = 1 nm, the limit potential for electron field emission is −109 Vm−1 × 10−9 m = −1 Volt. If they are 10 nm in size, then the potential is 10 V and for 0.1 μm, 100 V. Which means that the −34 V potential would not be applicable to very small particles. Note also that the charging time and the size of the particle keep an inverse proportion, such that τ ∼ r−1, which leads to longer times as the particles become smaller.
 Even taking into account those two considerations, the electron capture is nevertheless an efficient mechanism for these very small particles and the charging times are still small. In fact, using again equation (2), for a 10 nm particle and an equilibrium potential of 10 V, the charging time would be 6.1 s. For the smallest particle, 1 nm and a 1 V equilibrium potential, the charging time is of the order of 61 s. It is necessary to stress that since this 10V potential might be only an upper limit for charging purposes. Let us look at the Debye length λD = 740(kT/n) cm, where n is the particle density in cm−3 and T is the temperature in kelvins. For our kT = 10 eV and n = 18000 cm−3 plasma, λD = 17.4 cm. This means that the Debye spheres overlap, since the distance between the component particles of the ionospheric plasma is smaller than ∼0.04 cm, supposing sulphur dust particles and a gas ejection rate of ∼1011g/s [Ip, 1996; Spencer et al., 1997; Cataldo et al., 2002]. At the end, and due to this overlapping, there is the possibility that each dust particle might collect very few electrons indeed meaning that in such case only a fraction of the dust particles that reach the ionosphere get charged. The minimum charge might be just one electron which would mean around ∼−0.1 Volt surface potential for a 10 nm particle. Even this potential, as we will see in the next section, would be enough for dragging purposes.
3. Escape From Io
 The grains that could be dragged by the Jovian plasma must be those for which the electric force FE due to the induced co-rotational electric field of the Jovian magnetosphere is greater than the gravitational pull of Io FG. The limit, for calculation purposes may be written through the equilibrium equation:
 This last expression is the gravitational pull that the grains experience at the top of the plumes of height H. On the other hand:
is the electric force on the charge due to the co-rotational electric field = c−1(p × Ω) × = 0.113 Vm−1; where p is the position vector of the particle in jovicentric coordinates, Ω is the angular velocity of Jupiter (=1.74 × 10−4 rad/s), c is the speed of light (=3 × 108 m/s) and is the intensity of the magnetic field. For a dipolar configuration B = B0(RJ/rJ)3, with B0 = 4.2 × 10−4T and rJ = 5.20AU, the distance from Io to Jupiter in astronomical units.
 Since the charge of the grain may be expressed through the surface potential just like Q = 4πɛ0rϕ and the mass of the grain as mg = (4/3)πr3ρ, then, with a proper substitution in equation (3) and factorizing all constant terms, the grain maximum radius that could be dragged by the plasma among those injected from the plumes is:
with RIo(=1815 ± 5 km) and H in km, ϕ in volts and ρ in g/cm3.
 Let us suppose grains driven to the top of Io's tallest plume Pele of H = 460 km as maximum height. From equation (6), the maximum escape radii for this plume are: For SO2, 0.24 μm; for S, 0.21 μm and for Mg2SiO4, 0.17 μm. For the same plume, but taking now the lower limit of the plume (H = 300 km), the values would be 0.22 μm, 0.20 μm and 0.15 μm respectively.
 This result is generalized for all plumes in Figure 1 where some plumes are set as reference. The largest grains that can escape from Io have maximum radii in the interval ∼0.14–0.24 μm. The average values for all plumes are presented in Table 1 as well.
Table 1. Average Value of the Maximum Escape Radii From Io's Plumes
 If we consider that dust particles may collect a minimum charge, that is, only one electron, dust particle radii may be given modifying equation (6) as follows:
again RIo and H in kilometers and ρ in g cm−3.
 In this case, for Pele plume we would have: For SO2, 1.32 nm; for S, 1.23 nm and for Mg2SiO4, 1.05 nm.
 Of course, the dust particle radii, in comparison to our former calculation seem to be considerably reduced (Table 2). These results are generalized plotting equation (7) (Figure 2).
Table 2. Average Value of the Maximum Escape Radii From Io's Plumes Considering a Minimum Surface Charge Equivalent to Only One Electron or ∼−0.1 Volt Surface Potential
 Apparently only those very small particles from the volcanic ejections of Io may escape. Nevertheless, based on the whole analysis, the dust particles that escape might be smaller than 0.1 μm, but larger than 1 nm; that is, the actual typical radius could be simply 10 nm which is an average in orders of magnitude.
 Negatively charged dust particles may revert their charge to a positive electric potential and thus, be pushed away from Jupiter by the jovian co-rotational electric field [Horanyi et al., 1997]. Krüger et al.  calculate the minimum radius of grains (supposed spherical with ρ = 1.5 g/cm3) that escape from the Jovian system with the gyration of a charged grain along Jupiter's magnetic field through rmin ≈ 0.001 μm, and the radius of largest grain that can escape from Jupiter through the ratio L of the Lorentz force generated by the Jovian magnetic field and the gravitational pull of Jupiter experienced by the grain or in a simplified expression as rmax ≈ . For both cases they obtained 8 and 200 nm respectively.
4. Conclusions and Discussion
 Our interest along this work was to define the dominant charging mechanism for the dust particles, to calculate the equilibrium potential of the grains and the interval in which this charging takes place, as well as, the dependence of the grain size and surface potential to the escape from Io.
 First, it was found that the electron capture is by far a more efficient charging mechanism than any other in the Ionian ionosphere, mainly due to the particle density and temperature of the plasma, that allow particles reach a negative equilibrium potential in, at the most, a couple of seconds. There is also an important dependence to the grain sizes and thus to the composition of the grains.
 The average maximum escape radius of those grains that collect charges and are able to be dragged by the surrounding plasma, is ∼187 nm if we take the largest possible equilibrium potential. If the smallest potential is taken, the average radius is around 1.13 nm. If we consider the average of both last values in orders of magnitude, a typical radii of 10 nm is drawn. Interestingly, these dust sizes match very well the values required for the particles to escape from the Jovian system. In fact the largest particles supposing maximum potential that are swept from the ionosphere of Io are ∼10% larger, in radius, than those larger particles that are pushed away from the plasma torus.
 I am quite grateful to Prof. Dr. Eberhard Grün whose advise improved greatly this work. This work was supported by DGAPA IN104003-3 and CONACyT 40601-F grants.