In the vicinity of Enceladus, a geologically active moon of Saturn, the modeled spacecraft potential is significantly more negative than indicated by the Cassini Langmuir probe measurements. To understand this potential difference, we introduce two new dust–related charging currents: a) the dust ram current; and b) the dust impact–plasma current, in addition to the customary collection of electrons and ions, and photoemission. Our results show that these dust currents are important at high relative speeds between the spacecraft and the dust, or in regions with a low plasma-to-dust ratio, and can lead to reduced spacecraft charging.
 The potential of a SC in space is determined by the conditions of the ambient plasma and the solar radiation [Whipple, 1981]. The potential of a SC adjusts towards its equilibrium value where the sum of the charging currents is zero. The equilibrium SC surface potential can be estimated using the conventional Orbital Motion Limited (OML) theory [Mott-Smith and Langmuir, 1926]. The interaction between plasma particles and an embedded object can also be modeled by particle-in-cell (PIC) simulations, which provide more detail about the properties of the plasma sheath around the object.
 Using PIC simulations, Olson et al. [2010b] modeled the potential of the Cassini spacecraft in the plasma disk near Enceladus. These authors found the modeled SC potential to be more negative than expected, based on the Cassini Langmuir probe measurements [Olson et al., 2010b, Table III] (Table 2). This potential difference was too large to be explained by adjusting the photoemission induced by the solar UV.
 In a dust–rich environment the plasma charging currents can be altered, and additional dust-related currents can become significant. The amount of electrons bound to negatively charged dust grains may become significant and result in a reduction of the electron density [Havnes et al., 1990; Morooka et al., 2011]. The potential of the dust–plasma cloud decreases as the dust density increases [Goertz and Ip, 1984; Havnes et al., 1990]. The decrease in potential difference between the dust–plasma cloud and the SC reduces the collection of the free electrons, for example. Furthermore, due to large relative speeds between the dust and the SC, the incoming flux of charged dust to the SC may constitute significant charging currents [Morooka et al., 2011].
 In this work we focus on the SC potential calculation in a dust–rich environment, such as Enceladus' plume and in Saturn's E ring. In addition to the customary electron/ion collection currents and the photoelectron emission, we include “dust currents”, and estimate their effects on the SC charging.
2. Spacecraft Charging Calculation
 To simplify the problem, we consider a spherical and electrically conductive SC in a Maxwellian plasma. Due to its large surface area, the SC will swiftly charge to its local equilibrium value. The currents are calculated using the OML theory, which is an acceptable approximation even for an object that is larger than the Debye length [Willis et al., 2010].
2.1. Dust-Poor Condition
 In regions of low dust density, a SC is charged by collecting ambient plasma (ion and electron collection currents -Ji and Je) and emitting photoelectrons (Jν) [Grard, 1973; Whipple, 1981; Horányi, 1996; Graps et al., 2008]. In the plasma torus near Enceladus, the electron density is around 10 to 100 cm−3 and the temperature is relatively low (kTe < 5eV) [Gustafsson and Wahlund, 2010; Cravens et al., 2011]. The charging is dominated by the electron collection and leads to a negatively charged SC. The secondary electron emission from the SC can be ignored as kTe is low.
2.2. Dust-Rich Condition
 The presence of high–density charged dust particles can significantly change the potential of the dust–plasma cloud (ϕc), as well as the charging of an embedded object. This effect can be formulated as a function of the P parameter [Havnes et al., 1990]
where ne,0 is the background electron density (in cm−3), sd is the dust radius, and Nd = ∫ nd(sd)dsd is the total dust density. For simplicity, we adopt a power–law dust size distribution nddsd = nd0 ⋅ (sd/1μm)−μ, where nd0 is the density of 1 μm dust and μ is the power–law slope. The total dust density strongly depends on the chosen μ and smin. The slope derived from in-situ measurements ranges from 4.0 to 6.4 [Kempf et al., 2008; Kurth et al., 2006]. Here we choose μ = 5, smax = 10 μm, and smin = 0.01 μm (smaller particles are sputtered or ejected from the system and have very short lifetimes [Hsu et al., 2011]). In this work, we do not consider the nanodust detected by the Cassini Plasma Spectrometer [Jones et al., 2009; Hill et al., submitted manuscript, 2012].
 The P parameter is the ratio of the charge density in the dust component, calculated for isolated dust particles, relative to the background electron density. Considering the charge equilibrium of the plasma and the embedded dust [Havnes et al., 1990], Figure 1 shows the normalized ϕc (dashed line) and the potential differences between it and an embedded dust particle (black solid line, Φd = ϕc − ϕd, ϕd is the dust potential) as a function of P. When P ≪ 1, the dust density is low enough that each dust can be seen as an isolated object in the plasma and ϕd does not depend on the Nd. However, when P > 0.01, ϕc starts to decrease from zero to ∼ − 2.5 and Φd reaches zero (i.e., ϕd ≃ ϕc). The change of ϕc modifies the charging baseline as well as the ion/electron density in the dust–plasma cloud and therefore needs to be considered for current calculation in dust–rich conditions. The electron and ion densities at given ϕc are ne = ne,0 ⋅ exp(eϕc/kTe) and ni = ni,0 ⋅ exp(−eϕc/kTi), where ni,0 = ne,0 and kTi = kTe. Hence, in high dust density regions, the charging currents are functions of P.
 In addition, due to the relative motion between the charged dust and the SC, intense dust fluxes will lead to two other SC charging currents. They are (a) the dust ram current and (b) the dust impact–plasma current.
2.2.1. Dust Ram Current
 The dust ram current is the delivery of dust charges to the SC [see also Morooka et al., 2011, equation (A3)]. It can be written as
where Qd(sd) = 4πϵ0sdΦd is the dust charge, fd(sd, vrel) = πssc2 ⋅ vrel(sd) ⋅ nd(sd) is the dust flux to the SC, and ssc is the SC radius.
 The speed of an E ring dust particle depends on its charge–to–mass ratio due to the influence of Lorentz force [Horányi, 1996, equation (24)]. The relative speed between the SC and the dust, vrel, is thus a function of grain sizes in the E ring, i.e., smaller particles have a higher speed than the larger ones. On the contrary, during the traversals through Enceladus' plume, most grains that the SC encountered just emerged from the vents and have the same speed as the moon. In this case vrel is independent of the grain size. The kinetic energy of a dust grain is sufficiently high in all cases so that Jd,ram does not depend on the SC potential.
2.2.2. Dust Impact–Plasma Current
 The second dust current stems from the collection of the plasma produced from dust–SC impacts. The plasma generation through hyper-velocity dust impacts is a well–known though complicated process [Drapatz and Michel, 1974]. Through the impact, the kinetic energy of the incident dust grain is converted to transform the dust and part of the SC material into small debris, vapor, and plasma. It is the collection of the impact plasma that provides an additional current to the SC.
 An empirical relation for the impact plasma production yield (Y, in Coulomb) has been derived from the laboratory experiments [Dietzel et al., 1973; Göller and Grün, 1989]: Y = 5 ⋅ 10−4 ⋅ md ⋅ vrelβ, where md is the dust mass in gram, vrel is in km s−1, and β ranges from 3.5 to 5.5 [Dietzel et al., 1973, Figure 5]. We adopt β = 3.5 in the following calculations. The impact plasma flux (Simp, in Coulomb per second) is found to be
where fdm(sd, ϱd, vrel) is the dust mass flux and ϱd = 103 kg m−3 is the dust bulk density.
 The collection of the transient impact plasma depends on its energy distribution and the SC potential. The dust impact produces the same amount of ions and electrons, whose energy distributions are Maxwellian with a characteristic temperature, kTimp, of ∼ 1eV. The dust impact–plasma current becomes
where χe,iimp = ∓eΦsc/kTimp and Φsc = ϕsc − ϕc. The intensity of the dust currents depends strongly on the slope of the dust size distribution nddsd ∝ sd−μ. The chosen slope, μ = 5, implies that both dust currents are dominated by the sub–micron dust population.
3. Results and Discussion
 Taking all currents into account, Figure 1 shows two Φsc profiles for different levels of photoemission as a function of P. The dust–SC speed is 8 km s−1 and is size–independent here. The photoemission efficiency (κ of Horányi ) for the SC2 is 0.5, five times higher than for the SC1 and the dust. The photoemission current for the SC2 is about 240 nA m−2, comparable to the Cassini Langmuir probe (200 to 300 nA m−2). For the SC1 we set the spacecraft–to–plasma speed (vps) to zero (i.e., an isotropic ion current), and in SC2 to 30 km s−1 [Olson et al., 2010b, Table III]. When P ≪ 1, the SC1 potential (ΦSC1) is identical to the dust, while the SC2 (ΦSC2) is about 0.8 V less negative due to the higher photoemission and SC–plasma speed. The difference between the SC1 and SC2 is smaller than the potential difference between the measured SC potential and the PIC simulation results by Olson et al. [2010b] (no photoemission), which suggests that the photoemission is not responsible for this discrepancy. With increasing P, both SC potential profiles start to deviate from the dust potential due to the contributions of the dust currents. Table 1 lists the relative intensity of each current at the equilibrium potential for SC2 at P = 10−3, 10−1, and 10 (blue squares in Figure 1).
Table 1. Spacecraft Charging Condition at Different Pa
P = 10−3
P = 10−1
P = 10
ne,0 = 102 cm−3, kTe = 1 eV, vrel = 8 km s−1, vps = 30 km s−1. The density of dust larger than 1 m (Nd,>1μm), the dust potential (Φd), the SC2 potential (Φsc), the cloud potential (ϕc), and current intensity ratios are shown at P = 10−3, 10−1, and 10 (blue squares in Figure 1). Currents are normalized to the random collection current of plasma electron (Je∗ = πssc2ne,0, where me is the electron mass.) to show their relative importance at different P. Notice that at P= 10 the dust impact–plasma currents are many times stronger than the collection of ambient plasma, i.e., the SC charging is governed by the impact-produced plasma. Parameters adopted in the calculation are shown at the bottom of the table.
1.1 ⋅ 10−4
1.1 ⋅ 10−2
1.1 ⋅ 100
 At P = 10−3, the dust currents barely have any influence. At P = 10−1, the dust impact–plasma currents become stronger due to the higher dust density, but remain small compared to other currents. Since the SC remains negatively charged, more ions than electrons from the dust impact plasma are collected. Because of Jd,impi, the SC is charged less negatively. At P = 10, the SC charging is governed by the impact–produced plasma. Strong dust impact–plasma currents tend to minimize the potential difference between the SC and the cloud, leading to Φsc ∼ 0V.
Figure 2 shows the normalized SC potential as a function of the total dust (Nd) and electron (ne,0) number densities, providing an overview of the SC charging condition during Enceladus flybys and ring plane crossings (RPXs) with different flyby speeds. The relative speeds, vrel, are around 8, 13, and 18 km s−1 for the three panels, respectively. Here we consider the dust size dependence of vrel only for Figures 2b and 2c. Figure 2a is basically the same as the SC2 in Figure 1 and Table 1. All of these represent the SC charging condition during a 8 km s−1 Enceladus flyby. Note that we do not include the nanodust population found in the plume of Enceladus, which may have additional contribution to the SC charging during plume traversals.
Figures 2a–2c show that the influence of dust currents on the SC potential increases with the dust density, resulting in a less negative SC potential due to the collection of impact plasma ions. Because of the contributions from the dust and the photoemission, in regions of low plasma densities the SC may even be charged slightly more positively than the plasma cloud. The effect of increasing vrel on the potential of the Cassini spacecraft during Enceladus flybys or RPXs can be seen by comparing Figures 2a–2c. With the same dust and electron density, the SC potential is shifted positively toward the cloud potential with higher vrel.
 To verify our model, we adopt the Cassini Langmuir probe plasma and the Cosmic Dust Analyzer dust density measurements with given impact speeds to calculate the SC potential. Our result is compared to the PIC simulations and the Cassini measurements in Table 2 (Langmuir probe and the PIC simulation results are from Table III of Olson et al. [2010b]). Unfortunately, except for the E2 flyby (2005 day 195), the SC attitude was not favorable for the dust measurements, so we adopt the dust density given by Kempf et al.  for the other two cases.
Table 2. Comparison of the SC Potential From the Cassini Langmuir Probe Measurements (ϕLP), PIC Code Simulation (ϕPIC, Without Photoemission), and This Work (ϕOML, Without Dust Effect and Photoemission; ϕν, With Photoemission; and ϕD, With Both Effects)a
The electron density (ne), temperature (kTe) derived from the Langmuir probe, the relative speed between the Cassini and the Keplerian motion (Vsc−kep), and the dust number density (Nd) are also shown. With the absence of dust and the photoemission, our results (ϕOML) are consistent with the PIC simulations, which are much lower than the measurements. With both effects, our model (ϕD) can better reproduce the Cassini Langmuir probe results.
 With our OML approach, the SC potential is calculated with and without dust effects. When the photoemission and the dust effects are omitted, our results (ϕOML) match the PIC simulation (ϕPIC) and, both ϕOML and ϕPIC are much more negative than indicated by the Langmuir probe results (ϕLP). Adding photoemission (ϕν) only raises the potential slightly. However, after including both effects, our modeled SC potential (ϕD) becomes less negative and is comparable to the measured SC potential. This suggests that the dust impact–plasma currents, i.e., the collection of the dust impact plasma generated at the SC surface, have a significant effect on the SC charging when traversing dust-rich regions. The corresponding SC potentials for these three measurements are also marked inFigures 2a–2c, respectively.
 There are two implications regarding in–situ thermal plasma measurements due to the presence of the impact–generated plasma. Firstly, as described above, the SC potential may be modified and thus alter the collection of the ambient thermal plasma. Secondly, similar to photoelectrons, impact–generated plasma may be collected by the plasma probes and not distinguishable from the ambient plasma. A better characterization of the dust–SC impact plasma will improve our understanding of the in–situ measurements as well as the dust–plasma interactions in dusty environments.
 In this work we discuss the SC charging in dust–rich environments and compare our results with PIC simulations and measurements near Enceladus. In addition to the customary charging currents, two types of dust currents are considered in our model. We showed that the SC potential can become less negative due to ion collection from the dust–SC impact plasma. This provides a natural explanation for the difference between the SC potentials derived from PIC simulations and from in–situ Cassini measurements close to Enceladus. The dust–SC impact generated plasma behaves like a potential damper, which reduces the potential difference between the SC and the ambient dust–plasma cloud. The influence of the dust impacts on the SC charging needs to be taken into account for the analysis of the SC charging as well as in–situ thermal plasma measurements in the dense part of the E ring and in the plume of Enceladus. A self-consistent analysis of the Langmuir probe plasma measurements, and its estimates for the SC potential in dust rich regions will be addressed in future works.
 The authors acknowledge support from the Cassini project. We thank the anonymous reviewers for the useful comments.
 The Editor thanks the anonymous reviewers for their assistance in evaluating this paper.