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1 Originally published as: Yaroshenko, V., Lühr, H. (2014): Random dust charge fluctuations in the near-enceladus plasma. - Journal of Geophysical Research, 119, 8, p DOI:

2 JournalofGeophysicalResearch: SpacePhysics RESEARCH ARTICLE Key Points: Stochastic dust charge fluctuations are studied in the light of Cassini data Modeling predicts coexistence of negative/positive fluctuating dust charges Results might explain the available CAPS measurements Correspondence to: V. V. Yaroshenko, Citation: Yaroshenko, V. V., and H. Lühr (2014), Random dust charge fluctuations in the near-enceladus plasma, J. Geophys. Res. Space Physics, 119, , doi:. Received 24 MAR 2014 Accepted 17 JUL 2014 Accepted article online 23 JUL 2014 Published online 4 AUG 2014 Random dust charge fluctuations in the near-enceladus plasma V. V. Yaroshenko 1 and H. Lühr 1 1 Deutsches GeoForschungsZentrum, Potsdam, Germany Abstract Stochastic dust charge fluctuations have been studied in the light of Cassini data on the near-enceladus plasma environment. Estimates of fluctuation time scales showed that this process can be of importance for the grains emanating from the icy moon. The analytical modeling predicts that in the dust-loaded Enceladus plasma a majority of the grains acquires fluctuating negative charges, but there might appear a minority of positively charged particles. The probability of this effect mostly depends on the ratio of the dust/plasma number densities. Our findings appear to be supported by the available Cassini Plasma Spectrometer measurements of the charged grain distributions during E3 and E5 plume flybys. The theoretical results can also provide new insights into the intricate process of particle dynamics in the inner magnetosphere. 1. Introduction The Cassini flyby in July 2005 revealed the geologically active nature of the icy moon Enceladus. The orbiter approach provided a good view of the south polar region, and the Composite Infrared Spectrometer (CIRS) instrument detected anomalous thermal emission from the south pole [Spencer et al., 2006] which correlated with four prominent warm fractures on the moon surface tiger stripes seen in detail by the Imaging Science Subsystem (ISS) cameras [Porcoetal., 2006]. Near closest approach, Ultraviolet Imaging Spectrograph (UVIS) observed a stellar occultation and detected water vapor over the south pole [Hansen et al., 2006], but not at equatorial latitudes. Modeling of the occultation data gave a 200 kg/s escape rate of water molecules from Enceladus. Simultaneously, the Cosmic Dust Analyzer (CDA) detected a large concentration of micron-sized dust grains above the south pole [Spahn et al., 2006;Kempf et al., 2008]. Combining these data, it became clear that Enceladus south pole was actively erupting plumes of ice grains, water vapor, and other gases, probably from the tiger stripe fractures, to supply both the E ring and the extensive neutral torus around the moon s orbit. The dust plumes were seen directly at high resolution in high-phase Imaging Science Subsystem (ISS) images taken on 27 November 2005, which revealed multiple individual jets [Porco et al., 2006]. Furthermore, it became clear that plumes of neutral water vapor react with Saturn s corotating plasma, loading the magnetosphere with fresh and cold ions and decelerating the corotating plasma [Tokaretal., 2006]. The close Cassini flyby in March 2008 (E3) provided an opportunity to penetrate the plume much more deeply than in July 2005, resulting in high-quality Ion and Neutral Mass Spectrometer measurements of the gas composition, and CIRS obtained a high-resolution view of the thermal emission along the tiger stripes. During E3 flyby, the Cassini plasma spectrometer (CAPS) saw signatures due to the impact of tiny dust particles with the instrument [Jones et al., 2009], with flux spike locations consistent with the jet locations determined by Spitale and Porco [2007]. The grains were interpreted as ice particles of a few nanometers in diameter. It was observed that nanograins form spatially separated particle streams and thus can have both negative and positive charges [Jones et al., 2009]. Recent interpretations of the CAPS measurements, during the flybys E3 and E5, provided additional quantitative data on the dust particles in the plume: Hill et al. [2012] reported number densities up to 10 3 cm 3 for negatively and up to 1cm 3 for positively charged nanograins at the time of the closest approach to the south polar axis of the Enceladus plume. One can conclude therefore that the neutral gas, plasma, and tiny dust grains are likely the main constituents of the southerly extensive moon s plume. One of the basic problems associated with dust in space is the grain charging. There is the continuing modeling of the charging process for many different plasma compositions and radiative conditions, so only some immediately relevant references will be given below. In the Saturn s magnetosphere, including the Enceladus proximity, this process has been extensively studied both theoretically and numerically [Horányi, YAROSHENKO AND LÜHR American Geophysical Union. All Rights Reserved. 6190

3 1996; Horanyi et al., 2004; Graps et al., 2008; Hsu et al., 2013] as well as experimentally applying the CDA data [Kempf et al., 2006]. Incorporating the main constraints of the parameter space from the CAPS measurements in the near-enceladus plasma [Sittler et al., 2006, 2008, Tokaretal., 2008, 2009], Yaroshenko et al. [2014] studied equilibrium dust charges in the dense dust limit relevant for the plume. This treatment showed that a predominant mechanism of grain charging is a collection of plasma electrons and ions. In the dense plume the corotating flow is significantly slowed down, and thermal plasma is mainly responsible for the plasma fluxes to the grain surface and its equilibrium negative charge. Moreover, in the dust-loaded regions, values of the grain charge can be significantly decreased compared to the equilibrium charge of an isolated grain. Dust charging in plasmas has, however, other ramifications. The grain charges are not fixed but can fluctuate. This aspect introduces a crucial difference between charged grains and electron/ion species. There are two reasons for fluctuating dust charges. First, the charging of the grains depends on the local plasma characteristics and thus their temporal or spatial perturbations will ultimately affect grain charges. As an example, we mention fluctuations of the dust charges in Saturn s rings arising due to the grain transition through the solar terminator discussed by Bliokh et al. [1995]. The second reason for charge fluctuations is the discrete nature of charge carriers. Electrons and ions absorbed or emitted by the grain surface randomly provide stochastically fluctuating dust charges (contrary to coherent fluctuations caused by the first reason). Such random dust charge fluctuations exist always even in a steady state uniform plasma, but their studies have mainly been limited to the approximation of an isolated grain. Here we mention studies important in a space plasma context. Morfill et al. [1980], assumed that the number of charges, residing on a dust grain varies randomly according to Poisson statistics, providing the root-mean-square (RMS) of the charge fluctuations σ Z = Z, where Z is the equilibrium dust grain charge in units of electron charges. Later, a numerical simulation of random charge fluctuations by Cui and Goree [1994] yield a time series of the charge acquired by a single grain in plasmas. It was found that the RMS of the charge fluctuations is given by σ Z = 0.5 Z for a wide range of plasma and grain parameters, if the mean charge Z 1. Several papers by Matsoukas and Russell [1995, 1996] then discussed an analytical model of stochastic charge fluctuations carried by a dust particle in a stationary undisturbed low-temperature plasma. Later, [Khrapak et al., 1999] included some emission processes and discussed the amplitude of random charge fluctuations on a single grain. In the light of the CAPS observational data on the nanograins in the plume (high dust density, simultaneous existence of negative and positive grain charges, domination of grain charges Z 1 [Jones et al., 2009;Hill et al., 2012]), it is now straightforward to assume that the dust fluctuation problem might be an important issue in describing the physics of this region. For example, the dynamics of dust particles with fluctuating charges is more complicated than that governed by fixed equilibrium charges. This will inevitably affect dust structures in the direct vicinity of the moon and determine their time variability. Another possible consequence of dust charge fluctuations might be the modification of the grain-grain interactions, which increase the likelihood of dust-dust collisions thus triggering the agglomeration processes important for the formation of the E ring particles. Finally, the stochastic dust charging might be an important issue for the interpretation of the CAPS data, which initially contain the charge-to-mass ratios and require some assumptions on the charging state to determine the particle size. In this paper, we model stochastic charge fluctuations and determine their static and dynamic characteristics in the limit of dense dust clouds relevant for the near Enceladus plasma and inside the moon s plume. Considering the dust charging due to collection of electrons and ions from the ambient plasma, a temporal autocorrelation function of the dust charge fluctuations is derived, thus providing both the magnitude and characteristic time scales of dust charge fluctuations. We analyze the resulting charge distributions, discuss the possible observational signatures of this effect, and compare them with the available Cassini data. 2. Analytical Modeling The surface of a dust grain immersed in a plasma will continually be bombarded by incident plasma electrons and ions. Some of the plasma particles can be captured by or will stick onto the dust surface, charging the grain at random intervals. The instantaneous dust charge Z(t) = q d (t) e is governed by a conventional equation dz dt = J = J e J i, (1) YAROSHENKO AND LÜHR American Geophysical Union. All Rights Reserved. 6191

4 where J e and J i are the electron and ion currents to the grain surface. For Maxwellian plasma species, the plasma fluxes to the negatively charged particles are given by the orbital-motion-limited approximation [Horányi, 1996] J e = ( ) eφ 8πa 2 n e v Te exp, (2) J i = ( 8πa 2 n i v Ti 1 eφ ). (3) T i Here the ions are assumed to be singly charged and a and φ refer to the grain radius and dust floating potential, n e(i) and T e(i) stand for the electron (ion) number density and temperature, and v Te(i) = T e(i) m e(i) denotes the thermal electron (ion) velocity. A steady state dust charge Z is reached when the net current to the grain surface is zero, i.e., J = J e J i = 0. In the case of an isolated grain (n e n i ), the current balance leads to a standard equation for the potential φ independent of the plasma density and the grain size. In dense dust configurations a negative dust charge density might provide a significant imbalance in the electron-ion plasma populations, which has to be taken into account in the initial plasma currents given by equations (2) and (3). Considering the dust particles as heavy negative ions carrying the charge Z e, we assume that they contribute to the plasma quasi-neutrality condition similar to the electrons and ions. Then an imbalance in the electron/ion density reads T e n e n i = 1 Z n d n i, (4) where n d denotes the dust number density of the negatively charged grains. Combining the current balance equation, J = 0, and charge neutrality condition (4) allows in the case of T e T i 1 to give a closed form representation for the average dust charge number Z without having to rely on numerical methods, viz., ( ) ni ni B Z n i n d LambertW n d D e n d D D (5) with notations D = at e e 2 and B = v Ti v Te. The quantity LambertW(x) denotes the Lambert W function, which has recently been applied to the problem of dust charging [Dubinov and Dubinova, 2005; Cousens et al., 2012]. One finds immediately two limiting solutions for the particle charges: (a) when n d n i, Z D lnb (approximation of an individual grain) and (b) increasing the dust density to the critical value n d n i and assuming D > 1 yields a solution Z (1 B)n i n d that is explicitly independent of the grain size and only weakly dependent of the plasma thermal temperatures. Assuming that the grain charge is a random function of time, we put Z(t) Z +ΔZ with ΔZ denoting small fluctuations from the equilibrium charge (ΔZ Z ). The charging equation (1) can then be presented as ΔZ t +ΔZ τ q = f(t). (6) As we show below, the quantity τ q, a relaxation time for small charge deviations from the equilibrium value Z, determines a fundamental time scale of dust charge fluctuations. The stochastic forcing term f(t) accounts for the fluctuating character of the dust charging due to random sequences of the electron and ion emission/absorption. The time intervals between the successive acts of absorption or emission are also random variables. It is reasonable to assume that the electron and ion emission/absorption have equal probabilities, so that on average, f(t) = 0. Moreover, the random process f(t) might be assumed to be a stationary one, whose autocorrelation function depends only on a time difference, namely, < f(t)f(t ) >= B(t t ). Equation (6) represents the Langevin equation describing the motion of Brownian particle in the one-dimensional case, where ΔZ plays the role of a particle velocity and f(t) of a random force arising from the atom or molecular random bombardment (Brownian force). Hence, the charging process is decomposed into a rapidly varying stochastic term f(t) accounting for the fluctuating part of the dust charge and a slowly varying deterministic term given by ΔZ τ q. The discussion of possible solutions is now reminiscent of what happens with a free Brownian particle when the values of τ q and f(t) are given. YAROSHENKO AND LÜHR American Geophysical Union. All Rights Reserved. 6192

5 A time constant τ q is an inverse of the characteristic steady state charging frequency [Khrapak et al., 1999], so that τ q = ( J Z) 1 Z= Z. (7) Combining the plasma fluxes given by equations (2) and (3) with the plasma quasi-neutrality equation (4) yields explicitly τ q 1 8πa2 n i V Ti [ ], (8) n d + D 1 n i Z n d where once again we have used the condition T e T i 1 relevant for the magnetospheric plasma at the Enceladus orbital distances [Sittler et al., 2006]. Note that contrary to the case of an isolated grain, the time scale τ q depends on the dust charge density ( Z n d ), associated with a certain grain size a. The latter immediately modifies the size dependence τ q a 1 relevant for a single grain approximation. This effect might have some consequences in the case of a dust size distribution, but this paper is mainly focused on monodisperse dust. The meaning of the time scale τ q becomes clear upon solving the differential equation (6) and calculating ΔZ(t). Using the variation of constants methods, one obtains ( ΔZ(t) =ΔZ(0) exp t ) t ( ) t t + exp f ( t ) dt, (9) τ q 0 τ q where ΔZ(0) = ΔZ(t = 0) is an initial deviation of a grain charge from equilibrium. Calculating then the average ΔZ(t) leads to ΔZ(t) =ΔZ(0) exp ( t τ q ). (10) One immediately finds that initial fluctuations ΔZ(0) approach ΔZ(t) 0 at t, and the time scale for this process is the relaxation time τ q. This quantity can be thus considered as a long time. Another temporal scale, τ f, is defined as an inverse frequency of the electron/ion absorption or emission, = J i + J e [Khrapak et al., 1999], yielding τ 1 f D τ f 2 [ 8πa 2 n i V Ti D + Z Te ( )]. (11) T i D The time scale of electron/ion absorptions represents a rather short time providing τ f τ q.itis now straightforward to introduce the autocorrelation function of the random forcing term in the form f(t)f(t ) = τ 1 δ(t t ) and using equation (9), obtain the autocorrelation function of the stochastic dust f charge fluctuations ΔZ(t)ΔZ(t ) = τ q (2τ f ) exp ( ( t t ) τ q ). (12) Here for simplicity we put ΔZ(0) =0, assuming that initially a grain has an equilibrium charge. Equation (12) solves the dynamic part of the charge fluctuation problem showing that the time scale of dust charge fluctuations is the relaxation time τ q, given by equation (8). The variance of the random fluctuations, however, is given by a ratio of the two characteristic times σ 2 Z = ΔZ(t) 2 = τ q 2τ f. (13) To determine the static properties of the charge fluctuations, i.e., the density distribution function of the fluctuations, we focus on the expression (9) for ΔZ = Z Z. As seen, the fluctuation ΔZ at each moment t is a sum (integral) of a large number of independent acts of absorption/emission of plasma particles (electrons and ions) over the period t. The number of these events at the time scale of τ q is also large enough, and the random force f(t) is a stationary process. According to the central limit theorem, the function ΔZ(t) will tend then to be normally distributed at the times t τ q. This results in the Gaussian probability distribution W(Z) = 1 2πσZ exp ( ) (Z Z )2 characterizing the random dust charge fluctuations in thermal plasmas. 2σ 2 Z (14) YAROSHENKO AND LÜHR American Geophysical Union. All Rights Reserved. 6193

6 3. Implications for the Enceladus Plasma and Plume We apply now the theoretical results to the plasma system consisting of thermal electrons (T e 1 3 ev) and water group ions (m i = 18 amu, T i ev) which mainly dominate in the plume region as well as negatively charged dust particles (nanograins). Based on CAPS measurements, the plasma density in the Enceladus plasma torus is in the range of n e n i cm 3 [Sittler et al., 2006; Tokaretal., 2009]. For the plume plasma the Langmuir Probe (LP) measurements of the Radio and Plasma Waves instrument revealed local increases in plasma density up to cm 3 and a significant electron depletion with n e n i [Morooka et al., 2011], while the electron temperature T e remained to be of 1 ev. In our estimates we therefore consider the kinetic temperatures T e 1 3 ev, T i 30 ev, and the densities n i 10 3 cm 3 for the core plasmas. The important parameters in description of grain charging are the dust characteristics, namely, the number densities, grain size, or dust size distribution [Yaroshenko et al., 2014], which often are difficult to quantify observationally. The Cosmic Dust Analyzer aboard the Cassini spacecraft consists of two independent subsystems, the high-rate detector (HRD), and the dust analyzer (DA) [Srama et al., 2004]. The HRD reliably registers particles a > 0.9μm[Kempf et al., 2008], which seem too large to be the main constituents of the dust plume. The much more sensitive DA cannot also resolve the plume dust because of dead time effects. On the other hand, the CAPS instrument can measure the grain charge-to-mass ratio. Assuming a charging state, the CAPS data can be transformed into the dust sizes [Jones et al., 2009]. During E3 and E5 flybys, CAPS detected a dense population of charged particles having mass-to-charge ratios up to the maximum detectable value m d q d 10 4 amu/e. These particles were interpreted as nanometer-sized (a 1 nm) water-ice grains carrying very small charges q d 1e. With the assumption of singly charged nanograins, Hill et al. [2012] deduced their maximum number density n d 10 3 cm 3. Furthermore, the analysis based on the interpretation of the Cassini LP measurements of the electron depletion and assumption of a strong domination of grains (a 30 nm) yielded a plume dust density n d 10 2 cm 3 [Shafiq et al., 2011]. A numerical model for the grain condensation and growth inside the moon fractures predicted a bimodal dust size distribution with a first peak located between a 10 nm and a 50 nm [Schmidt et al., 2008]. This peak is due to a nucleation burst within the water vapor ascending through the fractures in the Enceladus ice crust, happening at the narrowest part, where the gas is supersonic. Since there is a rather large uncertainty in the present data about the characteristic dust size in the plume, we consider the charge fluctuation problem for two grain sizes a 3 nm and a 30 nm as possible typical representatives of the plume dust particles. The first size is consistent with the nanograins inferred from the CAPS observations by Jones et al. [2009]. The larger particulates (a 30 nm) are associated with the plume modeling by Schmidt et al. [2008]. The size a 30 nm might be also considered using an analogy between the formation of the grains emanating from the icy moon Enceladus and of those formed in noctilucent clouds (NLC) in the polar summer mesopause region. Analysis of the different NLC types strongly indicates a very narrow size distribution of the mesospheric icy particles, with a preferential size typically of a 30 nm [see Baumgarten et al., 2010, and references therein]. Since in both environments the grain growth seems to occur under physically similar conditions, we also include a limiting case of a 30 nm. Finally, incorporating the effect of dust density (n d ) in the description of the dust charge fluctuations, we will keep the density ratio n d n i as a free parameter in the parameter space indicated in Figure 1. Here we show the functional dependence of n d, (cm 3 ) versus the grain size a, (nm) constrained by the condition that the dust production rate does not exceed 5 kg/s [Schmidt et al., 2008], while the neutral source rate is set to 200 kg/s (blue curve) as inferred from UVIS data by Hansen et al. [2008]. To include possible variability in the source rates we add a higher limit of the neutral source rate 1600 kg/s deduced from the Cassini magnetic field data by Saur et al. [2008]. In calculations, the density of ice and mass of neutrals are taken to be 934 kg/m 3 and 18 amu, respectively. The region lying below the appropriate curve in Figure 1 expresses admissible n d (cm 3 ) for a given a (nm). It is seen that as the dust size increases from a = 3 nm to a = 30 nm, the admissible n d decreases significantly the range of possible n d n i. Thus, possible variations of the density ratio for nanograins a 3 nm cover a range n d n i (0.1,1), while for larger particles a 30 nm the interval converges to n d n i (0.01, 0.1). In Figure 2 we show variations of the time scales involved in the fluctuation problem (τ q, equation (8) and τ f, equation (11)) with the relative dust number density n d n i for the grains a = 3 nm and 30 nm at YAROSHENKO AND LÜHR American Geophysical Union. All Rights Reserved. 6194

7 various T 10 4 e. All quantities are normalized as τ q t 0 and τ f t 0, where t 0 = T e 8πae 2 n i V Ti defines the relaxation time in the case of an 10 3 isolated grain (n d n i 0). It can be seen that the short time τ f t 0 (dashed curves) depends 10 2 very weakly on the dust density, remaining practically constant τ f t 0 1 (2D), while 10 the long time τ q t 0 (solid curves) strongly decreases with growth of the dust population. As a result, already at n d n i 0.1, the relaxation time for the dust charge fluctuations τ q grain size, nm approaches the order of the time scales for the Figure 1. The admissible parameter space in terms of grain size electron/ion absorption τ f and the used representation of the δ-correlated fluctuations (nm) and dust number density (cm 3 ) lying below the curves is constrained by the condition that the dust production rate ( f(t)f (t ) = τ 1 δ (t t )) might not be a f does not exceed 5 kg/s [Schmidt et al., 2008], and the neutral good approximation anymore. This means that sourcerateissetat 200 kg/s (blue curve) [Hansen et al., 2006] or 1600 kg/s (red curve) [Saur et al., 2008]. at n d n i 0.1, we can consider the results only qualitatively. As seen from Figure 2, the growth of the electron temperature T e or the particle size a can also significantly decrease the fluctuation time τ q, but, in this case, the time scale τ f also decreases accordingly. Figure 3 illustrates variations of the fluctuation level σ Z Z = τ q ( ) 2 Z τ f versus the density ratio n d n i. As expected at the very low dust density, n d n i 0, the amplitude of fluctuations σ Z Z remain close to 0.5, confirming the results found by Cui and Goree [1994] in the approximation of an individual grain. A growth of n d n i, however, can significantly reduce the level of dust charge fluctuations. This eventually affects the resulting dust charge distribution given by equation (14). In Figure 4, we illustrate modifications of W(Z) for various n d n i ratios relevant for the Enceladus plume. A contour plot of the enhanced dust density is quantitatively consistent with the CAPS measurements of the grain population during E3 and E5 plume flybys [Hill et al., 2012]. In the insertion, we plot the charge distribution W(Z) for various relative dust densities n d n i of nanograins with a = 3 nm, at T e = 3 ev and T i = 30 ev. Assuming that the ion density within the plume remains practically constant n i cm 3 [Tokar et al., 2009], the presented color-coded dust charge distributions in the insertion of Figure 4 are relevant within the three (internal) contour ranges of the plume dust population. It is seen that increasing the dust density shifts the function W(Z) to less negative charges and simultaneously decreasing the spread of the charge distribution. At the same time the probability of acquiring a zero charge (Z 0) increases practically exponentially with the relative dust (nd ) density W(0) n i B 1 e n i 2Bn d. Therefore, one can speculate that at higher n d, i.e., in denser parts of the plume, conditions might appear favoring zero or even positive (in our notations Z < 0) dust charges. The latter could provide dynamically small amounts of neutral and positively charged grains arising only due to stochastic charge fluctuations (note, in steady state all grains would be negatively charged to the same value Z = Z ). The quantity W(0) and thus both, neutral and positive, Figure 2. Variations of the normalized fluctuation time τ q t 0 dust populations in such a scenario are strongly with n d n i ratio for fixed grain sizes at different electron temperatures: a = 3 nm, T e = 1 ev (solid gray curve); a = 3 nm, dependent on the value n d n i ratio and weakly dependent on the plasma kinetic temperatures T e = 3 ev (solid green curve); and a = 30 nm, T e = 1 ev (solid red curve). Dashed lines represent respective dependencies τ f t 0 versus n d n i. Other plasma parameters used in the ity W(0) is only indirectly, via admissible n d n i, (B T i T e ). It is interesting, that the probabil- calculations are m i = 18 amu and T i = 30 ev. dependent on the grain size. dust density, cm -3 YAROSHENKO AND LÜHR American Geophysical Union. All Rights Reserved. 6195

8 Figure 3. Normalized level (amplitude) of stochastic charge fluctuations σ Z Z versus relative dust density n d n i : T e = 1 ev and a = 30 nm (red curve) and T e = 3 ev and a = 3 nm (green curve). Other parameters are as in Figure 2. Figure 4. Contour plot of the dust density in a (ρ, z) cylindrical coordinate system (aligned with the polar axis of Enceladus) quantitatively consistent with the CAPS measurements during the E3 and E5 plume flybys [Hill et al., 2012]. In the insertion: modifications of the probability charge distribution W(Z) at different dust densities: n d n i = 0.15 (light blue curve), 0.35 (blue curve), and 0.7 (dark blue curve). In the calculations, a = 3 nm and T e = 3 ev have been used. Other parameters are as in Figure 2. It is now worth speculating about possible observational signatures of the considered effect of grain charge fluctuations in the dusty loaded plasma of the Enceladus plume. Our findings suggest two qualitative conclusions that allow a direct comparison with Cassini observations: (i) the higher the density of negatively charged dust (a ratio n d n i ), the more probable and thus more numerous the positively charged grains seem to be, which arises due to the dust charge fluctuations; (ii) the maxima of number densities of negatively and positively charged dust have to coincide locally. Spatial profiles of the positively and negatively charged nanograins registered by CAPS in the plume during E3 and E5 flybys are shown in Figures 8 and 10 of Hill et al. [2012]. Close inspection of the CAPS data indicates that both our predictions are fully consistent with the measured grain characteristics. Although our results seem to be supported by Cassini observations, it is important to mention the limitations of our approach. First, charge fluctuations on a grain are assumed uncorrelated to fluctuations on other grains. This requires that the intergrain distances d (n d ) 1 3 are much larger than the effective length of the dust-plasma interactions, r int = Ze 2 min{t e, T i } [Khrapak et al., 1999]. Physically this condition means that the charging currents to the surfaces of different grains can flow practically independently, providing uncorrelated charge fluctuations. For the adopted in the plume plasma parameters d 0.1 cm, the maximal value of r int does not exceed 10 6 cm, thus always satisfying the condition r int d. There exists, however, another restriction associated with very high dust density which might lead to τ q <τ f as shown in Figure 2. In the light of this condition and because the plasma currents in equations (2) and (3) are defined only for negatively charged grains (Z > 0), we can draw here only qualitative but no quantitative conclusions about the occurrence of the dust grains with positive charges in the dense limit. Finally, note that though we do not include any ionization processes in our model, the physical reality is that electrons and ions continuously arrive at the dust particles and recombine on their surfaces, and so ionization (e.g., due to the solar UV radiation or electron impacts) must be taking place to maintain dynamically the plume plasma. The effect of plasma sink/sources in the plume is omitted here, for simplicity and analytical tractability. It is of interest to compare time scales of the dust charge fluctuations with the particle transport time through the plume. In Figure 5, we show a ratio of YAROSHENKO AND LÜHR American Geophysical Union. All Rights Reserved. 6196

9 fluctuation/transport time grain size, nm Figure 5. Comparison between the fluctuation τ q and transport time τ tr L V d for nanograins of different sizes. In calculations dust density is taken to be n d = 10 2 cm 3 and plasma density n i = 10 3 cm 3. Grain velocity is set 0.2 km/s (lower curve) and 0.8 km/s (upper curve). the fluctuation time τ q to the grain transport time τ tr as a function of the grain size in the limit of a fairly dense dust cloud, for n d n i = 0.1. The transport time can be roughly estimated as τ tr L V d. We put a plume size scale L 10 3 km and, for the grain speed V d, consider two limiting cases. The lower curve in Figure 5 illustrates the limit, when the dust expands at the gas escape speed V d 0.24 km/s. The upper curve is calculated for the higher dust velocity V d 0.8 km/s, which is close to the gas bulk velocity inferred from the Cassini UVIS data by Hansen et al. [2008] and the plume modeling by Smith et al. [2010]. Indeed, Hansen et al. [2008] revealed the vertical gas bulk velocity 1.5V Tg 0.6 km/s, while Smith et al. [2010] found a higher value 1.8V Tg 0.72 km/s, where the gas thermal velocity is expected to be around V Tg 0.4 km/s. As seen from Figure 5, time scales τ q in both cases are comparable or even shorter than the particle transport time for nanograins with a 3 nm. For a more dense dust configurations, the charge fluctuations will develop even faster. We can conclude therefore that the effect of random charge fluctuations on small dust particles likely occurs in the dense plume plasma of the Enceladus. Acknowledgments Data presented in Figures 2 5 are calculated using equations (8), (13), and (14) in the parameter space relevant for the near-enceladus plasma [see Yaroshenko et al., 2014, Tables 1 and 2 and references therein]. This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) under the grant YA 349/1-1 (Special Program, PlanetMag, SPP 1488). The authors also thank both reviewers for their helpful comments on the manuscript. Larry Kepko thanks Sascha Kempf and Mihaly Horanyi for their assistance in evaluating this paper. 4. Summary A description of random dust charge fluctuations has been given for grain ensembles. All statistic characteristics obtained in the limit of a dust-loaded plasma, relevant for the near-enceladus environment, are significantly modified compared to earlier works that considered an individual grain. Under similar ambient conditions, denser clouds admit narrower dust charge distributions than those obtained from an isolated grain model. At grain densities, typical for the Enceladus plume, the difference can be quite appreciable. These findings might be of importance for the interpretation of the CAPS data, which initially contain the charge-to-mass ratios and require some assumptions on the charging state to determine the particle size. The modeling in the dust-loaded Enceladus plume predicts that a majority of the grains acquires fluctuating negative charges, but there might appear a minority of positively charged particles without involving secondary or photo emission. The probability of the coexistence of two oppositely charged dust species is mainly determined by the ratio of the dust/plasma number densities. The obtained results appear to be supported by the CAPS measurements of the spatial profiles of the positively and negatively charged nanograins in the plume during E3 and E5 plume flybys [Hill et al., 2012]. Although our findings qualitatively match the available Cassini data, the model restrictions do not allow us to give quantitative estimates of the ratio between positive and negative dust particles. 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