An analytical investigation: Effect of solar wind on lunar photoelectron sheath

Abstract

The formation of a photoelectron sheath over the lunar surface and subsequent dust levitation, under the influence of solar wind plasma and continuous solar radiation, has been analytically investigated. The photoelectron sheath characteristics have been evaluated using the Poisson equation configured with population density contributions from half Fermi-Dirac distribution of the photoemitted electrons and simplified Maxwellian statistics of solar wind plasma; as a consequence, altitude profiles for electric potential, electric field, and population density within the photoelectron sheath have been derived. The expression for the accretion rate of sheath electrons over the levitated spherical particles using anisotropic photoelectron flux has been derived, which has been further utilized to characterize the charging of levitating fine particles in the lunar sheath along with other constituent photoemission and solar wind fluxes. This estimate of particle charge has been further manifested with lunar sheath characteristics to evaluate the altitude profile of the particle size exhibiting levitation. The inclusion of solar wind flux into analysis is noticed to reduce the sheath span and altitude of the particle levitation; the dependence of the sheath structure and particle levitation on the solar wind plasma parameters has been discussed and graphically presented.

Full text

An analytical investigation: Effect of solar wind on lunar photoelectron sheath
S. K. Mishra, and Shikha Misra
Citation: Physics of Plasmas 25, 023702 (2018);
View online: https://doi.org/10.1063/1.5021260
View Table of Contents: http://aip.scitation.org/toc/php/25/2
Published by the American Institute of Physics

An analytical investigation: Effect of solar wind on lunar photoelectron
sheath
S. K. Mishra
1,a)
and Shikha Misra
2,3
1
Extreme Light Infrastructure-Attosecond Light Pulse Source (ELI-ALPS), Szeged 6720, Hungary
2
Rakesh P. G. College, Pilani 333031, India
3
F-32, CSIR-CEERI Col. Pilani 333031, India
(Received 2 January 2018; accepted 18 January 2018; published online 5 February 2018)
The formation of a photoelectron sheath over the lunar surface and subsequent dust levitation,
under the influence of solar wind plasma and continuous solar radiation, has been analytically
investigated. The photoelectron sheath characteristics have been evaluated using the Poisson
equation configured with population density contributions from half Fermi-Dirac distribution of the
photoemitted electrons and simplified Maxwellian statistics of solar wind plasma; as a conse-
quence, altitude profiles for electric potential, electric field, and population density within the pho-
toelectron sheath have been derived. The expression for the accretion rate of sheath electrons over
the levitated spherical particles using anisotropic photoelectron flux has been derived, which has
been further utilized to characterize the charging of levitating fine particles in the lunar sheath
along with other constituent photoemission and solar wind fluxes. This estimate of particle charge
has been further manifested with lunar sheath characteristics to evaluate the altitude profile of the
particle size exhibiting levitation. The inclusion of solar wind flux into analysis is noticed to reduce
the sheath span and altitude of the particle levitation; the dependence of the sheath structure
and particle levitation on the solar wind plasma parameters has been discussed and graphically
presented. Published by AIP Publishing. https://doi.org/10.1063/1.5021260
I. INTRODUCTION
The observation of lunar horizon glow and streamers
over the moon surface by Apollo lunar missions
1
was the
first signature of the dusty environment on the moon where
sunlight scattering is primarily caused by charged dust par-
ticles originating from the lunar surface.
2
In the absence of
the atmosphere, the frequent impact of debris/asteroids over
the lunar surface generates fine dust exhibiting a broad size
distribution.
3
The moon surface and fine particles get
charged under direct exposure of solar radiation and wind
plasma, giving rise to the phenomena of sheath formation
and dust levitation. The direct observations
1–6
of dust over
the lunar surface via Surveyor spacecraft and by the Apollo-
17 orbiting command module and its analysis led to an intui-
tive understanding and theoretical development of electro-
static dust levitation.
3
In this context, multiple experimental/
theoretical/simulation investigations have been performed to
interpret the phenomenon of sheath formation and particle
levitation;
7–17
for instance, recent work
15
predicts the fine
nanometer sized (10 nm) particles’ levitation up to 10s of
meters while the sub-micron (100–250) nm grains can float
up to an altitude of (1–100) cm.
A significant analysis interpreting the photoelectron
sheath over the lunar surface was presented by Nitter et al.,
18
where the incoming and outgoing photoelectrons were con-
sistently taken into account to evaluate the sheath structure;
however, an oversimplified isotropic Maxwellian distribution
of emitted photoelectrons was considered. In order to include
the anisotropic feature of the photoemitted electrons in eval-
uating incoming/outgoing flux over levitated dust particles,
later Nitter’s
18
analysis was modified by including arbitrary
half Maxwellian distribution
12–17
of the photoemitted elec-
trons; however, this is still a simplified assumption of the
photoelectron distribution which essentially should be half
Fermi-Dirac energy distribution.
19,20
In a recent work, the
analysis has been further re-formulated by Sodha and
Mishra
21
by including the adequate half Fermi-Dirac statis-
tics of photoemitted electrons along with appropriate expres-
sions for photoemission flux; the sheath features over the
lunar surface are shown to be significantly different from
those predicted by half Maxwellian statistics. The consider-
ation of Maxwellian distribution
21
of the surface electrons
predicts a smaller photoemission rate with respect to FD sta-
tistics, which ultimately reduces the lunar surface potential
and the subsequent population density in the sheath; this
results in a larger sheath, weaker potential decay, and smaller
electric field in the sheath, which also infers a smaller alti-
tude of the particle levitation. In their analysis,
21
the expres-
sions for the electron accretion flux over levitated dust
particles have been derived considering the anisotropic flux
in the photoelectron sheath, which has been further used in
evaluating the size distribution of the levitating dust par-
ticles; the analysis was performed for the dominant extreme
ultraviolet (EUV) Lyman-aradiation. Later this analysis is
extended to a continuous solar spectrum in deriving an asym-
metric sheath formation around spherical objects orbiting in
near-earth space.
22
Although these earlier analyses
21–23
put
forward a significant physical understanding of the sheath
formation, the contribution from solar wind plasma in
a)
E-mail: nishfeb@gmail.com
1070-664X/2018/25(2)/023702/10/$30.00 Published by AIP Publishing.25, 023702-1
PHYSICS OF PLASMAS 25, 023702 (2018)

photoelectron sheath formation has been ignored; some of
the recent simulation studies
24–26
highlight the role of solar
wind plasma in the lunar photoelectron sheath and subse-
quent dust transport. This feature has been analytically
addressed in this work.
In this work, we take account of our recent analysis
21
to
formulate the sheath formation and particle levitation over
the moon surface irradiated by a continuous solar spectrum
and solar wind flux. Using the Fowlers approach of photo-
emission of electrons (from the moon surface and dust par-
ticles), anisotropic half Fermi-Dirac statistics for the
photoemitted electrons, simplified Maxwellian distribution
for solar wind plasma, continuous solar spectrum (black
body radiation at 5800 K
27
plus Lyman-aradiation
28
), and
balance of charge (over the moon surface and levitated dust
particles) along with the Poisson equation, the steady state
altitude dependence of the electric potential, electric field,
population density, and particle size on the lunar photoelec-
tron sheath has been investigated.
This manuscript has been organized as follows: The for-
mulation for the sheath structure over the lunar surface com-
posed of photoelectron and solar wind plasma population has
been derived in Sec. II; the computational scheme and
numerical results/discussion for the photoelectron sheath
over the lunar surface have been presented in Sec. III. The
levitation of dust particles along with the evaluation of con-
stituent photoemission flux, anisotropic flux in photoelectron
sheath, and solar wind flux has been discussed in Sec. IV;
based on this analysis, the numerical results for the dust
charging and their levitation within lunar photoelectron
sheath have been presented in Sec. V. A summary of the out-
come in Sec. VI concludes this paper.
II. SHEATH MODEL
The sheath formation in the proximity of the moon sur-
face is caused by dynamic equilibrium between the photo-
emission flux and return (accretion) current over its surface,
maintaining it at positive potential. Analytically, the sheath
characteristics may be modelled by evaluating the net plasma
(electron/ion) population density in the space over the lunar
surface and utilizing it in solving the Poisson equation. The
continuous solar wind and solar radiation (causing photo-
emission) incident over lunar surface may be considered as a
principal source for the plasma population. In order to ana-
lyze the sheath structure, we follow the approach adopted in
Ref. 21 and modify the analysis by including the additional
effects of solar wind plasma and white light (i.e., continuous
solar radiation spectrum) in determining the effective plasma
density in the space over the lunar surface. Concerning the
large curvature of the moon, its surface may be considered
as the horizontal (planar) surface for all the practical realiza-
tion. Further, due to the large distance between the sun and
moon, the lunar surface may be considered to be uniformly
irradiated via solar wind and solar spectrum. With these sim-
plifications, we next evaluate the contribution from solar
radiation and solar wind in determining the plasma popula-
tion density over the moon surface.
A. Photoelectron population
The solar radiation falling on the lunar surface (regolith)
includes the continuous solar spectrum
27
in addition to a
dominant Lyman-a(k121.57 nm) spectrum in the EUV
(extreme ultraviolet) regime;
28
the entire range of solar radi-
ation is considered to cause photoemission of the electrons
from the lunar regolith (and levitated particles). The continu-
ous radiation from the sun may adequately be approximated
as a black body radiating at temperature T
s
5800 K. The net
photon flux associated with the solar radiation reaching (nor-
mally) the sunlit surface of the moon and available for pho-
toemission may approximately be expressed as
27–29
Kinc ¼ðKcr þKLaÞ¼ ðem
e0
dKcr þKLa
"#
¼4p
c2
 e
300h

3r2
s
r2
d
!
ðem
e0
E2
expðE=kTsÞ1½
1
"
dEþKLa#;(1)
where rsð6:96 1010 cmÞis the radius of the radiating sur-
face of the sun, rdð 1:45 1013 cmÞrefers to the mean dis-
tance between the sun and moon, Tsis the temperature of the
radiating sun, E0ð¼ /þVÞand Emare the lower and upper
limits of the useful solar radiation spectrum, /and Vare the
work function and surface potential of the moon/dust surface
materials, KLa(¼31011 cm
2
s
1
) corresponds to the pho-
ton flux associated with Lyman-aradiation,
28
and e, k, and h
are the Planck constant, Boltzmann constant, and electronic
charge; here, Eis expressed in eV.
Following the approach adopted in Ref. 21, the momen-
tum distribution of the outgoing photoelectrons due to the
incident solar radiation, emitted per unit area per unit time
from the lunar plane just outside the surface (x¼0) having a
potential Vo(lunar surface), can be written as
d2nphðtoÞ¼ ðem
e0
vðeÞ
Uð1rÞ

FDðexþet1rÞdexdet
½dKcr
"
þvðeLaÞ
Uð1r;LaÞ
!
KLaFDðexþet1r;LaÞdexdet#;(2)
where vðeÞis the photoelectric efficiency of the surface mate-
rial, 1r¼ðeurÞ,1r;La¼ðeLaurÞ,ur¼ðe/r=kToÞ,
e¼E=kTo,eLa¼ELa=kTo,em¼Em=kTo,e0¼E0=kTo,
to¼eVo=kTo,FDðgÞ¼ð1þexp gÞ1refers to the Fermi-
Dirac distribution,
30
/rrefers to the work function of the lunar
regolith, Torefers to the temperature of the emitting surface,
exand etare the normal and parallel components of the photo-
electron energy (normalized with kTo), and Uð1Þ¼Ðexp 1
0X1
lnð1þXÞdX. Considering the spectral dependence of the
photoelectric yield of the emitting surface, Draine’s formula-
tion
31
describing its spectral dependence has been taken into
account. Algebraically, this relation can be expressed as
vðeÞ¼v0½1ðu=eÞ,wherev0is the maximum photoelec-
tric yield.
023702-2 S. K. Mishra and S. Misra Phys. Plasmas 25, 023702 (2018)

This electron flux traverses in space normal to the lunar
surface and gives rise to electron density population.
Consider a virtual plane parallel to the horizontal lunar sur-
face (at finite x) at potential V(tin normalized units).
Following the approach similar to Ref. 21, the momentum
distribution in reference to the virtual surface (at surface
potential V) corresponding to Eq. (2) can be written as
d2nphðtÞ¼ ðem
e0
vðeÞ
Uð1rÞ

FDðexþet1rtoþtÞdexdet
½dKcr
"
þvðeLaÞ
Uð1r;LaÞ
!
KLaFDðexþet1r;LatoþtÞdexdet#:
(3)
The emission flux across the virtual plane (at potential V)
may be determined by integrating the above expression [Eq.
(3)] over the parallel and normal energy components with
appropriate limits. The net electron flux available to cross
the virtual plane having normal energy exand exþdexmay
be written by integrating Eq. (3) over et2ð0;1Þ as
dnphðtÞ¼ ðem
e0
vðeÞ
Uð1rÞ

ln 1þexp ðex1rtoþtÞ½½
"
dexdKcrþvðeLaÞ
Uð1r;LaÞ
!
KLaln
1þexp ðex1r;LatoþtÞ

dex#:(4)
The outward photoelectron flux through the virtual plane
may be obtained by integrating Eq. (4) over normal energy
space ex2ð0;1Þ and may be expressed as
nphoðtÞ¼ ðem
e0
Uðtotþ1rÞ
Uð1rÞ

vðeÞdKcr
"
þUðtotþ1r;LaÞ
Uð1r;LaÞ
"#
vðeLaÞKLa#:(5)
The electrons having the normal energy ex<tð¼ eV=kToÞ
return back to the virtual plane and hence the inward electron
flux may be written by integrating Eq. (4) over normal
energy space ex2ð0;tÞas
nphiðtÞ¼ ðem
e0
Uðtotþ1rÞUðtoþ1rÞ
Uð1rÞ

vðeÞdKcr
"
þUðtotþ1LaÞUðtoþ1r;LaÞ
Uð1r;LaÞ
"#
vðeLaÞKLa:(6)
The net electron flux coming out from the virtual plane may
be expressed as
nphðtÞ¼ npho ðtÞnphi ðtÞ

¼ðem
e0
Uðtoþ1rÞ
Uð1rÞ

vðeÞdKcr
"
þUðtoþ1r;LaÞ
Uð1r;LaÞ
"#
vðeLaÞKLa#:(7)
From Eq. (7), it is evident that in the steady state, the net flux
of photoemitted electrons across all the virtual planes (all
t’s/x’s) over the lunar surface is independent of the potential
of the virtual surface (t) and is equivalent to the flux from
the regolith surface (at x¼0); it should be stated that the net
flux is a significant function of the steady state lunar surface
potential (to). Using Eq. (4), the electron densities associated
with the outward and inward fluxes and hence the net elec-
tron density due to photoemission through virtual plane can
be written as
npeoðtÞ¼ðm=2kToÞ1=2ð1
0
e1=2
xdnphðtÞ;(8)
npeiðtÞ¼ðm=2kToÞ1=2ðt
0
e1=2
xdnphðtÞ;(9)
and
npeðtÞ¼ nepo ðtÞþnepoðtÞ

¼ðm=2kToÞ1=2ð1
0
e1=2
xdnphðtÞþðt
0
e1=2
xdnphðtÞ

:
(10)
Equation (10) infers the electron density at any virtual plane
at vertical altitude (x) due to photoemission as a function of
steady state surface potential of lunar regolith and the tem-
perature of the emitting surface (T
o
). Next, we evaluate the
contribution of solar wind to the plasma density on these vir-
tual planes.
B. Contribution from solar wind plasma
The solar wind is primarily a continuous flow of plasma
comprising electrons, ions, and neutrals at high temperature.
The specification of solar wind plasma has been highlighted
in an elegant article by Mann et al.;
32
specific data used for
the calculations
33
have been given later in the text. For the
sake of simplicity in the analysis, the charged particles in the
solar wind plasma are considered to exhibit Maxwellian dis-
tribution of their energy. Considering the high temperature
operating regime and thermal speed of electrons in compari-
son to the flow speed, this assumption is a reasonable
approximation, however, marginally applicable for the heavy
ions. With this notion, the momentum distribution of the
electrons in the solar wind plasma having momentum
between peand (peþdpe) can be written as
34,35
d2nse ¼nsoð2pmkTse Þ3=2ð2ppteÞexp p2
e=2mkTse

dpxedpte ;
(11)
where n
so
refers to the unperturbed peak density of the solar
wind plasma at mean operating temperature T
se
; rest of the
parameters have their usual meaning, mentioned before.
The momentum flux normal (x) on any planar surface at
positive potential V, over the lunar surface, can be expressed as
d2nse ¼nsoðpxe =mÞð1=2pmkTseÞ3=2ð2ppte Þ
exp ðp2
xe þp2
teÞ=2mkTse

dpxedpte
¼nsoðkTse =2pmÞ1=2exp ðexe þeteÞ½dexedete ;(12)
023702-3 S. K. Mishra and S. Misra Phys. Plasmas 25, 023702 (2018)

where exeð¼ p2
xe=2mkTse Þand eteð¼ p2
te=2mkTse Þinfer the
normal and parallel components of the electron energy nor-
malized with electron temperature. The momentum distribu-
tion in terms of normal energy may be obtained by
integrating Eq. (12) over ete 2ð0;1Þ space and may be
expressed as
dnse ¼nsoðkTse =2pmÞ1=2expðexeÞdexe :(13)
It should be noted that the normal energy of the electrons is
enhanced by tweð¼ eV=kTse ¼tTo=Tse Þas it approaches
the virtual plane at potential V; physically, this may be
included either in the distribution function or in the
limits of integration. Nonetheless, if one assumes that min-
imum energy of electrons in the solar wind is zero (no
additional potential barrier), then the electrons with
exe >0 contribute to the flux over the virtual plane and can
be written as
nse ¼ð1
0
dnse ¼nsoðkTse =2pmÞ1=2:(14)
The electron reaching the planar surface of potential V
has net normal energy equivalent to (exe twe), and
hence, the corresponding electron density at that level can
be written as
nwe ¼ðm=2kTseÞ1=2ð1
0
dnse=ðexe twe Þ1=2
hi
¼ðnso=2p1=2Þexpðtwe Þerfc ðtwe Þ1=2
hi
:(15)
Similarly, in the case of the ions, the momentum distri-
bution corresponding to the normal flux crossing the virtual
plane can be retrieved as
dnsi ¼nsoðkTsi =2pmiÞ1=2expðexiÞdexi;(16)
where exið¼ p2
xi=2mkTsi Þrefers to the normal energy compo-
nent of the ion in distribution.
In case the planar surface is at positive potential (V),
only those ions, which have energy larger than the surface
potential, i.e., exi >twið¼ eV=kTsi ¼tTo=Tsi Þ, will contrib-
ute and hence the net ion flux can be expressed as
nsi ¼ð1
twi
dnsi ¼nsoðkTsi =2pmiÞ1=2expðtwiÞ:(17)
Further, the ion reaching the planar surface of potential
Vhas net normal energy equivalent to (exi þtwi ), and hence,
the corresponding ion density at that level at potential Vcan
be written as
nwi ¼ðmi=2kTsiÞ1=2ð1
twi
dnsi=ðexi þtwi Þ1=2
hi
¼ðnso=2Þexpðtwi Þ:(18)
The two terms derived in Eqs. (15) and (18) infer the
density contribution of electrons and ions at any virtual
plane, in the formation of the sheath over the lunar surface,
respectively. Taking these estimates into account, we next
evaluate the sheath structure.
C. Sheath structure
The space evolution of the electric potential in the
sheath over the lunar regolith in the steady state may be
derived using the Poisson equation where the charge den-
sity at any altitude normal to the lunar surface is given by
adding all the density contributions from the solar wind
plasma and photoemission. Thus potential structure can be
written as
21
d2V
dx2

¼4pens)ðd2t=d12Þ¼ðnpe þnwe nwiÞ=ns0;
(19)
where 1¼ðx=kdÞ,kd¼ð4pns0e2=kToÞ1=2, and ns0¼ns
ðt¼toÞ.
In the above equation, the term ðnpe þnwe nwi Þns
refers to the net population density over the lunar surface
characterized by potential tat an altitude x. Equation (19)
can be solved simply by using the appropriate boundary con-
ditions. Multiplying both sides of Eq. (19) by ð2dt=d1Þand
using a suitable boundary conditions, viz.,dt=d1¼0 and
t¼0as1!1, one obtains
ðdt=d1Þ2¼2ðt
0
ðnpe þnwe nwiÞ=nco

dt¼cðtÞ½
2:(20)
Only the positive square root of ðdt=d1Þis physically tenable.
Integrating Eq. (20) with the boundary condition tð1Þ¼toas
1!0, one gets
d1¼dt
cðtÞ

)1¼ð
t
to
1=cðtÞ½dt:(21)
From the above analysis, the flux (n), potential (t), and
density (n
s
) have been derived as a function of (tot) and
hence the lunar altitude (1). The sheath structure in terms of
the altitude dependence of density, electric field, and electric
potential has been evaluated by solving the set of Eqs.
(19)–(21); the results are graphically illustrated in figures in
Sec. III.
III. NUMERICAL RESULTS FOR SHEATH FORMATION
OVER LUNAR SURFACE
A. Determination of lunar surface potential
In order to illustrate the conceptual basis, it is customary
to have notion of the lunar surface potential. As stated before,
the photoemission from the regolith surface and the solar
wind plasma (comprising electrons/protons) are understood to
maintain a plasma environment at the moon (sunlit) surface.
The lunar surface usually acquires positive potential under the
influence of continuous solar illumination and wind plasma.
21
The steady state potential of the lunar surface may be obtained
by balancing the flux associated with photoemission and solar
wind plasma; algebraically, this may be presented as
nph0þnic0;s¼nec0;s;(22)
023702-4 S. K. Mishra and S. Misra Phys. Plasmas 25, 023702 (2018)

where nph0refers to the photoemission current to the lunar
surface, i.e., nphðt¼toÞ, while n
ec0,s
and n
ic0,s
refer to the
electron/ion flux associated with solar wind plasma; the
expressions are as follows:
35
nec0;sðtoÞ¼nsoðkTse =2pmÞ1=2;(23)
nic0;sðtoÞ¼nsoðkTsi =2pmiÞ1=2expðto;wiÞ;(24)
nph0ðtoÞ¼ ðem
e0
Uðtoþ1rÞ
Uð1rÞ

vðeÞdKcr
"
þUðtoþ1r;LaÞ
Uð1r;LaÞ
"#
vðeLaÞKLa#:(25)
With the help of the above expressions [Eqs. (23)–(25)], Eq.
(22) has numerically been solved to obtain steady state
potential over the moon surface which has been further used
in determining the sheath structure.
B. Computational scheme and data
As a first step, the steady state potential V
o
of the lunar
surface under the influence of continuous solar radiation and
solar wind plasma has been obtained by using Eq. (22) with
a suitable set of parameters. For this known electric potential
V
o
(or to), Eq. (20) has been used to evaluate a physically
tenable positive value of parameter cðt¼toÞcorresponding
to the moon surface (i.e.,1¼0). After knowing the value of
tð¼ toÞand cðt¼toÞon the lunar surface (1¼0), Eq. (21)
has been solved numerically as an initial value problem to
determine the altitude profile of the electric potential (t)in
the lunar sheath; using the altitude profile of potential tð1Þin
the lunar sheath, consequently, the electric field (@1t) and
density (n
s
) profiles within the sheath have also been evalu-
ated. The dimensionless parameters like 1and tused in the
analysis may be transformed in real units by using the nor-
malization factors kdand ðkTo=eÞ.
The typical data
33
for solar wind plasma, i.e., nes nis
8:7cm
3and Tes 2Tis 1:4105K, have been used to
determine the potential of the sunlit surface of the moon.
Although we are using typical solar wind plasma parameters
for the illustration, it should be pointed out that the solar erup-
tions causing wind plasma are unpredictable and may vary
significantly. As per the SPIS (Spacecraft Plasma Interaction
Software) manual,
36
the solar wind plasma density may go up
to 23 cm
3
in reference to the NASA worst case. The sun is
considered as a black body object radiating at temperature
Ts¼5800 K; a white light solar spectrum ranging up to
extreme UV radiation (0 <e<eLa) and dominant Lyman
aradiation (with photon flux 31011=cm2s and wave-
length k¼121:57 nm) has been considered as the source of
electrons from the lunar surface and levitating dust particles.
Considering the region across a subpolar point and
limb, Grobman and Blank
37
predict a plausible range of the
regolith work function in the range of 4 V to 6 V; for the cal-
culation purpose following a recent investigation,
15
the
work function of the moon’s surface (/r) is taken to be
6.0 eV. The photoemission rate is assisted with Draine’s for-
mulation
31
in accounting for the spectral dependence of the
photo-efficiency; v0may take values
38,39
in the range of
0.1–0.5. Another important parameter is the temperature of
the lunar sunlit surface which may be evaluated by equating
the solar radiation absorbed by the surface to the power lost
by thermal radiation and emission cooling. Following the
essence of the Diviner Lunar Radiometer Experiment
40
on-
board the Lunar Reconnaissance Orbiter (LRO), a consistent
value of lunar surface temperature, i.e., To¼400 K, has been
taken for computations. The standard set of parameters taken for
computation is as follows: /r¼6:0eV, To¼400 K, v0¼0:5
(lunar surface), nes nis 8:7=cc,Tes 1:4105K, and
Tis 7104K(solar wind). The effect of various parameters
on the sheath structure (viz., electric potential/field and density)
around the lunar surface has been investigated by varying one
and keeping others the same.
C. Results and discussion
The inclusion of solar wind in the analysis might aid the
particle (electron/ion) flux and density to the lunar surface
(and sheath) in addition to the photoemission induced electron
cloud; intuitively in this case, decay in the lunar surface poten-
tial and smaller sheath is anticipated. On the other hand, due
to the large effective barrier (i.e., work function plus positive
potential) for photoemission from the lunar regolith, the con-
tinuous solar spectrum marginally contributes to the EUV
regime with respect to dominant Lyman-aphotons; however,
the analysis and calculations take both the contributions into
account. As a first step, we evaluate the physical parameters
(potential, electric field, and density) on the lunar surface. The
dependence of steady state potential of the sunlit lunar surface
on the electron (ion) density associated with the solar wind
plasma for varying values of photoelectric efficiency (v
0
)is
illustrated in Fig. 1(a); for the evaluation of surface potential,
Eq. (22) is used. It is noticed that depending on solar wind
flux (n
so
) and photoelectric yield (v
0
) of the material, the lunar
surface acquires a positive potential in the range V0
ð3:54:5ÞV. The decrease in the lunar surface potential with
increasing n
so
is primarily a consequence of increasing electron
accretion flux. The increase in surface potential with the photo-
electric yield may be understood in terms of increasing charg-
ing current due to photoemission of the electrons from the lunar
surface. The subsequent electric field (E
o
) and effective popula-
tion density (n
s0
) on the lunar surface (i.e., x¼0) are illustrated
in Figs. 1(b) and 1(c), respectively; the population density is a
contribution from the photoemission flux in the steady state
(Fig. 1(a), positive potential), while the electric field is in
conformance with the Poisson equation [i.e., Eq. (20)]as
ðdt=d1Þ1!0. Using these parameters as initial conditions in
solving the Poisson equation for the lunar sheath [Eq. (21)], the
characteristic sheath features have been evaluated.
The altitude profile of the electric potential, electric
field, and population density over the lunar surface is illus-
trated in Fig. 2for different values of n
so
. The sheath poten-
tial is noticed to span over an altitude of 100 cm in the
absence of solar wind, while its width gradually decays with
increasing wind plasma density; for instance, the sheath
reduces to 70 cm for n
so
8.7 cm
3
. This behaviour may
be attributed to the contribution of solar wind flux in
023702-5 S. K. Mishra and S. Misra Phys. Plasmas 25, 023702 (2018)

determining the lunar surface potential and increase in
effective population density. It is also noticed that the elec-
tric potential falls steeply for low values of xand then drops
slowly with increasing x. The electric field profile which is
primarily the derivative of the potential profile is depicted
in Fig. 2(b). It is noticed that as the potential gradient
becomes smaller for large x[Fig. 2(a)], the field strength
(E
s
) changes its trend with respect to wind plasma density
(around x45 cm) and terminates (i.e., becomes zero) at
smaller altitude for large n
so
. The subsequent density (n
s
)
corresponding to the potential structure [Fig. 2(a)]isdis-
played in Fig. 2(c) and is a consequence of reduced incom-
ing/outgoing electron flux with sheath potential V.From
this set of calculations, the solar wind plasma is shown to
influence the lunar surface features and hence the sheath
formation which ultimately reduces the sheath span. After
knowing the potential/field structure in the lunar sheath, we
next examine the levitation of fine dust particles in the
sheath region.
FIG. 1. Dependence of (1a) potential (V
o
), (1b) electric field strength
(E
o
), and (1c) population density (n
s0
) on the lunar surface (x¼0) as a
function of solar wind plasma density (n
so
); the computations correspond
to parameters stated in text, while the colour labels refer to different val-
ues of v0.
FIG. 2. The altitude profile of (2a) sheath potential (V), (2b) electric field
strength (E
s
), and (2c) electron population density (n
s
) over the lunar sur-
face; computations correspond to the parameters stated in text and the colour
labels in figure refer to different values of n
so
.
023702-6 S. K. Mishra and S. Misra Phys. Plasmas 25, 023702 (2018)

IV. DUST PARTICLE LEVITATION IN LUNAR SHEATH
The sheath structure over the lunar surface is character-
ized by the altitude profile of the electric field, originated
through the positively charged lunar surface, as described in
Sec. III. This electric field may support the levitation of
positively charged fine particulates in the sheath due to the
balance of gravity over the moon surface with the electro-
static force. In the literature,
15
the lunar dust is character-
ized by the work function smaller than the regolith
(/d4 eV) and may acquire finite positive potential (V
s
)
on account of the dominant photoemission process. The
equilibrium condition for the dust levitation in the lunar
sheath can be expressed as
21
qV þmdgmx¼qVo)mdgm¼qE;(26)
where q(¼aVs) is the charge on the spherical dust particle of
radius a,andmdð¼ 4pqda3=3Þand qdrefer to the mass and
density of the dust particle, respectively, while gm¼ðge=6Þ
163:5cm=s2is the gravitational acceleration over the moon
surface. Equation (26) can be rewritten as
a¼ð3=4pÞðkTo=eÞ2=qdgm
hi
1=2
t1=2
sðtotÞ1=2x1=2¼a0ðxÞ:
(27)
Further, for a<a0, the particles move upwards, while for
a>a0, they move downwards until they attain an equilibrium
position. In order to analyze the levitation, one needs to evalu-
ate the charge (potential) on the dust particles which is eventu-
ally the consequence of dynamic equilibrium between the
accretion of sheath electron and solar wind plasma along with
photoemission flux over the dust particle surface. The flux bal-
ance over the spherical dust particles can be expressed as
NphðtstÞþNic;sðtstÞ¼Nec ðtstÞþNec;sðtstÞ;
(28)
where N
ph
refers to the rate of photoelectron emission from
the spherical surface at an electric potential (VsV) with
respect to the surrounding layer at electric potential V, and
N
ec
corresponds to the accretion of the sheath electrons over
the spherical surface, while N
ec.s
(N
ic.s
) corresponds to the
accretion current associated with solar wind electrons (ions).
Next, we evaluate/define the constituent fluxes.
A. Photoemission flux
The photoemission flux from the levitated spherical dust
particles operating at finite temperature (T
d
) under the influ-
ence of the continuous solar radiation can be written as
35
NphðtstÞ¼pa2ðem
e0
ðtsd tdÞln 1þexpð1dþtsd tdÞ½þUð1dþtsd tdÞ
Uð1dÞ

vdðeÞdKcr
"
þðtsd tdÞln 1þexpð1d;Laþtsd tdÞ

þUð1d;Laþtsd tdÞ
Uð1d;LaÞ
"#
vdðeLaÞKLa#;(29)
where 1d¼ðeudÞðTo=TdÞ,1d;La¼1dðe¼eLaÞ,tsd
¼tsðTo=TdÞ,td¼tðTo=TdÞ,ud¼ðe/d=kTdÞ, and vdrefers
to the photoelectric yield of the dust material.
B. Accretion current due to sheath electrons
The sheath electrons contribute to the electron flux
accretion over the levitating dust particles. Under the notion
of steady state equilibrium of the lunar sheath (comprising
electrons, dust particles, and moon surface) and illustration
of conceptual basis, as a simplification, the sheath electrons
are taken to be at the same temperature equivalent to the
lunar surface/dust particles. Considering the levitated dust
particles to be at positive potential (V
s
), the effective colli-
sion cross section of the accreting electrons can be put in the
form
35
rðex;etÞ¼pa21ðtstÞ=ðexþetÞ½:(30)
Using the momentum distribution of electrons in the
lunar sheath as half Fermi-Dirac statistics,
31
the rate of elec-
tron accretion on the spherical particle due to outward flux
may be written as
21
Neco ¼1
2ð1
0ð1
0
rðex;etÞd2nph
¼pa2
2

ðem
e0
vðeÞ
Uð1rÞ

ð1
0
e1tst
e

FDðe1rtoþtÞdeÞdKcrþvðeLaÞ
Uð1r;LaÞ
!
KLa
ð
1
0
e1tst
e

FDðe1r;LatoþtÞde:(31)
Following the approach similar to Sec. II A, the electrons
having radial energy less than dust potential ex<ðtÞwill
return back to the particle, and hence, the rate of electron
accretion on account of the inward flux can be expressed as
Neci ¼Neco pa2
2

ðem
e0
vðeÞ
Uð1rÞ

ð1
0
e1tst
et

FDðe1rtoÞdeÞdKcrvðeLaÞ
Uð1r;LaÞ
!
KLa
ð
1
0
e1tst
et

FDðe1r;LatoÞde:(32)
023702-7 S. K. Mishra and S. Misra Phys. Plasmas 25, 023702 (2018)

The net accretion current associated with sheath elec-
trons on a spherical grain at a layer characterized by an elec-
tric potential (V) can be expressed as
NecðtstÞ¼Neco þNeci :(33)
It is evident from the above derived expressions for accretion
current of the sheath electrons that it significantly depends
on the nature of electron energy distribution in the sheath.
21
C. Accretion flux associated with solar wind plasma
As discussed in an earlier section (Sec. II B), the solar
wind has been approximated by plasma comprising electrons
and ions at high temperature exhibiting Maxwellian statis-
tics. With this simplification, the collision cross section of
electrons and ions accreting over the positively charged dust
particle may be expressed as
35
reðexe;ete Þ¼pa21ðts;we tweÞ=ðexe þeteÞ

;(34a)
riðexi;eti Þ¼pa21þðts;wi twiÞ=ðexi þetiÞ

;(34b)
where ts;weð¼ tsTo=Tse Þ,ts;wið¼ tsTo=Tsi Þ, and ee;i¼ðexe;xi
þete;tiÞ.
Following the orbital motion limited (OML) based
approach
35
for determining the accretion current of the elec-
trons/ions over positively charged dust particles levitating at
sheath layer with potential V, the electron and ion accretion
current associated with the solar wind plasma may be
expressed as
35
Nec;sðtstÞ¼nsoðkTse =2pmÞ1=2pa2
ð1
0
ee1ðts;we tweÞ
ee

expðeeÞdee
¼nsoðkTse =2pmÞ1=2pa21ðts;we twe Þ

;(35)
Nic;sðtstÞ¼nsoðkTsi =2pmiÞ1=2pa2
ð1
0
ei1þðts;wi twiÞ
eiðts;wi twi Þ
!
exp ðeits;wi þtwiÞ

dei
¼nsoðkTsi =2pmiÞ1=2pa2expðts;wi twiÞ:(36)
With the help of expressions derived herein, Eqs. (27)
and (28) may numerically be solved to determine particle
charging and optimum particle size within the lunar sheath.
It should be mentioned here that the role of photoemission
from particles has been ignored in determining the sheath
structure over the lunar surface.
V. NUMERICAL RESULTS FOR FINE PARTICLE
LEVITATION IN THE SHEATH
Taking forward the knowledge of the photoelectron
sheath, i.e., altitude profile of the electric potential, electric
field, and population density (Fig. 2), we use tð1Þrelation
and expressions [Eqs. (29),(33),(35), and (36)] to solve Eq.
(28) numerically and evaluate the steady state mean surface
potential (ts) over the levitating dust grains in the lunar
sheath. In describing the charging of fine dust particles in the
sheath region, uniform potential theory is applied where the
photoemission and accretion (of sheath electrons/solar wind
plasma) are considered as dominant charging mechanisms.
For the sake of simplicity in analysis, Mie scattering
41
of the
radiation from levitating fine particles and size dependence
of the photoelectric yield
21
have been ignored, which is per-
tinent for the particles larger than 200 nm. The theory pre-
sented herein is also limited by the fact that the dust particle
number density is such that the photoelectron sheath struc-
ture remains unaffected. The uniform potential approach
yields an important consequence that the steady state poten-
tial is independent of the particle size,
42
and hence, charge
on the surface increases linearly with the radius (zs/a).
Manifesting the known value of particle charge (via potential
V
s
) on levitating grains with the electric field altitude profile
in the lunar sheath (E
s
), one may obtain the optimum particle
size as a function of lunar surface altitude using Eq. (27).
For the computations, the same set of solar wind and photon
flux data as in Sec. III B has been used. Draine’s formula-
tion
28
has been used to describe the spectral dependence of
the photoelectric efficiency of the levitating fine particles,
while considering the recent analysis,
30
their work function
(/d) and density (qd) are taken to be 4.0 eV and 3.0 g/cm
3
,
respectively. The parameters for dust particulates used in
these calculations, in addition to solar radiation/wind, are
as follows: /d¼4:0 eV, Td¼400 K, v0¼0:5, and qd
¼3g=cc. Next, we discuss the dust particle charging and
their levitation in the sheath for the standard case.
In contrast to the lunar regolith, due to the low work
function of the particulates, the charging is significantly
influenced by the continuous solar radiation spectrum
(h>/d). Further, due to the presence of solar wind plasma,
the particle potential (charge) is anticipated to decrease on
account of electron accretion flux. The dependence of the
steady state dust potential (V
s
) on the surface altitude (x)in
the presence (red) and absence (black) of the solar wind is
illustrated in Fig. 3(a). The decaying nature of the dust sur-
face potential may be attributed to the potential profile in the
photoelectron sheath [Fig. 2(a)]. The reduction in positive
potential in the presence of solar wind may be understood in
terms of enhanced electron flux over dust particles associated
with wind plasma. After knowing dust potential, the altitude
dependence of the maximum possible radius of the dust par-
ticle which can levitate in the presence of electrostatic sheath
field is illustrated in Fig. 3(b); this nature is in conformance
with the altitude profile of the sheath electric field [E
s
, Fig.
2(b)]. For this particular set of parameters, it is noticed that
the levitation terminates around 70 cm in the presence of
solar wind while it continues up to 100 cm when the effect
of solar wind is ignored. In addition, the optimum particle
size is little higher at lower altitude (x<40 cm) when the
solar wind is considered; however, in consistency with the
sheath electric field profile [Fig. 2(b)], its value drops rapidly
at higher lunar altitudes (x>45 cm). At lower altitudes, the
photoelectron sheath may hold submicron size particles; for
instance, 500 and 200 nm particles are predicted to float up
to 10 cm and 32 cm, respectively, in the photoelectron
sheath.
023702-8 S. K. Mishra and S. Misra Phys. Plasmas 25, 023702 (2018)

VI. SUMMARY AND CONCLUSIONS
In summary, a self-consistent analytical model describ-
ing the photoelectron sheath structure over the lunar surface
has been developed. The formulation includes the continuous
solar spectrum, solar flux plasma, and half Fermi-Dirac dis-
tribution of photoelectron velocities in evaluating the sheath
structure through the Poisson equation; considering the sim-
plified Maxwellian nature of solar wind plasma, the expres-
sions for its contribution to the electron/ion population
density has been derived. After a notion of the lunar sheath
structure, the levitation of fine dust particles over the moon
surface has been examined by taking adequate expressions
(derived herein) for anisotropic flux in the photoelectron
sheath, solar wind flux, and photoemission over dust par-
ticles into account. In deriving the sheath structure and dust
levitation phenomena, the steady state potential over the
lunar regolith and dust particulates has consistently been
determined by balancing the flux of electrons and ions over
respective surfaces. As an illustration of the conceptual
basis, the altitude profile of the electric potential/electric
field and population density in the lunar sheath has been
depicted; in particular, the role of solar wind plasma has
been identified. The analysis presented herein is of general in
nature and is applicable to any other resembling physical
scenario in nature and laboratory experiments. The solar
wind plasma is noticed to suppress the sheath width and
demonstrate high constituent electric field which eventually
leads to the levitation of large particles at lower altitudes.
The analysis presents a comprehensive model for the
structure of photoelectron sheath and levitation of dust,
based on sound physics. As the analysis takes account of few
simplifications, it is reasonable to comment on its applicabil-
ity; here, we discuss the limitations of the present analysis
and consequent effects. As a simplification, the electrons
(ions) in the solar wind plasma are considered to be of
Maxwellian in nature, which eventually should be character-
ized by shifting Maxwellian distribution of velocity (in par-
ticular ions); in such a case, the accretion current
43,44
decreases and the particles are supposed to gain higher posi-
tive potential, which may result in higher levitation altitude.
The observed solar wind is noted to comprise a variety of
electron (and ion) energy population; for instance, in an ele-
gant study,
45
the electrons in high speed solar wind are
ascribed to core (low energy) and halo and strahl (high
energy) populations. On account of different temperature/
population densities of the constituent core and halo/strahl,
electrons may characterize different accretion flux over the
lunar surface and dust particulates; the inclusion of such dis-
tributions might influence the lunar surface potential and
hence the sheath structure. Qualitatively, the increase
(decrease) in the electron population/temperature in the dis-
tribution ultimately leads to the larger (smaller) collection
current and reduced (high) surface potential which eventu-
ally may cause a smaller (larger) sheath span; similarly, the
charge of the levitated dust is also affected and one may
anticipate the lower (higher) altitude of levitation. It should
also be stated that though the analysis has been performed
for simplified Maxwellian statistics of the solar wind plasma
constituents, the analytical approach is equally applicable
and reiterated for any specified distribution. Further, in the
exotic cases (corresponding to extreme solar activities), the
solar wind plasma density is reported
46,47
to reach the order
of 100 cm
3
. On the basis of present analysis and under-
standing, under these extreme conditions when solar flux is
high, the lunar surface potential and the sheath span are
expected to reduce; subsequently, the levitating particle
charge may also be anticipated to be small and hence lower
altitude of levitation. Another concern is the Mie scattering
of radiation from smaller dust particulates
41
(2pa=k<10),
which might reduce the mean particle charge and hence the
optimum size of the dust particles for levitation. As another
simplification, the role of levitated particles in the lunar
sheath is ignored; the photoemitted electrons from dust may
lead to the increase in density population in the sheath which
in turn may reduce the sheath expansion. Further uniform
surface potential, which is primarily pertinent to conducting
substances, is taken in order to avoid the complexity associ-
ated with dielectric (non-conducting) materials. In this con-
text, another concern may be the surface inhomogeneity
which might lead to an inhomogeneous photoelectron sheath
FIG. 3. The altitude profile of (3a) the dust surface potential (V
s
) and (3b)
optimum particle size within lunar photoelectron sheath; computations cor-
respond to the parameters stated in text and the colour labels in figure refer
to different values of n
so
.
023702-9 S. K. Mishra and S. Misra Phys. Plasmas 25, 023702 (2018)

and hence irregular particle levitation/distribution. Although
these aspects limit the applicability of the presented results,
the analytical formulation gives a feasible solution (and scal-
ing) of lunar sheath features and puts forward a basis for fur-
ther advancement. This work may also be of practical
significance in designing test experiments in labs for future
lunar/space campaigns.
ACKNOWLEDGMENTS
This work was performed under ELI-ALPS Project (No.
GINOP-2.3.6-15-2015-00001) which is supported by the
European Union and co-financed by the European Regional
Development fund.
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