- Email: [email protected]
Electrostatic dust transport on the terminator of atmosphereless bodies
Journal Pre-proof Electrostatic dust transport on the terminator of atmosphereless bodies C. López-Sisterna, R.A. Gil-Hutton PII:
S0032-0633(18)30346-5
DOI:
https://doi.org/10.1016/j.pss.2019.104775
Reference:
PSS 104775
To appear in:
Planetary and Space Science
Received Date: 3 October 2018 Revised Date:
2 October 2019
Accepted Date: 8 October 2019
Please cite this article as: López-Sisterna, C., Gil-Hutton, R.A., Electrostatic dust transport on the terminator of atmosphereless bodies, Planetary and Space Science (2019), doi: https://doi.org/10.1016/ j.pss.2019.104775. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Electrostatic dust transport on the terminator of atmosphereless bodies C. L´opez-Sisternaa,∗, R. A. Gil-Huttona a
Grupo de Ciencias Planetarias, Departamento de Geof´ısica y Astronom´ıa, Facultad de Ciencias Exactas, F´ısicas y Naturales, Universidad Nacional de San Juan - CONICET, Av. Jos´e I. de la Roza 590 (O), J5402DCS Rivadavia, San Juan, Argentina
Abstract Small dust particles may levitate over the surface of atmosphereless bodies, but to study electrostatic dust transport during a complete rotational period it is necessary to evaluate the surface potential at the terminator zone. We simulate the photo-electron layer and the dust charging process according to previously used models for dust levitation, but we propose a modification of the models to add the effects produced by surface roughness for low solar elevation angles. The correction for surface roughness on the model showed that a positive value for the surface potential and the electric field appear at the terminator. We found that submicron particles could be in stable levitation for small gravity fields. Our results also showed that the finest regolith grains can be lost into space due to the electrostatic repulsion. Keywords: Asteroids: general, dusty plasma, levitation, photoelectron emission ∗
Corresponding author Email addresses: [email protected] (C. L´opez-Sisterna ), [email protected] (R. A. Gil-Hutton)
Preprint submitted to Planetary & Space Science
October 16, 2019
1
1. Introduction
2
The observation of the Lunar Horizon Glow by the Surveyor spacecraft
3
(Criswell, 1972, 1973; Rennilson and Criswell, 1974) and by the Apollo 17
4
mission (McCoy and Criswell, 1974; Zook and McCoy, 1991) led to the de-
5
velopment of an electrostatic dust levitation theory. During the last 40 years
6
several models arose trying to describe the physics for charging in space and
7
dusty plasma physics (Criswell, 1973; De and Criswell, 1977; Criswell and De,
8
1977; Whipple, 1981; Goertz, 1989; Nitter and Havnes, 1992; Nitter et al.,
9
1994; Colwell et al., 2005; Hughes et al., 2008), and some of these models have
10
been proposed to explain the unusual observations on asteroids (Lee, 1996;
11
Colwell et al., 2005; Miyamoto et al., 2007) and also hypothesized to occur on
12
comets (Mendis et al., 1981). In addition, different laboratory experiments
13
have confirmed dust levitation under conditions that resemble atmosphere-
14
less solar system environments (Sickafoose et al., 2002; Sheridan and Hayes,
15
2011; Wang et al., 2016).
16
The surfaces of these bodies in interplanetary space are exposed to solar
17
wind plasma and solar UV radiation; generally on the sunlit side photoelectric
18
charging exceeds plasma charging, resulting on a positive surface potential
19
(Mendis et al., 1981; Lee, 1996). These effects contribute to the generation of
20
two important currents over the surface, responsible for developing a photo-
21
electron sheath over the lit side. In general, the upper layer of the regolith
22
on a small body may be considered as an effective insulator (Sheridan et al., 2
23
1992; Lee, 1996; Kimura, 2016). Therefore, regolith dust particles will have
24
a non-zero charge while resting on the surface and, under certain conditions,
25
the electric force on sub-micrometer particles would exceed the gravitational
26
force causing dust particles to detach from the surface and levitate.
27
This electrostatic mechanism could be responsible for the redistribution
28
of regolith particles over the surface of asteroids (Veverka et al., 2001; Col-
29
well et al., 2005), but there are different arguments about how dust levitation
30
works near the terminator region because in a smooth surface the light is al-
31
most tangent to the surface. For example, Criswell and De (1977) described
32
what they called supercharging effect near the day-night boundary: the pho-
33
toelectrons recently ejected from the lit side are attracted by the positive
34
charge that has just turn into the night side of a rotating body increasing
35
the potential of the illuminated regions. Furthermore, Lee (1996) proposed
36
that the prime charging process to consider at the terminator is the photo-
37
electric effect. This author considered solar wind currents to be negligible
38
on the trailing side owing to the velocity aberration arising from the orbital
39
motion of the asteroid. More recently, Hartzell and Scheeres (2011) analyzed
40
the role of cohesive forces on sub-micrometer particles. They found that
41
taking into account cohesion, levitation may be possible only for a range of
42
particles size near the terminator of the Moon and asteroids. On the other
43
hand, Kimura et al. (2014) proposed that electrostatic forces on irregular
44
grains with rough hydroxylated surface could overcame cohesive forces near
45
the terminator. 3
46
The process responsible for detaching the particles from the surface is
47
strongly dependent on the illumination conditions. Moreover, near the ter-
48
minator the surface roughness produces areas strongly illuminated that could
49
reach high potential and should have a strong influence on the electrostatic
50
interaction of the particles with their surroundings. This effect produced
51
by roughness is not taken into account by the available models (for example,
52
Colwell et al., 2005) producing a negative surface potential near the termina-
53
tor which prevents the use of these models to simulate processes that involve
54
a complete rotation of the object.
55
In this paper we present a new way of estimating the effects of the electric
56
field close to the terminator taking into account the mean surface roughness
57
of objects with micro-gravity. We describe the electrostatic transport model
58
in section 2. In section 3 we discuss our results and in section 4 we summarize
59
the conclusions.
60
2. Electrostatic transport model
61
To simulate the photo-electron layer and the dust charging currents over
62
the surface, we initially follow the simplified monotonically-decreasing sheath
63
potential model described by Colwell et al. (2005) and Hughes et al. (2008)
64
which is briefly explained below.
65
The model assumes that the solar UV radiation strikes the surface and
66
release electrons of at least the same energy of the incoming photons, but at
67
the same time the solar wind hit the surface producing an electron current. 4
68
If the photoemission process results to be more effective than the solar wind
69
charging process, then the sunlit surface becomes positively charged. The
70
surface will continue charging until the sum of the currents becomes zero.
71
Then, the surface potential, φs , is determined by equating the photo-electron
72
(Ipe ) and the solar wind (Isw ) currents at the surface (Whipple, 1981; Colwell
73
et al., 2005):
Ipe Isw
−qe φs cos θsun , = πrd qe Ipe0 exp kB Tpe r kB Tsw qe φs 2 = πrd qe nsw 1+ , 2πme kB Tsw
2
(1) (2)
74
where rd is the grain radius, qe is the absolute value of the electron charge,
75
me is the mass of the electron, kB is the Boltzmann constant, and θsun is the
76
complementary angle of the solar elevation angle. The photo-electron and
77
the solar wind temperatures are Tpe and Tsw , respectively, for which we used
78
values of 2.2 eV (Willis et al., 1973) and 10 eV, where the second value is
79
well below the energy range where secondary electron production becomes
80
important.
81
In order to calculate the photo-electron current at the surface (Ipe ), we
82
assumed that it has the same photo-emissive properties as the lunar regolith
83
(Willis et al., 1973; Sternovsky et al., 2002), but we take into account that the
84
usual value for the emitted photoelectrons proposed byWillis et al. (1973) was
85
underestimated by a factor of 2.1 (Senshu et al., 2015). Then, the parameters 5
86
used for the emitted photo-electron flux for normal incidence of light, Ipe0 ,
87
and the electron density of the solar wind, nsw , are scaled to its value at 1 au:
Ipe0 nsw
5.88 × 109 electrons = , d2sun cm2 s 5 electrons = 2 . dsun cm−3
(3) (4)
88
Besides, the photo-electron sheath is a region of increased electron density
89
near the surface formed as a consequence of the release of photo-electrons
90
and its electric attraction to the positive surface. Then the photo-electron
91
density is defined by:
npe0 = 92
2Ipe0 cos θsun , vpe
(5)
where vpe , the characteristic photo-electron velocity, is: r vpe =
2kB Tpe ≈ 880 km/s. me
(6)
93
Then, an electric field appears as result of the potential difference between
94
the surface and this sheath. Grard and Tunaley (1971) defined the electric
95
field and the photo-electron density as function of height (z), above the
96
surface:
6
√ −1 2 2φs ∆z E (∆z) = , 1+ √ λD 2λD −2 ∆z npe (∆z) = npe0 1 + √ , 2λD 97
(7) (8)
where: s λD =
ε0 kB Tpe , npe qe2
(9)
98
is the Debye length at the surface, which indicates the width of the photo-
99
electron sheath, and ε0 is the permittivity in vacuum.
100
2.1. Charge and dynamics of dust particles
101
Dust grains also release electrons through the photoelectric effect, and col-
102
lect solar wind electrons and photo-electrons from the photo-electron sheath
103
in a similar way as happens on the surface. This process is time dependent
104
and results in a variation of the grain charge, Qd , produced by the currents
105
balance: dQd = Ipe,d − Ie,d − Isw,d , dt
(10)
106
where Ipe,d , Ie,d and Isw,d are the currents due to photo-emission from the
107
particle, to the collection of photo-electrons emitted by the surface, and to
108
the capture of solar wind electrons, respectively. These currents depend on
7
109
the grain charge, and in the case it is charged positively:
−qe φd Ipe,d = πrd 2 qe Ipe0 exp , kB Tpe r qe φd 8kB Tsw 2 1+ , Isw,d = πrd qe nsw πme kB Tsw r 8kB Tpe qe φ d 2 1+ , Ie,d = πrd qe npe πme kB Tpe
(11) (12) (13)
110
where φd is the electric potential of the grain. In the case of a negative
111
potential, the currents are given by:
Ipe,d = πrd 2 qe Ipe0 , r
8kB Tsw qe φ d Isw,d = πrd 2 qe nsw exp , πme kB Tsw r qe φd 8kB Tpe exp . Isw,d = πrd 2 qe npe πme kB Tpe
(14) (15) (16)
112
If the electrostatic and the gravitational forces are balanced, the dust
113
grain levitates through the photo-electron sheath, but if an unbalance be-
114
tween these two forces occurs the charged particle moves in the radial direc-
115
tion following the equation of motion: d2 r = Qd E(∆z) − gA md , dt2
116
(17)
where gA is the object gravity, md is the grain mass, and E(∆z) is defined 8
117
by the equation 7.
118
2.2. Illumination excess due to roughness
119
If we assume in this model that the surface of the object is smooth, then
120
close to the terminator the photo-electron density and the photo-electron
121
current fall very fast and the surface potential reaches negative values for
122
θsun = 90 ◦ . However, the surfaces of solar system objects are highly irregular
123
showing macroscopic and microscopic roughness. This roughness could be
124
important in the regions near the terminator because it produces small areas
125
with orientations that are not normal to the radial direction, and then they
126
could appear illuminated by the Sun (Criswell, 1972, 1973). These small
127
regions undergo the usual process of gain and loss of electrons as the object
128
rotates, but in this case the illuminated regions constantly lose electrons
129
because they are more likely to be collected by large dark areas when they
130
fall again to the surface. As a consequence of this particular effect, the electric
131
field on the surface would increase rather than decline in the terminator zone
132
producing a greater radial acceleration than that expected from a smooth
133
surface (Eq. 17).
134
Since the effects of a rough surface near the terminator are the result
135
of the existence there of small illuminated areas which also contribute to
136
produce the total brightness of the object, we decide to estimate the value
137
of the electric field near the terminator using a formulation similar to that
138
given by Lumme and Bowell (1981a,b) to calculate the phase function for
9
139
atmosphereless bodies. According to these authors, the normalized phase
140
function for atmosphereless bodies is given by:
Φ(α) = (1 − G)Φ1 (α) + GΦM (α),
(18)
141
where α is the phase angle, G is a parameter which measures the ratio be-
142
tween the brightness produced by multiple scattering and the total bright-
143
ness, Φ1 is the phase function for singly scattered light due to a single parti-
144
cle, shadowing and roughness, and ΦM is the phase function due to multiple
145
scattering between particles. The functions Φ(α), Φ1 (α) and ΦM (α) are nor-
146
malized to their values at α = 0 ◦ .
147
148
Following the formulation originally proposed by Lumme and Bowell (1981a), we have:
Φ1 (α) = ΦS (α, D)ΦR (α, ρ)ΦP (α, σ),
(19)
149
where φS , φR , and φP are the normalized phase functions due to shadowing,
150
roughness, and single particle scattering, respectively. These functions de-
151
pend on the volume fraction D (or the porosity, 1 − D), the relation between
152
the depth and the radius of a typical irregularity on the surface (the rough-
153
ness, ρ), and the ratio between backscattering and forward scattering (the
154
asymmetry factor, σ). On the other hand, ΦM is not difficult to calculate
155
because it is similar to the phase function of a Lambert reflector:
10
φM ≈
1 [sin(α) + (π − α)(cos(α)], π
0 6 α 6 π.
(20)
156
This formulation can be used to derive a factor that allow us to estimate
157
the surface brightness, which is proportional to the illumination received at
158
different illumination angles. Although in the original formulation by Lumme
159
and Bowell (1981a,b) the G parameter depends on the wavelength used to
160
measure it, in our model it is used as a measure of surface roughness. Thus,
161
if G and the phase functions for a particular object are known and we assume
162
that α ≡ 180 ◦ − 2θsun , then the function:
ψ(θsun ) = Φ(α) − GΦM (α),
(21)
163
is the measurement of the relative contribution of the brightness (or the
164
illumination at constant albedo) due to shadowing and roughness. Then, the
165
photo-electron current and the photo-electron density modified to take into
166
account this effect are:
Ipe,r npe0,r
−qe φs = πrd qe Ipe0 exp [cos θsun (1 − ζ) + ζψ], kB Tpe 2Ipe0 [sin θsun (1 − ζ) + ζψ] = , vpe 2
(22) (23) (24)
167
where:
11
ζ=
2 (θ0 −θsun ) , for θsun > θ0 ; θ0 0,
168
(25)
otherwise;
and θ0 is defined as ψ(θ0 ) = 0.
169
The surface electric potential as function of the solar angle for a smooth
170
and rough surface are shown in fig. 1. These curves have a similar behavior
171
up to solar angles of ∼ 70◦ , but beyond this point the surface potential for
172
a smooth surface decrease to approximately φs ∼ −1.0 V at the terminator
173
while for an irregular surface it reaches a minimum at θsun = 85◦ and increase
174
again to reach at the terminator a surface potential of φs ∼ 2.0 V ( E(z =
175
0) ∼ 1 V /m). 6 Rough surface Smooth surface
Electric surface potential [V]
5
4
3
cc 2
1
0
-1 0
10
20
30
40
50
60
70
80
90
Solar incidence angle [Deg]
Figure 1: Variation of the surface electric potential with respect to solar-zenith angle at the equator of an spherical body.
12
176
3. Numerical results
177
To test the changes in the model to calculate the electrostatic dust trans-
178
port near the terminator, we made a computational code that simulates a
179
rotating spherical object with an arbitrary orientation of its axis of rotation.
180
The code allows to follow the particle trajectories integrating the equations
181
10 and 17 by Euler method. The parameters used to run the simulation are
182
listed in table 1. The value assumed for the porosity is usual for an aster-
183
oid and higher values (0.4 − 0.5) are found on few asteroids, like (25143)
184
Itokawa or (253) Mathilde. The roughness of 0.2 is in agreement with values
185
measured for some asteroids (Melosh, 1989; Veverka et al., 1999, 2000) and
186
the Moon (Pike, 1974). Since the parameter G changes from one asteroid to
187
another, we decided to use as an example the value of the magnitude system
188
(V-band) for (433) Eros.
189
As mentioned earlier, the motion of a dust particle due to levitation
190
will be affected by the gravity force, therefore we take three values for the
191
acceleration to study different scenarios: the first one corresponds to an
192
asteroid radius of less than a kilometer (e.g. (101955) Bennu, Nolan et al.,
193
2013), the second represents asteroids with radii of the order of 10 km (like
194
Eros, Yeomans et al., 2000), and the last value for gravity corresponds to
195
larger asteroids with radii of several kilometers (Kuzmanoski and Kovaˇcevi´c,
196
2002).
197
We run the simulation assuming that the object is at 1.8 au from the
198
Sun (neither too close where small particles would be lost into space nor far 13
199
away where they could not levitate), with a rotational period of 5 h, and two
200
orientations of the spin axis, one perpendicular to the orbital plane and the
201
other pole-on. Table 1: Initial parameters
Parameter distance to Sun, dsun asteroid gravity, gA rotational period, Ω G porosity, (1-D) surface roughness, ρ asymmetry factor, σ
Value 1.8 au 0.5 × 10−4 ; 0.5 × 10−2 ; 0.5 m/s2 5h 0.46 0.3 0.2 0
202
If we take a snapshot of the simulation, we will see the dependence of the
203
surface electric potential with respect to θsun (fig. 2). For the three profiles
204
(60◦ , 30◦ , and 0◦ in latitude and simetrical with respect to the equator) shown
205
in fig. 2, the potential is positive on the day side and at longitudes ±90◦ it is
206
possible to see the change in surface potential due to the model modification.
207
On the dark side, the region with longitudes between 90◦ and 270◦ from the
208
sub-solar point, the electric potential and the electric field were set equal to
209
zero. We also see how the maximum value for the surface potential change
210
for the different latitudes. Being lower at a latitude of 60◦ (top panel),
211
approximately 5.5 V at 30◦ (middle panel), and reaching a maximum value
212
of 6 V at the equator (bottom panel).
213
The charging mechanism of dust particles depends on the radius and
214
density of the grain, the surface properties, and the time-dependent solar
14
Latitude of 60° 6 5 4 3 2 1 0
Surface potential [V]
0
50
100
150
200
250
300
350
Latitude of 30° 6 5 4 3 2 1 0
cc
0
50
100
150
200
250
300
350
300
350
At the equator 6 5 4 3 2 1 0 0
50
100
150
200
250
Longitude [°] Figure 2: Profiles of the surface electric potential as function of longitude, for a spherical object with perpendicular spin axis to the orbital plane. In the top panel is shown the profile at a latitude of 60◦ , in the middle panel at 30◦ latitude, and in the bottom panel at 0◦ latitude. The sub-solar point is at zero degrees of latitude and zero degrees of longitude.
215
zenith angle at the location of the particle. In our simulations we use particles
216
initially deposited at the surface with electric charge and potential equal to
217
zero, radius between 0.1 and 1.1 µm, and density of 3.7 g/cm3 .
218
The most likely mechanism to give particles their initial velocities is by
219
micro-meteoroid bombardment which could break-up rocks and/or bedrock
220
into small pieces (Gr¨ un et al., 2011) and could also throw out small particles
221
by collisions. Due to this process some of the ejecta may escape the gravity
222
force of the asteroid but other particles may fall back onto the surface, so 15
223
small dust grains are formed continuously on the surface of atmosphereless
224
bodies and the micro-meteoroid bombardment provide them an initial veloc-
225
ity. In general, the escape velocity for an object with microgravity is of the
226
order of a few meters per second (Veverka et al., 2000; Fujiwara et al., 2006).
227
Extrapolating the results obtained experimentally by Wang et al. (2016) on
228
the maximum lofting height as function of the vertical initial speed, we found
229
for particles in asteroid surfaces a mean initial radial velocity of 0.4 m/s.
230
The dust grains were launched from the surface with random directions.
231
Once a grain lifts off the surface, the particle motion can be described as
232
either ballistic, escaping, or levitating. In the first case the trajectory of the
233
particle is governed by the gravity force and return to the surface in a short
234
time, being negligible the electrostatic interaction. In the second case, the
235
particle gains enough kinetic energy to escape into space. Essentially, all
236
grains launched with enough initial velocity cannot be slowed down regard-
237
less of its size. Finally, the condition for particles to levitate is when the
238
electrostatic and gravity forces are almost equal, opposite in direction and
239
the electrostatic force varies slightly in time.
240
We first simulate an object with a gravity acceleration of 0.005 m/s−2
241
and a five-hour rotational period. We found that the crossing time of the
242
photo-electron sheath for all particles with radii below 0.3 µm is very short
243
and they cannot accumulate enough charge to reverse their motion, resulting
244
in an escaping trajectory. By contrast, particles with radii between ∼ 0.3 µm
245
and 0.5 µm acquired enough charge to interact with the photoelectron sheath 16
246
until they are no longer on the lit side of the object. When these particles
247
approach the region near the terminator, the photoelectron density decreases
248
and the particles fall back onto the surface. At the day-night boundary dust
249
grains of this size are strongly repelled by the positively charged surface
250
(fig. 1), and as a result, the particles escape into interplanetary space (fig. 3). 850
0.3 µm 0.5 µm 0.7 µm 0.9 µm 1.1 µm
800 750 700 650
Height [meters]
600 550 500 450
cc
400 350 300 250 200 150 100 50 0 0
5000
10000
15000
20000
25000
30000
35000
Time [seconds]
Figure 3: Evolution of particles in time, for a rotating object with a size comparable to Eros and its spin axis perpendicular to the orbital plane. All particles are launch with an initial radial velocity of 0.4 m/s.
251
For larger particles (fig. 3) we find different scenarios according to their
252
size, their initial location, and their initial velocity vector. For example, we
253
found that for some particles with size between 0.7 − 0.9 µm the electric
254
force is almost balanced by the force of gravity allowing these particles to
255
go through the sheath and at the terminator they could fall to the surface
256
and remain at rest during the night. Others levitate while they are being
257
illuminated and at the terminator region, where the sheath has fewer pho-
17
258
toelectrons, the gravity force pull these particles down and as soon as they
259
reach the surface they are electrostatically repelled.The particles with radii
260
of 1.1 µm usually follow ballistic trajectories; but when these particles re-
261
main under sunlit areas they constantly lose electrons and this allows them
262
to levitate for longer periods (fig. 3).
263
If the initial velocity of the grains are set to zero we found that particles
264
with radii smaller than 0.3 µm are lost before they cross the terminator.
265
Then, it seems that in asteroids with radius similar to Eros the smallest
266
particles can be lost regardless of the initial velocity. On the other hand,
267
particles larger than 0.5 µm that were initially at rest get charge and levitate
268
while the solar phase angle is less than ∼ 90◦ . Then, on the night side the
269
grains fall back to the surface until they are on the day side again (fig. 4). 400
rd= 0.5 µm, Vi= 0 m/s rd= 0.5 µm, Vi= 0.4 m/s rd= 0.7 µm, Vi= 0 m/s rd= 0.7 µm, Vi= 0.4 m/s
350
Height [meters]
300
250
cc
200
150
100
50
0 0
5000
10000
15000
20000
25000
30000
35000
Time [seconds]
Figure 4: Height as function of time for two set of particles, with zero and non-zero initial radial velocity, on an Eros-like object with a perpendicular spin axis.
270
We also simulate particles under gravity accelerations of 0.5 × 10−4 and 18
271
0.5 m/s2 obtaining completely opposite results: in the first case all particles
272
smaller than 1.1 µm would be lost, while in the second case the dynamics of
273
the particles are governed by gravity and none of them can levitate.
274
In the case of a face-on object, i.e. a whole hemisphere been illuminated
275
permanently, we found that even the largest particles could escape the as-
276
teroid gravity if they receive an impulse in the zone close to the terminator.
277
The particles launch from the day side levitate while they are illuminated
278
(fig. 5), they get charged and are attracted by the photoelectron sheath while
279
those that reach the dark side hit the surface and stay there. 1000
0.5 µm 0.7 µm 0.9 µm 1.1 µm
950 900 850 800 750
Height [meters]
700 650 600 550
cc
500 450 400 350 300 250 200 150 100 50 0 0
5000
10000
15000
20000
25000
30000
35000
Time [seconds]
Figure 5: Height as a function of time for particles on an object surface, with a face-on configuration. All particles are launch with an initial radial velocity of 0.4 m/s.
280
4. Conclusions
281
Some previous works (De and Criswell, 1977; Criswell and De, 1977; Lee,
282
1996) suggested that the electric field at the terminator might be of several 19
283
kV m−1 due to the potential difference between the day and night sides.
284
Although on the model of De and Criswell (1977) and Criswell and De (1977),
285
a sharp boundary at the terminator is being assumed, this could mean that
286
the electric field would not be as intense in the day-night boundary as they
287
found.Mendis et al. (1981) argued that the potential is positive around the
288
sub-solar point and then it decreases to negative values near the terminator.
289
We found that the electric potential and the electric field decrease from the
290
sub-solar point up to a minimum (∼ θsun = 85◦ ,see fig. 5), and then at
291
the terminator the electric potential rise to a positive value. The change
292
in the behavior of the electric potential found in this work is the result of
293
the surface roughness contribution to the electric field and to the potential
294
at the terminator. The G and Φ1 are the only parameters, used in our
295
model, that depend on the rugosity of the surface and on the particular
296
properties of the surface of the objects. Only in the case of a rarely large
297
value of G (smooth surface) the model will show a negative surface potential
298
in the terminator zone. Taking into account the roughness contribution, we
299
can study and simulate the evolution of regolith particles during a complete
300
rotational period and evaluate which are the surface regions where the dust
301
grains could accumulate.
302
The results of the simulations made considering the roughness contri-
303
bution at the terminator agree with Lee (1996), Colwell et al. (2005) and
304
Miyamoto et al. (2007) in their conclusion that electrostatic dust transport
305
might be happening on some asteroids, although we are aware that the photo20
306
electron sheath structure depends on the plasma and surface properties (Nit-
307
ter et al., 1998; Poppe and Hor´anyi, 2010). Then our results support the
308
idea that on some small asteroids the finest regolith particles which receive
309
an initial velocity may escape due to the electrostatic force acting on them
310
(Yamamoto and Nakamura, 2000). We also find that on illuminated areas of
311
an asteroid a submicron dust grain would be on stable levitation (Lee, 1996;
312
Colwell et al., 2005). Moreover, particles with radii between 0.5 and 0.7 µm
313
might levitate, without any external impulse, after traveling from the dark
314
to the illuminated side of an asteroid and finding a positive surface potential
315
at the terminator.
316
It will be subject of future studies the correlation between our model
317
with data collected by rendezvous missions, like Hayabusha 2 (Ogawa et al.,
318
2019; Tachibana, 2019) and OSIRIS-REx (Lauretta et al., 2019). However,
319
L´opez-Sisterna et al. (2019) report polarimetric observations that could be
320
an observational evidence about the presence of an electrostatic levitation
321
mechanism on the surface of some asteroids.
322
acknowledgements
323
The authors thank Prof. Julio Fern´andez and an anonymous referee for
324
their review which led to an improvement of the paper, and gratefully ac-
325
knowledge financial support by CONICET through PIP 112-201501-00525
326
and San Juan National University by a CICITCA grant for the period 2018-
327
2019. 21
328
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• • •
Sub-micron particles may transport by electrostatic levitation over asteroid surfaces. Contribution of the surface roughness to the the illumination conditions near the terminator zone of an asteroid. Computational model for the photoelectron layer and the dust charging process.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
S0032-0633(18)30346-5
DOI:
https://doi.org/10.1016/j.pss.2019.104775
Reference:
PSS 104775
To appear in:
Planetary and Space Science
Received Date: 3 October 2018 Revised Date:
2 October 2019
Accepted Date: 8 October 2019
Please cite this article as: López-Sisterna, C., Gil-Hutton, R.A., Electrostatic dust transport on the terminator of atmosphereless bodies, Planetary and Space Science (2019), doi: https://doi.org/10.1016/ j.pss.2019.104775. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Electrostatic dust transport on the terminator of atmosphereless bodies C. L´opez-Sisternaa,∗, R. A. Gil-Huttona a
Grupo de Ciencias Planetarias, Departamento de Geof´ısica y Astronom´ıa, Facultad de Ciencias Exactas, F´ısicas y Naturales, Universidad Nacional de San Juan - CONICET, Av. Jos´e I. de la Roza 590 (O), J5402DCS Rivadavia, San Juan, Argentina
Abstract Small dust particles may levitate over the surface of atmosphereless bodies, but to study electrostatic dust transport during a complete rotational period it is necessary to evaluate the surface potential at the terminator zone. We simulate the photo-electron layer and the dust charging process according to previously used models for dust levitation, but we propose a modification of the models to add the effects produced by surface roughness for low solar elevation angles. The correction for surface roughness on the model showed that a positive value for the surface potential and the electric field appear at the terminator. We found that submicron particles could be in stable levitation for small gravity fields. Our results also showed that the finest regolith grains can be lost into space due to the electrostatic repulsion. Keywords: Asteroids: general, dusty plasma, levitation, photoelectron emission ∗
Corresponding author Email addresses: [email protected] (C. L´opez-Sisterna ), [email protected] (R. A. Gil-Hutton)
Preprint submitted to Planetary & Space Science
October 16, 2019
1
1. Introduction
2
The observation of the Lunar Horizon Glow by the Surveyor spacecraft
3
(Criswell, 1972, 1973; Rennilson and Criswell, 1974) and by the Apollo 17
4
mission (McCoy and Criswell, 1974; Zook and McCoy, 1991) led to the de-
5
velopment of an electrostatic dust levitation theory. During the last 40 years
6
several models arose trying to describe the physics for charging in space and
7
dusty plasma physics (Criswell, 1973; De and Criswell, 1977; Criswell and De,
8
1977; Whipple, 1981; Goertz, 1989; Nitter and Havnes, 1992; Nitter et al.,
9
1994; Colwell et al., 2005; Hughes et al., 2008), and some of these models have
10
been proposed to explain the unusual observations on asteroids (Lee, 1996;
11
Colwell et al., 2005; Miyamoto et al., 2007) and also hypothesized to occur on
12
comets (Mendis et al., 1981). In addition, different laboratory experiments
13
have confirmed dust levitation under conditions that resemble atmosphere-
14
less solar system environments (Sickafoose et al., 2002; Sheridan and Hayes,
15
2011; Wang et al., 2016).
16
The surfaces of these bodies in interplanetary space are exposed to solar
17
wind plasma and solar UV radiation; generally on the sunlit side photoelectric
18
charging exceeds plasma charging, resulting on a positive surface potential
19
(Mendis et al., 1981; Lee, 1996). These effects contribute to the generation of
20
two important currents over the surface, responsible for developing a photo-
21
electron sheath over the lit side. In general, the upper layer of the regolith
22
on a small body may be considered as an effective insulator (Sheridan et al., 2
23
1992; Lee, 1996; Kimura, 2016). Therefore, regolith dust particles will have
24
a non-zero charge while resting on the surface and, under certain conditions,
25
the electric force on sub-micrometer particles would exceed the gravitational
26
force causing dust particles to detach from the surface and levitate.
27
This electrostatic mechanism could be responsible for the redistribution
28
of regolith particles over the surface of asteroids (Veverka et al., 2001; Col-
29
well et al., 2005), but there are different arguments about how dust levitation
30
works near the terminator region because in a smooth surface the light is al-
31
most tangent to the surface. For example, Criswell and De (1977) described
32
what they called supercharging effect near the day-night boundary: the pho-
33
toelectrons recently ejected from the lit side are attracted by the positive
34
charge that has just turn into the night side of a rotating body increasing
35
the potential of the illuminated regions. Furthermore, Lee (1996) proposed
36
that the prime charging process to consider at the terminator is the photo-
37
electric effect. This author considered solar wind currents to be negligible
38
on the trailing side owing to the velocity aberration arising from the orbital
39
motion of the asteroid. More recently, Hartzell and Scheeres (2011) analyzed
40
the role of cohesive forces on sub-micrometer particles. They found that
41
taking into account cohesion, levitation may be possible only for a range of
42
particles size near the terminator of the Moon and asteroids. On the other
43
hand, Kimura et al. (2014) proposed that electrostatic forces on irregular
44
grains with rough hydroxylated surface could overcame cohesive forces near
45
the terminator. 3
46
The process responsible for detaching the particles from the surface is
47
strongly dependent on the illumination conditions. Moreover, near the ter-
48
minator the surface roughness produces areas strongly illuminated that could
49
reach high potential and should have a strong influence on the electrostatic
50
interaction of the particles with their surroundings. This effect produced
51
by roughness is not taken into account by the available models (for example,
52
Colwell et al., 2005) producing a negative surface potential near the termina-
53
tor which prevents the use of these models to simulate processes that involve
54
a complete rotation of the object.
55
In this paper we present a new way of estimating the effects of the electric
56
field close to the terminator taking into account the mean surface roughness
57
of objects with micro-gravity. We describe the electrostatic transport model
58
in section 2. In section 3 we discuss our results and in section 4 we summarize
59
the conclusions.
60
2. Electrostatic transport model
61
To simulate the photo-electron layer and the dust charging currents over
62
the surface, we initially follow the simplified monotonically-decreasing sheath
63
potential model described by Colwell et al. (2005) and Hughes et al. (2008)
64
which is briefly explained below.
65
The model assumes that the solar UV radiation strikes the surface and
66
release electrons of at least the same energy of the incoming photons, but at
67
the same time the solar wind hit the surface producing an electron current. 4
68
If the photoemission process results to be more effective than the solar wind
69
charging process, then the sunlit surface becomes positively charged. The
70
surface will continue charging until the sum of the currents becomes zero.
71
Then, the surface potential, φs , is determined by equating the photo-electron
72
(Ipe ) and the solar wind (Isw ) currents at the surface (Whipple, 1981; Colwell
73
et al., 2005):
Ipe Isw
−qe φs cos θsun , = πrd qe Ipe0 exp kB Tpe r kB Tsw qe φs 2 = πrd qe nsw 1+ , 2πme kB Tsw
2
(1) (2)
74
where rd is the grain radius, qe is the absolute value of the electron charge,
75
me is the mass of the electron, kB is the Boltzmann constant, and θsun is the
76
complementary angle of the solar elevation angle. The photo-electron and
77
the solar wind temperatures are Tpe and Tsw , respectively, for which we used
78
values of 2.2 eV (Willis et al., 1973) and 10 eV, where the second value is
79
well below the energy range where secondary electron production becomes
80
important.
81
In order to calculate the photo-electron current at the surface (Ipe ), we
82
assumed that it has the same photo-emissive properties as the lunar regolith
83
(Willis et al., 1973; Sternovsky et al., 2002), but we take into account that the
84
usual value for the emitted photoelectrons proposed byWillis et al. (1973) was
85
underestimated by a factor of 2.1 (Senshu et al., 2015). Then, the parameters 5
86
used for the emitted photo-electron flux for normal incidence of light, Ipe0 ,
87
and the electron density of the solar wind, nsw , are scaled to its value at 1 au:
Ipe0 nsw
5.88 × 109 electrons = , d2sun cm2 s 5 electrons = 2 . dsun cm−3
(3) (4)
88
Besides, the photo-electron sheath is a region of increased electron density
89
near the surface formed as a consequence of the release of photo-electrons
90
and its electric attraction to the positive surface. Then the photo-electron
91
density is defined by:
npe0 = 92
2Ipe0 cos θsun , vpe
(5)
where vpe , the characteristic photo-electron velocity, is: r vpe =
2kB Tpe ≈ 880 km/s. me
(6)
93
Then, an electric field appears as result of the potential difference between
94
the surface and this sheath. Grard and Tunaley (1971) defined the electric
95
field and the photo-electron density as function of height (z), above the
96
surface:
6
√ −1 2 2φs ∆z E (∆z) = , 1+ √ λD 2λD −2 ∆z npe (∆z) = npe0 1 + √ , 2λD 97
(7) (8)
where: s λD =
ε0 kB Tpe , npe qe2
(9)
98
is the Debye length at the surface, which indicates the width of the photo-
99
electron sheath, and ε0 is the permittivity in vacuum.
100
2.1. Charge and dynamics of dust particles
101
Dust grains also release electrons through the photoelectric effect, and col-
102
lect solar wind electrons and photo-electrons from the photo-electron sheath
103
in a similar way as happens on the surface. This process is time dependent
104
and results in a variation of the grain charge, Qd , produced by the currents
105
balance: dQd = Ipe,d − Ie,d − Isw,d , dt
(10)
106
where Ipe,d , Ie,d and Isw,d are the currents due to photo-emission from the
107
particle, to the collection of photo-electrons emitted by the surface, and to
108
the capture of solar wind electrons, respectively. These currents depend on
7
109
the grain charge, and in the case it is charged positively:
−qe φd Ipe,d = πrd 2 qe Ipe0 exp , kB Tpe r qe φd 8kB Tsw 2 1+ , Isw,d = πrd qe nsw πme kB Tsw r 8kB Tpe qe φ d 2 1+ , Ie,d = πrd qe npe πme kB Tpe
(11) (12) (13)
110
where φd is the electric potential of the grain. In the case of a negative
111
potential, the currents are given by:
Ipe,d = πrd 2 qe Ipe0 , r
8kB Tsw qe φ d Isw,d = πrd 2 qe nsw exp , πme kB Tsw r qe φd 8kB Tpe exp . Isw,d = πrd 2 qe npe πme kB Tpe
(14) (15) (16)
112
If the electrostatic and the gravitational forces are balanced, the dust
113
grain levitates through the photo-electron sheath, but if an unbalance be-
114
tween these two forces occurs the charged particle moves in the radial direc-
115
tion following the equation of motion: d2 r = Qd E(∆z) − gA md , dt2
116
(17)
where gA is the object gravity, md is the grain mass, and E(∆z) is defined 8
117
by the equation 7.
118
2.2. Illumination excess due to roughness
119
If we assume in this model that the surface of the object is smooth, then
120
close to the terminator the photo-electron density and the photo-electron
121
current fall very fast and the surface potential reaches negative values for
122
θsun = 90 ◦ . However, the surfaces of solar system objects are highly irregular
123
showing macroscopic and microscopic roughness. This roughness could be
124
important in the regions near the terminator because it produces small areas
125
with orientations that are not normal to the radial direction, and then they
126
could appear illuminated by the Sun (Criswell, 1972, 1973). These small
127
regions undergo the usual process of gain and loss of electrons as the object
128
rotates, but in this case the illuminated regions constantly lose electrons
129
because they are more likely to be collected by large dark areas when they
130
fall again to the surface. As a consequence of this particular effect, the electric
131
field on the surface would increase rather than decline in the terminator zone
132
producing a greater radial acceleration than that expected from a smooth
133
surface (Eq. 17).
134
Since the effects of a rough surface near the terminator are the result
135
of the existence there of small illuminated areas which also contribute to
136
produce the total brightness of the object, we decide to estimate the value
137
of the electric field near the terminator using a formulation similar to that
138
given by Lumme and Bowell (1981a,b) to calculate the phase function for
9
139
atmosphereless bodies. According to these authors, the normalized phase
140
function for atmosphereless bodies is given by:
Φ(α) = (1 − G)Φ1 (α) + GΦM (α),
(18)
141
where α is the phase angle, G is a parameter which measures the ratio be-
142
tween the brightness produced by multiple scattering and the total bright-
143
ness, Φ1 is the phase function for singly scattered light due to a single parti-
144
cle, shadowing and roughness, and ΦM is the phase function due to multiple
145
scattering between particles. The functions Φ(α), Φ1 (α) and ΦM (α) are nor-
146
malized to their values at α = 0 ◦ .
147
148
Following the formulation originally proposed by Lumme and Bowell (1981a), we have:
Φ1 (α) = ΦS (α, D)ΦR (α, ρ)ΦP (α, σ),
(19)
149
where φS , φR , and φP are the normalized phase functions due to shadowing,
150
roughness, and single particle scattering, respectively. These functions de-
151
pend on the volume fraction D (or the porosity, 1 − D), the relation between
152
the depth and the radius of a typical irregularity on the surface (the rough-
153
ness, ρ), and the ratio between backscattering and forward scattering (the
154
asymmetry factor, σ). On the other hand, ΦM is not difficult to calculate
155
because it is similar to the phase function of a Lambert reflector:
10
φM ≈
1 [sin(α) + (π − α)(cos(α)], π
0 6 α 6 π.
(20)
156
This formulation can be used to derive a factor that allow us to estimate
157
the surface brightness, which is proportional to the illumination received at
158
different illumination angles. Although in the original formulation by Lumme
159
and Bowell (1981a,b) the G parameter depends on the wavelength used to
160
measure it, in our model it is used as a measure of surface roughness. Thus,
161
if G and the phase functions for a particular object are known and we assume
162
that α ≡ 180 ◦ − 2θsun , then the function:
ψ(θsun ) = Φ(α) − GΦM (α),
(21)
163
is the measurement of the relative contribution of the brightness (or the
164
illumination at constant albedo) due to shadowing and roughness. Then, the
165
photo-electron current and the photo-electron density modified to take into
166
account this effect are:
Ipe,r npe0,r
−qe φs = πrd qe Ipe0 exp [cos θsun (1 − ζ) + ζψ], kB Tpe 2Ipe0 [sin θsun (1 − ζ) + ζψ] = , vpe 2
(22) (23) (24)
167
where:
11
ζ=
2 (θ0 −θsun ) , for θsun > θ0 ; θ0 0,
168
(25)
otherwise;
and θ0 is defined as ψ(θ0 ) = 0.
169
The surface electric potential as function of the solar angle for a smooth
170
and rough surface are shown in fig. 1. These curves have a similar behavior
171
up to solar angles of ∼ 70◦ , but beyond this point the surface potential for
172
a smooth surface decrease to approximately φs ∼ −1.0 V at the terminator
173
while for an irregular surface it reaches a minimum at θsun = 85◦ and increase
174
again to reach at the terminator a surface potential of φs ∼ 2.0 V ( E(z =
175
0) ∼ 1 V /m). 6 Rough surface Smooth surface
Electric surface potential [V]
5
4
3
cc 2
1
0
-1 0
10
20
30
40
50
60
70
80
90
Solar incidence angle [Deg]
Figure 1: Variation of the surface electric potential with respect to solar-zenith angle at the equator of an spherical body.
12
176
3. Numerical results
177
To test the changes in the model to calculate the electrostatic dust trans-
178
port near the terminator, we made a computational code that simulates a
179
rotating spherical object with an arbitrary orientation of its axis of rotation.
180
The code allows to follow the particle trajectories integrating the equations
181
10 and 17 by Euler method. The parameters used to run the simulation are
182
listed in table 1. The value assumed for the porosity is usual for an aster-
183
oid and higher values (0.4 − 0.5) are found on few asteroids, like (25143)
184
Itokawa or (253) Mathilde. The roughness of 0.2 is in agreement with values
185
measured for some asteroids (Melosh, 1989; Veverka et al., 1999, 2000) and
186
the Moon (Pike, 1974). Since the parameter G changes from one asteroid to
187
another, we decided to use as an example the value of the magnitude system
188
(V-band) for (433) Eros.
189
As mentioned earlier, the motion of a dust particle due to levitation
190
will be affected by the gravity force, therefore we take three values for the
191
acceleration to study different scenarios: the first one corresponds to an
192
asteroid radius of less than a kilometer (e.g. (101955) Bennu, Nolan et al.,
193
2013), the second represents asteroids with radii of the order of 10 km (like
194
Eros, Yeomans et al., 2000), and the last value for gravity corresponds to
195
larger asteroids with radii of several kilometers (Kuzmanoski and Kovaˇcevi´c,
196
2002).
197
We run the simulation assuming that the object is at 1.8 au from the
198
Sun (neither too close where small particles would be lost into space nor far 13
199
away where they could not levitate), with a rotational period of 5 h, and two
200
orientations of the spin axis, one perpendicular to the orbital plane and the
201
other pole-on. Table 1: Initial parameters
Parameter distance to Sun, dsun asteroid gravity, gA rotational period, Ω G porosity, (1-D) surface roughness, ρ asymmetry factor, σ
Value 1.8 au 0.5 × 10−4 ; 0.5 × 10−2 ; 0.5 m/s2 5h 0.46 0.3 0.2 0
202
If we take a snapshot of the simulation, we will see the dependence of the
203
surface electric potential with respect to θsun (fig. 2). For the three profiles
204
(60◦ , 30◦ , and 0◦ in latitude and simetrical with respect to the equator) shown
205
in fig. 2, the potential is positive on the day side and at longitudes ±90◦ it is
206
possible to see the change in surface potential due to the model modification.
207
On the dark side, the region with longitudes between 90◦ and 270◦ from the
208
sub-solar point, the electric potential and the electric field were set equal to
209
zero. We also see how the maximum value for the surface potential change
210
for the different latitudes. Being lower at a latitude of 60◦ (top panel),
211
approximately 5.5 V at 30◦ (middle panel), and reaching a maximum value
212
of 6 V at the equator (bottom panel).
213
The charging mechanism of dust particles depends on the radius and
214
density of the grain, the surface properties, and the time-dependent solar
14
Latitude of 60° 6 5 4 3 2 1 0
Surface potential [V]
0
50
100
150
200
250
300
350
Latitude of 30° 6 5 4 3 2 1 0
cc
0
50
100
150
200
250
300
350
300
350
At the equator 6 5 4 3 2 1 0 0
50
100
150
200
250
Longitude [°] Figure 2: Profiles of the surface electric potential as function of longitude, for a spherical object with perpendicular spin axis to the orbital plane. In the top panel is shown the profile at a latitude of 60◦ , in the middle panel at 30◦ latitude, and in the bottom panel at 0◦ latitude. The sub-solar point is at zero degrees of latitude and zero degrees of longitude.
215
zenith angle at the location of the particle. In our simulations we use particles
216
initially deposited at the surface with electric charge and potential equal to
217
zero, radius between 0.1 and 1.1 µm, and density of 3.7 g/cm3 .
218
The most likely mechanism to give particles their initial velocities is by
219
micro-meteoroid bombardment which could break-up rocks and/or bedrock
220
into small pieces (Gr¨ un et al., 2011) and could also throw out small particles
221
by collisions. Due to this process some of the ejecta may escape the gravity
222
force of the asteroid but other particles may fall back onto the surface, so 15
223
small dust grains are formed continuously on the surface of atmosphereless
224
bodies and the micro-meteoroid bombardment provide them an initial veloc-
225
ity. In general, the escape velocity for an object with microgravity is of the
226
order of a few meters per second (Veverka et al., 2000; Fujiwara et al., 2006).
227
Extrapolating the results obtained experimentally by Wang et al. (2016) on
228
the maximum lofting height as function of the vertical initial speed, we found
229
for particles in asteroid surfaces a mean initial radial velocity of 0.4 m/s.
230
The dust grains were launched from the surface with random directions.
231
Once a grain lifts off the surface, the particle motion can be described as
232
either ballistic, escaping, or levitating. In the first case the trajectory of the
233
particle is governed by the gravity force and return to the surface in a short
234
time, being negligible the electrostatic interaction. In the second case, the
235
particle gains enough kinetic energy to escape into space. Essentially, all
236
grains launched with enough initial velocity cannot be slowed down regard-
237
less of its size. Finally, the condition for particles to levitate is when the
238
electrostatic and gravity forces are almost equal, opposite in direction and
239
the electrostatic force varies slightly in time.
240
We first simulate an object with a gravity acceleration of 0.005 m/s−2
241
and a five-hour rotational period. We found that the crossing time of the
242
photo-electron sheath for all particles with radii below 0.3 µm is very short
243
and they cannot accumulate enough charge to reverse their motion, resulting
244
in an escaping trajectory. By contrast, particles with radii between ∼ 0.3 µm
245
and 0.5 µm acquired enough charge to interact with the photoelectron sheath 16
246
until they are no longer on the lit side of the object. When these particles
247
approach the region near the terminator, the photoelectron density decreases
248
and the particles fall back onto the surface. At the day-night boundary dust
249
grains of this size are strongly repelled by the positively charged surface
250
(fig. 1), and as a result, the particles escape into interplanetary space (fig. 3). 850
0.3 µm 0.5 µm 0.7 µm 0.9 µm 1.1 µm
800 750 700 650
Height [meters]
600 550 500 450
cc
400 350 300 250 200 150 100 50 0 0
5000
10000
15000
20000
25000
30000
35000
Time [seconds]
Figure 3: Evolution of particles in time, for a rotating object with a size comparable to Eros and its spin axis perpendicular to the orbital plane. All particles are launch with an initial radial velocity of 0.4 m/s.
251
For larger particles (fig. 3) we find different scenarios according to their
252
size, their initial location, and their initial velocity vector. For example, we
253
found that for some particles with size between 0.7 − 0.9 µm the electric
254
force is almost balanced by the force of gravity allowing these particles to
255
go through the sheath and at the terminator they could fall to the surface
256
and remain at rest during the night. Others levitate while they are being
257
illuminated and at the terminator region, where the sheath has fewer pho-
17
258
toelectrons, the gravity force pull these particles down and as soon as they
259
reach the surface they are electrostatically repelled.The particles with radii
260
of 1.1 µm usually follow ballistic trajectories; but when these particles re-
261
main under sunlit areas they constantly lose electrons and this allows them
262
to levitate for longer periods (fig. 3).
263
If the initial velocity of the grains are set to zero we found that particles
264
with radii smaller than 0.3 µm are lost before they cross the terminator.
265
Then, it seems that in asteroids with radius similar to Eros the smallest
266
particles can be lost regardless of the initial velocity. On the other hand,
267
particles larger than 0.5 µm that were initially at rest get charge and levitate
268
while the solar phase angle is less than ∼ 90◦ . Then, on the night side the
269
grains fall back to the surface until they are on the day side again (fig. 4). 400
rd= 0.5 µm, Vi= 0 m/s rd= 0.5 µm, Vi= 0.4 m/s rd= 0.7 µm, Vi= 0 m/s rd= 0.7 µm, Vi= 0.4 m/s
350
Height [meters]
300
250
cc
200
150
100
50
0 0
5000
10000
15000
20000
25000
30000
35000
Time [seconds]
Figure 4: Height as function of time for two set of particles, with zero and non-zero initial radial velocity, on an Eros-like object with a perpendicular spin axis.
270
We also simulate particles under gravity accelerations of 0.5 × 10−4 and 18
271
0.5 m/s2 obtaining completely opposite results: in the first case all particles
272
smaller than 1.1 µm would be lost, while in the second case the dynamics of
273
the particles are governed by gravity and none of them can levitate.
274
In the case of a face-on object, i.e. a whole hemisphere been illuminated
275
permanently, we found that even the largest particles could escape the as-
276
teroid gravity if they receive an impulse in the zone close to the terminator.
277
The particles launch from the day side levitate while they are illuminated
278
(fig. 5), they get charged and are attracted by the photoelectron sheath while
279
those that reach the dark side hit the surface and stay there. 1000
0.5 µm 0.7 µm 0.9 µm 1.1 µm
950 900 850 800 750
Height [meters]
700 650 600 550
cc
500 450 400 350 300 250 200 150 100 50 0 0
5000
10000
15000
20000
25000
30000
35000
Time [seconds]
Figure 5: Height as a function of time for particles on an object surface, with a face-on configuration. All particles are launch with an initial radial velocity of 0.4 m/s.
280
4. Conclusions
281
Some previous works (De and Criswell, 1977; Criswell and De, 1977; Lee,
282
1996) suggested that the electric field at the terminator might be of several 19
283
kV m−1 due to the potential difference between the day and night sides.
284
Although on the model of De and Criswell (1977) and Criswell and De (1977),
285
a sharp boundary at the terminator is being assumed, this could mean that
286
the electric field would not be as intense in the day-night boundary as they
287
found.Mendis et al. (1981) argued that the potential is positive around the
288
sub-solar point and then it decreases to negative values near the terminator.
289
We found that the electric potential and the electric field decrease from the
290
sub-solar point up to a minimum (∼ θsun = 85◦ ,see fig. 5), and then at
291
the terminator the electric potential rise to a positive value. The change
292
in the behavior of the electric potential found in this work is the result of
293
the surface roughness contribution to the electric field and to the potential
294
at the terminator. The G and Φ1 are the only parameters, used in our
295
model, that depend on the rugosity of the surface and on the particular
296
properties of the surface of the objects. Only in the case of a rarely large
297
value of G (smooth surface) the model will show a negative surface potential
298
in the terminator zone. Taking into account the roughness contribution, we
299
can study and simulate the evolution of regolith particles during a complete
300
rotational period and evaluate which are the surface regions where the dust
301
grains could accumulate.
302
The results of the simulations made considering the roughness contri-
303
bution at the terminator agree with Lee (1996), Colwell et al. (2005) and
304
Miyamoto et al. (2007) in their conclusion that electrostatic dust transport
305
might be happening on some asteroids, although we are aware that the photo20
306
electron sheath structure depends on the plasma and surface properties (Nit-
307
ter et al., 1998; Poppe and Hor´anyi, 2010). Then our results support the
308
idea that on some small asteroids the finest regolith particles which receive
309
an initial velocity may escape due to the electrostatic force acting on them
310
(Yamamoto and Nakamura, 2000). We also find that on illuminated areas of
311
an asteroid a submicron dust grain would be on stable levitation (Lee, 1996;
312
Colwell et al., 2005). Moreover, particles with radii between 0.5 and 0.7 µm
313
might levitate, without any external impulse, after traveling from the dark
314
to the illuminated side of an asteroid and finding a positive surface potential
315
at the terminator.
316
It will be subject of future studies the correlation between our model
317
with data collected by rendezvous missions, like Hayabusha 2 (Ogawa et al.,
318
2019; Tachibana, 2019) and OSIRIS-REx (Lauretta et al., 2019). However,
319
L´opez-Sisterna et al. (2019) report polarimetric observations that could be
320
an observational evidence about the presence of an electrostatic levitation
321
mechanism on the surface of some asteroids.
322
acknowledgements
323
The authors thank Prof. Julio Fern´andez and an anonymous referee for
324
their review which led to an improvement of the paper, and gratefully ac-
325
knowledge financial support by CONICET through PIP 112-201501-00525
326
and San Juan National University by a CICITCA grant for the period 2018-
327
2019. 21
328
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Sub-micron particles may transport by electrostatic levitation over asteroid surfaces. Contribution of the surface roughness to the the illumination conditions near the terminator zone of an asteroid. Computational model for the photoelectron layer and the dust charging process.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.