Charge fluctuations on the sunlit surface of airless bodies and their role in dust levitation

Icarus 338 (2020) 113538

Contents lists available at ScienceDirect

Icarus journal homepage: www.elsevier.com/locate/icarus

Charge fluctuations on the sunlit surface of airless bodies and their role in dust levitation Е.V. Rosenfeld a, b, А.V. Zakharov a, * a b

Space Research Institute, Russian Academy of Sciences, Profsoyuznaya str, 84/32, Moscow 117997, Russia Institute of Metal Physics, Ural Branch of Russian Academy of Sciences, Kovalevskaya str. 18, Yekaterinburg 620990, Russia

A R T I C L E I N F O

A B S T R A C T

Keywords: Lunar surface Regolith Charge fluctuations Dust particles Levitation

The most likely cause of levitation of dust particles above the illuminated surface of airless bodies is the elec­ trostatic mechanism. However, there are two essential problems to explain this effect. The first one is how a large electric charge (several units or even dozens of elementary charges e) which is required for levitation in the field of the electrical double layer, can be accumulated on a dust particle lying on the surface. The second problem is to find out the nature of the force that detaches dust particle from the surface overcoming the adhesive van der Waals force. The paper shows that the density of charge, which arises on a lit surface due to the photoeffect, is very small only when averaged over regions of a macroscopic size. On a submicron scale, the surface appears to be a collection of chaotic “spots” having positive or negative charges. These spots arise due to fluctuations in the fluxes of the photoelectrons taking off and falling back onto the surface, and the modulus of charge density within these spots is millions of times larger than the value of the charge density averaged over macroscopic regions. The number of positively charged spots is only slightly larger than the number of the negatively charged spots, which explains the smallness of the average charge density. The local charge density inside the spots is sufficient to allow the particles about 0.1 μm or less to get a charge sufficient to levitate them within the electric double layer at an altitude about 1 μm or more, where the field strength of the double layer is on the order of 10 Vm 1. The electrostatic field directly above the center of a charge spot is proportional to the local rather than average charge density and appears to be several orders of magnitude greater. Therefore, the force acting on a dust particle lying interior to a charge spot directly on the surface is proportional to the square of the local charge density and its magnitude can reach a tenth of pico-newton, which is probably sufficient for a particle to overcome the adhesive van der Waals force and take off.

1. Introduction Space bodies devoid of the atmosphere are subject to significant in­ fluence of the solar ultraviolet radiation and the solar wind in the absence of an intrinsic magnetic field. It was shown that the photoeffect is the dominant factor in the formation of the electric potential of the illuminated lunar surface outside the geotail [Opik and Singer, 1960; Singer and Walker, 1962, Manka and Michel, 1973; Colwell et al., 2007; Poppe et al., 2012 and others]. The upper layer of the regolith of the Moon and most of the small bodies is considered to be a good insulator [Carrier et al., 1991]. The formation of the surface electrical potential on airless bodies is a complicated process, since it occurs as a result of several factors, including the solar wind plasma, secondary electrons, and ultraviolet radiation of the Sun, causing the photoemission of

electrons. The contribution of each of these factors to the lunar surface potential mainly depends on solar wind parameters that vary in space and time. The dependence of the potential of the lunar surface on the solar wind flux and photoemission under various conditions was considered in [Grobman and Blank, 1969]. An estimate of the solar wind effect on the lunar surface potential and the exosphere was made e.g. in [Bernstain et al., 1963; Hurley et al., 2017]. It was shown that the electrons in the quiet solar wind are the main factor affecting on the lunar night side potential of the surface near the terminator. Low-energy photoelectrons above the surface and positive “holes” remaining on the surface form the double electric layer (DEL), a nearsurface (about 1 m) electric field of several Vm 1 [Manka, 1973; Manka and Michel, 1973; Freeman et al., 1973; Freeman and Ibrahim, 1975]. It is commonly believed that in this field, submicron and micron

* Corresponding author. E-mail address: [email protected] (А.V. Zakharov). https://doi.org/10.1016/j.icarus.2019.113538 Received 14 January 2019; Received in revised form 28 September 2019; Accepted 7 November 2019 Available online 19 November 2019 0019-1035/© 2019 Elsevier Inc. All rights reserved.

Е.V. Rosenfeld and А.V. Zakharov

Icarus 338 (2020) 113538

regolith dust particles are able to lift off the surface, forming, together with the surrounding plasma, a plasma-dust exosphere near the surface of such bodies. The direct evidence for such kind electrostatic processes acting on lunar dust particles was a set of images taken by the TV cameras of Surveyor 5, 6, 7 lunar landers [Criswell, 1973; Rennilson and Criswell, 1974]. These images show a distinct glow just above the lunar horizon, what was interpreted to be forward scattered sunlight from a cloud of dust particles levitated above the surface. Most of the problems associated with the formation of the photo­ electron layer field above the surface were largely solved by experi­ mental and theoretical works of De and Criswell, 1977; Criswell and De, 1977; Poppe and Hor� anyi, 2010; Dove et al., 2012; Stubbs et al., 2014; Wang et al., 2016a; Vaverka et al., 2016; Popel et al., 2016a, 2016b and others. Therefore, we will focus only on one aspect of the problem of the photoelectron layer formed above the illuminated surface: the fluctua­ tions in the charge density on the surface itself. In our opinion, it is these fluctuations play a fundamental role in the process of charging and taking-off of the particles lying on the surface. It is assumed that the surface potential decreases to the value of φplasma at a distance from the surface of the order of the Debye shielding radius λD. The value of λD for the solar wind plasma in the vicinity of the Earth is about 1�10 m [Stubbs et al., 2006, 2014]. Therefore, it should be expected that the electric field intensity at the surface is about 10 Vm 1 in the lunar illuminated hemisphere. This value is confirmed by in situ measurements on the lunar surface [Freeman et al., 1973]. The gravity force acting on a dust particle with a diameter of 100 nm on the lunar surface is about 3 � 10 18 N. Such, or a smaller dust particle ac­ quired a few elementary charges e, can only not levitate but also fly quite high, accelerating in the near-surface double layer [Stubbs et al., 2006]. However, a charge of at least 2e is required for levitation, if the dust particle has a slightly larger size and here the electrostatic theory of dust levitation faces the first fundamental problem. The matter is that to create a uniform field strength EDEL within the electric double layer (a capacitor) a charge density is required on the surface (positive electrode) hvie ¼ ε0 EDEL � 10

10

Cm 2 ⇒hvi � 109 m

2

the surface charge density, which occur when two stochastic flows of opposite charges having the same fluence rate fall on the surface. As a particular case, we will consider holes that appear on the surface when photoelectrons are knocked out, and photoelectrons that fall back to the surface. These positive and negative charges appear on the surface in a purely randomly manner, and if a certain number of electrons and holes appear in a small area of s, then most likely it will have a charge of a few e of one or another sign. At the same time, if the area of the region is sufficiently small, then the average number of excess charges on it, determined in accordance with (1), will be shνi. In particular, in our case, the average number of excess holes in the region with s ¼ 1 μm2 is 0.001, so that a photoelectron falling into it means the appearance of fluctuations with an amplitude a thousand times higher than the average value. On the other hand, in accordance with the law of large numbers, for a region with a large area into which a large number N > > 1 of electrons and holes randomly falls, the amplitude of fluctuations is pffiffiffi proportional to N. Since N ∝ s it is easy to understand that now the average value of the charge of the shνi region will be much larger than pffiffi the amplitude of its fluctuations, proportional to s. As a result, the surface is covered with fluctuating “charge spots” of different signs, the local charge density ν(r,t) inside of which is deter­ mined not by the macroscopic field strength EDEL averaged over the macroscopic surface regions, but by the density of the stochastic charge flows. Therefore, the ratio |ν(r)| > > |hνi| can be valid for small sites, and a dust particle that is inside such a charge spot acquires a charge proportional to ν(r) sufficient for its levitation. In addition, the local electric field directly above the surface of the charge spot is also propor­ tional to ν(r), which leads to the Coulomb force becoming proportional to ν2(r) and sufficient to overcome the adhesive van der Waals forces. Depending on what type of random charge flows fall on the surface (plasma, plasma þ charged particle beam, etc.), some peculiarities arise in this fluctuation process, which we considered in our previous work [Rosenfeld and Zakharov, 2018]. In this article, we will consider in detail the result of the impact of only solar ultraviolet radiation on the non-conductive surface of an airless body, since it is solar ultraviolet radiation that is the main cause of the appearance of the surface charge (at least in the midday regions) [Stubbs et al., 2014].

(1)

where hvi ¼ hvþ v i (angle brackets indicate mean value) is the mean number of excess elementary charges e per 1 m2. Therefore, on average, one excess elementary charge is located on an area of ~ 1000 μm2. So, the probability of detecting even one charged submicron dust particle seems to be extremely low, not to mention dust particles with several elementary charges. Thus, we come to a paradoxical conclusion: if the electrostatic field above the surface is small, then, in fact, there are no dust particles capable of flying, and if the considerable number of dust particles have a sufficient charge, the field above the surface should be millions of times stronger. The second problem is related to the adhesive van der Waals forces that hold the dust particles on the surface and/or bind them together. According to the estimates by [Li et al., 2006; Sheridan and Hayes, 2011], in the best case these forces in the order of the magnitude are not less than a piconewton, so that the dust particle with the charge 1e can fly alone in a field with a strength E~107 V m 1. Previous attempts to solve this problem [Flanagan and Goree, 2006; Sheridan and Hayes, 2011; Sheridan, 2013] based on calculations of charge fluctuations on solitary dust particles lying on a plane surface, did not lead to success. However, it should be noted that in the work [Flanagan and Goree, 2006] it was suggested that the force sufficient to overcome the adhesive force may occur as a result of the Coulomb repulsion of two closely lying dust particles on which there are charge fluctuations of one sign. Another possibility of an increase of a dust particle charge was suggested by Wang et al., 2016b; Zimmerman et al., 2016, Schwan et al., 2017. The role of cohesive forces in dust particles launching on the Moon was also discussed by (Hartzell and Scheeres, 2011]. The aim of this work is to study the microscopic-scale fluctuations of

2. The mean value and the instantaneous magnitude of the electrical charge of a small region on a sunlit surface So, we assume that the main source of the field above the illuminated surface of airless bodies is the double electric layer created by photo­ electrons that fly above the surface and positive “holes” remaining on the surface after the photoemission [Manka, 1973; Manka and Michel, 1973; Freeman et al., 1973; Freeman and Ibrahim, 1975] and assume that the field inside this layer is EDEL � 10 V m 1. Then, in accordance with (1), only hνi � 109 excess charges must be located on each m2 of the surface. However, this is too small for any noticeable number of dust particles to accumulate the charge of several e necessary for takeoff. Indeed, in this case, the average excess charge of a dust particle with diameter d, lying on the surface, is equal to 1 hqðdÞ i ¼ πd2 hνie � 8⋅108 d2 e 4

(2)

This means that hq(d)i can exceed one (!) elementary charge e only if the diameter of a dust particle is about 30 μm or more. For smaller dust particles, the probability (proportional to d2) to acquire this minimally possible charge e is much less: on average only one of the 105 dust particles (lying in the uppermost layer) with diameters d ¼ 100 nm and only one of the 107 particles with a diameter d ¼ 10 nm has a charge 1e. In practice, a submicron dust particle cannot obtain two or more elementary charges. Therefore, if the dust actually levitates under the action of the EDEL � 10 Vm 1 field, there must be some mechanism that allows a large number of dust particles lying on the surface to accu­ mulate the charge necessary to take off into such a weak electrostatic 2

Е.V. Rosenfeld and А.V. Zakharov

Icarus 338 (2020) 113538

field. One of the possibilities is to postulate that a dust particle lying on the surface should acquire such a charge that would create a potential φ on the surface of the particle, equal to the potential of the surface of the large body on which it lies, see e.g. [Walch et al., 1994] or the discussion of “isolated capacitors” and the “shared charge” models [Flanagan and Goree, 2006]. However, in the case when a charge arises on dust par­ ticles already lying on the surface, a similar does not occur. Indeed, a single sphere with a radius r embedded in a plasma will accumulate a negative charge as long as its surface will acquire the potential φ ¼ q/(4πε0r), which is determined only by the temperatures and masses of the plasma species and does not depend on r. Hence, the charge q of the sphere should be proportional to r and its occurrence leads to the appearance of an electric field E ¼ q/(4πε0r2)~r 1 above the surface. It is this field which equalizes the electron and ion currents to the sphere surface, and it practically disappears at the distance of a few r from the surface of the sphere. Therefore, this field on the surface itself, and with it the surface charge density σsph ¼ q/(4πr2) ¼ ε0E should be inversely proportional to the size of the sphere E ¼ φr 1. A similar situation arises in the case of a dust particle is in a vacuum, and the charge on it appears due to the photoelectric effect. However, in this case, the field above its surface will not equalizes the velocities of the incident ions and electrons, as it was under plasma conditions, but it will stop and return the emitted photoelectrons back. Now the dust particle potential φ will be equal to the energy of the emitted photo­ electron in electron volts. If a speck of dust lies on a dielectric surface in a plasma, the speeds of electrons and ions flying toward it are already the same, because they have passed the plasma sheath before approaching the speck of dust. Therefore, there is no reason to expect that a charge of the speck of dust is greater than on any adjacent surface with the same area. Similarly, in the case of a photoelectric effect, photoelectrons are knocked out of the dust particle with exactly the same probability as from neighboring surface areas, and the same double-layer field returns all of the ejected photoelectrons back. In other words, the excess charge, in any case, will be evenly distributed over the surface, and only a “shared charge” model should be used in calculating the charge of dust particles on the surface. So, we have to conclude that (i) only the formulas (1) and (2) should be used to study the charge accumulation per particle located on a surface and (ii) only about 109 (or something like that) surplus elementary charges are shared over every m2 in our case. Therefore, either there should be some other reason why dust particles at the very low average charge still obtain a charge sufficient for takeoff, or dust levitation should result from some non-electrostatic mechanism. As for an alternative to electrostatic mechanisms, the only example known to us is one considered in [Rosenfeld et al., 2016] related to the rise of dust particles with a diameter of the order of 10 nm from the surface due to thermal fluctuations. In that paper, it was shown that the “boiling layer” of dust with a thickness of a few decimeters can form above the hot (about 400 K [Vaniman et al., 1991]) surface of the Moon. In contrast to the electrostatic mechanism that is capable of lifting only charged dust particles, thermal fluctuations lift almost all fine particles lying on the illuminated and sufficiently hot surface. However, this mechanism can explain neither the levitation of larger dust particles nor especially higher-altitude levitation [Zook and McCoy, 1991; Stubbs et al., 2006]. Taking into account that the charge of anybody can vary only in a discrete manner by � e, it is natural to assume that this mechanism should be associated with fluctuations of the charge accumulated by the dust particle. A number of attempts to calculate these fluctuations have been made during last decades, see e.g. [Khrapak et al., 1999; Vaulina et al., 1999, Arnas et al., 2000; Robertson et al., 2003; Flanagan and Goree, 2006; Sheridan and Hayes, 2011]. In these works, the “isolated capacitors” model was used or the charge fluctuations of a solitary dust particle were considered so that they could use relatively simple methods based on the assumption that the mean value of particle charge

is large compared with its standard deviation (SD). A much more rele­ vant approach was used in the works by Sheridan (2013), in which charge fluctuations on dust particle lying on the surface was considered in the framework of the “shared charge” model. It was shown there that the SD of the charge fluctuations can be several orders of magnitude higher than the mean value if the latter is small enough. In particular, it was shown that submicron particle lying on a solid surface can reach tens or even hundreds of the elementary charge e. As we saw previously, this may be quite sufficient to lift a solitary particle in the double layer field, so this conclusion solves the problem of charge accumulation on small dust particles, which is sufficient for their soaring above the surface. However, another extremely important consequence of this result was left without attention in Sheridan (2013): since the average value of the dust particle charge is close to zero, not only the magnitude but also the sign of the dust particle charge should fluctuate. Its exceptional importance becomes quite apparent when you consider when a surface is irradiated with charged particle flows or light quants, they fall with equal probability both on the dust grains lying on this surface, and on the areas surrounding them. Consequently, similar charge fluctuations of different signs should appear in any small region (charge spots) on the surface. In this case, the size of charged spots can be much larger than the size of a single dust particle resided within it, and a huge repulsive Coulomb force can appear [Flanagan and Goree, 2006]. In this context, the natural questions arise: (i) what is the characteristic size of the area where the surface charge density does not change sign, (ii) what is the average value of the modulus of the charge density in these areas and (iii) what is the characteristic lifetime of these regions. Therefore, we will consider the fluctuations of charge density arising on a dielectric flat surface onto which two stochastic charge fluxes fall. The first one is the light quantum flux, which creates positively charged holes, and the second one is the flux of returning photoelectrons. We emphasize that this is just a special case of the problem of a charge distribution over a surface on which charges of different signs arise in a random manner. In other cases, the statement of a question can differ in details: the appearance of positive charges can be associated with the incident ion flux, or the average field above the surface can be oppositely directed, or the surface can have a finite conductivity or irregularities etc. 3. The force lifting a particle It is very simple to understand how charged spots arise in our case. During a short time after sunrise, it is extremely unlikely that the falling back photoelectron enters the atom with the “hole” that arose previously due to the photoeffect. Consequently, the total value of both the positive and negative charges accumulated on the surface is proportional to the duration of the illumination, and these opposite charges are randomly distributed over the surface without annihilating each other (repeat, we neglect the conductivity of the regolith [Carrier et al., 1991). In these conditions, the regions in which the local charge density of a particular sign (positive or negative) significantly exceeds the charge density mean value over the entire surface (Eq. (1)) must appear. It is natural to call these regions “charge spots”, and we will use this name. In this case, oppositely charged regions should alternate over the surface and if a small dust speck is inside such a spot, the sign of its charge will coincide with the sign of the charge on the surface in its nearest vicinity. So, if dust particle resides in the neighborhood of a spot center, the attractive Coulomb forces acting on it from neighboring spots of other sign are mainly horizontal, and the repulsive forces from the nearest neighbor­ hood are mainly vertical, which ensures a lifting force. We emphasize that the structure of the distribution of these spots on the surface must be similar to a fractal structure: inside any spot with a characteristic size much larger than the atomic scale, there must be many smaller spots. In such a structure, the density of the local surface charge Σ at a surface region centered in a point r will depend on the 3

Е.V. Rosenfeld and А.V. Zakharov

Icarus 338 (2020) 113538

characteristic size of this area d: Σ ¼ Σ(r, d). The average value of the surface charge density for the entire plane does not depend on either r or d: Σ(∞) ¼ hνie (1), but the smaller the size of the region over which the averaging is performed, the more likely the deviations from this value are. This purely mathematical conclusion is extremely important from the physical point of view. The value of the electric field at a certain height h over such a “spotted” surface is defined mainly by the charges lying on the surface inside the circle with a radius of two or three h, and the contributions of the more remote sites almost completely compen­ sate each other. Consequently, in such a system the field should vary sharply with h, which is of fundamental importance in studying the detachment of a particle from a surface. Indeed, if a particle with the charge q lies on a flat surface with homogeneous charge density Σ, then the force F ¼ qΣ= acts on the particle. However, in calculating the 2ε0 Coulomb force acting on a particle, lying on a surface covered by fluc­ tuating “charged spots” it is necessary to introduce some changes into this formula. In the simplest approximation, we can consider only the field that is formed by the charges located within the circular fluctuation with the diameter d around the dust particle with a radius R and neglect the field of all charge “spots” beyond this area, see Fig. 1. Then, the charge of the dust particle will be about πR2Σ(d), and the circular area of diameter d on the surface around this particle has the same charge density πR2Σ(d). If we assume that the charge of a speck of dust is concentrated at its center, then the force acting on it is 2 3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 Z 6 3 2 2 2 u πR Σ ðdÞ d2 2πxdx π R Σ ðdÞ 6 u 2 7 1 u F¼ (3) � �2 7 �32 ¼ 2pffi2ffi ε 6 4 5 4πε0 t 2 2 0 R þx R 1 þ d=2R

fountain mechanism [Stubbs et al., 2006] will enter into action and the dust grain will soar. Otherwise, it will most likely move to the nearest “spot” with a charge of the opposite sign and there it will fall back to the plane. If there is a constant horizontal component of the electric field (say, near the border of light and shadow), such small “jumps” should, apparently, cause dust transport along the surface. These purely qualitative considerations are given here only in order to clarify the meaning of the calculations below and the difference be­ tween our approach to the problem of charge fluctuations and the approach of the works [Sheridan and Hayes, 2011; Sheridan, 2013]. We emphasize once again that in the paper we do not attempt to calculate the trajectories of the flying particles, to determine the exact value of their maximum size, to take into account the influence of various external factors, etc. We simply try to propose a physically clear analytical method for the description of the dynamics of charge spots, which can be used further in the solution of completely different prob­ lems associated with the electrostatic mechanism of detachment of small particles adhering to a surface. 4. “Random walks” of the charge of surface areas or dust particles As mentioned above, for some time after the occurrence of the double layer the photoelectrons falling back to the surface almost certainly fall into neutral atoms and not into those atoms from which the electron was previously knocked out. This means that the positive and negative charges �e are distributed randomly over the surface with almost identical and proportional to the illumination time t average density

=

=

�

νþ ðtÞ � ν ðtÞ ¼ jph t e � 3⋅1013 t m

For example, if the size of the fluctuating region is twice the size of a dust particle d ¼ 4R, the force acting on the latter is approximately four times less than the force acting on it on a uniformly charged (with the same Σ(d)) plane, but this difference decreases rather rapidly with increasing ratio d/R. Thus, the force acting on the dust particle, lying on a non-condycting surface, is proportional to the square of the local charge density in the “spot” of size d, surrounding it: F ∝ Σ2(d). However, if the particle rises to a height h, the force becomes proportional to Σ(d)Σ(h), i.e. decreases with the height, since the mean charge density within a region should quickly decrease with an increase in its size. This means that inside a “spot” with a charge of any sign the dust particle can take off from the surface at the moment when the Coulomb force acting on it exceeds the powerful but short-range van der Waals force FVdW. The further fate of the particle rising above the surface de­ pends on the magnitude and sign of its charge. If the magnitude of the charge is sufficient for lifting, and EDEL will repel it from the surface, the

2

(4)

Here, jph is the density of the photocurrent in the lunar midday zone. In accordance with [Feuerbacher et al., 1972] jph equals jph � 5 μA m

2

� 3⋅1013 e m

2

s

1

(5)

It is clear that almost immediately after the start of illumination the values of densities ν� will be much higher than the average of the dif­ ference between them hνi (1) and will continue to grow linearly with time. On the other hand, after setting EDEL the time-averaged difference between the values of the positive and negative charges in any area of diameter d is constant: hnþ ðd; tÞ i

π

hn ðd; tÞ i ¼ hνid2 � 8⋅108 4

(6)

This value is absolutely insignificant compared to the background of the linearly increasing numbers of charges for each sign in this area

Fig. 1. A dust particle lies on a plain surface under UV irradiation. In the case of a uniform surface charge distribution (gray area) with a density Σ on the surface, a spherical particle of radius R, with the upper part having a charge q � πR2Σ, is repelled from the plane with a large force FC ∝ Σ2 (see the Eq. (5)). Fc - the Coulomb force, FvdW - the van der Waals force. 4

Е.V. Rosenfeld and А.V. Zakharov

Icarus 338 (2020) 113538

� hnþ ðd; tÞ i � hn ðd; tÞ i � hnðd; tÞ i ¼ πd2 jph t 4e � 2⋅1013 d2 t

it takes rather more time for larger and smaller particles. Although according to the formula (9), the amplitude of the charge fluctuations increases monotonically with time, the charge of any area or grain of dust cannot grow indefinitely. There must always be a mechanism to suppress fluctuations because of the action of which their lifetime and therefore their amplitude is limited. In our case, this is the Coulomb mechanism discussed in the next section. This mechanism re­ lates to the fact that the positively charged spots attract, and the nega­ tively charged ones repel the photoelectrons falling back on the surface. (It should be noted that it is not the only possible mechanism: in particular, if the plane has a non-zero conductivity, the surface current will also contribute to the attenuation of the fluctuations).

(7)

(t in seconds, d in meters). It means, that every second an additional number of positive and negative charges (7) falls on any site, and this number is 2,5 104 times higher than the average number (6) of excess charges on the site at the moment. Although the average values of nþ(d, t) and n (d, t) grow linearly with time, their instantaneous values change randomly, since the drop of both electrons and light quanta, like the emission of photoelectrons, are purely random processes. The appearance of a hole increases the charge of the area by e, and the fall of the photoelectron decreases its charge by e, which makes the process quite similar to the process of coin flipping there are “random walks” of the charge (see e.g. [Weiss and Rubin, 1982]). So, the charge of any small enough area must constantly change not only its magnitude but also the sign. One “step of walk”, i.e. change of a charge on �e, on average takes time � �2 � � δtðdÞ ¼ 4e πd2 j � 4⋅ 10 7 d s ph

5. Coulomb mechanism of attenuation of the charge fluctuations The linear with time increase of the density of positive and negative charges ν� (7) will occur only if the probabilities of knocking out an electron in the area and it falling back into the area are the same. This is true for a neutral area (or a dust particle), but when uncompensated charges appear, the situation changes. Ceteris paribus, the probability of an electron falling into a positively charged area must be higher than the probability of photoemission (charge independent), and vice versa for a negatively charged area. This means that the “walks” of the charge ceases to be random because a force tending to return the total charge of an area to zero appears. It is important to emphasize here that the case under consideration is radically different from the case of charge fluctuations on a solitary dust particle in a sufficiently dense plasma [e.g. Khrapak et al., 1999; Vaulina et al., 1999; Robertson et al., 2003; Flanagan and Goree, 2006; Sheridan and Hayes, 2011]. In the last case, it is usually assumed that the charge q is uniformly distributed over the entire surface of the spherical dust particle so that the potential is the same at all points. In addition, it is believed that the free path of electrons and ions is small compared to the characteristic size of the region in which the dust creates a field of noticeable magnitude. This means that the probability of hitting of a species with the surface of the dust particle is proportional to the � � Boltzmann factor exp �eφ=k T , that allows you to calculate the

(8)

This is of the order of 0.04 s for a micron area, 4 s for a 100 nm diameter area and about 6–7 min for a 10 nm diameter area. As a result, the charge of a particular sign should appear not on a small percentage, but on almost every dust particle with a diameter of 10 nm or more within a few minutes after sunrise and the photocurrent appearance. However, the total amount of dust particles with an excess of electrons is slightly less than with their deficiency, so the sums of charges of dust grains with different signs almost exactly compensate each other. It is just the reason why the average value of the charge of any area (or dust particle) (6) is very small in comparison with the charge of each sign (7) accumulated in this area during a time interval large in comparison with δt(d) (8). Further, since the values nþ(d, t) and n (d, t) change randomly, more and more noticeable fluctuations in the magnitude and the sign of the instantaneous value δn(d, t) ¼ nþ(d, t) n (d, t) of their difference should appear with time. It is well known (see, for example, [Van Kampen, 2007] and references therein) that for purely random walks the rootmean-square deviation increases in proportion to the square root of the “number of steps”. In our case, for an area (or for a single dust particle) with a diameter d, the number of steps taken in time t is the ratio 2 t=δtðdÞ or simply 2hn(d, t)i (7). Therefore, the root mean square

B

electron and ion currents to the surface of the dust particles. In our case, the extremely rarefied plasma is practically collisionless and the free path of its species (photoelectrons þ solar wind) is astro­ nomically large. Therefore, the Boltzmann distribution is not applicable when calculating current densities flowing to microscopically small spots randomly distributed over the charged surface. Therefore, it is necessary to directly consider the curvature of the trajectories of elec­ trons and ions when calculating the probability of their hitting the surface of a charge spot. In fact, this appears to be an attempt to make analytical calculations alike the particle in cell method. Of course, in this situation, the real computer simulation could give the most accurate description of the dynamics of surface charge fluctuations, but we try to derive principal laws of this process in an analytical form, using the methods of stochastic differential equations [Øksendal, 2000]. For this purpose, we will imagine a plane that contains randomly distributed oppositely charged spots with approximately the same total amounts of positive and negative charges. In general, we obtain some­ thing like a chessboard with crooked cells (areas) that have a positive or negative charge according to their color. Near the plane, at an altitude h that is less than or comparable to the characteristic cell size d, the electric field created by this charge distribution will be extremely inhomogeneous. At this height, only the charges, which belong to a single cell, are close enough to the observation point and above the positive cells, E is directed up, while above negative cells it is directed down. However, at a greater distance from the surface h> > d, a plu­ rality of neighboring areas with charges of different signs produce in the observation point the fields approximately equal in magnitude, but opposite in direction. As a result, as h increases, the random

(RMS) σ(d, t) defining the value of the number of excess elementary charges (of any sign!) averaged at the moment t over the regions (dust particles) of diameter d, is given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE ffiffi rD pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffi σ ðd; tÞ ¼ ½nþ ðd; tÞ n ðd; tÞ �2 � 2hnðd; tÞ i � 6⋅106 d t (9)

(time in seconds, diameter in meters). It is easy to estimate the time tlevit(d) required to accumulate a charge sufficient for levitation a dust particle of the diameter d in the field E ¼ 10 Vm 1: �

σ ðd; tlevit Þ � d3 ρgL 2eE

⇒

pffi d2 t � 3⋅1015 E

⇒ tlevit ðdÞ � 105 d4

(10)

(time in seconds, diameter in micrometers). So, in the purely random “walks” of a charge, for a particle with d � 0.1 μm, it will take a few seconds between the start of the illumination and the instant when it emits two photoelectrons and is ready to levitate. For a particle with d ¼ 200 nm which should obtain the charge 12e it will take about 3 min, and for d ¼ 1 μm it will take about the Earth day. Note that the formula (10) does not work for the smallest dust grains with d < 85 nm (remember: this is an evaluation for E ¼ 10 V m 1). In the absence of adhesive forces, such particles must emit a single photoelectron for take-off, and the probability of this is proportional to d2, see (8). This means that almost immediately after the appearance of the photocurrent with the density (5), dust particles with a diameter <80�90 nm are ready to levitate, and 5

Е.V. Rosenfeld and А.V. Zakharov

Icarus 338 (2020) 113538

. average, the same number of holes Sjph e is produced on any surface area S over time, but the number of electrons falling in this area depends on its charge. It is clear that the direction of the velocities of the lifting-off photoelectrons is randomly distributed around the direction of the sur­ face normal. Therefore, the electrons emitted from any neighboring area, but not too far removed, can fall on any site. This means that in the equilibrium state, the number of electrons which take off from any re­ gion, on average equals the number of electrons (not necessarily the same electrons) which fall back and the fluctuations in these flows arise independently of each other. For simplicity, we assume that the uniform electron flux falls strictly vertically at a height h > > d, and the electron trajectories are curved only near the surface where the field E|| is significant, see Fig. 2. Then . only SBN jph electrons will fall on the negatively charged area S within a e . second, and the remaining ðS SBN Þjph e electrons directed toward it will be deflected by the E|| field and will fall on neighboring positively charged sites. If, as we have assumed above, the areas of positively and negatively charged sites are the same, then as many extra electrons arrive in each positively charged region as those that do not reach the negatively charged area. Taking this into account, the rate of change in the number of elementary charges in any area is

contributions are more and more compensated and for h > > d the field turns into the field of a uniformly charged plane Eplane ¼ Q=2ε S where S 0

is the area of the plane, Q is its total charge. In the ideal case when all cells on the surface are squares, have the same size and have the same on the modulus charges, it can be shown that the intensity of the field created by the cells will decrease expo­ nentially with distance from the plane [Rosenfeld, 2000]. This means that the component of the field strength E|| parallel to the surface is maximal near the surface close to the boundaries between the oppositely charged areas. On the contrary, E|| practically disappears around the center of a spot and anywhere at altitudes much larger than the di­ mensions of the fluctuating areas. The component of the field perpen­ dicular to the plane E? is directed upwards above the positively charged cells, downwards over the negatively charged cells, and is maximum near the cell centers. With increasing h, the inhomogeneities of this component also disappear rapidly and it approaches a uniform value Eplane. The most important role of the E? component is that it creates the force that lifts particles from the surface. Additionally, it slows down or accelerates the electrons falling on the surface in the last segment of the flight. However, in the simplest approximation, this change of the ve­ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi locity can be neglected in comparison with the speed v ¼ 2eV=m that e the photoelectron returning to the surface obtains inside DEL (V is the potential difference within DEL). The E|| component of the field plays a much more important role in the curvature of electron trajectories near the surface. It is this compo­ nent of the field that deflects the trajectories of the falling electrons from the negatively charged spots and attracts them to positively charged areas or particles. Fig. 2 presents an idealized picture of photoelectron trajectories when they fall onto the surface. At high altitude h > > d electrons fall vertically along straight lines, which are called “sighting lines” in me­ chanics. The falling electrons will reach a negatively charged sites only if their sight lines pass through a “bottleneck” (BN) area, with a crosssection SBN. Otherwise, the field E|| will –– their trajectories suffi­ ciently that they will fall on the positively charged areas. Thus, the probability of an electron falling into an area depends on the charge of this area that changes its value by –e for each electron hit. At the same time, the probability of a photon falling into this area and resulting in the appearance of an additional charge þe does not depend on the magnitude and sign of the surface charge. Consequently, on

d δn ¼ dt

� jph ΔS esignðδnÞ;

ΔS ¼ S

Sbn

(11)

In order to obtain a rough estimate of the magnitude ΔS, let us as­ sume that the entire charge δq ¼ eδn of each square “cell” with the edge d is concentrated at its center. Then in the middle of the edge, which is the boundary of two “cells” with charges of different signs Ek � 2

e

πε0 d2

δn

(12)

where the coefficient 2 takes into account that two neighboring oppo­ sitely charged areas create equally directed fields E|| on the boundary. Assuming now that the field intensity of this component initially changes slowly with height, and then decrease sharply to zero at an altitude h, equal to a few d, the time during which the field (12) accel­ pffiffiffiffiffiffiffiffiffiffiffiffiffi erates an electron is of the order h=v � h m=2eV , and the distance by which it moves

Fig. 2. The uniform flow of mono-energetic photoelectron streams down to a plane where there are oppositely charged areas. Only those electrons whose sight lines pass through BNs can enter the negatively charged surface regions. The remaining electrons are deflected by the E|| field component and fall onto the positively charged areas. 6

Е.V. Rosenfeld and А.V. Zakharov

(13)

σ ∞ ðdÞ ¼ σ ðd; τÞ �

ΔS � π dΔr ¼

eh2 jδnj 2ε0 Vd

(14)

This is an extreme approximation, although the proper choice of parameter h should minimize its limitations. We emphasize, that we do not want exact quantitative results, but are interested in general quali­ tative conclusions. The approximation (14) is introduced because it al­ lows us to obtain an analytical formula. Substituting (14) into (11), we obtain the radioactive decay equation: jph h2 δn 2ε0 Vd

(15)

=

d δn ¼ dt

� 100 nm, this ratio is about 700,000. It is for this reason that any dust particle of a sufficiently small size lying on the surface can accumulate a charge sufficient for levitation (the result obtained by Sheridan (2013)), and this solves the first of the problems formulated in the Introduction.

6. The dependences of the charge fluctuation amplitude and the lifetime on the size of a fluctuating spot The Eq. (15) predicts that any initial excess charge magnitude will decrease exponentially with time: � � 2ε0 V mε0 � δnðtÞ ¼ δn0 exp (16) t=τ ; τ ¼ v0 2 d ¼ j p2 d ep2 j ph

7. Coulomb force raising a dust particle above the surface Of course, the Eq. (18) is semi-phenomenological, but it gives almost the correct result. A much more rigorous description of the stochastic process discussed above is given in Appendix A, and it is easy to see that pffiffiffi the Eqs. (18) and (A6) differ only in the numerical factor 2. Thus, it can be expected that the average number of excess charges e for the same values of the parameters V � 10 V and p2 � 10 on the dust particle with d~100 nm will be around 3, with d~1 μm – about 10, etc. If a grain has acquired a positive charge, in the field of the double layer EDEL � 10 Vm 1 it is risen above the surface by the lifting force σ ∞eEDEL. Equating the lifting force to the gravity, we get a rough estimate for the maximum size of the dust, capable of levitating (in fact, it’s the same reasoning that was applied to the Eq. (10)):

ph

It is important to note that the relaxation time τ is proportional to the square of the falling electron velocity v0, or, what is the same, the kinetic energy of the photoelectron at the moment when it lifts off the surface. The reason is that the higher the velocity, the shorter the time during which the field E|| deflects electrons, and therefore the smaller the dis­ tance that they will be deflected. This leads to an important conclusion: the higher the velocity of the incident electrons, the slower the charge fluctuations will be “dissolved”, and therefore the fluctuations can reach a higher maximum charge. It should be noted that the notation p ¼ h= is introduced in the Eq. d (16). This value shows how many times the height h, at which the fluctuating component E|| of the electric field disappears, is larger than the size of the fluctuating region d. Since h increases with increasing d, we can assume that the dependence p(d) is weaker than the dependence h(d). The parameter p has a value about a few units, but it is difficult to give a more precise value prior to computer simulation of the process. Thus, taking into account (5), we obtain for the relaxation time the expression V s; p2 d

1 ρgL dmax 3 ¼ 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vd EDEL πε0 e max 2p2

#15

" ⇒ dmax �

2πε0 eV ðpρgL Þ2

EDEL

2

� 130 nm (19)

We emphasize that this formula presents only an extremely rough estimate and does not define the exact maximum size of the flying particles, but the “boundary” size, after which the probability of a par­ ticle levitating starts to drop rapidly. As to the second problem mentioned in the Introduction, to deter­ mine the amplitude of surface charge density fluctuations Σ(d) in the case of stationary fluctuations σ∞(d) should be divided by the area πd2 4 : rffiffiffiffiffiffiffiffiffiffiffiffi 4σ ∞ ðdÞ 1 4ε0 eV 10 15 104 ΣðdÞ ¼ � 2 3 C m 2 � 1:2 3 e m 2 (20) e ¼ 3 2 2 πd2 π d d2 pd

(17)

=

τ � 4⋅103

(18)

The physical meaning of this expression is very simple. σ∞(d) de­ termines the variation of the number of excess charges δn ¼ nþ n from one area of diameter d to another area of the same diameter. As we pffiffi discussed in Section 3, the standard deviation σ is proportional to d t for “random walks” when the increase and decrease of the charge under fluctuations are purely random. If the regular relaxation process (16) limits the lifetime of the fluctuations, then the charge increases only during a time of the order of τ(d), and from the Eqs. (16) and (9) we obtain the formula (18), where the symbol ∞ indicates that this is a steady-state fluctuation mode that is set sometime after the start of illumination, when all transients have ended. It should be noted, that the ratio of the amplitude of the charge fluctuations σ ∞(D) in a small region (or on a dust particle) with the characteristic size D to the average value of the charge hq(d)ie 1 (2) in � such regions is about 2⋅10 5 d32 . For example, for dust particles with d

where eV is the energy of photoelectrons emitted from the regolith, d is the characteristic size of the fluctuating region in nanometers and p2~10. According to this formula, at V~10 V, the lifetime τ of fluctuations for d ~ 10 nm is several minutes or is of the order of tens of seconds if d ~ 100 nm, and is a few seconds for micron-size areas. It is quite natural that the lifetime of the fluctuating region decreases with the growth of its size, but the explicit dependence τ on d and on the energy of pho­ toelectrons is determined by a specific relaxation mechanism. Besides this relatively slow unidirectional change of charge with time, fluctuations will appear and rise due to the impacts of electrons on the surface and the photoelectron emissions which occur randomly. Therefore, the Eq. (16) should be combined with the Eq. (9), which determines the time dependence of the RMS of the number of excess charges in the region:

=

=

It is clear that at different points on an edge separating cells with charges of different signs, the field will be different, in particular, in the corners of the cells it disappears. To simplify the problem we can approximate the cells as circles with diameter d, and E|| on its boundary defined by (12). Then, ΔS is giving as:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi πε0 Vd 4 Vd � 1:3⋅10 p e p2

=

eh2 δn 2πε0 Vd2

=

Δr ¼ at2 2 �

Icarus 338 (2020) 113538

=

This means that the average local density of excess elementary charges hν(d)i on the area (or dust particle) with a diameter of d~100 nm is approximately 4 ⋅ 1014m 2 and as the diameter increases, this value will decrease ∝d 2 to the macroscopic average value hνi � 109m 2 (1). These two values different almost by six orders of magnitude and respectfully the strength of the field directly above the central part of a charge spot with a diameter of 100 nm rises up to Elocal~ 107 Vm 1. Now, inserting the Eq. (20) into (3) and assuming the dust speck diameter 2R is half the diameter of the fluctuating area d, we obtain the Coulomb force 3

=

7

Е.V. Rosenfeld and А.V. Zakharov

FðR � :25dÞ � :03

eV 10 20 � 2 pd d

Icarus 338 (2020) 113538

levitation of the lunar dust. In the Conclusion, we will present a quali­ tative discussion of the analytical treatment of charge fluctuations on a dielectric surface illuminated by UV radiation considered in this article. Two circumstances are crucial for understanding the role of charge fluctuations. On the one hand, the charge density averaged over any macroscopic area of the planet’s surface is extremely small (this value determines the intensity of the macroscopic electric field above the surface). In the midday regions of the moon’s surface, the average excess charge density should be about one e per 1000 μm2, see the Eq. (1). On the other hand, the huge intensity of sunlight leads to the fact that ~30 photoelectrons take off from every μm2 surface every second, and the same number falls back, see the Eq. (2). Since each photoelectron take-off/recovery changes the charge of the surface by �e, practically every speck of dust on the surface acquires a charge of one or another sign for a short period of time. The number of negatively charged dust particles is slightly less than the number of positively charged ones, which ensures the constancy of the average value of the charge density. It is here that the fundamental difference between the case under consideration and the charge fluctuations on a shallow solitary dust particle in the plasma is covered. Really, a solitary particle acquires many (hundreds or even thousands) electrons, and the charge fluctuations caused by the fall of new electrons / ions onto its surface are not significant. Their amplitude is proportional to the square root of the excess charge number and so is much less than the latter. On the other hand, the average number of excess charges on a small area of the flat surface is negligibly small compared to 1 while the appearance of any - positive or negative - charge of several e, or even several dozens of e on the area, is quite possible. This solves the first problem discussed in the Introduction - the accumulation of a charge sufficient for levitation. To solve the second problem discussed there and determine the source of the very large force that overcomes the adhesive forces, it is necessary to take into account the stochastic nature of the processes of emission and return of photoelectrons. In the simplest case, when we imagine a surface completely covered with specks of dust, we must recognize that not only the magnitude but also the sign of the charge of any of them, is largely a random variable. In particular, this means that the signs of the charges of two adjacent dust particles do not necessarily have to be the opposite. On the contrary, two or more adjacent dust particles can quite easily acquire charges of the same sign, so that the entire surface, regardless of whether it is covered with dust or not, should be covered with charge spots of different shapes and sizes. It is clear at the same time that the larger the size of such a spot, the less is the probability of its appearance, so there should be a characteristic size of the charge spot, determined by the conditions of the appearance of charges of different signs on the surface. Moreover, the appearance of such charge spots is no way connected with the presence or absence of dust on the surface. They should occur in all cases when positive and negative charges occur randomly on the surface, for example, on a solid dielectric surface in contact with a plasma cloud. In this case, near the center of such a charge spot above the surface, there must exist an electric field perpendicular to the sur­ face, the direction of which is determined by the sign, and the strength value is proportional to the local density of the surface charge Σ inside the spot. On the illuminated side of the Moon, the value of Σ is several orders of magnitude higher than the average value of the charge density across the surface. Therefore, the electrostatic force acting on a dust particle lying near the center of a spot and having the same sign of the charge can be millions of times greater than the force of gravity acting on it and, apparently, is able to overcome the van der Waals force. As a result, a speck of dust with a charge of any sign is able to break away from the surface. However, as the particle rises, the electrostatic field strength de­ creases rapidly, and already at a height comparable to the characteristic size of the charge spot, only the average field of the double layer acts on the dust particle. It raises and accelerates dust particles with a charge of one sign (the fountain mechanism by Stubbs et al., 2006), and dust

(21)

Its value for a particle with a size d ~ 10 7 m is about 0.1 pN. Therefore, although the value of F is six orders of magnitude greater than the magnitude of the gravity acting on a particle of this size in the lunar condition, it is too small in comparison with the estimates by Flanagan and Goree (2006); Sheridan and Hayes (2011); Sheridan (2013). In this regard, following to Sheridan and Hayes (2011) it necessary to draw attention to two circumstances. On the one hand, the strength of the local field Elocal~107 Vm 1 obtained by us is rather large. It would be strange to expect that hundreds of times stronger fields may appear on the surface due to some reason, otherwise this would lead to fields capable of ripping atoms away from solids. On the other hand, it should probably be expected that the surface of natural dust particles has extremely irregular shapes with sharp edges. This means that the area of contact between the grains hardly correlates with their size. If this is so, then the measured values of the van der Waals force, carried out for particles with a smooth surface, can be very highly overestimated for lunar dust particles [Li et al., 2006; Sheridan and Hayes, 2011]. It is easy to see that the mechanism of the occurrence of the lifting force considered above applies both to perfectly flat surfaces and extremely uneven surfaces characteristic of lunar regolith and other airless bodies. When calculating the force acting on a dust speck lying in the center of the flat “charge ring” surrounding it (Eq. (3)), we consider that the speck charge concentrates in its center located at a height R above the plane (see Fig. 1). If we assume that the charge concentrates at the point of contact of the dust particle with the plane (which, of course, is absurd), then no force will act on it at all. If we additionally take into account, that one part of the electrons/ holes inside “the charge ring” lies above the plane of the latter and the other part below it, then nothing will change fundamentally. As before, the interaction with the nearest neighboring charges will be the main contribution to the force acting on the dust particle. If all these charges are on the dust specks lying above our “central” speck, then, of course, the downward force will act on the last. However, the force directed upwards will act on all those dust particles that lie above the neigh­ boring ones. It is easy to estimate the order of magnitude of this force. Let us consider two dust particles neighboring in the plane of the surface with the same charges q and diameters d, one of which lies slightly higher than the other. Then the significant part of the force of their Coulomb repulsion effects the first of them. At d �0.1 μm and q�10e, these dust particles repel with the force F¼

q2 2:5⋅10 36 � 1010 ¼ 2:5 pN 4πε0 d2 10 14

which coincides in order of magnitude with the above estimates. For most dust particles, a part of the neighbors will lie above, and the other part will lie below them. The result of the addition of several closes in modulus, but multidirectional forces will have approximately the same order of magnitude. However, dust particles that are on the tops of “cones” consisting of particles of the same charge will be affected by a significantly higher lifting force. Thus, taking into account the surface roughness of the dust layer within “the charge spot” introduces an additional factor that increases the spread in the magnitudes of the lifting forces acting on them. Probably, any reliable estimate of this factor can be obtained only using numerical calculations. However, this factor cannot change in any way either the mechanism of the appear­ ance of “charge spots” on the surface, or the order of magnitude of the average value of the lifting force acting inside these spots. 8. Conclusions So, charge fluctuations on non-conducting surfaces play a funda­ mentally important role for the work of the electrostatic mechanism to takeoff and levitation dust particles on airless bodies, in particular, the 8

Е.V. Rosenfeld and А.V. Zakharov

Icarus 338 (2020) 113538

adhesive van der Waals force.

particles with a charge of another sign fall back to the surface with some shift. Maybe, it’s just the reason for dust transport along the surface. The analytical approach that we used in this paper and numerical estimates made on the basis of the results of this analysis shows that local charge fluctuations on the illuminated surface of the Moon are able to create the lifting force, the value of which (about 0.1 pN) is close to the lower boundary of the available magnitude estimation of the

Acknowledgments The research was carried out using funds of the Russian Science Foundation (project RNF N�17-12-01458). The authors are grateful Prof. Thomas Duxbury for his help in the preparation of this paper.

Appendix A Stochastic differential equations are used to describe systems in which the increment dx of a physical quantity x in time dt is determined not only by the usual smooth processes occurring in the system but also by fluctuations: (A1)

dx ¼ aðx; tÞdt þ fluctðx; t; dtÞ

The first term describes the usual smooth change occurring at a rate a(x, t) proportional to dt. Considering the Wiener process, i.e. a continuous random walk where the duration and length of the step tend to zero, one obtains pffiffiffiffi fluctðx; t; dtÞ ¼ σ ðx; tÞδW; δW ¼ ε dt (A2) where f is the standard deviation (SD) and ε is a Gaussian random variable with a zero mean and a unit variance. The solution of the Eqs. (A1) and (A2) is not one curve, but an array of an infinite number of curves. Each curve describes the behavior of one of the (identical) systems of the ensemble originally located in the same initial states and evolved under the same conditions. In this case, measuring the value of x for each system at time t, we obtain the Gaussian distribution with the mean hx(t)i and varianceσ2(t). The value of x for each individual system will then differ from hx(t)i by a random amount εσ(t) so that the graphs of the values of x (t) for different systems (in our case – charged spots) will lie near the line hx(t)i, most inside a neighborhood with a width of 2σ(t). The theory of stochastic differential equations [Øksendal, 2000] allows us to calculate the time dependences of the mean value δn(t) and SD σ (t) pffiffiffiffi knowing the time dependences of the coefficients a(δn, t) and b(δn, t) before dt and before the infinitesimal increment of the Wiener noise ε dt in the equation pffiffiffiffi dðδnÞ ¼ aðδn; tÞdt þ bðδn; tÞε dt (A3) This equation is not an ordinary differential equation, it has a formal nature, and a(δn, t) and b(δn, t) represent the rates of “regular” and “sto­ chastic” change of δn at the time t, respectively. In our case, these quantities have already been determined (see the Eqs. (16) and (9)): p2 jph d ðδn hδni Þ � 2ε0 V rffiffiffiffiffiffiffi πjph bðn; tÞ ¼ d � 7⋅106 d: 2e

aðn; tÞ ¼

2:5⋅105

p2 d ðδn V

hδni Þ; (A4)

Substituting these results in (A3), one can neglect the extremely small average value of the number of elementary charges hδni ¼ hqðdÞ i= (see (6) e and the text relating to this formula) in comparison with its instantaneous value δn(t). As a result, we obtain a standard equation describing the Ornstein-Uhlenbeck process dðδnÞ ¼

δn

dt

τ

(A5)

þ σ0 dW:

The solution of this equation [Øksendal, 2000] has the form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffi � � 2 t=τ ε; t=τ þ σ∞ 1 exp s s ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi rffiffiffiffiffiffiffiffi 1 τðdÞ πε0 Vd Vd 4 � 10 � 104 d 2 : ¼ σ∞ ðdÞ ¼ σ0 ðdÞ 2 2ep2 p2 δnðtÞ ¼ δn0 exp

(A6)

=

Accordingly, the spectral function of the fluctuations (that is, the Fourier transform of the autocorrelation function) is a Lorentz function Z 1 ∞ 1 σ 20 SðωÞ � covðtÞexpðiωtÞdt ¼ (A7) π ∞ π ω2 þ τ 2 Here we use the standard notation covðt1 ; t2 Þ ¼ hðδnðt1 Þ

hδnðt1 Þ i Þðδnðt2 Þ

hδnðt2 Þ i Þ i

given that cov(t1, t2) ¼ cov (t1 t2) in the case of a stationary process. In our case, the meaning of the spectral function S(ω) is very simple. Since the function δn(t) (A6) describes the time dependence of the excess charge in the area under consideration, its sign changes with time and δn(t) can be represented as a sum of sinusoids with different periods T ¼ 2π= ,

ω

and S(ω) determines the probability that a fluctuation with frequency ω will be represented in this expansion. Thus, S(ω) determines the characteristic range of frequencies of charge oscillations in the area. It can be seen from the Eq. (A7) that all low-frequency oscillations with a period exceeding 2πτ (d) are approximately equally probable, but the probability of more rapid oscillations falls off rather sharply with frequency. 9

Е.V. Rosenfeld and А.V. Zakharov

Icarus 338 (2020) 113538

References

Poppe, A., Hor� anyi, M., 2010. Simulations of the photoelectron sheath and dust levitation on the lunar surface. J. Geophys. Res. 115, A08106 https://doi.org/ 10.1029/2010JA015286. Poppe, A.R., Piquette, M., Likhanskii, A., Hor� anyi, M., 2012. The effect of surface topography on the lunar photoelectron sheath and electrostatic dust transport. Icarus 221, 135–146. Rennilson, J.J., Criswell, D.R., 1974. Surveyor observations of lunar horizon-glow. Moon 10, 121–142. Robertson, S., Gulbis, A.A.S., Colwell, J., Hor� anyi, M., 2003. Dust grain charging and levitation in a weakly collisional sheath. Phys. Plasmas 10, 3874. Rosenfeld, E.V., 2000. Calculation of the field of a lattice point magnetic dipoles. Phys. Solid State 42 (#9), 1680–1687. Rosenfeld, E.V., Zakharov, A.V., 2018. Dust shedding from a dielectric surface in plasma as a result of charge fluctuations. Phys. Plasmas 25, 103703. https://doi.org/ 10.1063/1.5029562. Rosenfeld, E.V., Korolev, A.V., Zakharov, A.V., 2016. Lunar nanodust: is it a borderland between powder and gas? Adv. Space Res. 58, 560–563. Schwan, J., Wang, X., Hsu, H.-W., Grun, E., Horanyi, M., 2017. The charge state of electrostatically transported dust on regolith surface. Geophys. Res. Lett. 44, 3059–3065. https://doi.org/10.1002/2017GL072909. Sheridan, T.E., 2013. Charging time for dust grain on surface exposed to plasma. J. Appl. Phys. 113, 143304. Sheridan, T.E., Hayes, A., 2011. Charge fluctuations for particles on a surface exposed to plasma. Appl. Phys. Lett. 98, 091501. Singer, S.F., Walker, E.H., 1962. Electrostatic dust transport on the lunar surface. Icarus 1, 112–120. Stubbs, T.J., Vondrak, R.R., Farrell, W.M., 2006. A dynamic fountain model for lunar dust. Adv. Space Res. 37, 59–66. https://doi.org/10.1016/j.asr.2005.04.048. Stubbs, T.J., Farrell, W.M., Halekas, J.S., Burchill, J.K., Collier, M.R., Zimmerman, M.I., Vondrak, R.R., Delory, G.T., Pfaff, R.F., 2014. Dependence of lunar surface charging on solar wind plasma conditions and solar irradiation. Planet. Space Sci. 90, 10–27. Van Kampen, N.G., 2007. Stochastic Processes in Physics and Chemistry, Revised and Enlarged Edition. Elsevier, North-Holland, Amsterdam. Vaniman, В., Reedy, R., Heiken, G., Olhoeft, G., Mendell, W., 1991. The lunar environment. In: Heiken, G.H., Vaniman, D.T., French, B.M. (Eds.), Lunar Sourcebook. Cambridge Univ. Press, pp. 27–60. http://www.lpi.usra.edu/publicatio ns/books/lunar_sourcebook/. Vaulina, O.S., Nefedov, A.P., Petrov, O.F., Khrapak, S.A., 1999. Role of stochastic fluctuations in the charge on macroscopic particles in dusty plasmas. J. Exp. Theor. Phys. 88 (6), 1130–1136. � ankov� Vaverka, J., Richterov� a, I., Pavlu, J., Safr� a, J., N� eme�cek, Z., 2016. Lunar surface and dust grain potentials during the Earth’s magnetosphere crossing. Astrophys. J. 825, 1331-10. https://doi.org/10.3847/0004-637X/825/2/1331. Walch, B., Horanyi, M., Robertson, S., 1994. Measurement of the charging of individual dust grains in a plasma. IEEE Trans. Plasma Sci. 22 (2), 97–102. Wang, X., Pilewskie, J., Hsu, H.-W., Hor� anyi, M., 2016. Plasma potential in the sheaths of electron-emitting surfaces in space. Geophys. Res. Lett. 43, 525–531. https://doi. org/10.1002/2015GL067175. Wang, X., Schwan, J., Hsu, H.-W., Grün, E., Hor� anyi, M., 2016. Dust charging and transport on airless planetary bodies. Geophys. Res. Lett. 43, 6103–6110. https:// doi.org/10.1002/2016GL069491. Weiss, G.H., Rubin, R.J., 1982. Random walks: theory and selected applications. In: Prigogine, I., Rice, S.A. (Eds.), Advances in Chemical Physics, 52, pp. 363–505. https://doi.org/10.1002/9780470142769.ch5. Zimmerman M. I., Farrell W. M., Hartzell C. M., Wang X., Horanyi M., Harley D.M., Hibbits K., 2016. Grain-scale supercharging and break down on airless regoliths, American Geophysical Union, die: https://doi.org/10.1002/2016JE005049. Zook, H.A., McCoy, J.E., 1991. Large-scale lunar horizon glow and a high altitude lunar dust exosphere. Geophys. Res. Lett. 18, 2117–2120.

Arnas, C., Mikikian, M., Bachet, G., Doveil, F., 2000. Sheath modification in the presence of dust particles. Phys. Plasmas 7, 4418. Bernstain, W., Fredricks, R.W., Vogl, J.L., Fowler, W.A., 1963. The lunar atmosphere and the solar wind. Icarus 2, 233–248. Carrier, I.I.I.W.D., Olhoeft, G.R., Mendell, W., 1991. Physical properties of the lunar surface. In: Heiken, G.H., Vaniman, D.T., French, B.M. (Eds.), Lunar Sourcebook. Cambridge Univ. Press, p. 475. http://www.lpi.usra.edu/publications/books/luna r_sourcebook/. Colwell, J.E., Batiste, S., Horanyi, M., Robertson, S., Sture, S., 2007. Lunar surface: dust dynamics and regolith mechanics. Rev. Geophys. 45, RG2006. Criswell, D.R., 1973. Horizon-glow and the motion of lunar dust. In: Grard, R.J.L. (Ed.), Photon and Particle Interactions With Surfaces in Space. Springer, New York, pp. 545–556. Criswell, D.R., De, B.R., 1977. Intense localized photoelectric charging in the lunar sunset terminator region. 2. Supercharging at the progression of sunset. J. Geophys. Res. 82 (7), 1005–1007. De, B.R., Criswell, D.R., 1977. Intense localized photoelectric charging in the lunar sunset terminator region. 1. Development of potentials and fields. J. Geophys. Res. 82 (7), 999–1004. Dove, A., Horanyi, M., Wang, X., Piquette, M., Poppe, A.R., Rebertson, S., 2012. Experimental study of a photoelectron sheath. Phys. Plasmas 19, 043502. https:// doi.org/10.1063/1.3700170. Feuerbacher, B., Anderegg, M., Fitton, B., Laude, L.D., Willis, R.F., Grard, R.J.L., 1972. Photoemission from lunar surface fines and the lunar photoelectron sheath. Geochim. Cosmochim. Acta 3 (Suppl. 3), 2655–2663. Flanagan, T.M., Goree, J., 2006. Dust release from surfaces exposed to plasma. Phys. Plasmas 13, 123504. Freeman, J.W., Ibrahim, M., 1975. Lunar electric fields, surface potential and associated plasma sheaths. The Moon 14, 103–114. Freeman, J.W., Fenner, M.A., Hills, H.K., 1973. The electric potential of the moon in the solar wind. J. Geophys. Res. 78, 4560–4567. Grobman, W.D., Blank, J.L., 1969. Electrostatic potential distribution of the sunlit lunar surface. J. Geophys. Res. 74 (16), 3943–3951. Hartzell, C., Scheeres, D.J., 2011. The role of cohesive forces in particle launching on the moon and asteroids. Planet. Space Sci. 59, 1758–1768. Hurley, D.M., Cook, J.C., Retherford, K.D., Greathouse, T., Gladston, G.R., Mandt, K., Grava, C., Kaufmann, D., Hendrix, A., Feldman, P.D., Pryor, W., Stickle, A., Killen, R. M., Stern, S.A., 2017. Contributions of solar wind and micrometeoroids to molecular hydrogen in the lunar exosphere. Icarus 283, 31–37. Khrapak, S.A., Nefedov, A.P., Petrov, O.F., Vaulina, O.S., 1999. Dynamical properties of random charge fluctuations in a dusty plasma with different charging mechanisms. Phys. Rev. E 59 (5), 6017. Li, Q., Rudolph, V., Peukert, W., 2006. London-van der Waals adhesiveness of rough particles. Powder Technol. 161, 248–255. Manka, R.H., 1973. Plasma and potential at the lunar surface. In: Grard, R.J.L. (Ed.), Photon and Particle Interactions with Surfaces in Space. D. Reidel Publishing Co, Dordrecht, Holland, pp. 347–361. Manka, R.H., Michel, F.C., 1973. Lunar ion energy spectra and surface potential. Proceed. of the Fourth Lunar Sci. Conf. 3 (Suppl. 4, Geochimica et Cosmochimica Acta), 2897–2908. Øksendal, B., 2000. Stochastic Differential Equations. Springer-Verlag, Heidelberg, New York. Opik, E.J., Singer, S.F., 1960. Escape of gases from the moon. J. Geophys. Res. 65, 3065–3070. Popel, S.I., Zelenyi, L.M., Atamaniuk, B., 2016. Plasma in the region of the lunar terminator. Plasma Phys. Rep. 42 (5), 543–548. Popel, S.I., Zelenyi, L.M., Atamaniuk, B., 2016. Dusty plasma in the region of the lunar terminator. Plasma Phys. Rep. 42 (5), 543–548.

10